]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Remove directories for tutorials that aren't ready yet.
authorbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 24 Jun 2010 22:48:59 +0000 (22:48 +0000)
committerbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Thu, 24 Jun 2010 22:48:59 +0000 (22:48 +0000)
git-svn-id: https://svn.dealii.org/branches/releases/Branch-6-3@21352 0785d39b-7218-0410-832d-ea1e28bc413d

54 files changed:
deal.II/examples/step-26/Makefile [deleted file]
deal.II/examples/step-26/doc/intro.dox [deleted file]
deal.II/examples/step-26/doc/results.dox [deleted file]
deal.II/examples/step-26/doc/step-26.surface.png [deleted file]
deal.II/examples/step-26/doc/step-26.volume.png [deleted file]
deal.II/examples/step-26/step-26.cc [deleted file]
deal.II/examples/step-26/surface-points [deleted file]
deal.II/examples/step-32/Makefile [deleted file]
deal.II/examples/step-32/doc/builds-on [deleted file]
deal.II/examples/step-32/doc/intro.dox [deleted file]
deal.II/examples/step-32/doc/kind [deleted file]
deal.II/examples/step-32/doc/results.dox [deleted file]
deal.II/examples/step-32/doc/step-32.2d.grid.2100.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.partition.2100.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.t_vs_vmax.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0000.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0100.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0200.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0300.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0400.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0500.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0600.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0700.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0800.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.0900.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1000.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1100.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1200.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1300.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1400.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1500.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1600.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1700.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1800.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.1900.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.2000.png [deleted file]
deal.II/examples/step-32/doc/step-32.2d.temperature.2100.png [deleted file]
deal.II/examples/step-32/doc/step-32.3d.cube.0.png [deleted file]
deal.II/examples/step-32/doc/step-32.3d.cube.1.png [deleted file]
deal.II/examples/step-32/doc/step-32.beta.2d.png [deleted file]
deal.II/examples/step-32/doc/step-32.beta_cr.2d.png [deleted file]
deal.II/examples/step-32/doc/tooltip [deleted file]
deal.II/examples/step-32/step-32.cc [deleted file]
deal.II/examples/step-37/Makefile [deleted file]
deal.II/examples/step-37/doc/builds-on [deleted file]
deal.II/examples/step-37/doc/intro.dox [deleted file]
deal.II/examples/step-37/doc/kind [deleted file]
deal.II/examples/step-37/doc/results.dox [deleted file]
deal.II/examples/step-37/doc/tooltip [deleted file]
deal.II/examples/step-37/step-37.cc [deleted file]
deal.II/examples/step-42/Makefile [deleted file]
deal.II/examples/step-42/doc/intro.dox [deleted file]
deal.II/examples/step-42/doc/results.dox [deleted file]
deal.II/examples/step-42/step-42.cc [deleted file]

diff --git a/deal.II/examples/step-26/Makefile b/deal.II/examples/step-26/Makefile
deleted file mode 100644 (file)
index 6afd6b0..0000000
+++ /dev/null
@@ -1,154 +0,0 @@
-# $Id: Makefile 11779 2005-11-23 15:54:25Z wolf $
-
-
-# For the small projects Makefile, you basically need to fill in only
-# four fields.
-#
-# The first is the name of the application. It is assumed that the
-# application name is the same as the base file name of the single C++
-# file from which the application is generated.
-target = $(basename $(shell echo step-*.cc))
-
-# The second field determines whether you want to run your program in
-# debug or optimized mode. The latter is significantly faster, but no
-# run-time checking of parameters and internal states is performed, so
-# you should set this value to `on' while you develop your program,
-# and to `off' when running production computations.
-debug-mode = on
-
-
-# As third field, we need to give the path to the top-level deal.II
-# directory. You need to adjust this to your needs. Since this path is
-# probably the most often needed one in the Makefile internals, it is
-# designated by a single-character variable, since that can be
-# reference using $D only, i.e. without the parentheses that are
-# required for most other parameters, as e.g. in $(target).
-D = ../../
-
-
-# The last field specifies the names of data and other files that
-# shall be deleted when calling `make clean'. Object and backup files,
-# executables and the like are removed anyway. Here, we give a list of
-# files in the various output formats that deal.II supports.
-clean-up-files = *gmv *gnuplot *gpl *eps *pov
-
-
-
-
-#
-#
-# Usually, you will not need to change anything beyond this point.
-#
-#
-# The next statement tell the `make' program where to find the
-# deal.II top level directory and to include the file with the global
-# settings
-include $D/common/Make.global_options
-
-
-# Since the whole project consists of only one file, we need not
-# consider difficult dependencies. We only have to declare the
-# libraries which we want to link to the object file, and there need
-# to be two sets of libraries: one for the debug mode version of the
-# application and one for the optimized mode. Here we have selected
-# the versions for 2d. Note that the order in which the libraries are
-# given here is important and that your applications won't link
-# properly if they are given in another order.
-#
-# You may need to augment the lists of libraries when compiling your
-# program for other dimensions, or when using third party libraries
-libs.g   = $(lib-deal2-3d.g) \
-          $(lib-lac.g)      \
-           $(lib-base.g)
-libs.o   = $(lib-deal2-3d.o) \
-          $(lib-lac.o)      \
-           $(lib-base.o)
-
-
-# We now use the variable defined above which switch between debug and
-# optimized mode to select the set of libraries to link with. Included
-# in the list of libraries is the name of the object file which we
-# will produce from the single C++ file. Note that by default we use
-# the extension .g.o for object files compiled in debug mode and .o for
-# object files in optimized mode (or whatever the local default on your
-# system is instead of .o).
-ifeq ($(debug-mode),on)
-  libraries = $(target).g.$(OBJEXT) $(libs.g)
-else
-  libraries = $(target).$(OBJEXT) $(libs.o)
-endif
-
-
-# Now comes the first production rule: how to link the single object
-# file produced from the single C++ file into the executable. Since
-# this is the first rule in the Makefile, it is the one `make' selects
-# if you call it without arguments.
-$(target) : $(libraries)
-       @echo ============================ Linking $@
-       @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS)
-
-
-# To make running the application somewhat independent of the actual
-# program name, we usually declare a rule `run' which simply runs the
-# program. You can then run it by typing `make run'. This is also
-# useful if you want to call the executable with arguments which do
-# not change frequently. You may then want to add them to the
-# following rule:
-run: $(target)
-       @echo ============================ Running $<
-       @./$(target)$(EXEEXT)
-
-
-# As a last rule to the `make' program, we define what to do when
-# cleaning up a directory. This usually involves deleting object files
-# and other automatically created files such as the executable itself,
-# backup files, and data files. Since the latter are not usually quite
-# diverse, you needed to declare them at the top of this file.
-clean:
-       -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files)
-
-
-# Since we have not yet stated how to make an object file from a C++
-# file, we should do so now. Since the many flags passed to the
-# compiler are usually not of much interest, we suppress the actual
-# command line using the `at' sign in the first column of the rules
-# and write the string indicating what we do instead.
-./%.g.$(OBJEXT) :
-       @echo ==============debug========= $(<F)
-       @$(CXX) $(CXXFLAGS.g) -c $< -o $@
-./%.$(OBJEXT) :
-       @echo ==============optimized===== $(<F)
-       @$(CXX) $(CXXFLAGS.o) -c $< -o $@
-
-
-# The following statement tells make that the rules `run' and `clean'
-# are not expected to produce files of the same name as Makefile rules
-# usually do.
-.PHONY: run clean
-
-
-# Finally there is a rule which you normally need not care much about:
-# since the executable depends on some include files from the library,
-# besides the C++ application file of course, it is necessary to
-# re-generate the executable when one of the files it depends on has
-# changed. The following rule to created a dependency file
-# `Makefile.dep', which `make' uses to determine when to regenerate
-# the executable. This file is automagically remade whenever needed,
-# i.e. whenever one of the cc-/h-files changed. Make detects whether
-# to remake this file upon inclusion at the bottom of this file.
-#
-# If the creation of Makefile.dep fails, blow it away and fail
-Makefile.dep: $(target).cc Makefile \
-              $(shell echo $D/*/include/*/*.h)
-       @echo ============================ Remaking $@
-       @$D/common/scripts/make_dependencies  $(INCLUDE) -B. $(target).cc \
-               > $@ \
-         || (rm -f $@ ; false)
-       @if test -s $@ ; then : else rm $@ ; fi
-
-
-# To make the dependencies known to `make', we finally have to include
-# them:
-include Makefile.dep
-
-
diff --git a/deal.II/examples/step-26/doc/intro.dox b/deal.II/examples/step-26/doc/intro.dox
deleted file mode 100644 (file)
index c59cfdf..0000000
+++ /dev/null
@@ -1,2 +0,0 @@
-<a name="Intro"></a>
-<h1>Introduction</h1>
diff --git a/deal.II/examples/step-26/doc/results.dox b/deal.II/examples/step-26/doc/results.dox
deleted file mode 100644 (file)
index f4c6fee..0000000
+++ /dev/null
@@ -1 +0,0 @@
-<h1>Results</h1>
diff --git a/deal.II/examples/step-26/doc/step-26.surface.png b/deal.II/examples/step-26/doc/step-26.surface.png
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diff --git a/deal.II/examples/step-26/doc/step-26.volume.png b/deal.II/examples/step-26/doc/step-26.volume.png
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diff --git a/deal.II/examples/step-26/step-26.cc b/deal.II/examples/step-26/step-26.cc
deleted file mode 100644 (file)
index 24fb18a..0000000
+++ /dev/null
@@ -1,775 +0,0 @@
-/* $Id$ */
-/* Author: Wolfgang Bangerth, University of Heidelberg, 1999 */
-
-/*    $Id$       */
-/*                                                                */
-/*    Copyright (C) 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 by the deal.II authors */
-/*                                                                */
-/*    This file is subject to QPL and may not be  distributed     */
-/*    without copyright and license information. Please refer     */
-/*    to the file deal.II/doc/license.html for the  text  and     */
-/*    further information on this license.                        */
-
-                                 // @sect3{Include files}
-
-                                // The first few (many?) include
-                                // files have already been used in
-                                // the previous example, so we will
-                                // not explain their meaning here
-                                // again.
-#include <grid/tria.h>
-#include <dofs/dof_handler.h>
-#include <grid/grid_generator.h>
-#include <grid/tria_accessor.h>
-#include <grid/tria_iterator.h>
-#include <dofs/dof_accessor.h>
-#include <fe/fe_q.h>
-#include <dofs/dof_tools.h>
-#include <fe/fe_values.h>
-#include <base/quadrature_lib.h>
-#include <base/function.h>
-#include <numerics/vectors.h>
-#include <numerics/matrices.h>
-#include <lac/vector.h>
-#include <lac/full_matrix.h>
-#include <lac/sparse_matrix.h>
-#include <lac/solver_cg.h>
-#include <lac/precondition.h>
-
-#include <numerics/data_out.h>
-#include <fstream>
-#include <iostream>
-
-                                // This is new, however: in the previous
-                                // example we got some unwanted output from
-                                // the linear solvers. If we want to suppress
-                                // it, we have to include this file and add a
-                                // single line somewhere to the program (see
-                                // the main() function below for that):
-#include <base/logstream.h>
-
-
-#include <algorithm>
-#include <numeric>
-#include <grid/tria_boundary.h>
-
-                                // The last step is as in all
-                                // previous programs:
-using namespace dealii;
-
-class PointCloudSurface : public StraightBoundary<3>
-{
-  public:
-                                    /**
-                                     * Constructor.
-                                     */
-    PointCloudSurface (const std::string &filename);
-    
-                                    /**
-                                     * Let the new point be the
-                                     * arithmetic mean of the two
-                                     * vertices of the line.
-                                     *
-                                     * Refer to the general
-                                     * documentation of this class
-                                     * and the documentation of the
-                                     * base class for more
-                                     * information.
-                                     */
-    virtual Point<3>
-    get_new_point_on_line (const Triangulation<3>::line_iterator &line) const;
-
-                                    /**
-                                     * Let the new point be the
-                                     * arithmetic mean of the four
-                                     * vertices of this quad and the
-                                     * four midpoints of the lines,
-                                     * which are already created at
-                                     * the time of calling this
-                                     * function.
-                                     *
-                                     * Refer to the general
-                                     * documentation of this class
-                                     * and the documentation of the
-                                     * base class for more
-                                     * information.
-                                     */
-    virtual Point<3>
-    get_new_point_on_quad (const Triangulation<3>::quad_iterator &quad) const;
-
-                                    /**
-                                     * Gives <tt>n=points.size()</tt>
-                                     * points that splits the
-                                     * StraightBoundary line into
-                                     * $n+1$ partitions of equal
-                                     * lengths.
-                                     *
-                                     * Refer to the general
-                                     * documentation of this class
-                                     * and the documentation of the
-                                     * base class.
-                                     */
-    virtual void
-    get_intermediate_points_on_line (const Triangulation<3>::line_iterator &line,
-                                    std::vector<Point<3> > &points) const;
-
-                                    /**
-                                     * Gives <tt>n=points.size()=m*m</tt>
-                                     * points that splits the
-                                     * p{StraightBoundary} quad into
-                                     * <tt>(m+1)(m+1)</tt> subquads of equal
-                                     * size.
-                                     *
-                                     * Refer to the general
-                                     * documentation of this class
-                                     * and the documentation of the
-                                     * base class.
-                                     */
-    virtual void
-    get_intermediate_points_on_quad (const Triangulation<3>::quad_iterator &quad,
-                                    std::vector<Point<3> > &points) const;
-
-                                    /**
-                                     * A function that, given a point
-                                     * <code>p</code>, returns the closest
-                                     * point on the surface defined by the
-                                     * input file. For the time being, we
-                                     * simply return the closest point in the
-                                     * point cloud, rather than doing any
-                                     * sort of interpolation.
-                                     */
-    Point<3> closest_point (const Point<3> &p) const;    
-  private:
-    std::vector<Point<3> > point_list;
-};
-
-
-PointCloudSurface::PointCloudSurface (const std::string &filename)
-{
-                                  // first read in all the points
-  {
-    std::ifstream in (filename.c_str());
-    AssertThrow (in, ExcIO());
-    
-    while (in)
-      {
-       Point<3> p;
-       in >> p;
-       point_list.push_back (p);
-      }
-
-    AssertThrow (point_list.size() > 1, ExcIO());
-  }
-  
-                                  // next fit a linear model through the data
-                                  // cloud to rectify it in a local
-                                  // coordinate system
-                                  //
-                                  // the first step is to move the center of
-                                  // mass of the points to the origin
-  {
-    const Point<3> c_o_m = std::accumulate (point_list.begin(),
-                                           point_list.end(),
-                                           Point<3>()) /
-                            point_list.size();
-    for (unsigned int i=0; i<point_list.size(); ++i)
-      point_list[i] -= c_o_m;
-  }
-
-                                  // next do a least squares fit to the
-                                  // function ax+by. this leads to the
-                                  // following equations:
-  
-                                  // min f(a,b) = sum_i (zi-a xi - b yi)^2 / 2
-                                  //
-                                  // f_a = sum_i (zi - a xi - b yi) xi = 0
-                                  // f_b = sum_i (zi - a xi - b yi) yi = 0
-                                  //
-                                  // f_a = (sum_i zi xi) - (sum xi^2) a - (sum xi yi) b = 0
-                                  // f_a = (sum_i zi yi) - (sum xi yi) a - (sum yi^2) b = 0
-  {
-    double A[2][2] = {{0,0},{0,0}};
-    double B[2] = {0,0};
-
-    for (unsigned int i=0; i<point_list.size(); ++i)
-      {
-       A[0][0] += point_list[i][0] * point_list[i][0];
-       A[0][1] += point_list[i][0] * point_list[i][1];
-       A[1][1] += point_list[i][1] * point_list[i][1];
-
-       B[0] += point_list[i][0] * point_list[i][2];
-       B[1] += point_list[i][1] * point_list[i][2];
-      }
-
-    const double det = A[0][0]*A[1][1]-2*A[0][1];
-    const double a = (A[1][1] * B[0] - A[0][1] * B[1]) / det;
-    const double b = (A[0][0] * B[1] - A[0][1] * B[0]) / det;
-
-
-                                    // with this information, we can rotate
-                                    // the points so that the corresponding
-                                    // least-squares fit would be the x-y
-                                    // plane
-    const Point<2> gradient_direction
-      = Point<2>(a,b) / std::sqrt(a*a+b*b);
-    const Point<2> orthogonal_direction
-      = Point<2>(-b,a) / std::sqrt(a*a+b*b);
-
-    const double stretch_factor = std::sqrt(1.+a*a+b*b);
-    
-    for (unsigned int i=0; i<point_list.size(); ++i)
-      {
-                                        // we can do that by, for each point,
-                                        // first subtract the points in the
-                                        // plane:
-       point_list[i][2] -= a*point_list[i][0] + b*point_list[i][1];
-
-                                      // we made a mistake here, though:
-                                      // we've shrunk the plan in the
-                                      // direction parallel to the
-                                      // gradient. we will have to correct
-                                      // for this:
-       const Point<2> xy (point_list[i][0],
-                          point_list[i][1]);
-       const double grad_distance = xy * gradient_direction;
-       const double orth_distance = xy * orthogonal_direction;
-
-                                        // we then have to stretch the points
-                                        // in the gradient direction. the
-                                        // stretch factor is defined above
-                                        // (zero if the original plane was
-                                        // already the xy plane, infinity if
-                                        // it was vertical)
-       const Point<2> new_xy
-         = (grad_distance * stretch_factor * gradient_direction +
-            orth_distance * orthogonal_direction);
-       point_list[i][0] = new_xy[0];
-       point_list[i][1] = new_xy[1];   
-      }
-  }
-}
-
-
-Point<3>
-PointCloudSurface::closest_point (const Point<3> &p) const
-{
-  double distance = p.distance (point_list[0]);
-  Point<3> point = point_list[0];
-  
-  for (std::vector<Point<3> >::const_iterator i=point_list.begin();
-       i != point_list.end(); ++i)
-    {
-      const double d = p.distance (*i);
-      if (d < distance)
-       {
-         distance = d;
-         point = *i;
-       }
-    }
-
-  return point;
-}
-
-  
-Point<3>
-PointCloudSurface::
-get_new_point_on_line (const Triangulation<3>::line_iterator &line) const
-{
-  return closest_point (StraightBoundary<3>::get_new_point_on_line (line));
-}
-
-
-
-Point<3>
-PointCloudSurface::
-get_new_point_on_quad (const Triangulation<3>::quad_iterator &quad) const
-{
-  return closest_point (StraightBoundary<3>::get_new_point_on_quad (quad));
-}
-
-
-
-void
-PointCloudSurface::
-get_intermediate_points_on_line (const Triangulation<3>::line_iterator &line,
-                                std::vector<Point<3> > &points) const
-{
-  StraightBoundary<3>::get_intermediate_points_on_line (line,
-                                                       points);
-  for (unsigned int i=0; i<points.size(); ++i)
-    points[i] = closest_point(points[i]);
-}
-
-
-
-void
-PointCloudSurface::
-get_intermediate_points_on_quad (const Triangulation<3>::quad_iterator &quad,
-                                std::vector<Point<3> > &points) const
-{
-  StraightBoundary<3>::get_intermediate_points_on_quad (quad,
-                                                       points);
-  for (unsigned int i=0; i<points.size(); ++i)
-    points[i] = closest_point(points[i]);
-}
-
-
-
-PointCloudSurface pds("surface-points");
-
-
-
-
-
-
-
-
-                                 // @sect3{The <code>LaplaceProblem</code> class template}
-
-                                // This is again the same
-                                // <code>LaplaceProblem</code> class as in the
-                                // previous example. The only
-                                // difference is that we have now
-                                // declared it as a class with a
-                                // template parameter, and the
-                                // template parameter is of course
-                                // the spatial dimension in which we
-                                // would like to solve the Laplace
-                                // equation. Of course, several of
-                                // the member variables depend on
-                                // this dimension as well, in
-                                // particular the Triangulation
-                                // class, which has to represent
-                                // quadrilaterals or hexahedra,
-                                // respectively. Apart from this,
-                                // everything is as before.
-template <int dim>
-class LaplaceProblem 
-{
-  public:
-    LaplaceProblem ();
-    void run ();
-    
-  private:
-    void make_grid_and_dofs ();
-    void assemble_system ();
-    void solve ();
-    void output_results () const;
-
-    Triangulation<dim>   triangulation;
-    FE_Q<dim>            fe;
-    DoFHandler<dim>      dof_handler;
-
-    SparsityPattern      sparsity_pattern;
-    SparseMatrix<double> system_matrix;
-
-    Vector<double>       solution;
-    Vector<double>       system_rhs;
-};
-
-
-                                 // @sect3{Right hand side and boundary values}
-
-
-
-
-template <int dim>
-class BoundaryValues : public Function<dim> 
-{
-  public:
-    BoundaryValues () : Function<dim>() {}
-    
-    virtual double value (const Point<dim>   &p,
-                         const unsigned int  component = 0) const;
-};
-
-
-
-template <int dim>
-double BoundaryValues<dim>::value (const Point<dim> &p,
-                                  const unsigned int /*component*/) const 
-{
-  return std::max(p[dim-1], -5.);
-}
-
-
-
-                                 // @sect3{Implementation of the <code>LaplaceProblem</code> class}
-
-                                 // Next for the implementation of the class
-                                 // template that makes use of the functions
-                                 // above. As before, we will write everything
-                                 // as templates that have a formal parameter
-                                 // <code>dim</code> that we assume unknown at the time
-                                 // we define the template functions. Only
-                                 // later, the compiler will find a
-                                 // declaration of <code>LaplaceProblem@<2@></code> (in
-                                 // the <code>main</code> function, actually) and
-                                 // compile the entire class with <code>dim</code>
-                                 // replaced by 2, a process referred to as
-                                 // `instantiation of a template'. When doing
-                                 // so, it will also replace instances of
-                                 // <code>RightHandSide@<dim@></code> by
-                                 // <code>RightHandSide@<2@></code> and instantiate the
-                                 // latter class from the class template.
-                                 //
-                                 // In fact, the compiler will also find a
-                                 // declaration <code>LaplaceProblem@<3@></code> in
-                                 // <code>main()</code>. This will cause it to again go
-                                 // back to the general
-                                 // <code>LaplaceProblem@<dim@></code> template, replace
-                                 // all occurrences of <code>dim</code>, this time by
-                                 // 3, and compile the class a second
-                                 // time. Note that the two instantiations
-                                 // <code>LaplaceProblem@<2@></code> and
-                                 // <code>LaplaceProblem@<3@></code> are completely
-                                 // independent classes; their only common
-                                 // feature is that they are both instantiated
-                                 // from the same general template, but they
-                                 // are not convertible into each other, for
-                                 // example, and share no code (both
-                                 // instantiations are compiled completely
-                                 // independently).
-
-
-                                 // @sect4{LaplaceProblem::LaplaceProblem}
-
-                                // After this introduction, here is the
-                                // constructor of the <code>LaplaceProblem</code>
-                                // class. It specifies the desired polynomial
-                                // degree of the finite elements and
-                                // associates the DoFHandler to the
-                                // triangulation just as in the previous
-                                // example program, step-3:
-template <int dim>
-LaplaceProblem<dim>::LaplaceProblem () :
-                fe (1),
-               dof_handler (triangulation)
-{}
-
-
-                                 // @sect4{LaplaceProblem::make_grid_and_dofs}
-
-                                // Grid creation is something
-                                // inherently dimension
-                                // dependent. However, as long as the
-                                // domains are sufficiently similar
-                                // in 2D or 3D, the library can
-                                // abstract for you. In our case, we
-                                // would like to again solve on the
-                                // square [-1,1]x[-1,1] in 2D, or on
-                                // the cube [-1,1]x[-1,1]x[-1,1] in
-                                // 3D; both can be termed
-                                // <code>hyper_cube</code>, so we may use the
-                                // same function in whatever
-                                // dimension we are. Of course, the
-                                // functions that create a hypercube
-                                // in two and three dimensions are
-                                // very much different, but that is
-                                // something you need not care
-                                // about. Let the library handle the
-                                // difficult things.
-                                //
-                                // Likewise, associating a degree of freedom
-                                // with each vertex is something which
-                                // certainly looks different in 2D and 3D,
-                                // but that does not need to bother you
-                                // either. This function therefore looks
-                                // exactly like in the previous example,
-                                // although it performs actions that in their
-                                // details are quite different if <code>dim</code>
-                                // happens to be 3. The only significant
-                                // difference from a user's perspective is
-                                // the number of cells resulting, which is
-                                // much higher in three than in two space
-                                // dimensions!
-template <int dim>
-void LaplaceProblem<dim>::make_grid_and_dofs ()
-{
-  GridGenerator::hyper_cube (triangulation, -30, 30);
-
-  for (unsigned int f=0; f<GeometryInfo<dim>::faces_per_cell; ++f)
-    if (triangulation.begin()->face(f)->center()[2] > 15)
-      {
-       triangulation.begin()->face(f)->set_boundary_indicator (1);
-       for (unsigned int i=0; i<GeometryInfo<dim>::lines_per_face; ++i)
-         triangulation.begin()->face(f)->line(i)->set_boundary_indicator (1);
-       break;
-      }
-  triangulation.set_boundary (1, pds);
-  
-  
-  for (unsigned int v=0; v<GeometryInfo<dim>::vertices_per_cell; ++v)
-    if (triangulation.begin()->vertex(v)[2] > 0)
-      triangulation.begin()->vertex(v)
-       = pds.closest_point (Point<3>(triangulation.begin()->vertex(v)[0],
-                                     triangulation.begin()->vertex(v)[1],
-                                     0));
-       
-  for (unsigned int i=0; i<4; ++i)
-    {
-      for (typename Triangulation<dim>::active_cell_iterator
-            cell = triangulation.begin_active();
-          cell != triangulation.end(); ++cell)
-       for (unsigned int f=0; f<GeometryInfo<dim>::faces_per_cell; ++f)
-         if (cell->face(f)->boundary_indicator() == 1)
-           cell->set_refine_flag ();
-      
-      triangulation.execute_coarsening_and_refinement ();
-
-      std::cout << "Refinement cycle " << i << std::endl
-               << "   Number of active cells: "
-               << triangulation.n_active_cells()
-               << std::endl
-               << "   Total number of cells: "
-               << triangulation.n_cells()
-               << std::endl;
-
-    }
-  
-  
-  dof_handler.distribute_dofs (fe);
-
-  std::cout << "   Number of degrees of freedom: "
-           << dof_handler.n_dofs()
-           << std::endl;
-
-  sparsity_pattern.reinit (dof_handler.n_dofs(),
-                          dof_handler.n_dofs(),
-                          dof_handler.max_couplings_between_dofs());
-  DoFTools::make_sparsity_pattern (dof_handler, sparsity_pattern);
-  sparsity_pattern.compress();
-
-  system_matrix.reinit (sparsity_pattern);
-
-  solution.reinit (dof_handler.n_dofs());
-  system_rhs.reinit (dof_handler.n_dofs());
-}
-
-
-                                 // @sect4{LaplaceProblem::assemble_system}
-
-                                // Unlike in the previous example, we
-                                // would now like to use a
-                                // non-constant right hand side
-                                // function and non-zero boundary
-                                // values. Both are tasks that are
-                                // readily achieved with a only a few
-                                // new lines of code in the
-                                // assemblage of the matrix and right
-                                // hand side.
-                                //
-                                // More interesting, though, is the
-                                // way we assemble matrix and right
-                                // hand side vector dimension
-                                // independently: there is simply no
-                                // difference to the 
-                                // two-dimensional case. Since the
-                                // important objects used in this
-                                // function (quadrature formula,
-                                // FEValues) depend on the dimension
-                                // by way of a template parameter as
-                                // well, they can take care of
-                                // setting up properly everything for
-                                // the dimension for which this
-                                // function is compiled. By declaring
-                                // all classes which might depend on
-                                // the dimension using a template
-                                // parameter, the library can make
-                                // nearly all work for you and you
-                                // don't have to care about most
-                                // things.
-template <int dim>
-void LaplaceProblem<dim>::assemble_system () 
-{  
-  MatrixTools::create_laplace_matrix (dof_handler,
-                                     QGauss<dim>(2),
-                                     system_matrix);
-  system_rhs = 0;
-  
-  std::map<unsigned int,double> boundary_values;
-  VectorTools::interpolate_boundary_values (dof_handler,
-                                           0,
-                                           BoundaryValues<dim>(),
-                                           boundary_values);
-  MatrixTools::apply_boundary_values (boundary_values,
-                                     system_matrix,
-                                     solution,
-                                     system_rhs);
-}
-
-
-                                 // @sect4{LaplaceProblem::solve}
-
-                                // Solving the linear system of
-                                // equations is something that looks
-                                // almost identical in most
-                                // programs. In particular, it is
-                                // dimension independent, so this
-                                // function is copied verbatim from the
-                                // previous example.
-template <int dim>
-void LaplaceProblem<dim>::solve () 
-{
-                                  // NEW
-  SolverControl           solver_control (dof_handler.n_dofs(),
-                                         1e-12*system_rhs.l2_norm());
-  SolverCG<>              cg (solver_control);
-
-  PreconditionSSOR<> preconditioner;
-  preconditioner.initialize(system_matrix, 1.2);
-
-  cg.solve (system_matrix, solution, system_rhs,
-           preconditioner);
-}
-
-
-                                 // @sect4{LaplaceProblem::output_results}
-
-                                // This function also does what the
-                                // respective one did in step-3. No changes
-                                // here for dimension independence either.
-                                 //
-                                 // The only difference to the previous
-                                 // example is that we want to write output in
-                                 // GMV format, rather than for gnuplot (GMV
-                                 // is another graphics program that, contrary
-                                 // to gnuplot, shows data in nice colors,
-                                 // allows rotation of geometries with the
-                                 // mouse, and generates reasonable
-                                 // representations of 3d data; for ways to
-                                 // obtain it see the ReadMe file of
-                                 // deal.II). To write data in this format, we
-                                 // simply replace the
-                                 // <code>data_out.write_gnuplot</code> call by
-                                 // <code>data_out.write_gmv</code>.
-                                 //
-                                 // Since the program will run both 2d and 3d
-                                 // versions of the laplace solver, we use the
-                                 // dimension in the filename to generate
-                                 // distinct filenames for each run (in a
-                                 // better program, one would check whether
-                                 // `dim' can have other values than 2 or 3,
-                                 // but we neglect this here for the sake of
-                                 // brevity).
-template <int dim>
-void LaplaceProblem<dim>::output_results () const
-{
-  DataOut<dim> data_out;
-
-  data_out.attach_dof_handler (dof_handler);
-  data_out.add_data_vector (solution, "solution");
-
-  data_out.build_patches ();
-
-  std::ofstream output (dim == 2 ?
-                       "solution-2d.gmv" :
-                       "solution-3d.gmv");
-  data_out.write_gmv (output);
-}
-
-
-
-                                 // @sect4{LaplaceProblem::run}
-
-                                 // This is the function which has the
-                                // top-level control over
-                                // everything. Apart from one line of
-                                // additional output, it is the same
-                                // as for the previous example.
-template <int dim>
-void LaplaceProblem<dim>::run () 
-{
-  std::cout << "Solving problem in " << dim << " space dimensions." << std::endl;
-  
-  make_grid_and_dofs();
-  assemble_system ();
-  solve ();
-  output_results ();
-}
-
-
-                                 // @sect3{The <code>main</code> function}
-
-                                // And this is the main function. It also
-                                // looks mostly like in step-3, but if you
-                                // look at the code below, note how we first
-                                // create a variable of type
-                                // <code>LaplaceProblem@<2@></code> (forcing the
-                                // compiler to compile the class template
-                                // with <code>dim</code> replaced by <code>2</code>) and run a
-                                // 2d simulation, and then we do the whole
-                                // thing over in 3d.
-                                //
-                                // In practice, this is probably not what you
-                                // would do very frequently (you probably
-                                // either want to solve a 2d problem, or one
-                                // in 3d, but not both at the same
-                                // time). However, it demonstrates the
-                                // mechanism by which we can simply change
-                                // which dimension we want in a single place,
-                                // and thereby force the compiler to
-                                // recompile the dimension independent class
-                                // templates for the dimension we
-                                // request. The emphasis here lies on the
-                                // fact that we only need to change a single
-                                // place. This makes it rather trivial to
-                                // debug the program in 2d where computations
-                                // are fast, and then switch a single place
-                                // to a 3 to run the much more computing
-                                // intensive program in 3d for `real'
-                                // computations.
-                                //
-                                // Each of the two blocks is enclosed in
-                                // braces to make sure that the
-                                // <code>laplace_problem_2d</code> variable goes out
-                                // of scope (and releases the memory it
-                                // holds) before we move on to allocate
-                                // memory for the 3d case. Without the
-                                // additional braces, the
-                                // <code>laplace_problem_2d</code> variable would only
-                                // be destroyed at the end of the function,
-                                // i.e. after running the 3d problem, and
-                                // would needlessly hog memory while the 3d
-                                // run could actually use it.
-                                 //
-                                 // Finally, the first line of the function is
-                                 // used to suppress some output.  Remember
-                                 // that in the previous example, we had the
-                                 // output from the linear solvers about the
-                                 // starting residual and the number of the
-                                 // iteration where convergence was
-                                 // detected. This can be suppressed through
-                                 // the <code>deallog.depth_console(0)</code> call.
-                                 //
-                                 // The rationale here is the following: the
-                                 // deallog (i.e. deal-log, not de-allog)
-                                 // variable represents a stream to which some
-                                 // parts of the library write output. It
-                                 // redirects this output to the console and
-                                 // if required to a file. The output is
-                                 // nested in a way so that each function can
-                                 // use a prefix string (separated by colons)
-                                 // for each line of output; if it calls
-                                 // another function, that may also use its
-                                 // prefix which is then printed after the one
-                                 // of the calling function. Since output from
-                                 // functions which are nested deep below is
-                                 // usually not as important as top-level
-                                 // output, you can give the deallog variable
-                                 // a maximal depth of nested output for
-                                 // output to console and file. The depth zero
-                                 // which we gave here means that no output is
-                                 // written. By changing it you can get more
-                                 // information about the innards of the
-                                 // library.
-int main () 
-{
-  deallog.depth_console (0);
-  {
-    LaplaceProblem<3> laplace_problem_3d;
-    laplace_problem_3d.run ();
-  }
-  
-  return 0;
-}
diff --git a/deal.II/examples/step-26/surface-points b/deal.II/examples/step-26/surface-points
deleted file mode 100644 (file)
index 06c17bc..0000000
+++ /dev/null
@@ -1,199202 +0,0 @@
-                100.000702 64.444870 18.640993
-                100.001732 58.902710 15.242172
-                100.003227 61.463951 16.076944
-                100.006111 61.650089 16.086910
-                100.008110 57.082504 14.420845
-                100.008537 62.520744 16.962706
-                10.001202 56.855850 -7.629524
-                10.001554 58.377510 -6.702730
-                100.017540 59.961662 16.179766
-                100.019104 61.271141 16.155931
-                100.019402 65.486839 19.577337
-                100.020836 65.672371 19.579966
-                100.021027 64.623596 18.729914
-                100.022263 67.759712 21.275852
-                100.022972 59.083233 15.336061
-                100.023674 57.265400 14.488837
-                10.002565 46.178841 -11.539805
-                100.030266 65.854836 19.618811
-                100.031487 60.144032 16.241335
-                10.003348 46.376587 -11.537080
-                100.034447 64.804901 18.787405
-                100.034851 66.712395 20.490171
-                100.035156 60.329926 16.253040
-                100.035728 63.752766 17.941080
-                100.035797 61.078152 16.238670
-                10.003754 55.872288 -7.660773
-                100.038864 60.515800 16.265440
-                100.040543 57.447708 14.564336
-                100.042076 61.824642 16.249334
-                100.042709 60.701557 16.279892
-                100.043449 60.888432 16.278502
-                100.045784 62.694252 17.131380
-                10.004656 51.236549 -9.581793
-                100.050140 59.261642 15.459518
-                100.053658 57.631260 14.620840
-                100.058922 64.981941 18.896219
-                100.061073 66.887749 20.606680
-                100.061508 62.875450 17.199224
-                10.007089 44.035114 -12.545167
-                100.074310 57.812305 14.713459
-                100.074501 63.057632 17.254875
-                100.075668 67.923332 21.517727
-                100.076317 66.022865 19.827442
-                100.083374 67.064568 20.704491
-                100.087883 59.436562 15.631111
-                100.090866 61.994549 16.472155
-                100.094849 63.917332 18.211142
-                100.099335 63.235420 17.365744
-                100.101952 57.991070 14.839401
-                100.105995 67.241211 20.804350
-                100.115295 65.146477 19.153843
-                10.011702 53.559151 -8.649872
-                100.122711 62.170326 16.615950
-                10.012673 48.515648 -10.602343
-                100.128403 66.188332 20.063452
-                10.012884 45.982822 -11.592372
-                100.132256 59.609089 15.835029
-                100.134087 58.168442 14.984985
-                10.013704 58.184128 -6.760799
-                100.139801 64.087044 18.414787
-                100.140167 67.413055 20.957655
-                100.142906 63.406101 17.565250
-                10.014887 41.884617 -13.574464
-                100.152702 58.349888 15.068696
-                100.164810 62.342213 16.808329
-                100.167046 58.532776 15.130359
-                100.168304 59.784328 15.998121
-                100.175049 66.355789 20.274775
-                100.177879 64.259201 18.587055
-                100.181488 65.307114 19.456329
-                100.184837 58.714401 15.210201
-                100.191704 59.963554 16.104029
-                100.193550 67.576942 21.199200
-                100.200775 63.571430 17.829426
-                100.201126 64.436836 18.690355
-                100.201767 60.147324 16.146217
-                100.202255 60.521412 16.139502
-                100.202278 60.334347 16.143690
-                100.204582 60.707813 16.145515
-                10.020597 51.042603 -9.657804
-                100.207863 58.894363 15.313117
-                100.207886 62.513699 17.004513
-                100.208588 60.893494 16.160715
-                100.211090 65.481773 19.588947
-                100.211334 66.527336 20.436777
-                100.212578 61.079308 16.174498
-                100.213661 61.266121 16.174107
-                100.213852 64.618408 18.744150
-                100.222664 61.450199 16.211346
-                100.226418 65.661911 19.654709
-                100.229668 64.798683 18.814240
-                100.230621 59.074234 15.416235
-                10.023313 55.680271 -7.751896
-                100.233643 62.691345 17.119911
-                100.242561 66.700706 20.576151
-                100.244522 62.874283 17.165791
-                100.249741 67.739487 21.454386
-                100.255127 59.253548 15.525524
-                100.257378 61.625057 16.370146
-                100.259109 63.736416 18.096378
-                100.259888 63.055580 17.232111
-                10.025993 48.320618 -10.667483
-                100.261971 64.972626 18.959612
-                100.261978 65.834084 19.814993
-                100.264511 66.877670 20.673069
-                10.027419 57.991074 -6.824481
-                10.028183 53.365910 -8.728141
-                100.285469 59.430782 15.664100
-                100.286530 67.054581 20.770348
-                100.293571 61.799412 16.533905
-                100.297737 67.905190 21.671587
-                100.300491 63.227432 17.418148
-                10.030302 43.840439 -12.656449
-                100.304710 66.003288 20.008574
-                10.030633 50.650280 -9.713679
-                100.308914 67.231293 20.868673
-                100.312363 61.979786 16.616913
-                10.031318 50.847610 -9.711220
-                100.315308 63.901958 18.353388
-                100.316589 59.607716 15.804996
-                100.326180 65.134163 19.252838
-                10.032937 45.788334 -11.688521
-                100.332375 62.159595 16.706976
-                10.033606 50.453648 -9.731759
-                100.337791 59.787792 15.900807
-                10.033841 41.688644 -13.666801
-                100.349197 59.971184 15.948731
-                100.349609 66.171486 20.211603
-                10.035273 48.124882 -10.715206
-                100.353600 60.156902 15.964468
-                100.353783 64.074005 18.527214
-                100.355919 60.343220 15.971826
-                100.362778 62.335705 16.844276
-                100.365723 60.527111 16.012901
-                100.368736 67.392578 21.141171
-                100.371986 63.387829 17.746969
-                100.380569 60.709240 16.076942
-                100.382980 64.249420 18.658401
-                100.385620 65.297318 19.524466
-                10.038827 55.487228 -7.825045
-                100.391273 66.340820 20.400236
-                100.393356 62.511681 16.981792
-                100.397354 60.890511 16.152081
-                100.398338 64.430000 18.725111
-                100.412346 67.560310 21.338470
-                100.413185 62.691410 17.069660
-                100.413963 64.610344 18.793886
-                100.414383 61.071777 16.226948
-                100.417580 65.471024 19.668270
-                100.419914 66.515213 20.528437
-                100.428154 61.254124 16.285997
-                100.428268 62.872841 17.135338
-                100.429314 63.553230 18.009125
-                100.440521 64.786514 18.912792
-                100.450424 61.433376 16.386810
-                100.450836 66.688744 20.665277
-                100.452202 65.643532 19.823942
-                100.459595 67.726433 21.552071
-                100.460121 63.047966 17.280224
-                10.046725 53.173069 -8.815616
-                100.476738 63.722160 18.225605
-                100.479301 61.610283 16.517199
-                100.482414 66.861786 20.807068
-                100.485802 64.955414 19.118967
-                100.486633 65.816032 19.979284
-                100.500282 61.789925 16.610022
-                10.050298 47.930092 -10.788013
-                100.505249 67.038269 20.908285
-                10.050729 57.800552 -6.932127
-                100.508743 67.891586 21.775076
-                100.511139 61.973000 16.657358
-                100.518394 65.989471 20.122162
-                100.519402 63.892666 18.419939
-                100.526924 62.154377 16.726427
-                100.536896 67.211075 21.049841
-                100.539734 63.205482 17.647655
-                100.546112 65.118332 19.394560
-                10.054816 41.492802 -13.768233
-                100.556450 64.065178 18.587433
-                100.557022 62.330544 16.862539
-                10.055902 50.260883 -9.836891
-                100.559540 66.159065 20.308157
-                100.580254 64.242577 18.693752
-                100.582199 62.508389 16.975050
-                100.584984 67.377037 21.268320
-                100.593361 66.331436 20.460648
-                100.595100 65.285439 19.617983
-                100.596848 64.422562 18.767702
-                100.599976 63.369892 17.924480
-                10.060064 45.594830 -11.816424
-                100.602661 62.687897 17.066004
-                100.617241 67.549377 21.413292
-                10.062499 55.296185 -7.934549
-                100.625099 66.504646 20.602114
-                100.628227 64.596909 18.909344
-                10.063089 52.979675 -8.893394
-                100.637566 65.454918 19.811062
-                100.640221 62.860977 17.238491
-                100.640991 63.541233 18.110840
-                10.065509 43.647335 -12.821613
-                100.657967 67.718102 21.597040
-                100.660622 66.676132 20.761944
-                100.669029 64.767555 19.094362
-                100.675201 65.626183 19.981592
-                100.681999 63.712471 18.297752
-                10.068612 57.608555 -7.014975
-                100.693291 66.848648 20.909086
-                100.702637 65.801338 20.104006
-                100.706421 67.883476 21.817226
-                10.071006 41.296379 -13.848843
-                100.710907 63.021862 17.564184
-                100.718437 64.934700 19.320145
-                100.718605 63.885216 18.464123
-                100.727142 67.020576 21.061739
-                100.727859 65.977272 20.216726
-                10.074036 47.736797 -10.900516
-                100.748070 64.060532 18.597046
-                100.758804 63.190907 17.783720
-                100.762878 65.103668 19.522800
-                100.764748 66.148476 20.383770
-                100.766052 67.190308 21.237358
-                100.774612 64.236740 18.717054
-                10.077521 50.067905 -9.938993
-                100.788422 67.366730 21.336634
-                10.079432 52.786213 -8.970445
-                100.794388 66.322449 20.516512
-                100.803253 63.361057 17.987076
-                100.807381 65.272423 19.725117
-                10.080865 45.400352 -11.916655
-                100.809898 64.409622 18.877279
-                100.821480 67.538635 21.485559
-                100.830070 66.494041 20.676613
-                10.083317 51.998894 -9.007666
-                10.083881 57.415810 -7.086842
-                100.840202 63.533844 18.155346
-                100.852249 64.579720 19.070475
-                100.856773 67.709534 21.644390
-                100.865883 66.665298 20.838100
-                100.876511 63.706776 18.319969
-                10.088021 41.099964 -13.932439
-                10.088846 55.105656 -8.055913
-                10.091293 47.542305 -10.983271
-                100.913208 63.879433 18.487032
-                10.092822 45.204388 -11.976561
-                10.093253 52.198055 -9.047166
-                100.938004 64.056465 18.598368
-                10.094789 47.345417 -11.005172
-                10.097180 52.593037 -9.054398
-                10.101335 49.875271 -10.050872
-                10.101338 52.396889 -9.078419
-                10.101428 43.454197 -12.989997
-                10.104673 51.806442 -9.108394
-                10.104776 56.240906 -7.202295
-                10.106004 43.058563 -13.023809
-                10.106209 40.903622 -14.021354
-                10.106443 43.256744 -13.018799
-                10.107475 49.284637 -10.094850
-                10.108162 49.482048 -10.092410
-                10.108521 54.913441 -8.147561
-                10.108848 49.679455 -10.089946
-                10.109529 45.009094 -12.057730
-                10.113775 47.151134 -11.096390
-                10.119234 56.441257 -7.263066
-                10.119496 49.089668 -10.154770
-                10.120212 42.862339 -13.094607
-                10.120967 54.719475 -8.207774
-                10.126995 40.707516 -14.123263
-                10.129342 54.524544 -8.249127
-                10.129992 44.814354 -12.156454
-                10.131824 57.231525 -7.303281
-                10.133741 46.956947 -11.191963
-                10.141135 48.896404 -10.256976
-                10.142178 40.510750 -14.199347
-                10.142604 42.667080 -13.202826
-                10.143004 46.760921 -11.238953
-                10.144761 59.148823 -6.428877
-                10.145706 54.331444 -8.326755
-                10.151984 51.619274 -9.324196
-                10.154605 44.620121 -12.273109
-                10.155189 56.056789 -7.430306
-                10.156584 56.647480 -7.424881
-                10.157012 40.313831 -14.273122
-                10.157655 54.137295 -8.384105
-                10.159718 51.029572 -9.374467
-                10.160200 51.226746 -9.371095
-                10.160631 46.566216 -11.324110
-                10.160782 51.424007 -9.368896
-                10.163677 57.042923 -7.448198
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-                99.646721 67.953148 21.282005
-                99.647003 60.152271 16.225624
-                99.648308 55.258430 13.672398
-                99.650833 64.814514 18.768990
-                99.652206 46.171989 11.161873
-                99.653259 55.444912 13.690613
-                99.655563 52.675159 12.863125
-                99.657150 57.829891 14.550034
-                99.657341 60.335968 16.268286
-                99.657951 55.631405 13.709031
-                99.658134 49.521297 12.035785
-                99.662163 63.944847 17.966507
-                99.663559 59.270191 15.436463
-                99.670372 60.518623 16.325350
-                99.672295 67.086357 20.553312
-                99.672356 55.814995 13.772834
-                9.967287 46.765934 -11.362288
-                99.676163 61.826759 16.319927
-                99.676476 61.639446 16.326239
-                99.677025 46.355408 11.276357
-                99.678154 49.704842 12.126154
-                99.678459 64.990402 18.891361
-                99.678764 52.857082 12.968238
-                99.682739 58.009441 14.665938
-                99.683723 55.999474 13.821873
-                99.684677 63.068680 17.212458
-                99.684692 62.010883 16.356123
-                99.689262 60.699333 16.408377
-                9.969228 42.473640 -13.347318
-                99.693718 49.889362 12.195580
-                99.694557 61.445915 16.415703
-                99.695580 62.194210 16.401867
-                99.696785 59.446690 15.586424
-                99.698669 56.182899 13.886822
-                9.969870 56.650982 -7.493515
-                99.698967 46.539383 11.375873
-                99.699883 53.039444 13.063558
-                99.703629 50.075207 12.237954
-                99.703842 67.259514 20.693920
-                99.707115 64.114761 18.169964
-                99.708092 60.880039 16.490471
-                99.709480 66.398819 19.871624
-                99.710129 50.261768 12.266268
-                99.711929 58.187855 14.797300
-                99.712898 50.449242 12.275712
-                99.713303 53.223831 13.122275
-                99.713432 46.724712 11.441700
-                99.714333 61.252003 16.511179
-                99.716484 63.244011 17.355257
-                99.717651 56.365017 13.972044
-                99.720406 65.160713 19.080559
-                99.720451 46.911633 11.471093
-                99.721748 50.635265 12.313705
-                99.721863 62.372028 16.518671
-                99.722137 47.099640 11.475466
-                99.723083 61.061897 16.556679
-                99.724823 47.287373 11.485889
-                99.725098 53.408573 13.173983
-                9.972901 49.297783 -10.401513
-                99.732597 50.820770 12.360997
-                99.733810 47.473824 11.523961
-                99.736221 58.367630 14.906514
-                99.738083 53.592991 13.231247
-                99.738632 67.431252 20.849514
-                99.740166 59.619621 15.784365
-                99.741112 56.545719 14.078215
-                99.744827 51.005898 12.414829
-                99.748260 64.285950 18.355743
-                99.750237 63.418510 17.506527
-                99.750748 53.777451 13.287013
-                99.752007 47.658245 11.607283
-                99.756638 51.191124 12.466690
-                9.975694 58.763874 -6.581076
-                99.757698 58.548355 15.001878
-                99.763763 53.961853 13.343053
-                99.764069 56.726494 14.181993
-                99.765373 47.843674 11.666517
-                99.769669 66.560982 20.144121
-                99.773026 51.375130 12.539430
-                99.774788 48.029949 11.707224
-                99.776604 62.539536 16.768358
-                99.777786 67.601074 21.024618
-                99.778877 58.729004 15.097206
-                99.778961 54.145523 13.410700
-                99.779541 63.594540 17.637705
-                99.780701 56.909203 14.254284
-                9.978109 48.904022 -10.435844
-                99.783615 48.216225 11.746490
-                99.784859 65.322182 19.373913
-                99.786179 59.791653 15.993851
-                9.978797 49.101543 -10.433401
-                9.978950 51.428375 -9.462043
-                99.790642 64.456520 18.546741
-                99.791763 54.329811 13.466501
-                99.793678 51.558002 12.632773
-                99.798546 57.091431 14.334145
-                99.800453 58.909546 15.193002
-                9.980282 53.945919 -8.500159
-                99.804565 54.514091 13.522948
-                99.811157 51.741570 12.712579
-                99.811234 67.773140 21.174026
-                99.813103 65.869919 19.489609
-                9.981438 42.276978 -13.409284
-                99.815964 57.273689 14.412080
-                99.817383 54.698288 13.579415
-                99.818306 63.766842 17.814167
-                99.820717 66.726616 20.375055
-                99.822548 51.926708 12.762508
-                99.822990 61.651234 16.120476
-                99.823120 59.089565 15.294690
-                99.823212 59.966427 16.162848
-                99.824684 65.679184 19.547922
-                99.825340 64.629860 18.702715
-                99.827423 65.491928 19.565371
-                99.830322 54.882442 13.636503
-                99.830429 57.456932 14.474803
-                99.832527 61.460609 16.170282
-                99.832924 52.112068 12.807611
-                99.833939 66.047981 19.580568
-                99.834114 61.834469 16.168802
-                99.834457 62.705620 17.033882
-                99.838333 52.298687 12.829450
-                99.841652 55.067017 13.686017
-                99.844131 60.146545 16.255472
-                99.845673 52.484783 12.859732
-                99.845757 64.808525 18.792540
-                99.847221 57.639309 14.549195
-                99.850555 55.252335 13.722189
-                99.852867 59.267277 15.428600
-                99.854424 60.330162 16.299297
-                99.854683 66.898903 20.527138
-                99.854736 62.014442 16.259949
-                99.859528 55.437515 13.760698
-                99.860222 67.938652 21.395058
-                99.862007 52.668484 12.932811
-                99.862282 61.263046 16.313772
-                9.986426 44.230129 -12.446037
-                99.865013 60.513702 16.343775
-                99.866707 57.820873 14.635386
-                99.868690 62.880207 17.187683
-                99.869034 64.986069 18.894987
-                99.874374 63.932655 18.069038
-                99.876442 55.620350 13.836384
-                99.877594 60.696529 16.397547
-                99.878815 67.075081 20.633181
-                99.882744 52.850967 13.026515
-                99.883293 61.068710 16.415190
-                9.988397 48.708549 -10.487426
-                99.886414 60.880543 16.435129
-                99.887962 62.189857 16.409904
-                99.889366 63.059582 17.278975
-                99.890221 59.442287 15.599425
-                99.891731 55.803642 13.904469
-                99.892960 58.000145 14.754607
-                9.989303 58.570900 -6.644942
-                99.896713 66.209381 19.866161
-                99.906609 55.987053 13.969194
-                99.906761 53.032536 13.135685
-                99.907623 67.249283 20.761419
-                99.913513 63.237652 17.386446
-                99.917328 65.153847 19.113947
-                99.920311 64.102112 18.277418
-                99.920998 56.170506 14.032515
-                99.923592 58.177933 14.894255
-                99.926003 53.215267 13.222380
-                9.992927 53.751884 -8.561370
-                99.935822 56.353779 14.098366
-                99.940186 67.421852 20.907030
-                99.941437 53.399075 13.289879
-                99.941696 62.357807 16.654789
-                99.943237 59.612103 15.841970
-                99.944984 63.412937 17.527386
-                99.945381 58.358532 14.990673
-                99.950729 66.374039 20.111197
-                99.953911 53.583557 13.344302
-                99.955971 56.535397 14.189234
-                9.996245 46.573074 -11.498395
-                99.962662 58.540436 15.067961
-                99.963776 64.272331 18.474363
-                99.973526 56.717819 14.266535
-                9.997534 42.080753 -13.489339
-                99.981133 67.590904 21.090704
-                99.981483 63.586147 17.693605
-                99.981560 58.721859 15.151418
-                99.984962 65.314011 19.422552
-                99.985909 59.785046 16.037029
-                99.990257 56.900345 14.340770
-                99.995491 66.542198 20.312881
diff --git a/deal.II/examples/step-32/Makefile b/deal.II/examples/step-32/Makefile
deleted file mode 100644 (file)
index 770b78a..0000000
+++ /dev/null
@@ -1,178 +0,0 @@
-# $Id: Makefile,v 1.4 2006/02/10 17:53:05 wolf Exp $
-
-
-# For the small projects Makefile, you basically need to fill in only
-# four fields.
-#
-# The first is the name of the application. It is assumed that the
-# application name is the same as the base file name of the single C++
-# file from which the application is generated.
-target = $(basename $(shell echo step-*.cc))
-
-# The second field determines whether you want to run your program in
-# debug or optimized mode. The latter is significantly faster, but no
-# run-time checking of parameters and internal states is performed, so
-# you should set this value to `on' while you develop your program,
-# and to `off' when running production computations.
-debug-mode = off
-
-
-# As third field, we need to give the path to the top-level deal.II
-# directory. You need to adjust this to your needs. Since this path is
-# probably the most often needed one in the Makefile internals, it is
-# designated by a single-character variable, since that can be
-# reference using $D only, i.e. without the parentheses that are
-# required for most other parameters, as e.g. in $(target).
-D = ../../
-
-
-# The last field specifies the names of data and other files that
-# shall be deleted when calling `make clean'. Object and backup files,
-# executables and the like are removed anyway. Here, we give a list of
-# files in the various output formats that deal.II supports.
-clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk
-
-
-
-
-#
-#
-# Usually, you will not need to change anything beyond this point.
-#
-#
-# The next statement tell the `make' program where to find the
-# deal.II top level directory and to include the file with the global
-# settings
-include $D/common/Make.global_options
-
-################################################################
-# This example program will only work if Trilinos is installed. If this
-# is not the case, then simply redefine the main targets to do nothing
-ifneq ($(USE_CONTRIB_TRILINOS),yes)
-default run clean:
-       @echo
-       @echo "==========================================================="
-       @echo "=   This program cannot be compiled without Trilinos. Make="
-       @echo "=   sure you have Trilinos installed and detected during  ="
-       @echo "=   configuration of deal.II                              ="
-       @echo "==========================================================="
-       @echo
-
-else
-#
-################################################################
-
-
-
-
-# Since the whole project consists of only one file, we need not
-# consider difficult dependencies. We only have to declare the
-# libraries which we want to link to the object file, and there need
-# to be two sets of libraries: one for the debug mode version of the
-# application and one for the optimized mode. Here we have selected
-# the versions for 2d. Note that the order in which the libraries are
-# given here is important and that your applications won't link
-# properly if they are given in another order.
-#
-# You may need to augment the lists of libraries when compiling your
-# program for other dimensions, or when using third party libraries
-libs.g   = $(lib-deal2-2d.g) \
-          $(lib-deal2-3d.g) \
-          $(lib-lac.g)      \
-           $(lib-base.g)
-libs.o   = $(lib-deal2-2d.o) \
-          $(lib-deal2-3d.o) \
-          $(lib-lac.o)      \
-           $(lib-base.o)
-
-
-# We now use the variable defined above which switch between debug and
-# optimized mode to select the set of libraries to link with. Included
-# in the list of libraries is the name of the object file which we
-# will produce from the single C++ file. Note that by default we use
-# the extension .g.o for object files compiled in debug mode and .o for
-# object files in optimized mode (or whatever the local default on your
-# system is instead of .o).
-ifeq ($(debug-mode),on)
-  libraries = $(target).g.$(OBJEXT) $(libs.g)
-else
-  libraries = $(target).$(OBJEXT) $(libs.o)
-endif
-
-
-# Now comes the first production rule: how to link the single object
-# file produced from the single C++ file into the executable. Since
-# this is the first rule in the Makefile, it is the one `make' selects
-# if you call it without arguments.
-$(target) : $(libraries)
-       @echo ============================ Linking $@
-       @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS)
-
-
-# To make running the application somewhat independent of the actual
-# program name, we usually declare a rule `run' which simply runs the
-# program. You can then run it by typing `make run'. This is also
-# useful if you want to call the executable with arguments which do
-# not change frequently. You may then want to add them to the
-# following rule:
-run: $(target)
-       @echo ============================ Running $<
-       @./$(target)$(EXEEXT)
-
-
-
-# As a last rule to the `make' program, we define what to do when
-# cleaning up a directory. This usually involves deleting object files
-# and other automatically created files such as the executable itself,
-# backup files, and data files. Since the latter are not usually quite
-# diverse, you needed to declare them at the top of this file.
-clean:
-       -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files)
-
-
-# Since we have not yet stated how to make an object file from a C++
-# file, we should do so now. Since the many flags passed to the
-# compiler are usually not of much interest, we suppress the actual
-# command line using the `at' sign in the first column of the rules
-# and write the string indicating what we do instead.
-./%.g.$(OBJEXT) :
-       @echo ==============debug========= $(<F)
-       @$(CXX) $(CXXFLAGS.g) -c $< -o $@
-./%.$(OBJEXT) :
-       @echo ==============optimized===== $(<F)
-       @$(CXX) $(CXXFLAGS.o) -c $< -o $@
-
-
-
-# The following statement tells make that the rules `run' and `clean'
-# are not expected to produce files of the same name as Makefile rules
-# usually do.
-.PHONY: run clean
-
-
-# Finally there is a rule which you normally need not care much about:
-# since the executable depends on some include files from the library,
-# besides the C++ application file of course, it is necessary to
-# re-generate the executable when one of the files it depends on has
-# changed. The following rule to created a dependency file
-# `Makefile.dep', which `make' uses to determine when to regenerate
-# the executable. This file is automagically remade whenever needed,
-# i.e. whenever one of the cc-/h-files changed. Make detects whether
-# to remake this file upon inclusion at the bottom of this file.
-#
-# If the creation of Makefile.dep fails, blow it away and fail
-Makefile.dep: $(target).cc Makefile \
-              $(shell echo $D/*/include/*/*.h)
-       @echo ============================ Remaking $@
-       @$D/common/scripts/make_dependencies  $(INCLUDE) -B. $(target).cc \
-               > $@ \
-         || (rm -f $@ ; false)
-       @if test -s $@ ; then : else rm $@ ; fi
-
-
-# To make the dependencies known to `make', we finally have to include
-# them:
-include Makefile.dep
-
-
-endif  # USE_CONTRIB_TRILINOS
diff --git a/deal.II/examples/step-32/doc/builds-on b/deal.II/examples/step-32/doc/builds-on
deleted file mode 100644 (file)
index 11e000d..0000000
+++ /dev/null
@@ -1 +0,0 @@
-step-31 step-17
diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox
deleted file mode 100644 (file)
index 1b29e53..0000000
+++ /dev/null
@@ -1,920 +0,0 @@
-<br>
-
-<i>This program was contributed by Martin Kronbichler and Wolfgang
-Bangerth.
-<br>
-This material is based upon work partly supported by the National
-Science Foundation under Award No. EAR-0426271 and The California Institute of
-Technology. Any opinions, findings, and conclusions or recommendations
-expressed in this publication are those of the author and do not
-necessarily reflect the views of the National Science Foundation or of The
-California Institute of Technology.
-</i>
-
-
-<a name="Intro"></a>
-<h1>Introduction</h1>
-
-This program does pretty much exactly what step-31 already
-does: it solves the Boussinesq equations that describe the motion of a fluid
-whose temperature is not in equilibrium. As such, all the equations we have
-described in step-31 still hold: we solve the same partial
-differential equation, using the same finite element scheme, the same time
-stepping algorithm, and the same stabilization method for the temperature
-advection-diffusion equation. As a consequence, you may first want to
-understand that program before you work on the current one.
-
-The difference between step-31 and the current program is that
-here we want to do things in %parallel, using both the availability of many
-machines in a cluster (with parallelization based on MPI) as well as many
-processor cores within a single machine (with parallelization based on
-threads). This program's main job is therefore to introduce the changes that are
-necessary to utilize the availability of these %parallel compute resources.
-
-In addition to these changes, we also use a slightly different
-preconditioner, and we will have to make a number of changes that have
-to do with the fact that we want to solve a <i>realistic</i> problem
-here, not a model problem. The latter, in particular, will require
-that we think about scaling issues as well as what all those
-parameters and coefficients in the equations under consideration
-actually mean. We will discuss first the issues that affect changes in
-the mathematical formulation and solver structure, then how to
-parallelize things, and finally the actual testcase we will consider.
-
-
-<h3> Using the "right" pressure </h3>
-
-In step-31, we used the following Stokes model for the
-velocity and pressure field:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  -\rho \; \beta \; T \mathbf{g},
-  \\
-  \nabla \cdot {\mathbf u} &=& 0.
-@f}
-The right hand side of the first equation appears a wee bit
-unmotivated. Here's how things should really be. On the right side, we
-need the external forces that act on the fluid, which we assume are
-given by gravity only. In the current case, we assume that the fluid
-does expand slightly for the purposes of this gravity force, but not
-enough that we need to modify the incompressibility condition (the
-second equation). What this means is that for the purpose of the right
-hand side, we can assume that $\rho=\rho(T)$. An assumption that may
-not be entirely justified is that we can assume that the changes of
-density as a function of temperature are small, leading to an
-expression of the form $\rho(T) = \rho_{\text{ref}}
-[1-\beta(T-T_{\text{ref}})]$, i.e. the density equals
-$\rho_{\text{ref}}$ at reference temperature and decreases linearly as
-the temperature increases (as the material expands). The force balance
-equation then looks properly written like this:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  \rho_{\text{ref}} [1-\beta(T-T_{\text{ref}})] \mathbf{g}.
-@f}
-Now note that the gravity force results from a gravity potential as
-$\mathbf g=-\nabla \varphi$, so that we can re-write this as follows:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  -\rho_{\text{ref}} \; \beta\; T\; \mathbf{g}
-  -\rho_{\text{ref}} [1+\beta T_{\text{ref}}] \nabla\varphi.
-@f}
-The second term on the right is time independent, and so we could
-introduce a new "dynamic" pressure $p_{\text{dyn}}=p+\rho_{\text{ref}}
-[1+\beta T_{\text{ref}}] \varphi=p_{\text{total}}-p_{\text{static}}$
-with which the Stokes equations would read:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p_{\text{dyn}} &=&
-  -\rho_{\text{ref}} \; \beta \; T \; \mathbf{g},
-  \\
-  \nabla \cdot {\mathbf u} &=& 0.
-@f}
-This is exactly the form we used in step-31, and it was
-appropriate to do so because all changes in the fluid flow are only
-driven by the dynamic pressure that results from temperature
-differences.
-
-On the other hand, we will here use the form of the Stokes equations
-that considers the total pressure instead:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  \rho(T)\; \mathbf{g},
-  \\
-  \nabla \cdot {\mathbf u} &=& 0.
-@f}
-This way we can plot the pressure in our program in such a way that it
-actually shows the total pressure that includes the effects of
-temperature differences as well as the static pressure of the
-overlying rocks. Since the pressure does not appear any further in any
-of the other equations, whether to use one or the other is more a
-matter of taste than of correctness. The flow field is exactly the
-same, but we get a pressure that we can now compare with values that
-are given in geophysical books as those that hold at the bottom of the
-earth mantle, for example.
-
-A second reason to do this is discussed in the results section and
-concerns possible extensions to the model we use here. It has to do
-with the fact that while the temperature equation we use here does not
-include a term that contains the pressure. It should, however:
-rock, like gas, heats up as you compress it. Consequently,
-material that rises up cools adiabatically, and cold material that
-sinks down heats adiabatically. We discuss this further below.
-
-
-
-
-<h3> The scaling of discretized equations </h3>
-
-Remember that we want to solve the following set of equations:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  \rho(T) \mathbf{g},
-  \\
-  \nabla \cdot {\mathbf u} &=& 0,
-  \\
-  \frac{\partial T}{\partial t}
-  +
-  {\mathbf u} \cdot \nabla T
-  -
-  \nabla \cdot \kappa \nabla T &=& \gamma,
-@f}
-augmented by appropriate boundary and initial conditions. As discussed
-in step-31, we will solve this set of equations by
-solving for a Stokes problem first in each time step, and then moving
-the temperature equation forward by one time interval.
-
-The problem under consideration in this current section is with the
-Stokes problem: if we discretize it as usual, we get a linear system
-@f{eqnarray*}
-  M \; X
-  =
-  \left(\begin{array}{cc}
-    A & B^T \\ B & 0
-  \end{array}\right)
-  \left(\begin{array}{c}
-    U \\ P
-  \end{array}\right)
-  =
-  \left(\begin{array}{c}
-    F_U \\ 0
-  \end{array}\right)
-  =
-  F
-@f}
-which in this program we will solve with a BiCGStab solver. BiCGStab
-iterates until the residual of these linear equations is below a
-certain tolerance, i.e. until
-@f[
-  \left\|
-  \left(\begin{array}{c}
-    F_U - A U^{(k)} - B P^{(k)}
-    \\
-    B^T U^{(k)}
-  \end{array}\right)
-  \right\|
-  < \text{Tol}.
-@f]
-This does not make any sense from the viewpoint of physical units: the
-quantities involved here have physical units so that the first part of
-the residual has units $\frac{\text{Pa}}{\text{m}}
-\text{m}^{\text{dim}}$ (most easily established by considering the
-term $(\nabla \cdot \mathbf v, p)_{\Omega}$ and considering that the
-pressure has units $\text{Pa}=\frac{\text{kg}}{\text{m\; s}^2}$ and
-the integration yields a factor of $\text{m}^{\text{dim}}$), whereas
-the second part of the residual has units
-$\frac{\text{m}^{\text{dim}}}{\text{s}}$. Taking the norm
-of this residual vector would yield a quantity with units
-$\text{m}^{\text{dim}-1} \sqrt{\left(\text{Pa}\right)^2 +
-       \left(\frac{\text{m}}{\text{s}}\right)^2}$. This,
-quite obviously, does not make sense, and we should not be surprised
-that doing so is eventually going to come back hurting us.
-
-So why is this an issue here, but not in step-31? The
-reason back there is that everything was nicely balanced: velocities
-were on the order of one, the pressure likewise, the viscosity was
-one, and the domain had a diameter of $\sqrt{2}$. As a result, while
-non-sensical, nothing bad happened. On the other hand, as we will explain
-below, things here will not be that simply scaled: $\eta$ will be around
-$10^{21}$, velocities on the order of $10^{-8}$, pressure around $10^8$, and
-the diameter of the domain is $10^7$. In other words, the order of magnitude
-for the first equation is going to be
-$\eta\text{div}\varepsilon(\mathbf u) \approx 10^{21} \frac{10^{-8}}{(10^7)^2}
-\approx 10^{-1}$, whereas the second equation will be around
-$\text{div}{\mathbf u}\approx \frac{10^{-8}}{10^7} \approx 10^{-15}$. Well, so
-what this will lead to is this: if BiCGStab wants to make the residual small,
-it will almost entirely focus on the first set of equations because they are
-so much bigger, and ignore the divergence equation that describes mass
-conservation. That's exactly what happens: unless we set the tolerance to
-extremely small values, the resulting flow field is definitely not divergence
-free. As an auxiliary problem, it turns out that it is difficult to find a
-tolerance that always works; in practice, one often ends up with a tolerance
-that requires 30 or 40 iterations for most time steps, and 10,000 for some
-others.
-
-So what's a numerical analyst to do in a case like this? The answer is to
-start at the root and first make sure that everything is mathematically
-consistent first. In our case, this means that if we want to solve the system
-of Stokes equations jointly, we have to scale them so that they all have the
-same physical dimensions. In our case, this means multiplying the second
-equation by something that has units $\frac{\text{Pa\; s}}{\text{m}}$; one
-choice is to multiply with $\frac{\eta}{L}$ where $L$ is a typical lengthscale
-in our domain. Using the %numbers above, this factor is around $10^{14}$, which
-just so happens to be the order of magnitude that would make the two equations
-numerically about the same. So, we now get this for the Stokes system:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=&
-  \rho(T) \; \mathbf{g},
-  \\
-  \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0.
-@f}
-The trouble with this is that the result is not symmetric any more (we have
-$\frac{\eta}{L} \nabla \cdot$ at the bottom left, but not its transpose
-operator at the top right). This, however, can be cured by introducing a
-scaled pressure $\hat p = \frac{L}{\eta}p$, and we get the scaled equations
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) +
-  \nabla \left(\frac{\eta}{L} \hat p\right) &=&
-  \rho(T) \; \mathbf{g},
-  \\
-  \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0.
-@f}
-This is now symmetric. Obviously, we can easily recover the original pressure
-$p$ from the scaled pressure $\hat p$ that we compute as a result of this
-procedure.
-
-In the program below, we will introduce a factor
-<code>EquationData::pressure_scaling</code> that corresponds to
-$\frac{\eta}{L}$, and we will use this factor in the assembly of the system
-matrix and preconditioner. We will recover the unscaled pressure in the
-<code>output_results</code> function.
-
-
-<h3> Changes to the Stokes preconditioner </h3>
-
-In this tutorial program, we apply a variant of the preconditioner used in
-step-31. That preconditioner was built to operate on the
-system matrix <i>M</i> in block form such that the product matrix
-@f{eqnarray*}
-  P^{-1} M
-  =
-  \left(\begin{array}{cc}
-    A^{-1} & 0 \\ S^{-1} B A^{-1} & -S^{-1}
-  \end{array}\right)
-  \left(\begin{array}{cc}
-    A & B^T \\ B & 0
-  \end{array}\right)
-@f}
-is of a form that Krylov-based iterative solvers like GMRES can solve in a
-few iterations. We then replaced the exact inverse of <i>A</i> by the action
-of an AMG preconditioner $\tilde{A}$ based on a vector Laplace matrix,
-approximated the Schur complement $S = B A^{-1} B^T$ by a mass matrix $M_p$
-on the pressure space and wrote an <tt>InverseMatrix</tt> class for
-implementing the action of $M_p^{-1}\approx S^{-1}$ on vectors. In the
-InverseMatrix class, we used a CG solve with an incomplete Cholesky (IC)
-preconditioner for performing the inner solves.
-
-An observation one can make is that we use just the action of a
-preconditioner for approximating the velocity inverse $A^{-1}$ (and the
-outer GMRES iteration takes care of the approximate character of the
-inverse), whereas we use a more or less <i>exact</i> inverse for $M_p^{-1}$,
-realized by a fully converged CG solve. What we change here is to skip that
-<i>exact</i> inverse matrix and replace it &ndash; as usual &ndash; by the
-action of a preconditioner only. This works, as we will demonstrate
-below. For efficiency reasons, we want to avoid increasing the number of
-iterations for the block solve. Keep in mind that most of the time in the
-solution of the matrix system is the application of the AMG preconditioner
-(about half the time of the total solve), and the application of matrix
-<i>A</i> (about one third of the total solve time). This means that we
-really do not want to do those operations more often when we remove the
-inner solve on the Schur complement approximation. It turns out that the
-Trilinos IC preconditioner would not fulfill this requirement, however, the
-Trilinos ILU preconditioner does. It does even better than so &mdash; it
-decreases the iteration count for large 3D problems. The reason for that
-decrease is that we avoid some errors that CG introduces: Even a converged
-solve has still some residual. That is a problem because that small error
-interferes with the outer iterative solver, probably because a CG solver
-does some nonlinear operations by weighting vectors by some inner products.
-
-Except the simplification in the preconditioner, we replaced the GMRES
-solver by BiCGStab. This is merely to demonstrate that GMRES is not the only
-possible option for a saddle point system like the Stokes
-equations. BiCGStab harmonizes nicely with the ILU preconditioner on a
-pressure mass matrix as approximation for $S^{-1}$, so it is at least as
-good as GMRES in this example. Keep in mind the discussion in the results
-section of the step-22 tutorial program, where we observed
-that BiCGStab does <i>not</i> like inner solves with CG, which made us
-prefer GMRES in step-31.
-
-As a final note, let us remark that in step-31 we computed the
-Schur complement $S=B A^{-1} B^T$ by approximating
-$-\text{div}(-\eta\Delta)^{-1}\nabla \approx \frac 1{\eta} \mathbf{1}$. Now,
-however, we have re-scaled the $B$ and $B^T$ operators. So $S$ should now
-approximate
-$-\frac{\eta}{L}\text{div}(-\eta\Delta)^{-1}\nabla \frac{\eta}{L} \approx
-\left(\frac{\eta}{L}\right)^2 \frac 1{\eta} \mathbf{1}$.
-This is exactly the operator we use to approximate $S$.
-
-
-
-<h3> Changes to the artificial viscosity stabilization </h3>
-
-As in step-31, we will use an artificial viscosity of
-the form
-@f{eqnarray*}
-  \nu_\alpha(T)|_K
-  =
-  \beta
-  \|\mathbf{u}\|_{L^\infty(K)}
-  \min\left\{
-    h_K,
-    h_K^\alpha
-    \frac{\|R_\alpha(T)\|_{L^\infty(K)}}{c(\mathbf{u},T)}
-  \right\}
-@f}
-in this problem, where
-$c(\mathbf{u},T) =
- c_R\ \|\mathbf{u}\|_{L^\infty(\Omega)} \ \mathrm{var}(T)
- \ |\mathrm{diam}(\Omega)|^{\alpha-2}$ (for the meaning of the various
-terms in these formulas, see step-31. In the results
-section of that program, we have discussed our choice for $c_R$ and
-how we arrived at the value used there mostly by accident, and in more
-detail how $\beta$ was chosen. For the
-current program, we want to go about this issue a bit more
-systematically for both parameters,
-using the same line of reasoning with which we chose two other
-parameters in our discretization, $c_k$ and $\beta$, in the results
-section of step-31. In particular, remember that we would like to make
-the artificial viscosity as small as possible while keeping it as large as
-necessary. To see what is happening, note that below we will impose
-boundary conditions for the temperature between 973 and 4273 Kelvin,
-and initial conditions are also chosen in this range; because there
-are no %internal heat sources or sinks, the temperature should
-consequently always be in this range, barring any %internal
-oscillations. If the minimal temperature drops below 973 Kelvin, then
-we need to add stabilization by either increasing $\beta$ or
-decreasing $c_R$.
-
-As we did in step-31, we first determine an optimal value of $\beta$
-by using the "traditional" formula
-@f{eqnarray*}
-  \nu_\alpha(T)|_K
-  =
-  \beta
-  \|\mathbf{u}\|_{L^\infty(K)}
-    h_K,
-@f}
-which we know to be stable if only $\beta$ is large enough. Doing a
-couple hundred time steps (on a coarser mesh than the one shown in the
-program, and with a different viscosity that affects transport
-velocities and therefore time step sizes) in 2d will produce the
-following graph:
-
-@image html step-32.beta.2d.png
-
-As can be seen, values $\beta \le 0.05$ are too small whereas
-$\beta=0.052$ appears to work, at least to the time horizon shown
-here. As a remark on the side, there are at least two questions one
-may wonder here: First, what happens at the time when the solution
-becomes unstable? Looking at the graphical output, we can see that
-with the unreasonably coarse mesh chosen for these experiments, around
-time $t=10^{15}$ seconds the plumes of hot material that have been
-rising towards the cold outer boundary and have then spread sideways
-are starting to get close to each other, squezzing out the cold
-material inbetween. This creates a layer of cells into which fluids
-flows from two opposite sides and flows out toward a third, apparently
-a scenario that then produce these instabilities without sufficient
-stabilization. Second: In step-31, we used
-$\beta=0.015\cdot\text{dim}$; why does this not work here? The answer
-to this is not entirely clear -- stabilization parameters are
-certainly known to depend on things like the shape of cells, for which
-we had square in step-31 but have trapezoids in the current
-program. Whatever the exact cause, we at least have a value of
-$\beta$, namely 0.052 for 2d, that works for the current program.
-
-With this value fixed, we can go back to the original formula for the
-viscosity $\nu$ and play with the constant $c_R$, making it as large
-as possible in order to make $\nu$ as small as possible. This gives us
-a picture like this:
-
-@image html doc/step-32.beta_cr.2d.png
-
-Consequently, $c_R=0.1$ would appear to be the right value.
-
-
-
-<h3> Parallelization on clusters </h3>
-
-Parallelization of scientific codes across multiple machines in a cluster of
-computers is almost always done using the Message Passing Interface
-(MPI). This program is no exception to that, and it follows the
-step-17 and step-18 programs in this.
-
-MPI is a rather awkward interface to program with, and so we usually try to
-not use it directly but through an interface layer that abstracts most of the
-MPI operations into a friendlier interface. In the two programs mentioned
-above, this was achieved by using the PETSc library that provides support for
-%parallel linear algebra in a way that almost completely hides the MPI layer
-under it. PETSc is powerful, providing a large number of functions that deal
-with matrices, vectors, and iterative solvers and preconditioners, along with
-lots of other stuff, most of which runs quite well in %parallel. It is,
-however, a few years old already, written in C, and generally not quite as
-easy to use as some other libraries. As a consequence, deal.II also has
-interfaces to Trilinos, a library similar to PETSc in its aims and with a lot
-of the same functionality. It is, however, a project that is several years
-younger, is written in C++ and by people who generally have put a significant
-emphasis on software design. We have already used Trilinos in
-step-31, and will do so again here, with the difference that we
-will use its %parallel capabilities.
-
-deal.II's Trilinos interfaces encapsulate pretty much everything Trilinos
-provides into wrapper classes (in namespace TrilinosWrappers) that make the
-Trilinos matrix, vector, solver and preconditioner classes look very much the
-same as deal.II's own implementations of this functionality. However, as
-opposed to deal.II's classes, they can be used in %parallel if we give them the
-necessary information. As a consequence, there are two Trilinos classes that
-we have to deal with directly (rather than through wrappers), both of which
-are part of Trilinos' Epetra library of basic linear algebra and tool classes:
-<ul>
-<li> The Epetra_Comm class is an abstraction of an MPI "communicator", i.e.
-  it describes how many and which machines can communicate with each other.
-  Each distributed object, such as a sparse matrix or a vector for which we
-  may want to store parts on different machines, needs to have a communicator
-  object to know how many parts there are, where they can be found, and how
-  they can be accessed.
-
-  In this program, we only really use one communicator object -- based on the
-  MPI variable <code>MPI_COMM_WORLD</code> -- that encompasses <i>all</i>
-  processes that work together. It would be perfectly legitimate to start a
-  process on $N$ machines but only store vectors on a subset of these by
-  producing a communicator object that only encompasses this subset of
-  machines; there is really no compelling reason to do so here, however. As a
-  second note, while we use <code>MPI_COMM_WORLD</code> as communicator in the
-  program's source code, every time we create a Trilinos object in the wrapper
-  classes in namespace TrilinosWrappers, we don't use the given communicator
-  but instead create a new and unique communicator that happens to have the
-  same machines but has a distinct communicator ID. This way, we can ensure
-  that all communications that have to do with this, say, sparse matrix really
-  only occur on a channel associated with only this one object, while all
-  other objects communicate on other channels of their own. This helps in
-  debugging, and may also allow some communications to be reordered for better
-  %parallel performance because they can be told apart by their communicator
-  number, not just their relative timing.
-
-<li> The Epetra_Map class is used to describe which elements of a vector or which
-  rows of a matrix reside on a given machine that is part of a
-  communicator. To create such an object, you need to know (i) the total
-  number of elements or rows, (ii) the number of elements or rows you want to
-  store on the current machine, and (iii) which communicator enumerates the
-  machines that we want this matrix or vector be stored on. We will set up
-  these maps (which we call <code>partitioners</code> in our code, since we
-  believe this is a better word) in the
-  <code>BoussinesqFlowProblem::setup_dofs</code> function below and then hand
-  it to every %parallel object we create.
-</ul>
-
-The only other things specific to programming using MPI that we will use in
-this program are the following facilities deal.II provides:
-<ul>
-<li> We need to subdivide the domain into subdomains, each of which will
-  represent the cells that one of the processors coupled by MPI shall consider
-  its own and work on. This is done using the
-  GridTools::partition_triangulation function.
-<li> In order to know which elements of a vector or rows of a matrix shall be
-  stored on each of the processors, we need to know how many degrees of
-  freedom each of the owners of certain subdomains call their own. This is
-  conveniently returned by the DoFTools::count_dofs_with_subdomain_association
-  function.
-</ul>
-The rest of the program is almost completely agnostic about the fact that we
-don't store all objects completely locally. There will be a few points where
-we can not use certain programming techniques (though without making explicit
-reference to MPI or parallelization) or where we need access to <i>all</i>
-elements of a vector and therefore need to <i>localize</i> its elements
-(i.e. create a vector that has all its elements stored on the current
-machine), but we will comment on these locations as we get to them in the
-program code.
-
-
-<h3> Parallelization within individual nodes of a cluster </h3>
-
-The second strategy to parallelize a program is to make use of the fact that
-most computers today have more than one processor that all have access to the
-same memory. In other words, in this model, we don't explicitly have to say
-which pieces of data reside where -- all of the data we need is directly
-accessible and all we have to do is split <i>processing</i> this data between
-the available processors. We will then couple this with the MPI
-parallelization outlined above, i.e. we will have all the processors on a
-machine work together to, for example, assemble the local contributions to the
-global matrix for the cells that this machine actually "owns" but not for
-those cells that are owned by other machines. We will use this strategy for
-four kinds of operations we frequently do in this program: assembly of the
-Stokes and temperature matrices, assembly of the matrix that forms the Stokes
-preconditioner, and assembly of the right hand side of the temperature system.
-
-All of these operations essentially look as follows: we need to loop over all
-cells for which <code>cell-@>subdomain_id()</code> equals the index our
-machine has within the communicator object used for all communication
-(i.e. essentially <code>MPI_COMM_WORLD</code>, as explained above), on each
-cell we need to assemble the local contributions to the global matrix or
-vector, and then we have to copy each cell's contribution into the global
-matrix or vector. Note that the first part of this (the loop) defines a range
-of iterators on which something has to happen. The second part, assembly of
-local contributions is something that takes the majority of CPU time in this
-sequence of steps, is a typical example of things that can be done in
-%parallel: each cell's contribution is entirely independent of all other cells'
-contributions. The third part, copying into the global matrix, must not happen
-in %parallel since we are modifying one object and so several threads can not
-at the same time read an existing matrix element, add their contribution, and
-write the sum back into memory without danger of producing a <a
-href="http://en.wikipedia.org/wiki/Race_condition">race condition</a>.
-
-deal.II has a class that is made for exactly this workflow: WorkStream. Its
-use is extensively documented in the module on @ref threads (in the section
-on @ref MTWorkStream "the WorkStream class") and we won't repeat here the
-rationale and detailed instructions laid out there, though you will want to
-read through this module to understand the distinction between scratch space
-and per-cell data. Suffice it to mention that we need the following:
-
-- An iterator range for those cells on which we are supposed to work. This is
-  provided by the FilteredIterator class which acts just like every other cell
-  iterator in deal.II with the exception that it skips all cells that do not
-  satisfy a particular predicate (i.e. a criterion that evaluates to true or
-  false). In our case, the predicate is whether a cell has the correct
-  subdomain id.
-
-- A function that does the work on each cell for each of the tasks identified
-  above, i.e. functions that assemble the local contributions to Stokes matrix
-  and preconditioner, temperature matrix, and temperature right hand
-  side. These are the
-  <code>BoussinesqFlowProblem::local_assemble_stokes_system</code>,
-  <code>BoussinesqFlowProblem::local_assemble_stokes_preconditioner</code>,
-  <code>BoussinesqFlowProblem::local_assemble_temperature_matrix</code>, and
-  <code>BoussinesqFlowProblem::local_assemble_temperature_rhs</code> functions in
-  the code below. These four functions can all have several instances of each
-  running in %parallel.
-
-- %Functions that copy the result of the previous ones into the global object
-  and that run sequentially to avoid race conditions. These are the
-  <code>BoussinesqFlowProblem::copy_local_to_global_stokes_system</code>,
-  <code>BoussinesqFlowProblem::copy_local_to_global_stokes_preconditioner</code>,
-  <code>BoussinesqFlowProblem::copy_local_to_global_temperature_matrix</code>, and
-  <code>BoussinesqFlowProblem::copy_local_to_global_temperature_rhs</code>
-  functions.
-
-We will comment on a few more points in the actual code, but in general
-their structure should be clear from the discussion in @ref threads.
-
-The underlying technology for WorkStream identifies "tasks" that need to be
-worked on (e.g. assembling local contributions on a cell) and schedules
-these tasks automatically to available processors. WorkStream creates these
-tasks automatically, by splitting the iterator range into suitable chunks,
-but as outlined in @ref threads, one can also create tasks explicitly. We
-use this in one place in the program, namely where we set up the Stokes
-system and preconditioner matrices as well as the temperature matrix. These
-are independent operations that, if enough processors are available, can be
-worked on in parallel (if not enough processors are available -- because the
-system has only one, or because the others are working on something else for
-us -- then these tasks will be worked on sequentially). Consequently,
-<code>BoussinesqFlowProblem::setup_dofs</code> creates tasks for the three
-calls to <code>BoussinesqFlowProblem::setup_stokes_matrix</code>,
-<code>BoussinesqFlowProblem::setup_stokes_preconditioner</code>, and
-<code>BoussinesqFlowProblem::setup_temperature_matrices</code> that are then
-scheduled to available resources. There is one problem with this, however:
-if we have more than one MPI process running in parallel, then all of these
-processes need to communicate in a certain order and that requires that the
-various <code>setup_*</code> functions can't run in parallel. To make
-things worse, even if there is only a single MPI process, we still
-have to make a few calls to the MPI runtime (for example to set up
-communicators for the various matrices and vectors; all of these
-communicators contain only a single processor, but we still have to
-call MPI functions for this). Unfortunately, most MPI implementations
-are not thread-safe, and we can't call MPI functions from multiple
-threads at once. For these setup operations, we will therefore only be
-able to make use of multiple processor cores within the same machine
-if we are not running under the control of MPI -- a condition we can
-check using the Utilities::System::job_supports_mpi() function.
-
-
-<h3> Implementation details </h3>
-
-One detail worth discussing before we show the actual implementation
-is how to deal with distributed vs. localized vectors. When we build
-the linear systems, the various matrices and right hand side vectors
-are always built distributed, i.e. each processor builds contributions
-for a subset of cells and adds them to the global matrix objects;
-Trilinos then makes sure that the right entries end up on the right
-processors so that each processor stores a part of these distributed
-objects.
-
-When we then come around to solving linear systems, the solution
-vectors will also be distributed. On the other hand, in later steps,
-processors will need to be able to access random elements for various
-tasks. For example, processor zero needs to have access to all
-elements in order to produce output; all processors need to have
-access to temperature and Stokes solution data on the cells they own
-to build right hand side vectors for the next time step; and all
-processors need to access data on the cells they own as well as their
-neighbors (whether these neighbors are owned by the same process or
-another one) in order to compute the jump of the gradient across cell
-interfaces in the Kelly error estimator when computing refinement
-information.
-
-In other words, for some operations we will have to exchange
-information between processors. We could try to be smart and really
-only exchange that data that we really need. This would probably be
-important if we were to run this program on thousands of processors,
-as then storing <i>all</i> elements of a solution vector may become
-impractical. On the other hand, deal.II currently has a number of
-other bottlenecks that make sure that thousands of processors is not
-possible anyway. Consequently, here we opt for the simplest choice:
-solve linear systems using distributed vectors, and immediately
-afterwards each processor requests a localized copy of the solution
-vector containing all elements that we will use henceforth for all
-operations. The distributed vector is no longer necessary at this
-point and is, in fact, deallocated again immediately.
-
-
-<h3> The testcase </h3>
-
-The setup for this program is mildly reminiscent of the problem we wanted to
-solve in the first place (see the introduction of step-31):
-convection in the earth mantle. As a consequence, we choose the following
-data, all of which appears in the program in units of meters and seconds (the
-SI system) even if we list them here in other units.
-
-As a reminder, let us again state the equations we want to solve are these:
-@f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) +
-  \nabla \left( \frac{\eta}{L} \hat p\right) &=&
-  \rho(T) \mathbf{g},
-  \\
-  \frac{\eta}{L} \nabla \cdot {\mathbf u} &=& 0,
-  \\
-  \frac{\partial T}{\partial t}
-  +
-  {\mathbf u} \cdot \nabla T
-  -
-  \nabla \cdot \kappa \nabla T &=& \gamma,
-@f}
-augmented by boundary and initial conditions. We then have to choose data for
-the following quantities:
-<ul>
-  <li>The domain is an annulus (in 2d) or a spherical shell (in 3d) with inner
-  and outer radii that match that of the earth: the total radius of the earth
-  is 6371km, with the mantle starting at a depth of around 35km (just under
-  the solid earth <a target="_top"
-  href="http://en.wikipedia.org/wiki/Crust_(geology)">crust</a> composed of
-  <a target="_top"
-  href="http://en.wikipedia.org/wiki/Continental_crust">continental</a> and <a
-  target="_top" href="http://en.wikipedia.org/wiki/Oceanic_crust">oceanic
-  plates</a>) to a depth of 2890km (where the
-  <a target="_top" href="http://en.wikipedia.org/wiki/Outer_core">outer earth
-  core</a> starts). The radii are therefore $R_0=(6371-2890)\text{km},
-  R_1=(6371-35)\text{km}$. This domain is conveniently generated using the
-  GridGenerator::hyper_shell() function, and we use a HyperShellBoundary
-  objects for the inner and outer boundary.
-
-  These two %numbers also provide us with the lengthscale $L$ we need for the
-  pressure scaling. We choose $L=R_1-R_0$.
-
-  <li>At the interface between crust and mantle, the temperature is between
-  500 and 900 degrees Celsius, whereas at its bottom it is around 4000 degrees
-  Celsius (see, for example, <a target="_top"
-  href="http://en.wikipedia.org/wiki/Mantle_(geology)">this Wikipedia
-  entry</a>). In Kelvin, we therefore choose $T_0=(4000+273)\text{K},
-  T_1=(500+273)\text{K}$ as boundary conditions at the inner and outer edge.
-
-  In addition to this, we also have to specifiy some initial conditions for
-  the temperature field. The real temperature field of the earth is quite
-  complicated as a consequence of the convection that has been going on for
-  more than four billion years -- in fact, it is the properties of this
-  temperature distribution that we want to explore with programs like this
-  (or, to be honest, more sophisticated versions of it). As a consequence, we
-  don't really have anything useful to offer here, but we can hope that if we
-  start with something and let things run for a while that the exact initial
-  conditions don't matter that much any more. The initial temperature field we
-  use here is given in terms of the radius by
-  @f{align*}
-    s &= \frac{r-R_0}{R_1-R_0}, \\
-    T(r) &= T_0(1-s)^2 + T_1[1-(1-s)^2].
-  @f}
-  This profile is quadratic and matches the boundary conditions at the inner
-  and outer radii. We will see in the results section that this is an
-  entirely unphysical temperature field (though it will make for
-  interesting images) as the equilibrium state for the temperature
-  will be an almost constant temperature with boundary layers at the
-  inner and outer boundary.
-
-  <li>The right hand side of the temperature equation contains the rate of
-  %internal heating $\gamma$. The earth does heat naturally through three mechanisms:
-  radioactive decay, chemical separation (heavier elements sink to the bottom,
-  lighter ones rise to the top; the countercurrents dissipate emergy equal to
-  the loss of potential energy by this separation process), and heat release
-  by crystallization of liquid metal as the solid inner core of the earth
-  grows. None of these processes are overly significant in the earth mantle, and
-  so we assume that the %internal heating can be set to zero. We
-  neglect one internal heat source, namely adiabatic heating here,
-  which will lead to a surprising temperature field. This point is
-  commented on in detail in the results section below.
-
-  <li>For the velocity we choose as boundary conditions $\mathbf{v}=0$ at the
-  inner radius (i.e. the fluid sticks to the earth core) and
-  $\mathbf{n}\cdot\mathbf{v}=0$ at the outer radius (i.e. the fluid flows
-  tangentially along the bottom of the earth crust). Neither of these is
-  physically overly correct: certainly, on both boundaries, fluids can flow
-  tangentially, but they will incur a shear stress through friction against
-  the medium at the other side of the interface (the metallic core and the
-  crust, respectively). Such a situation could be modeled by a Robin-type
-  boundary condition. To make things worse, the medium on the other side is
-  in motion as well, so the shear stress would, in the simplest case, be
-  proportional to the <i>velocity difference</i>, leading to a boundary
-  condition of the form
-  @f{align*}
-    \mathbf{n}\cdot [2\eta \varepsilon(\mathbf v)]
-    &=
-    s \mathbf{n} \times [\mathbf v - \mathbf v_0],
-    \\
-    \mathbf{n} \cdot \mathbf v &= 0,
-  @f}
-  with a proportionality constant $s$. Rather than going down this route,
-  however, we rather stick with the choice of zero (stick) and tangential
-  flow boundary conditions.
-
-  As a side note of interest, we may also have chosen tangential flow
-  conditions on both inner and outer boundary. That has a significant
-  drawback, however: it leaves the velocity not uniquely defined. The reason
-  is that all velocity fields $\hat{\mathbf v}$ that correspond to a solid
-  body rotation around the center of the domain satisfy $\mathrm{div}
-  \varepsilon(\hat{\mathbf v})=0, \mathrm{div} \hat{\mathbf v} = 0$, and
-  $\mathbf{n} \cdot \hat{\mathbf v} = 0$. As a consequence, if $\mathbf v$
-  satisfies equations and boundary conditions, then so does $\mathbf v +
-  \hat{\mathbf v}$. That's certainly not a good situation that we would like
-  to avoid. The traditional way to work around this is to pick an arbitrary
-  point on the boundary and call this your fixed point by choosing the
-  velocity to be zero in all components there.
-
-  <li>To first order, the gravity vector always points downward. The question for
-  a body as big as the earth is just: where is "up". The answer of course is
-  "radially inward, towards the center of the earth". So at the surface of the
-  earth, we have
-  @f[
-    \mathbf g
-    =
-    -9.81 \frac{\text{m}}{\text{s}^2} \frac{\mathbf x}{\|\mathbf x\|},
-  @f]
-  where $9.81 \frac{\text{m}}{\text{s}^2}$ happens to be the average gravity
-  acceleration at the earth surface. But in the earth interior, the question
-  becomes a bit more complicated: at the center of the earth, for example, you
-  have matter pulling equally hard in all directions, and so $\mathbf g=0$. In
-  between, the net force is described as follows: let us define the
-  <a target="_top"
-  href="http://en.wikipedia.org/wiki/Potential_energy#Gravitational_potential_energy">gravity
-  potential</a> by
-  @f[
-    \varphi(\mathbf x)
-    =
-    \int_{\text{earth}}
-    -G \frac{\rho(\mathbf y)}{\|\mathbf x-\mathbf y\|}
-    \ \text{d}y,
-  @f]
-  then $\mathbf g(\mathbf x) = -\nabla \varphi(\mathbf x)$. If we assume that
-  the density $\rho$ is constant throughout the earth, we can produce an
-  analytical expression for the gravity vector (don't try to integrate above
-  equation somehow -- it leads to elliptic integrals; a simpler way is to
-  notice that $-\Delta\varphi(\mathbf x) = 4\pi G \rho
-  \chi_{\text{earth}}(\mathbf x)$ and solving this
-  partial differential equation in all of ${\mathbb R}^3$ exploiting the
-  radial symmetry):
-  @f[
-    \mathbf g(\mathbf x) =
-    \left\{
-      \begin{array}{ll}
-        -\frac{4}{3}\pi G \rho \|\mathbf x\| \frac{\mathbf x}{\|\mathbf x\|}
-       & \text{for} \ \|\mathbf x\|<R_1, \\
-        -\frac{4}{3}\pi G \rho R^3 \frac{1}{\|\mathbf x\|^2}
-        \frac{\mathbf x}{\|\mathbf x\|}
-       & \text{for} \ \|\mathbf x\|\ge R_1
-      \end{array}
-    \right.
-  @f]
-  The factor $-\frac{\mathbf x}{\|\mathbf x\|}$ is the unit vector pointing
-  radially inward. Of course, within this problem, we are only interested in
-  the branch that pertains to within the earth, i.e. $\|\mathbf
-  x\|<R_1$. In
-  the program, we therefore only consider the expression
-  @f[
-    \mathbf g(\mathbf x) =
-        -\frac{4}{3}\pi G \rho \|\mathbf x\| \frac{\mathbf x}{\|\mathbf x\|}
-        =
-        -\frac{4}{3}\pi G \rho \mathbf x
-       =
-       - 9.81 \frac{\mathbf x}{R_1} \frac{\text{m}}{\text{s}^2},
-  @f]
-  where we can infer the last expression because we know Earth's gravity at
-  the surface (where $\|x\|=R_1$).
-
-  One can derive a more general expression by integrating the
-  differential equation for $\varphi(r)$ in the case that the density
-  distribution is radially symmetric, i.e. $\rho(\mathbf
-  x)=\rho(\|\mathbf x\|)=\rho(r)$. In that case, one would get
-  @f[
-    \varphi(r)
-    = 4\pi G \int_0^r \frac 1{s^2} \int_0^t t^2 \rho(t) \; ds \; dt.
-  @f]
-
-
-  There are two problems with this, however: (i) The Earth is not homogenous,
-  i.e. the density $\rho$ depends on $\mathbf x$; in fact it is not even a
-  function that only depends on the radius $r$. In reality, gravity therefore
-  does not always decrease as we get deeper: because the earth core is so much
-  denser than the mantle, gravity actually peaks at around $10.7
-  \frac{\text{m}}{\text{s}^2}$ at the core mantle boundary (see <a
-  target="_top" href="http://en.wikipedia.org/wiki/Earth's_gravity">this
-  article</a>). (ii) The density, and by
-  consequence the gravity vector, is not even constant in time: after all, the
-  problem we want to solve is the time dependent upwelling of hot, less dense
-  material and the downwelling of cold dense material. This leads to a gravity
-  vector that varies with space and time, and does not always point straight
-  down.
-
-  In order to not make the situation more complicated than necessary, we'll
-  here just go with the constant density model above.
-
-  <li>The density of the earth mantle varies spatially, but not by very
-  much. $\rho_{\text{ref}}=3300 \frac{\text{kg}}{\text{m}^3}$ is a relatively good average
-  value for the density at reference temperature $T_{\text{ref}}=293$ Kelvin.
-
-  <li>The thermal expansion coefficient $\beta$ also varies with depth
-  (through its dependence on temperature and pressure). Close to the surface,
-  it appears to be on the order of $\beta=45\cdot 10^{-6} \frac 1{\text{K}}$,
-  whereas at the core mantle boundary, it may be closer to $\beta=10\cdot
-  10^{-6} \frac 1{\text{K}}$. As a reasonable value, let us choose
-  $\beta=2\cdot 10^{-5} \frac 1{\text{K}}$. The density as a function
-  of temperature is then
-  $\rho(T)=[1-\beta(T-T_{\text{ref}})]\rho_{\text{ref}}$.
-
-  <li>The second to last parameter we need to specify is the viscosity
-  $\eta$. This is a tough one, because rocks at the temperatures and pressure
-  typical for the earth mantle flow so slowly that the viscosity can not be
-  determined accurately in the laboratory. So how do we know about the
-  viscosity of the mantle? The most commonly used route is to consider that
-  during and after ice ages, ice shields form and disappear on time scales
-  that are shorter than the time scale of flow in the mantle. As a
-  consequence, continents slowly sink into the earth mantle under the added
-  weight of an ice shield, and they rise up again slowly after the ice shield
-  has disappeared again (this is called <a target="_top"
-  href="http://en.wikipedia.org/wiki/Postglacial_rebound"><i>postglacial
-  rebound</i></a>). By measuring the speed of this rebound, we can infer the
-  viscosity of the material that flows into the area vacated under the
-  rebounding continental plates.
-
-  Using this technique, values around $\eta=10^{21} \text{Pa\; s}
-  = 10^{21} \frac{\text{N\; s}}{\text{m}^2}
-  = 10^{21} \frac{\text{kg}}{\text{m\; s}}$ have been found as the most
-  likely, though the error bar on this is at least one order of magnitude.
-
-  While we will use this value, we again have to caution that there are many
-  physical reasons to assume that this is not the correct value. First, it
-  should really be made dependent on temperature: hotter material is most
-  likely to be less viscous than colder material. In reality, however, the
-  situation is even more complex. Most rocks in the mantle undergo phase
-  changes as temperature and pressure change: depending on temperature and
-  pressure, different crystal configurations are thermodynamically favored
-  over others, even if the chemical composition of the mantle were
-  homogenous. For example, the common mantle material MgSiO<sub>3</sub> exists
-  in its <a target="_top"
-  href="http://en.wikipedia.org/wiki/Perovskite_(structure)">perovskite
-  structure</a> throughout most of the mantle, but in the lower mantle the
-  same substance is stable only as <a targe="_top"
-  href="http://en.wikipedia.org/wiki/Postperovskite">post-perovskite</a>. Clearly,
-  to compute realistic viscosities, we would not only need to know the exact
-  chemical composition of the mantle and the viscosities of all materials, but
-  we would also have to compute the thermodynamically most stable
-  configurations for all materials at each quadrature point. This is at the
-  time of writing this program not a feasible suggestion.
-
-  <li>Our last material parameter is the thermal diffusivity $\kappa$, which
-  is defined as $\kappa=\frac{k}{\rho c_p}$ where $k$ is the thermal
-  conductivity, $\rho$ the density, and $c_p$ the specific heat. For
-  this, the literature indicates that it increases from around $0.7$ in the
-  upper mantle to around $1.7 \frac{\text{mm}^2}{\text{s}}$ in the lower
-  mantle, though the exact value
-  is not really all that important: heat transport through convection is
-  several orders of magnitude more important than through thermal
-  conduction. It may be of interest to know that perovskite, the most abundant
-  material in the earth mantle, appears to become transparent at pressures
-  above around 120 GPa (see, for example, J. Badro et al., Science 305,
-  383-386 (2004)); in the lower mantle, it may therefore be that heat
-  transport through radiative transfer is more efficient than through thermal
-  conduction.
-
-  In view of these considerations, let us choose
-  $\kappa=1 \frac{\text{mm}^2}{\text{s}} =10^{-6} \frac{\text{m}^2}{\text{s}}$
-  for the purpose of this program.
-</ul>
-
-All of these pieces of equation data are defined in the program in the
-<code>EquationData</code> namespace. When run, the program produces
-long-term maximal velocities around 10-40 centimeters per year (see
-the results section below), approximately the physically correct order
-of magnitude. We will set the end time to 1 billion years.
-
diff --git a/deal.II/examples/step-32/doc/kind b/deal.II/examples/step-32/doc/kind
deleted file mode 100644 (file)
index e62f4e7..0000000
+++ /dev/null
@@ -1 +0,0 @@
-fluids
diff --git a/deal.II/examples/step-32/doc/results.dox b/deal.II/examples/step-32/doc/results.dox
deleted file mode 100644 (file)
index 53536d0..0000000
+++ /dev/null
@@ -1,380 +0,0 @@
-<h1>Results</h1>
-
-When run, the program simulates convection in 3d in much the same way
-as step-31 did, though with an entirely different testcase.
-
-
-<h3>Comparison of results with step-31</h3>
-
-Before we go to this testcase, however, let us show a few results from a
-slightly earlier version of this program that was solving exactly the
-testcase we used in step-31, just that we now solve it in parallel and with
-much higher resolution. We show these results mainly for comparison.
-
-Here are two images that show this higher resolution if we choose a 3d
-computation in <code>main()</code> and if we set
-<code>initial_refinement=3</code> and
-<code>n_pre_refinement_steps=4</code>. At the time steps shown, the
-meshes had around 72,000 and 236,000 cells, for a total of 2,680,000
-and 8,250,000 degrees of freedom, respectively, more than an order of
-magnitude more than we had available in step-31:
-
-<table align="center" border="1" cellspacing="3" cellpadding="3">
-  <tr>
-    <td>
-        @image html "step-32.3d.cube.0.png" "" width=80%
-    </td>
-  </tr>
-  <tr>
-    <td>
-        @image html "step-32.3d.cube.1.png" "" width=80%
-    </td>
-  </tr>
-</table>
-
-The computation was done on a subset of 50 processors of the Brazos
-cluster at Texas A&amp;M University.
-
-
-<h3>Results for a 2d circular shell testcase</h3>
-
-
-If we run the program as shown above, the output will look roughly like this,
-producing the final part of the output after some 2 days when run on 10
-processors:
-
-<code>
-<pre>
-Number of active cells: 12288 (on 6 levels)
-Number of degrees of freedom: 162432 (99840+12672+49920)
-
-Timestep 0:  t=0 years
-   Assembling...
-   Rebuilding Stokes preconditioner...
-   Solving...
-   14 BiCGStab iterations for Stokes subsystem.
-   Maximal velocity: 1.80187 cm/year
-   Time step: 607937 years
-   15 CG iterations for temperature
-   Temperature range: 973 4273.27
-
-Number of active cells: 17988 (on 7 levels)
-Number of degrees of freedom: 240504 (147888+18672+73944)
-
-Timestep 0:  t=0 years
-   Assembling...
-   Rebuilding Stokes preconditioner...
-   Solving...
-   3 BiCGStab iterations for Stokes subsystem.
-   Maximal velocity: 1.52763 cm/year
-   Time step: 357565 years
-   17 CG iterations for temperature
-   Temperature range: 973 4274.08
-
-Number of active cells: 34716 (on 8 levels)
-Number of degrees of freedom: 474550 (291896+36706+145948)
-
-Timestep 0:  t=0 years
-   Assembling...
-   Rebuilding Stokes preconditioner...
-   Solving...
-   3 BiCGStab iterations for Stokes subsystem.
-   Maximal velocity: 2.97931 cm/year
-   Time step: 91574.3 years
-   17 CG iterations for temperature
-   Temperature range: 973 4277.31
-
-Timestep 1:  t=91574.3 years
-   Assembling...
-   Solving...
-   2 BiCGStab iterations for Stokes subsystem.
-   Maximal velocity: 2.86431 cm/year
-   Time step: 95251.1 years
-   18 CG iterations for temperature
-   Temperature range: 973 4277.31
-
-[...]
-
-Timestep 53906:  t=9.99985e+08 years
-   Assembling...
-   Solving...
-   4 BiCGStab iterations for Stokes subsystem.
-   Maximal velocity: 12.2898 cm/year
-   Time step: 22199.6 years
-   18 CG iterations for temperature
-   Temperature range: 973 4273.07
-
-
-+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start    |  1.87e+05s |            |
-|                                             |            |            |
-| Section                         | no. calls |  wall time | % of total |
-+---------------------------------+-----------+------------+------------+
-| Assemble Stokes system          |     53909 |  1.06e+04s |       5.7% |
-| Assemble temperature matrices   |      5393 |       913s |      0.49% |
-| Assemble temperature rhs        |     53909 |  1.24e+04s |       6.6% |
-| Build Stokes preconditioner     |      5393 |  4.52e+03s |       2.4% |
-| Solve Stokes system             |     53909 |  6.94e+04s |        37% |
-| Solve temperature system        |     53909 |  2.34e+04s |        13% |
-| Postprocessing                  |      2157 |  5.51e+03s |       2.9% |
-| Refine mesh structure, part 1   |      5392 |  4.71e+04s |        25% |
-| Refine mesh structure, part 2   |      5392 |  1.02e+03s |      0.55% |
-| Setup dof systems               |      5393 |  1.11e+04s |       5.9% |
-+---------------------------------+-----------+------------+------------+
-</pre>
-</code>
-
-As can be seen here, we spend most of the compute time in assembling linear
-systems, refining the mesh, and in particular in solving the Stokes and
-temperature linear systems.
-
-The 50% spent on solving the linear systems are affected in large part
-because the Brazos cluster has a relatively slow ethernet interconnect. A
-cluster with a faster interconnect, for example using infiniband, should do
-better in this regard. On the other hand, there is little hope to do better
-with assembling the linear systems, though one could do significantly better
-with estimating the error by making sure that each processor only estimates
-the error on those cells it owns.
-
-
-The program writes output every 25th time step, but we won't show all
-2100 or so images this produces. Rather, let us only show the output
-from every 2500th time step here, even though this does, of course,
-not do full justice to the dynamics that are going on:
-<table>
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.0000.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0100.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0200.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.0300.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0400.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0500.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.0600.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0700.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.0800.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.0900.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1000.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1100.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.1200.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1300.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1400.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.1500.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1600.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1700.png
-    <td>
-  </tr>
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.1800.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.1900.png
-    <td>
-
-    <td>
-      @image html step-32.2d.temperature.2000.png
-    <td>
-  </tr>
-
-
-  <tr>
-    <td>
-      @image html step-32.2d.temperature.2100.png
-    <td>
-
-    <td>
-      @image html step-32.2d.grid.2100.png
-    <td>
-
-    <td>
-      @image html step-32.2d.partition.2100.png
-    <td>
-  </tr>
-</table>
-
-The last two images show the grid as well as the partitioning of the
-mesh for the last timestep shown into the 10 subdomains used for this
-computation. The full dynamics are really only visible by looking at
-an animation. <a
-href="http://www.math.tamu.edu/~bangerth/images/pictures/convection-outward/step-32.2d.convection.gif">At
-this site</a> is such an animation. Beware that this animation is
-about 20MB large, though it is well worth watching due to its almost
-artistic quality.
-
-If you watch the movie, you'll see that the convection pattern goes
-through several stages: First, it gets rid of the instable temperature
-layering with the hot material overlaid by the dense cold
-material. After this great driver is removed and we have a sort of
-stable situation, a few blobs start to separate from the hot boundary
-layer at the inner ring and rise up, with a few cold fingers also
-dropping down from the outer ring. During this phase, the solution
-remains mostly symmetric, reflecting the 12-fold symmetry of the
-original mesh. In a final phase, the fluid enters vigorous chaotic
-stirring in which all symmetries are lost. This is a pattern that then
-continues to dominate flow.
-
-These different phases can also be identified if we look at the
-maximal velocity as a function of time in the simulation:
-
-@image html step-32.2d.t_vs_vmax.png
-
-Here, the velocity (shown in centimeters per year) becomes very large,
-to the order of several meters per year) at the beginning when the
-temperature layering is instable. It then calms down to relatively
-small values before picking up again in the chaotic stirring
-regime. There, it remains in the range of 10-40 centimeters per year,
-quite within the physically expected region.
-
-
-
-<a name="extensions"></a>
-<h3>Possibilities for extensions</h3>
-
-Apart from the various possibilities for extensions already outlined
-in the step-31, here are a few more ideas:
-
-<ul>
-  <li> The temperature field we get in our simulations after a while
-  is mostly constant with boundary layers at the inner and outer
-  boundary, and streamers of cold and hot material mixing
-  everything. Yet, this doesn't match our expectation that things
-  closer to the earth core should be hotter than closer to the
-  surface. The reason is that the energy equation we have used does
-  not include a term that describes adiabatic cooling and heating:
-  rock, like gas, heats up as you compress it. Consequently, material
-  that rises up cools adiabatically, and cold material that sinks down
-  heats adiabatically. The correct temperature equation would
-  therefore look somewhat like this:
-  @f{eqnarray*}
-    \frac{D T}{Dt}
-    -
-    \nabla \cdot \kappa \nabla T &=& \gamma + \tau\frac{Dp}{Dt},
-  @f}
-  or, expanding the advected derivative $\frac{D}{Dt} =
-  \frac{\partial}{\partial t} + \mathbf u \cdot \nabla$:
-  @f{eqnarray*}
-    \frac{\partial T}{\partial t}
-    +
-    {\mathbf u} \cdot \nabla T
-    -
-    \nabla \cdot \kappa \nabla T &=& \gamma +
-    \tau\left\{\frac{\partial
-    p}{\partial t} + \mathbf u \cdot \nabla p \right\}.
-  @f}
-  In other words, as pressure increases in a rock volume
-  ($\frac{Dp}{Dt}>0$) we get an additional heat source, and vice
-  versa.
-
-  The time derivative of the pressure is a bit awkward to
-  implement. If necessary, one could approximate using the fact
-  outlined in the introduction that the pressure can be decomposed
-  into a dynamic component due to temperature differences and the
-  resulting flow, and a static component that results solely from the
-  static pressure of the overlying rock. Since the latter is much
-  bigger, one may approximate $p\approx p_{\text{static}}=-\rho_{\text{ref}}
-  [1+\beta T_{\text{ref}}] \varphi$, and consequently
-  $\frac{Dp}{Dt} \approx \left\{- \mathbf u \cdot \nabla \rho_{\text{ref}}
-  [1+\beta T_{\text{ref}}]\varphi\right\} = \rho_{\text{ref}}
-  [1+\beta T_{\text{ref}}] \mathbf u \cdot \mathbf g$.
-  In other words, if the fluid is moving in the direction of gravity
-  (downward) it will be compressed and because in that case $\mathbf u
-  \cdot \mathbf g > 0$ we get a positive heat source. Conversely, the
-  fluid will cool down if it moves against the direction of gravity.
-
-  Implementing this requires one additional step, however. As mentioned in the
-  introduction, we use a rather simplified model for gravity in which the
-  gravity force diminishes with depth, as if Earth had a homogenous
-  density. That isn't the case, however: the earth core is much denser than
-  the mantle, and gravity actually tops out at the core
-  mantle boundary. A consequence of this is that our computations predict
-  pressures around 70 GPa at the core mantle boundary, whereas in reality the
-  value is closer to 140 GPa. With a pressure wrong by so much, we can't
-  expect compression heating to be accurate.
-
-  A more realistic model for the gravity vector in the program would take the
-  spatially variable density into account, and that wouldn't actually be
-  terribly complicated: by integrating the PDE for the gravity potential under
-  the assumption that $\rho(\mathbf x)=\rho(r)$, we get
-  @f[
-    \varphi(r) = 4\pi G \int_0^r \frac 1{t^2} \int_0^t s^2 \rho(s) \; ds \; dt,
-  @f]
-  and consequently for the gravity vector
-  @f[
-    \mathbf g = - 4\pi G \frac 1{r^2} \left(
-      \int_0^r s^2 \rho(s) \; ds \right)
-    \frac{\mathbf x}{\|\mathbf x\|}.
-  @f]
-  This expression reduces to the one we use for the case that the density is
-  constant, but a more complete model would, for example, assume that the
-  density varies with the radius (in the simplest case it could be constant in
-  various layers). In either case, it can relatively easily be evaluated if
-  for non-trivial models of $\rho(r)$. Of course, a really complete model
-  would consider that $\rho$ can also vary in the tangential direction, for
-  example in a time dependent way as a consequence of the thermal expansion of
-  rocks as a result of the convection. Taking into account this self
-  gravitational effect of convection would be much harder, however.
-</ul>
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diff --git a/deal.II/examples/step-32/doc/step-32.beta.2d.png b/deal.II/examples/step-32/doc/step-32.beta.2d.png
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diff --git a/deal.II/examples/step-32/doc/step-32.beta_cr.2d.png b/deal.II/examples/step-32/doc/step-32.beta_cr.2d.png
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diff --git a/deal.II/examples/step-32/doc/tooltip b/deal.II/examples/step-32/doc/tooltip
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+++ /dev/null
@@ -1 +0,0 @@
-A parallel Boussinesq flow for thermal convection.
diff --git a/deal.II/examples/step-32/step-32.cc b/deal.II/examples/step-32/step-32.cc
deleted file mode 100644 (file)
index 9ca65c2..0000000
+++ /dev/null
@@ -1,3275 +0,0 @@
-//TODO: - adjust stopping criteria for solvers
-//      - better refinement at the start?
-//      - check solver stability
-
-
-/* $Id$ */
-/* Author: Martin Kronbichler, Uppsala University,
-           Wolfgang Bangerth, Texas A&M University 2008, 2009 */
-/*                                                                */
-/*    Copyright (C) 2008, 2009, 2010 by the deal.II authors */
-/*                                                                */
-/*    This file is subject to QPL and may not be  distributed     */
-/*    without copyright and license information. Please refer     */
-/*    to the file deal.II/doc/license.html for the  text  and     */
-/*    further information on this license.                        */
-
-                                // @sect3{Include files}
-
-                                // We include the functionality
-                                // of these well-known deal.II
-                                // library files and some C++
-                                // header files.
-#include <base/quadrature_lib.h>
-#include <base/logstream.h>
-#include <base/function.h>
-#include <base/utilities.h>
-#include <base/conditional_ostream.h>
-#include <base/work_stream.h>
-#include <base/timer.h>
-
-#include <lac/full_matrix.h>
-#include <lac/solver_bicgstab.h>
-#include <lac/solver_cg.h>
-#include <lac/constraint_matrix.h>
-#include <lac/block_sparsity_pattern.h>
-#include <lac/trilinos_block_vector.h>
-#include <lac/trilinos_sparse_matrix.h>
-#include <lac/trilinos_block_sparse_matrix.h>
-#include <lac/trilinos_precondition.h>
-
-#include <grid/tria.h>
-#include <grid/grid_generator.h>
-#include <grid/tria_accessor.h>
-#include <grid/tria_iterator.h>
-#include <grid/filtered_iterator.h>
-#include <grid/tria_boundary_lib.h>
-#include <grid/grid_tools.h>
-#include <grid/grid_refinement.h>
-
-#include <dofs/dof_handler.h>
-#include <dofs/dof_renumbering.h>
-#include <dofs/dof_accessor.h>
-#include <dofs/dof_tools.h>
-
-#include <fe/fe_q.h>
-#include <fe/fe_dgq.h>
-#include <fe/fe_system.h>
-#include <fe/fe_values.h>
-#include <fe/mapping_q1.h>
-
-#include <numerics/vectors.h>
-#include <numerics/matrices.h>
-#include <numerics/data_out.h>
-#include <numerics/error_estimator.h>
-#include <numerics/solution_transfer.h>
-
-#include <fstream>
-#include <iostream>
-#include <sstream>
-
-                                // This is the only include file that is new:
-                                // We use an IndexSet to describe the
-                                // %parallel partitioning of vectors and
-                                // matrices.
-#include <base/index_set.h>
-
-
-                                // Next, we import all deal.II
-                                // names into global namespace:
-using namespace dealii;
-
-                                // @sect3{Equation data}
-
-                                // In the following namespace, we define the
-                                // various pieces of equation data. All of
-                                // these are exhaustively discussed in the
-                                // description of the testcase in the
-                                // introduction:
-namespace EquationData
-{
-  const double eta                   = 1e21;
-  const double kappa                 = 1e-6;
-  const double reference_density     = 3300;
-  const double reference_temperature = 293;
-  const double expansion_coefficient = 2e-5;
-
-  const double R0      = 6371000.-2890000.;
-  const double R1      = 6371000.-  35000.;
-
-  const double T0      = 4000+273;
-  const double T1      =  700+273;
-
-  const double year_in_seconds  = 60*60*24*365.2425;
-  const double end_time         = 1e9 * year_in_seconds;
-
-  const double pressure_scaling = eta / (R1-R0);
-
-
-  double density (const double temperature)
-  {
-    return (reference_density *
-           (1 - expansion_coefficient * (temperature -
-                                         reference_temperature)));
-  }
-
-
-  template <int dim>
-  Tensor<1,dim> gravity_vector (const Point<dim> &p)
-  {
-    return -9.81 * p / R1;
-  }
-
-
-  template <int dim>
-  class TemperatureInitialValues : public Function<dim>
-  {
-    public:
-      TemperatureInitialValues () : Function<dim>(1) {}
-
-      virtual double value (const Point<dim>   &p,
-                           const unsigned int  component = 0) const;
-
-      virtual void vector_value (const Point<dim> &p,
-                                Vector<double>   &value) const;
-  };
-
-
-
-  template <int dim>
-  double
-  TemperatureInitialValues<dim>::value (const Point<dim>  &p,
-                                       const unsigned int) const
-  {
-    const double r = p.norm();
-    const double h = R1-R0;
-
-    const double s = (r-R0)/h;
-
-    return T1+(T0-T1)*((1-s)*(1-s));
-  }
-
-
-  template <int dim>
-  void
-  TemperatureInitialValues<dim>::vector_value (const Point<dim> &p,
-                                              Vector<double>   &values) const
-  {
-    for (unsigned int c=0; c<this->n_components; ++c)
-      values(c) = TemperatureInitialValues<dim>::value (p, c);
-  }
-
-
-
-  template <int dim>
-  class TemperatureRightHandSide : public Function<dim>
-  {
-    public:
-      TemperatureRightHandSide () : Function<dim>(1) {}
-
-      virtual double value (const Point<dim>   &p,
-                           const unsigned int  component = 0) const;
-
-      virtual void vector_value (const Point<dim> &p,
-                                Vector<double>   &value) const;
-  };
-
-
-
-  template <int dim>
-  double
-  TemperatureRightHandSide<dim>::value (const Point<dim>  &p,
-                                       const unsigned int component) const
-  {
-    return 0;
-  }
-
-
-  template <int dim>
-  void
-  TemperatureRightHandSide<dim>::vector_value (const Point<dim> &p,
-                                              Vector<double>   &values) const
-  {
-    for (unsigned int c=0; c<this->n_components; ++c)
-      values(c) = TemperatureRightHandSide<dim>::value (p, c);
-  }
-}
-
-
-
-                                // @sect3{Linear solvers and preconditioners}
-
-                                // In comparison to step-31, we did one
-                                // change in the linear algebra of the
-                                // problem: We exchange the
-                                // <code>InverseMatrix</code> that
-                                // previously held the approximation of the
-                                // Schur complement by a preconditioner
-                                // only (we will choose ILU in the
-                                // application code below), as discussed in
-                                // the introduction. This trick we already
-                                // did for the velocity block - the idea of
-                                // this is that the solver iterations on
-                                // the block system will eventually also
-                                // make the approximation for the Schur
-                                // complement good. If the preconditioner
-                                // we're using is good enough, there will
-                                // be no increase in the outer iteration
-                                // count compared to using converged solves
-                                // for the inverse matrices of velocity and
-                                // Schur complement. All we need to do for
-                                // implementing that change is to give the
-                                // respective variable in the
-                                // BlockSchurPreconditioner class another
-                                // name.
-namespace LinearSolvers
-{
-  template <class PreconditionerA, class PreconditionerMp>
-  class BlockSchurPreconditioner : public Subscriptor
-  {
-    public:
-      BlockSchurPreconditioner (
-       const TrilinosWrappers::BlockSparseMatrix  &S,
-       const PreconditionerMp                     &Mppreconditioner,
-       const PreconditionerA                      &Apreconditioner);
-
-      void vmult (TrilinosWrappers::MPI::BlockVector       &dst,
-                 const TrilinosWrappers::MPI::BlockVector &src) const;
-
-    private:
-      const SmartPointer<const TrilinosWrappers::BlockSparseMatrix> stokes_matrix;
-      const PreconditionerMp &mp_preconditioner;
-      const PreconditionerA  &a_preconditioner;
-      mutable TrilinosWrappers::MPI::Vector tmp;
-  };
-
-
-
-  template <class PreconditionerA, class PreconditionerMp>
-  BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::
-  BlockSchurPreconditioner(const TrilinosWrappers::BlockSparseMatrix &S,
-                          const PreconditionerMp                    &Mppreconditioner,
-                          const PreconditionerA                     &Apreconditioner)
-                 :
-                 stokes_matrix     (&S),
-                 mp_preconditioner (Mppreconditioner),
-                 a_preconditioner  (Apreconditioner),
-                 tmp               (stokes_matrix->block(1,1).row_partitioner())
-  {}
-
-
-
-  template <class PreconditionerA, class PreconditionerMp>
-  void BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::vmult (
-    TrilinosWrappers::MPI::BlockVector       &dst,
-    const TrilinosWrappers::MPI::BlockVector &src) const
-  {
-    a_preconditioner.vmult (dst.block(0), src.block(0));
-    stokes_matrix->block(1,0).residual(tmp, dst.block(0), src.block(1));
-    tmp *= -1;
-    mp_preconditioner.vmult (dst.block(1), tmp);
-  }
-}
-
-
-
-                                // @sect3{Definition of assembly data structures}
-                                //
-                                // As described in the introduction, we will
-                                // use the WorkStream mechanism discussed in
-                                // the @ref threads module to parallelize
-                                // operations among the processors of a
-                                // single machine. The WorkStream class
-                                // requires that data is passed around in two
-                                // kinds of data structures, one for scratch
-                                // data and one to pass data from the
-                                // assembly function to the function that
-                                // copies local contributions into global
-                                // objects.
-                                //
-                                // The following namespace (and the two
-                                // sub-namespaces) contains a collection of
-                                // data structures that serve this purpose,
-                                // one pair for each of the four operations
-                                // discussed in the introduction that we will
-                                // want to parallelize. Each
-                                // assembly routine gets two sets of data: a
-                                // Scratch array that collects all the
-                                // classes and arrays that are used for the
-                                // calculation of the cell contribution, and
-                                // a CopyData array that keeps local matrices
-                                // and vectors which will be written into the
-                                // global matrix. Whereas CopyData is a
-                                // container for the final data that is
-                                // written into the global matrices and
-                                // vector (and, thus, absolutely necessary),
-                                // the Scratch arrays are merely there for
-                                // performance reasons &mdash; it would be
-                                // much more expensive to set up a FEValues
-                                // object on each cell, than creating it only
-                                // once and updating some derivative data.
-                                //
-                                // Using the program in step-31, we have
-                                // four assembly routines. One for the
-                                // preconditioner matrix of the Stokes
-                                // system, one for the Stokes matrix and
-                                // right hand side, one for the
-                                // temperature matrices and one for the
-                                // right hand side of the temperature
-                                // equation. We organize the scratch
-                                // arrays and a CopyData arrays for each
-                                // of those four assembly components
-                                // using a <code>struct</code>
-                                // environment.
-                                //
-                                // Regarding the Scratch array, each
-                                // struct is equipped with a constructor
-                                // that create an FEValues object for a
-                                // @ref FiniteElement "finite element", a
-                                // @ref Quadrature "quadrature formula"
-                                // and some
-                                // @ref UpdateFlags "update flags".
-                                // Moreover, we manually
-                                // implement a copy constructor (since
-                                // the FEValues class is not copyable by
-                                // itself), and provide some additional
-                                // vector fields that are used to improve
-                                // performance of assembly.
-namespace Assembly
-{
-  namespace Scratch
-  {
-    template <int dim>
-    struct StokesPreconditioner
-    {
-       StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
-                             const Quadrature<dim>    &stokes_quadrature,
-                             const UpdateFlags         update_flags);
-       StokesPreconditioner (const StokesPreconditioner &data);
-
-       FEValues<dim>               stokes_fe_values;
-
-       std::vector<Tensor<2,dim> > grad_phi_u;
-       std::vector<double>         phi_p;
-    };
-
-    template <int dim>
-    StokesPreconditioner<dim>::
-    StokesPreconditioner (const FiniteElement<dim> &stokes_fe,
-                         const Quadrature<dim>    &stokes_quadrature,
-                         const UpdateFlags         update_flags)
-                   :
-                   stokes_fe_values (stokes_fe, stokes_quadrature,
-                                     update_flags),
-                   grad_phi_u (stokes_fe.dofs_per_cell),
-                   phi_p (stokes_fe.dofs_per_cell)
-    {}
-
-
-
-    template <int dim>
-    StokesPreconditioner<dim>::
-    StokesPreconditioner (const StokesPreconditioner &scratch)
-                   :
-                   stokes_fe_values (scratch.stokes_fe_values.get_fe(),
-                                     scratch.stokes_fe_values.get_quadrature(),
-                                     scratch.stokes_fe_values.get_update_flags()),
-                   grad_phi_u (scratch.grad_phi_u),
-                   phi_p (scratch.phi_p)
-    {}
-
-
-
-                                    // Observe that we derive the
-                                    // StokesSystem scratch array from the
-                                    // StokesPreconditioner array. We do this
-                                    // because all the objects that are
-                                    // necessary for the assembly of the
-                                    // preconditioner are also needed for the
-                                    // actual matrix system and right hand
-                                    // side, plus some extra data. This makes
-                                    // the program more compact. Note also
-                                    // that the assembly of the Stokes system
-                                    // and the temperature right hand side
-                                    // further down requires data from
-                                    // temperature and velocity,
-                                    // respectively, so we actually need two
-                                    // FEValues objects for those two cases.
-    template <int dim>
-    struct StokesSystem : public StokesPreconditioner<dim>
-    {
-       StokesSystem (const FiniteElement<dim> &stokes_fe,
-                     const Quadrature<dim>    &stokes_quadrature,
-                     const UpdateFlags         stokes_update_flags,
-                     const FiniteElement<dim> &temperature_fe,
-                     const UpdateFlags         temperature_update_flags);
-
-       StokesSystem (const StokesSystem<dim> &data);
-
-       FEValues<dim>  temperature_fe_values;
-
-       std::vector<Tensor<1,dim> >          phi_u;
-       std::vector<SymmetricTensor<2,dim> > grads_phi_u;
-       std::vector<double>                  div_phi_u;
-
-       std::vector<double>                  old_temperature_values;
-    };
-
-
-    template <int dim>
-    StokesSystem<dim>::
-    StokesSystem (const FiniteElement<dim> &stokes_fe,
-                 const Quadrature<dim>    &stokes_quadrature,
-                 const UpdateFlags         stokes_update_flags,
-                 const FiniteElement<dim> &temperature_fe,
-                 const UpdateFlags         temperature_update_flags)
-                   :
-                   StokesPreconditioner<dim> (stokes_fe, stokes_quadrature,
-                                              stokes_update_flags),
-                   temperature_fe_values (temperature_fe, stokes_quadrature,
-                                          temperature_update_flags),
-                   phi_u (stokes_fe.dofs_per_cell),
-                   grads_phi_u (stokes_fe.dofs_per_cell),
-                   div_phi_u (stokes_fe.dofs_per_cell),
-                   old_temperature_values (stokes_quadrature.n_quadrature_points)
-    {}
-
-
-    template <int dim>
-    StokesSystem<dim>::
-    StokesSystem (const StokesSystem<dim> &scratch)
-                   :
-                   StokesPreconditioner<dim> (scratch),
-                   temperature_fe_values (scratch.temperature_fe_values.get_fe(),
-                                          scratch.temperature_fe_values.get_quadrature(),
-                                          scratch.temperature_fe_values.get_update_flags()),
-                   phi_u (scratch.phi_u),
-                   grads_phi_u (scratch.grads_phi_u),
-                   div_phi_u (scratch.div_phi_u),
-                   old_temperature_values (scratch.old_temperature_values)
-    {}
-
-
-
-    template <int dim>
-    struct TemperatureMatrix
-    {
-       TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
-                          const Quadrature<dim>    &temperature_quadrature);
-       TemperatureMatrix (const TemperatureMatrix &data);
-
-       FEValues<dim>               temperature_fe_values;
-
-       std::vector<double>         phi_T;
-       std::vector<Tensor<1,dim> > grad_phi_T;
-    };
-
-    template <int dim>
-    TemperatureMatrix<dim>::
-    TemperatureMatrix (const FiniteElement<dim> &temperature_fe,
-                      const Quadrature<dim>    &temperature_quadrature)
-                   :
-                   temperature_fe_values (temperature_fe, temperature_quadrature,
-                                          update_values    | update_gradients |
-                                          update_JxW_values),
-                   phi_T (temperature_fe.dofs_per_cell),
-                   grad_phi_T (temperature_fe.dofs_per_cell)
-    {}
-
-
-    template <int dim>
-    TemperatureMatrix<dim>::
-    TemperatureMatrix (const TemperatureMatrix &scratch)
-                   :
-                   temperature_fe_values (scratch.temperature_fe_values.get_fe(),
-                                          scratch.temperature_fe_values.get_quadrature(),
-                                          scratch.temperature_fe_values.get_update_flags()),
-                   phi_T (scratch.phi_T),
-                   grad_phi_T (scratch.grad_phi_T)
-    {}
-
-
-    template <int dim>
-    struct TemperatureRHS
-    {
-       TemperatureRHS (const FiniteElement<dim> &temperature_fe,
-                       const FiniteElement<dim> &stokes_fe,
-                       const Quadrature<dim>    &quadrature);
-       TemperatureRHS (const TemperatureRHS &data);
-
-       FEValues<dim>               temperature_fe_values;
-       FEValues<dim>               stokes_fe_values;
-
-       std::vector<double>         phi_T;
-       std::vector<Tensor<1,dim> > grad_phi_T;
-
-       std::vector<Tensor<1,dim> > old_velocity_values;
-       std::vector<Tensor<1,dim> > old_old_velocity_values;
-
-       std::vector<double>         old_temperature_values;
-       std::vector<double>         old_old_temperature_values;
-       std::vector<Tensor<1,dim> > old_temperature_grads;
-       std::vector<Tensor<1,dim> > old_old_temperature_grads;
-       std::vector<double>         old_temperature_laplacians;
-       std::vector<double>         old_old_temperature_laplacians;
-
-       std::vector<double>         gamma_values;
-    };
-
-    template <int dim>
-    TemperatureRHS<dim>::
-    TemperatureRHS (const FiniteElement<dim> &temperature_fe,
-                   const FiniteElement<dim> &stokes_fe,
-                   const Quadrature<dim>    &quadrature)
-                   :
-                   temperature_fe_values (temperature_fe, quadrature,
-                                          update_values    |
-                                          update_gradients |
-                                          update_hessians  |
-                                          update_quadrature_points |
-                                          update_JxW_values),
-                   stokes_fe_values (stokes_fe, quadrature,
-                                     update_values),
-                   phi_T (temperature_fe.dofs_per_cell),
-                   grad_phi_T (temperature_fe.dofs_per_cell),
-
-                   old_velocity_values (quadrature.n_quadrature_points),
-                   old_old_velocity_values (quadrature.n_quadrature_points),
-
-                   old_temperature_values (quadrature.n_quadrature_points),
-                   old_old_temperature_values(quadrature.n_quadrature_points),
-                   old_temperature_grads(quadrature.n_quadrature_points),
-                   old_old_temperature_grads(quadrature.n_quadrature_points),
-                   old_temperature_laplacians(quadrature.n_quadrature_points),
-                   old_old_temperature_laplacians(quadrature.n_quadrature_points),
-
-                   gamma_values (quadrature.n_quadrature_points)
-    {}
-
-
-    template <int dim>
-    TemperatureRHS<dim>::
-    TemperatureRHS (const TemperatureRHS &scratch)
-                   :
-                   temperature_fe_values (scratch.temperature_fe_values.get_fe(),
-                                          scratch.temperature_fe_values.get_quadrature(),
-                                          scratch.temperature_fe_values.get_update_flags()),
-                   stokes_fe_values (scratch.stokes_fe_values.get_fe(),
-                                     scratch.stokes_fe_values.get_quadrature(),
-                                     scratch.stokes_fe_values.get_update_flags()),
-                   phi_T (scratch.phi_T),
-                   grad_phi_T (scratch.grad_phi_T),
-
-                   old_velocity_values (scratch.old_velocity_values),
-                   old_old_velocity_values (scratch.old_old_velocity_values),
-
-                   old_temperature_values (scratch.old_temperature_values),
-                   old_old_temperature_values (scratch.old_old_temperature_values),
-                   old_temperature_grads (scratch.old_temperature_grads),
-                   old_old_temperature_grads (scratch.old_old_temperature_grads),
-                   old_temperature_laplacians (scratch.old_temperature_laplacians),
-                   old_old_temperature_laplacians (scratch.old_old_temperature_laplacians),
-
-                   gamma_values (scratch.gamma_values)
-    {}
-  }
-
-
-                                  // The CopyData arrays are similar to the
-                                  // Scratch arrays. They provide a
-                                  // constructor, a copy operation, and
-                                  // some arrays for local matrix, local
-                                  // vectors and the relation between local
-                                  // and global degrees of freedom (a.k.a.
-                                  // <code>local_dof_indices</code>).
-  namespace CopyData
-  {
-    template <int dim>
-    struct StokesPreconditioner
-    {
-       StokesPreconditioner (const FiniteElement<dim> &stokes_fe);
-       StokesPreconditioner (const StokesPreconditioner &data);
-
-       FullMatrix<double>          local_matrix;
-       std::vector<unsigned int>   local_dof_indices;
-    };
-
-    template <int dim>
-    StokesPreconditioner<dim>::
-    StokesPreconditioner (const FiniteElement<dim> &stokes_fe)
-                   :
-                   local_matrix (stokes_fe.dofs_per_cell,
-                                 stokes_fe.dofs_per_cell),
-                   local_dof_indices (stokes_fe.dofs_per_cell)
-    {}
-
-
-
-    template <int dim>
-    StokesPreconditioner<dim>::
-    StokesPreconditioner (const StokesPreconditioner &data)
-                   :
-                   local_matrix (data.local_matrix),
-                   local_dof_indices (data.local_dof_indices)
-    {}
-
-
-
-    template <int dim>
-    struct StokesSystem : public StokesPreconditioner<dim>
-    {
-       StokesSystem (const FiniteElement<dim> &stokes_fe);
-       StokesSystem (const StokesSystem<dim> &data);
-
-       Vector<double> local_rhs;
-    };
-
-
-    template <int dim>
-    StokesSystem<dim>::
-    StokesSystem (const FiniteElement<dim> &stokes_fe)
-                   :
-                   StokesPreconditioner<dim> (stokes_fe),
-                   local_rhs (stokes_fe.dofs_per_cell)
-    {}
-
-
-    template <int dim>
-    StokesSystem<dim>::
-    StokesSystem (const StokesSystem<dim> &data)
-                   :
-                   StokesPreconditioner<dim> (data),
-                   local_rhs (data.local_rhs)
-    {}
-
-
-
-    template <int dim>
-    struct TemperatureMatrix
-    {
-       TemperatureMatrix (const FiniteElement<dim> &temperature_fe);
-       TemperatureMatrix (const TemperatureMatrix &data);
-
-       FullMatrix<double>          local_mass_matrix;
-       FullMatrix<double>          local_stiffness_matrix;
-       std::vector<unsigned int>   local_dof_indices;
-    };
-
-    template <int dim>
-    TemperatureMatrix<dim>::
-    TemperatureMatrix (const FiniteElement<dim> &temperature_fe)
-                   :
-                   local_mass_matrix (temperature_fe.dofs_per_cell,
-                                      temperature_fe.dofs_per_cell),
-                   local_stiffness_matrix (temperature_fe.dofs_per_cell,
-                                           temperature_fe.dofs_per_cell),
-                   local_dof_indices (temperature_fe.dofs_per_cell)
-    {}
-
-
-    template <int dim>
-    TemperatureMatrix<dim>::
-    TemperatureMatrix (const TemperatureMatrix &data)
-                   :
-                   local_mass_matrix (data.local_mass_matrix),
-                   local_stiffness_matrix (data.local_stiffness_matrix),
-                   local_dof_indices (data.local_dof_indices)
-    {}
-
-
-    template <int dim>
-    struct TemperatureRHS
-    {
-       TemperatureRHS (const FiniteElement<dim> &temperature_fe);
-       TemperatureRHS (const TemperatureRHS &data);
-
-       Vector<double>              local_rhs;
-       std::vector<unsigned int>   local_dof_indices;
-        FullMatrix<double>          matrix_for_bc;
-    };
-
-    template <int dim>
-    TemperatureRHS<dim>::
-    TemperatureRHS (const FiniteElement<dim> &temperature_fe)
-                   :
-                   local_rhs (temperature_fe.dofs_per_cell),
-                   local_dof_indices (temperature_fe.dofs_per_cell),
-                   matrix_for_bc (temperature_fe.dofs_per_cell,
-                                  temperature_fe.dofs_per_cell)
-    {}
-
-
-    template <int dim>
-    TemperatureRHS<dim>::
-    TemperatureRHS (const TemperatureRHS &data)
-                   :
-                   local_rhs (data.local_rhs),
-                   local_dof_indices (data.local_dof_indices),
-                   matrix_for_bc (data.matrix_for_bc)
-    {}
-  }
-}
-
-
-
-                                // @sect3{The <code>BoussinesqFlowProblem</code> class template}
-                                //
-                                // This is the declaration of the main
-                                // class. It is very similar to
-                                // step-31. Following the @ref
-                                // MTWorkStream "task-based parallelization"
-                                // paradigm, we split all the
-                                // assembly routines into two parts: a
-                                // first part that can do all the
-                                // calculations on a certain cell without
-                                // taking care of other threads, and a
-                                // second part (which is writing the
-                                // local data into the global matrices
-                                // and vectors) which can be entered by
-                                // only one thread at a time. In order to
-                                // implement that, we provide functions
-                                // for each of those two steps for all
-                                // the four assembly routines that we use
-                                // in this program.
-                                //
-                                // The <code>pcout</code> (for <i>%parallel
-                                // <code>std::cout</code></i>) object is used
-                                // to simplify writing output: each MPI
-                                // process can use this to generate output as
-                                // usual, but since each of these processes
-                                // will produce the same output it will just
-                                // be replicated many times over; with the
-                                // ConditionalOStream class, only the output
-                                // generated by one task will actually be
-                                // printed to screen, whereas the output by
-                                // all the other threads will simply be
-                                // forgotten.
-                                //
-                                // In a bit of naming confusion, you will
-                                // notice below that some of the variables
-                                // from namespace TrilinosWrappers are
-                                // taken from namespace
-                                // TrilinosWrappers::MPI (such as the right
-                                // hand side vectors) whereas others are
-                                // not (such as the various matrices). For
-                                // the matrices, we happen to use the same
-                                // class names for %parallel and sequential
-                                // data structures, i.e. all matrices will
-                                // actually be considered %parallel
-                                // below. On the other hand, for vectors,
-                                // only those from namespace
-                                // TrilinosWrappers::MPI are actually
-                                // distributed. In particular, we will
-                                // frequently have to query velocities and
-                                // temperatures at arbitrary quadrature
-                                // points; consequently, rather than
-                                // "localizing" a vector whenever we need a
-                                // localized vector, we solve linear
-                                // systems in %parallel but then immediately
-                                // localize the solution for further
-                                // processing. The various
-                                // <code>*_solution</code> vectors are
-                                // therefore filled immediately after
-                                // solving their respective linear system
-                                // in %parallel.
-                                //
-                                // The only other new data member is
-                                // <code>computing_timer</code>. Its class
-                                // type, TimerOutput, can be used to
-                                // conveniently account for compute time
-                                // spent in certain "sections" of the code
-                                // that are repeatedly entered. For
-                                // example, we will enter (and leave)
-                                // sections for Stokes matrix assembly and
-                                // would like to accumulate the run time
-                                // spent in this section over all time
-                                // steps. At the end of the program, the
-                                // destructor of the TimerOutput class will
-                                // automatically produce a nice summary of
-                                // the times spent in all the sections. For
-                                // this output, one can choose whether wall
-                                // clock or CPU times are to be printed, as
-                                // well as whether we want to produce
-                                // output every time we leave a section --
-                                // which would be quite a lot of additional
-                                // output -- or just in the end of the
-                                // program (this choice is made in the
-                                // from this variable in the results
-                                // section of this tutorial program.
-template <int dim>
-class BoussinesqFlowProblem
-{
-  public:
-    BoussinesqFlowProblem ();
-    void run ();
-
-  private:
-    void setup_dofs ();
-    void assemble_stokes_preconditioner ();
-    void build_stokes_preconditioner ();
-    void assemble_stokes_system ();
-    void assemble_temperature_matrix ();
-    void assemble_temperature_system (const double maximal_velocity);
-    void project_temperature_field ();
-    double get_maximal_velocity () const;
-    std::pair<double,double> get_extrapolated_temperature_range () const;
-    void solve ();
-    void output_results ();
-    void refine_mesh (const unsigned int max_grid_level);
-
-    double
-    compute_viscosity(const std::vector<double>          &old_temperature,
-                     const std::vector<double>          &old_old_temperature,
-                     const std::vector<Tensor<1,dim> >  &old_temperature_grads,
-                     const std::vector<Tensor<1,dim> >  &old_old_temperature_grads,
-                     const std::vector<double>          &old_temperature_laplacians,
-                     const std::vector<double>          &old_old_temperature_laplacians,
-                     const std::vector<Tensor<1,dim> >  &old_velocity_values,
-                     const std::vector<Tensor<1,dim> >  &old_old_velocity_values,
-                     const std::vector<double>          &gamma_values,
-                     const double                        global_u_infty,
-                     const double                        global_T_variation,
-                     const double                        cell_diameter) const;
-
-
-    ConditionalOStream                  pcout;
-
-    Triangulation<dim>                  triangulation;
-    double                              global_Omega_diameter;
-
-    const unsigned int                  stokes_degree;
-    FESystem<dim>                       stokes_fe;
-    DoFHandler<dim>                     stokes_dof_handler;
-    ConstraintMatrix                    stokes_constraints;
-
-    TrilinosWrappers::BlockSparseMatrix stokes_matrix;
-    TrilinosWrappers::BlockSparseMatrix stokes_preconditioner_matrix;
-
-    TrilinosWrappers::BlockVector       stokes_solution;
-    TrilinosWrappers::BlockVector       old_stokes_solution;
-    TrilinosWrappers::MPI::BlockVector  stokes_rhs;
-
-
-    const unsigned int                  temperature_degree;
-    FE_Q<dim>                           temperature_fe;
-    DoFHandler<dim>                     temperature_dof_handler;
-    ConstraintMatrix                    temperature_constraints;
-
-    TrilinosWrappers::SparseMatrix      temperature_mass_matrix;
-    TrilinosWrappers::SparseMatrix      temperature_stiffness_matrix;
-    TrilinosWrappers::SparseMatrix      temperature_matrix;
-
-    TrilinosWrappers::Vector            temperature_solution;
-    TrilinosWrappers::Vector            old_temperature_solution;
-    TrilinosWrappers::Vector            old_old_temperature_solution;
-    TrilinosWrappers::MPI::Vector       temperature_rhs;
-
-
-    double time_step;
-    double old_time_step;
-    unsigned int timestep_number;
-
-    std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionAMG> Amg_preconditioner;
-    std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionILU> Mp_preconditioner;
-    std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionIC>  T_preconditioner;
-
-    bool rebuild_stokes_matrix;
-    bool rebuild_stokes_preconditioner;
-    bool rebuild_temperature_matrices;
-    bool rebuild_temperature_preconditioner;
-
-    TimerOutput computing_timer;
-
-    void setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning);
-    void setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning);
-    void setup_temperature_matrices (const IndexSet &temperature_partitioning);
-
-    void
-    local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                         Assembly::Scratch::StokesPreconditioner<dim> &scratch,
-                                         Assembly::CopyData::StokesPreconditioner<dim> &data);
-
-    void
-    copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data);
-
-
-    void
-    local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                 Assembly::Scratch::StokesSystem<dim>  &scratch,
-                                 Assembly::CopyData::StokesSystem<dim> &data);
-
-    void
-    copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data);
-
-
-    void
-    local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                      Assembly::Scratch::TemperatureMatrix<dim>  &scratch,
-                                      Assembly::CopyData::TemperatureMatrix<dim> &data);
-
-    void
-    copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data);
-
-
-
-    void
-    local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
-                                   const double                   global_max_velocity,
-                                   const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                   Assembly::Scratch::TemperatureRHS<dim> &scratch,
-                                   Assembly::CopyData::TemperatureRHS<dim> &data);
-
-    void
-    copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data);
-};
-
-
-                                // @sect3{BoussinesqFlowProblem class implementation}
-
-                                // @sect4{BoussinesqFlowProblem::BoussinesqFlowProblem}
-                                //
-                                // The constructor of the problem is very
-                                // similar to the constructor in
-                                // step-31. What is different is the
-                                // %parallel communication: Trilinos uses a
-                                // message passing interface (MPI) for data
-                                // distribution. When entering the
-                                // BoussinesqFlowProblem class, we have to
-                                // decide how the parallization is to be
-                                // done. We choose a rather simple strategy
-                                // and let all processors that are running
-                                // the program work together, specified by
-                                // the communicator
-                                // <code>comm_world()</code>. Next, we
-                                // create some modified output stream as we
-                                // already did in step-18. In MPI, all the
-                                // processors run the same program
-                                // individually (they simply operate on
-                                // different chunks of data and exchange
-                                // some part of that data from time to
-                                // time). Next, we need to initialize the
-                                // <code>pcout</code> object in order to
-                                // print the user information only on one
-                                // processor. The implementation of this
-                                // idea is to check the process number when
-                                // <code>pcout</code> gets a true argument,
-                                // and it uses the <code>std::cout</code>
-                                // stream for output. If we are one
-                                // processor five, for instance, then we
-                                // will give a <code>false</code> argument
-                                // to <code>pcout</code>, which means that
-                                // the output of that processor will not be
-                                // printed anywhere.
-                                //
-                                // Finally, we enter the preferred options
-                                // for the TimerOutput object to its
-                                // constructor. We restrict the output to
-                                // the <code>pcout</code> stream (processor
-                                // 0), and then we specify that we want to
-                                // get a summary table in the end of the
-                                // program which shows us wallclock times
-                                // (as opposed to CPU times).
-template <int dim>
-BoussinesqFlowProblem<dim>::BoussinesqFlowProblem ()
-                :
-               pcout (std::cout,
-                      (Utilities::System::
-                       get_this_mpi_process(MPI_COMM_WORLD)
-                       == 0)),
-
-               triangulation (Triangulation<dim>::maximum_smoothing),
-
-                stokes_degree (1),
-                stokes_fe (FE_Q<dim>(stokes_degree+1), dim,
-                          FE_Q<dim>(stokes_degree), 1),
-               stokes_dof_handler (triangulation),
-
-               temperature_degree (2),
-               temperature_fe (temperature_degree),
-                temperature_dof_handler (triangulation),
-
-                time_step (0),
-               old_time_step (0),
-               timestep_number (0),
-               rebuild_stokes_matrix (true),
-               rebuild_stokes_preconditioner (true),
-               rebuild_temperature_matrices (true),
-               rebuild_temperature_preconditioner (true),
-
-               computing_timer (pcout, TimerOutput::summary,
-                                TimerOutput::wall_times)
-{}
-
-
-
-                                // @sect4{The BoussinesqFlowProblem helper functions}
-                                //
-                                // Except two small details, this
-                                // function is the very same as in
-                                // step-31. The first detail is
-                                // actually common to all functions
-                                // that implement loop over all cells
-                                // in the triangulation: When
-                                // operating in %parallel, each
-                                // processor only works on a chunk of
-                                // cells. This chunk of cells is
-                                // identified via a so-called
-                                // <code>subdomain_id</code>, as we
-                                // also did in step-18. All we need
-                                // to change is hence to perform the
-                                // cell-related operations only on
-                                // the process with the correct
-                                // ID. The second difference is the
-                                // way we calculate the maximum
-                                // value. Before, we could simply
-                                // have a <code>double</code>
-                                // variable that we checked against
-                                // on each quadrature point for each
-                                // cell. Now, we have to be a bit
-                                // more careful since each processor
-                                // only operates on a subset of
-                                // cells. What we do is to first let
-                                // each processor calculate the
-                                // maximum among its cells, and then
-                                // do a global communication
-                                // operation called
-                                // <code>MaxAll</code> that searches
-                                // for the maximum value among all
-                                // the maximum values of the
-                                // individual processors. MPI
-                                // provides such a call, but it's
-                                // even simpler to use the respective
-                                // function of the MPI
-                                // communicator object since that
-                                // will do the right thing even if we
-                                // work without MPI and on a single
-                                // machine only. The call to
-                                // <code>MaxAll</code> needs three
-                                // arguments, namely the local
-                                // maximum (input), a field for the
-                                // global maximum (output), and an
-                                // integer value one that says that
-                                // we only work on one double.
-template <int dim>
-double BoussinesqFlowProblem<dim>::get_maximal_velocity () const
-{
-  const QIterated<dim> quadrature_formula (QTrapez<1>(),
-                                          stokes_degree+1);
-  const unsigned int n_q_points = quadrature_formula.size();
-
-  FEValues<dim> fe_values (stokes_fe, quadrature_formula, update_values);
-  std::vector<Tensor<1,dim> > velocity_values(n_q_points);
-
-  const FEValuesExtractors::Vector velocities (0);
-
-  double max_local_velocity = 0;
-
-  typename DoFHandler<dim>::active_cell_iterator
-    cell = stokes_dof_handler.begin_active(),
-    endc = stokes_dof_handler.end();
-  for (; cell!=endc; ++cell)
-    if (cell->subdomain_id() ==
-       Utilities::System::get_this_mpi_process(MPI_COMM_WORLD))
-      {
-       fe_values.reinit (cell);
-       fe_values[velocities].get_function_values (stokes_solution,
-                                                  velocity_values);
-
-       for (unsigned int q=0; q<n_q_points; ++q)
-         max_local_velocity = std::max (max_local_velocity,
-                                        velocity_values[q].norm());
-      }
-
-  double max_velocity = 0.;
-#ifdef DEAL_II_COMPILER_SUPPORTS_MPI
-  MPI_Allreduce (&max_local_velocity, &max_velocity, 1, MPI_DOUBLE,
-                MPI_MAX, MPI_COMM_WORLD);
-#else
-  max_velocity = max_local_velocity;
-#endif
-
-  return max_velocity;
-}
-
-
-
-                                // Again, this is only a slightly
-                                // modified version of the respective
-                                // function in step-31. What is new is
-                                // that each processor works on its
-                                // partition of cells, and gets a minimum
-                                // and maximum temperature on that
-                                // partition. Two global communication
-                                // steps synchronize the data among the
-                                // processors.
-template <int dim>
-std::pair<double,double>
-BoussinesqFlowProblem<dim>::get_extrapolated_temperature_range () const
-{
-  const QIterated<dim> quadrature_formula (QTrapez<1>(),
-                                          temperature_degree);
-  const unsigned int n_q_points = quadrature_formula.size();
-
-  FEValues<dim> fe_values (temperature_fe, quadrature_formula,
-                           update_values);
-  std::vector<double> old_temperature_values(n_q_points);
-  std::vector<double> old_old_temperature_values(n_q_points);
-
-  if (timestep_number != 0)
-    {
-      double min_local_temperature = (1. + time_step/old_time_step) *
-                                    old_temperature_solution.linfty_norm()
-                                    +
-                                    time_step/old_time_step *
-                                    old_old_temperature_solution.linfty_norm(),
-            max_local_temperature = -min_local_temperature;
-
-      typename DoFHandler<dim>::active_cell_iterator
-       cell = temperature_dof_handler.begin_active(),
-       endc = temperature_dof_handler.end();
-      for (; cell!=endc; ++cell)
-       if (cell->subdomain_id() ==
-           Utilities::System::get_this_mpi_process(MPI_COMM_WORLD))
-         {
-           fe_values.reinit (cell);
-           fe_values.get_function_values (old_temperature_solution,
-                                          old_temperature_values);
-           fe_values.get_function_values (old_old_temperature_solution,
-                                          old_old_temperature_values);
-
-           for (unsigned int q=0; q<n_q_points; ++q)
-             {
-               const double temperature =
-                 (1. + time_step/old_time_step) * old_temperature_values[q]-
-                 time_step/old_time_step * old_old_temperature_values[q];
-
-               min_local_temperature = std::min (min_local_temperature,
-                                                 temperature);
-               max_local_temperature = std::max (max_local_temperature,
-                                                 temperature);
-             }
-         }
-
-      double min_temperature, max_temperature;
-#ifdef DEAL_II_COMPILER_SUPPORTS_MPI
-      MPI_Allreduce (&max_local_temperature, &max_temperature, 1, MPI_DOUBLE,
-                    MPI_MAX, MPI_COMM_WORLD);
-      MPI_Allreduce (&min_local_temperature, &min_temperature, 1, MPI_DOUBLE,
-                    MPI_MIN, MPI_COMM_WORLD);
-#else
-      min_temperature = min_local_temperature;
-      max_temperature = max_local_temperature;
-#endif
-
-      return std::make_pair(min_temperature, max_temperature);
-    }
-  else
-    {
-      double min_local_temperature = old_temperature_solution.linfty_norm(),
-            max_local_temperature = -min_local_temperature;
-
-      typename DoFHandler<dim>::active_cell_iterator
-       cell = temperature_dof_handler.begin_active(),
-       endc = temperature_dof_handler.end();
-      for (; cell!=endc; ++cell)
-       if (cell->subdomain_id() ==
-           Utilities::System::get_this_mpi_process(MPI_COMM_WORLD))
-         {
-           fe_values.reinit (cell);
-           fe_values.get_function_values (old_temperature_solution,
-                                          old_temperature_values);
-
-           for (unsigned int q=0; q<n_q_points; ++q)
-             {
-               const double temperature = old_temperature_values[q];
-
-               min_local_temperature = std::min (min_local_temperature,
-                                                 temperature);
-               max_local_temperature = std::max (max_local_temperature,
-                                                 temperature);
-             }
-         }
-
-      double min_temperature, max_temperature;
-#ifdef DEAL_II_COMPILER_SUPPORTS_MPI
-      MPI_Allreduce (&max_local_temperature, &max_temperature, 1, MPI_DOUBLE,
-                    MPI_MAX, MPI_COMM_WORLD);
-      MPI_Allreduce (&min_local_temperature, &min_temperature, 1, MPI_DOUBLE,
-                    MPI_MIN, MPI_COMM_WORLD);
-#else
-      min_temperature = min_local_temperature;
-      max_temperature = max_local_temperature;
-#endif
-
-      return std::make_pair(min_temperature, max_temperature);
-    }
-}
-
-
-
-                                // The function that calculates the
-                                // viscosity is purely local, so this is
-                                // the same code as in step-31.
-template <int dim>
-double
-BoussinesqFlowProblem<dim>::
-compute_viscosity (const std::vector<double>          &old_temperature,
-                  const std::vector<double>          &old_old_temperature,
-                  const std::vector<Tensor<1,dim> >  &old_temperature_grads,
-                  const std::vector<Tensor<1,dim> >  &old_old_temperature_grads,
-                  const std::vector<double>          &old_temperature_laplacians,
-                  const std::vector<double>          &old_old_temperature_laplacians,
-                  const std::vector<Tensor<1,dim> >  &old_velocity_values,
-                  const std::vector<Tensor<1,dim> >  &old_old_velocity_values,
-                  const std::vector<double>          &gamma_values,
-                  const double                        global_u_infty,
-                  const double                        global_T_variation,
-                  const double                        cell_diameter) const
-{
-  const double beta = 0.026 * dim;
-  const double alpha = 1;
-
-  if (global_u_infty == 0)
-    return 5e-3 * cell_diameter;
-
-  const unsigned int n_q_points = old_temperature.size();
-
-  double max_residual = 0;
-  double max_velocity = 0;
-
-  for (unsigned int q=0; q < n_q_points; ++q)
-    {
-      const Tensor<1,dim> u = (old_velocity_values[q] +
-                              old_old_velocity_values[q]) / 2;
-
-      const double dT_dt = (old_temperature[q] - old_old_temperature[q])
-                          / old_time_step;
-      const double u_grad_T = u * (old_temperature_grads[q] +
-                                  old_old_temperature_grads[q]) / 2;
-
-      const double kappa_Delta_T = EquationData::kappa
-                                  * (old_temperature_laplacians[q] +
-                                     old_old_temperature_laplacians[q]) / 2;
-
-      const double residual
-       = std::abs((dT_dt + u_grad_T - kappa_Delta_T - gamma_values[q]) *
-                  std::pow((old_temperature[q]+old_old_temperature[q]) / 2,
-                           alpha-1.));
-
-      max_residual = std::max (residual,        max_residual);
-      max_velocity = std::max (std::sqrt (u*u), max_velocity);
-    }
-
-  if (timestep_number == 0)
-    return beta * max_velocity * cell_diameter;
-  else
-    {
-      Assert (old_time_step > 0, ExcInternalError());
-
-      const double c_R = 0.11;
-      const double global_scaling = c_R * global_u_infty * global_T_variation *
-                                   std::pow(global_Omega_diameter, alpha - 2.);
-
-      return (beta *
-             max_velocity *
-             std::min (cell_diameter,
-                       std::pow(cell_diameter,alpha) * max_residual /
-                       global_scaling));
-    }
-}
-
-
-
-                                // This function is new compared to
-                                // step-31. What is does is to re-implement
-                                // the library function
-                                // <code>VectorTools::project()</code> for
-                                // an MPI-based parallelization, a function
-                                // we used for generating an initial vector
-                                // for temperature based on some initial
-                                // function. The library function only
-                                // works with shared memory but doesn't
-                                // know how to utilize multiple machines
-                                // coupled through MPI to compute the
-                                // projected solution. If run with
-                                // more than one MPI process, this would
-                                // mean that each processor projects the
-                                // whole field, which is clearly not very
-                                // efficient. The details of a
-                                // <code>project()</code> function are not
-                                // very difficult. All we do is to use a
-                                // mass matrix and put the evaluation of
-                                // the initial value function on the right
-                                // hand side. The mass matrix for
-                                // temperature we can simply generate using
-                                // the respective assembly function, so all
-                                // we need to do here is to create the
-                                // right hand side and do a CG solve. The
-                                // assembly function does a loop over all
-                                // cells and evaluates the function in the
-                                // <code>EquationData</code> namespace, and
-                                // does this only on cells pertaining to
-                                // the respective processor. The
-                                // implementation of this assembly differs
-                                // from the assembly we do for the
-                                // principal assembly functions further
-                                // down (which include thread-based
-                                // parallelization with the WorkStream
-                                // concept). Here we chose to keep things
-                                // simple (keeping in mind that this function
-                                // is also only called at the beginning of
-                                // the program, not every time step), and
-                                // generating that right hand
-                                // side is cheap anyway so we won't even
-                                // notice that this part is not parallized
-                                // by threads.
-                                //
-                                // Regarding the implementation of
-                                // inhomogeneous Dirichlet boundary
-                                // conditions: Since we use the temperature
-                                // ConstraintMatrix, we can apply the
-                                // boundary conditions directly when
-                                // building the respective matrix and right
-                                // hand side. In this case, the boundary
-                                // conditions are inhomogeneous, which
-                                // makes this procedure somewhat
-                                // tricky. Remember that we get the matrix
-                                // from some other function. However, the
-                                // correct imposition of boundary
-                                // conditions needs the matrix data we work
-                                // on plus the right hand side
-                                // simultaneously, since the right hand
-                                // side is created by Gaussian elimination
-                                // on the matrix rows. In order to not
-                                // introduce the matrix assembly at this
-                                // place, but still having the matrix data
-                                // available, we choose to create a dummy
-                                // matrix <code>matrix_for_bc</code> that
-                                // we only fill with data when we need it
-                                // for imposing boundary conditions. These
-                                // positions are exactly those where we
-                                // have an inhomogeneous entry in the
-                                // ConstraintMatrix. There are only a few
-                                // such positions (on the boundary dofs),
-                                // so it is still much cheaper to use this
-                                // function than to create the full matrix
-                                // here. To implement this, we ask the
-                                // constraint matrix whether the dof under
-                                // consideration is inhomogeneously
-                                // constraint. In that case, we generate
-                                // the respective matrix column that we
-                                // need for creating the correct right hand
-                                // side. Note that this (manually
-                                // generated) matrix entry needs to be
-                                // exactly the entry that we would fill the
-                                // matrix with &mdash; otherwise, this will
-                                // not work.
-template <int dim>
-void BoussinesqFlowProblem<dim>::project_temperature_field ()
-{
-  assemble_temperature_matrix ();
-
-  QGauss<dim> quadrature(temperature_degree+2);
-  UpdateFlags update_flags = UpdateFlags(update_values   |
-                                        update_quadrature_points |
-                                        update_JxW_values);
-  FEValues<dim> fe_values (temperature_fe, quadrature, update_flags);
-
-  const unsigned int dofs_per_cell = fe_values.dofs_per_cell,
-                    n_q_points    = fe_values.n_quadrature_points;
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-  Vector<double> cell_vector (dofs_per_cell);
-  FullMatrix<double> matrix_for_bc (dofs_per_cell, dofs_per_cell);
-
-  typename DoFHandler<dim>::active_cell_iterator
-    cell = temperature_dof_handler.begin_active(),
-    endc = temperature_dof_handler.end();
-
-  std::vector<double> rhs_values(n_q_points);
-
-  TrilinosWrappers::MPI::Vector
-    rhs (temperature_mass_matrix.row_partitioner()),
-    solution (temperature_mass_matrix.row_partitioner());
-
-  for (; cell!=endc; ++cell)
-    if (cell->subdomain_id() ==
-       Utilities::System::get_this_mpi_process(MPI_COMM_WORLD))
-      {
-       cell->get_dof_indices (local_dof_indices);
-       fe_values.reinit(cell);
-
-       EquationData::TemperatureInitialValues<dim>().value_list
-         (fe_values.get_quadrature_points(), rhs_values);
-
-       cell_vector = 0;
-       matrix_for_bc = 0;
-       for (unsigned int point=0; point<n_q_points; ++point)
-         for (unsigned int i=0; i<dofs_per_cell; ++i)
-           {
-             cell_vector(i) += rhs_values[point] *
-                               fe_values.shape_value(i,point) *
-                               fe_values.JxW(point);
-             if (temperature_constraints.is_inhomogeneously_constrained(local_dof_indices[i]))
-               {
-                 for (unsigned int j=0; j<dofs_per_cell; ++j)
-                   matrix_for_bc(j,i) += fe_values.shape_value(i,point) *
-                                         fe_values.shape_value(j,point) *
-                                         fe_values.JxW(point);
-               }
-           }
-
-       temperature_constraints.distribute_local_to_global (cell_vector,
-                                                           local_dof_indices,
-                                                           rhs,
-                                                           matrix_for_bc);
-      }
-
-  rhs.compress ();
-
-  SolverControl solver_control(5*rhs.size(), 1e-12*rhs.l2_norm());
-  SolverCG<TrilinosWrappers::MPI::Vector> cg(solver_control);
-
-  TrilinosWrappers::PreconditionSSOR preconditioner_mass;
-  preconditioner_mass.initialize(temperature_mass_matrix, 1.3);
-
-  cg.solve (temperature_mass_matrix, solution, rhs, preconditioner_mass);
-
-  old_temperature_solution = solution;
-  temperature_constraints.distribute (old_temperature_solution);
-}
-
-
-
-
-                                // @sect4{The BoussinesqFlowProblem setup functions}
-
-                                // The following three functions set
-                                // up the Stokes matrix, the matrix
-                                // used for the Stokes
-                                // preconditioner, and the
-                                // temperature matrix. The code is
-                                // mostly the same as in step-31, but
-                                // it has been broken out into three
-                                // functions of their own for
-                                // simplicity, but also so that they
-                                // can easily be run in %parallel on
-                                // multiple threads (unless we are
-                                // running with MPI, in which case
-                                // this is not possible, as explained
-                                // in the introduction).
-                                //
-                                // The main functional difference
-                                // between the code here and that in
-                                // step-31 is that the matrices we
-                                // want to set up are distributed
-                                // across multiple processors. Since
-                                // we still want to build up the
-                                // sparsity pattern first for
-                                // efficiency reasons, we could
-                                // continue to build the
-                                // <i>entire</i> sparsity pattern as
-                                // a
-                                // BlockCompressedSimpleSparsityPattern,
-                                // as we did in step-31. However,
-                                // that would be inefficient: every
-                                // processor would build the same
-                                // sparsity pattern, but only
-                                // initialize a small part of the
-                                // matrix using it.
-                                //
-                                // Rather, we use an object of type
-                                // TrilinosWrappers::BlockSparsityPattern,
-                                // which is (obviously) a wrapper
-                                // around a sparsity pattern object
-                                // provided by Trilinos. The
-                                // advantage is that the Trilinos
-                                // sparsity pattern class can
-                                // communicate across multiple
-                                // processors: if this processor
-                                // fills in all the nonzero entries
-                                // that result from the cells it
-                                // owns, and every other processor
-                                // does so as well, then at the end
-                                // after some MPI communication
-                                // initiated by the
-                                // <code>compress()</code> call, we
-                                // will have the globally assembled
-                                // sparsity pattern available with
-                                // which the global matrix can be
-                                // initialized.
-                                //
-                                // The only other change we need to
-                                // make is to tell the
-                                // DoFTools::make_sparsity_pattern
-                                // function that it is only supposed
-                                // to work on a subset of cells,
-                                // namely the ones whose
-                                // <code>subdomain_id</code> equals
-                                // the number of the current
-                                // processor, and to ignore all other
-                                // cells.
-                                //
-                                // This strategy is replicated across
-                                // all three of the following
-                                // functions.
-                                //
-                                // Note that Trilinos matrices store the
-                                // information contained in the sparsity
-                                // patterns, so we can safely release the
-                                // <code>sp</code> variable once the matrix
-                                // has been given the sparsity structure.
-template <int dim>
-void BoussinesqFlowProblem<dim>::
-  setup_stokes_matrix (const std::vector<IndexSet> &stokes_partitioning)
-{
-  stokes_matrix.clear ();
-
-  TrilinosWrappers::BlockSparsityPattern sp (stokes_partitioning,
-                                            MPI_COMM_WORLD);
-
-  Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
-
-  for (unsigned int c=0; c<dim+1; ++c)
-    for (unsigned int d=0; d<dim+1; ++d)
-      if (! ((c==dim) && (d==dim)))
-       coupling[c][d] = DoFTools::always;
-      else
-       coupling[c][d] = DoFTools::none;
-
-  DoFTools::make_sparsity_pattern (stokes_dof_handler, coupling, sp,
-                                  stokes_constraints, false,
-                                  Utilities::System::
-                                  get_this_mpi_process(MPI_COMM_WORLD));
-  sp.compress();
-
-  stokes_matrix.reinit (sp);
-}
-
-
-
-template <int dim>
-void BoussinesqFlowProblem<dim>::
-  setup_stokes_preconditioner (const std::vector<IndexSet> &stokes_partitioning)
-{
-  Amg_preconditioner.reset ();
-  Mp_preconditioner.reset ();
-
-  stokes_preconditioner_matrix.clear ();
-
-  TrilinosWrappers::BlockSparsityPattern sp (stokes_partitioning,
-                                            MPI_COMM_WORLD);
-
-  Table<2,DoFTools::Coupling> coupling (dim+1, dim+1);
-  for (unsigned int c=0; c<dim+1; ++c)
-    for (unsigned int d=0; d<dim+1; ++d)
-      if (c == d)
-       coupling[c][d] = DoFTools::always;
-      else
-       coupling[c][d] = DoFTools::none;
-
-  DoFTools::make_sparsity_pattern (stokes_dof_handler, coupling, sp,
-                                  stokes_constraints, false,
-                                  Utilities::System::
-                                  get_this_mpi_process(MPI_COMM_WORLD));
-  sp.compress();
-
-  stokes_preconditioner_matrix.reinit (sp);
-}
-
-
-template <int dim>
-void BoussinesqFlowProblem<dim>::
-  setup_temperature_matrices (const IndexSet &temperature_partitioner)
-{
-  T_preconditioner.reset ();
-  temperature_mass_matrix.clear ();
-  temperature_stiffness_matrix.clear ();
-  temperature_matrix.clear ();
-
-  TrilinosWrappers::SparsityPattern sp (temperature_partitioner,
-                                       MPI_COMM_WORLD);
-  DoFTools::make_sparsity_pattern (temperature_dof_handler, sp,
-                                  temperature_constraints, false,
-                                  Utilities::System::
-                                  get_this_mpi_process(MPI_COMM_WORLD));
-  sp.compress();
-
-  temperature_matrix.reinit (sp);
-  temperature_mass_matrix.reinit (sp);
-  temperature_stiffness_matrix.reinit (sp);
-}
-
-
-
-                                // The remainder of the setup function
-                                // (after splitting out the three functions
-                                // above) mostly has to deal with the
-                                // things we need to do for parallelization
-                                // across processors. In particular, at the
-                                // top it calls
-                                // GridTools::partition_triangulation to
-                                // subdivide all cells into subdomains of
-                                // roughly equal size and roughly minimal
-                                // interface length (using METIS). We then
-                                // distribute degrees of freedom for Stokes
-                                // and temperature DoFHandler objects, and
-                                // re-sort them in such a way that all
-                                // degrees of freedom associated with
-                                // subdomain zero come before all those
-                                // associated with subdomain one, etc. For
-                                // the Stokes part, this entails, however,
-                                // that velocities and pressures become
-                                // intermixed, but this is trivially solved
-                                // by sorting again by blocks; it is worth
-                                // noting that this latter operation leaves
-                                // the relative ordering of all velocities
-                                // and pressures alone, i.e. within the
-                                // velocity block we will still have all
-                                // those associated with subdomain zero
-                                // before all velocities associated with
-                                // subdomain one, etc. This is important
-                                // since we store each of the blocks of
-                                // this matrix distributed across all
-                                // processors and want this to be done in
-                                // such a way that each processor stores
-                                // that part of the matrix that is roughly
-                                // equal to the degrees of freedom located
-                                // on those cells that it will actually
-                                // work on. Note how we set boundary
-                                // conditions on the temperature by using
-                                // the ConstraintMatrix object.
-                                //
-                                // After this, we have to set up the
-                                // various partitioners (of type
-                                // <code>IndexSet</code>, see the
-                                // introduction) that describe which parts
-                                // of each matrix or vector will be stored
-                                // where, then call the functions that
-                                // actually set up the matrices
-                                // (concurrently if not using MPI
-                                // but sequentially otherwise, as explained
-                                // in the introduction), and at the end also
-                                // resize the various vectors we keep
-                                // around in this program. We given those
-                                // vectors the correct size using the
-                                // aforementioned Epetra_Map. Most of the
-                                // vectors are actually localized, i.e.,
-                                // they store all dofs in the problem on
-                                // each processor. In that case, the only
-                                // information that is used is the global
-                                // size. This is different for the two
-                                // right hand side vectors, which are
-                                // distributed ones, see also the class
-                                // declaration.
-                                //
-                                // Note how this function enters and leaves
-                                // a timed section so that we can get a
-                                // time report at the end of the
-                                // program. Note also the use of the
-                                // <code>pcout</code> variable: to every
-                                // process it looks like we can write to
-                                // screen, but only the output of the first
-                                // processor actually ends up somewhere. We
-                                // could of course have achieved the same
-                                // effect by writing to
-                                // <code>std::cout</code> but would then
-                                // have had to guard every access to that
-                                // stream by something like <code>if
-                                // (Utilities:: System::
-                                // get_this_mpi_process
-                                // (MPI_COMM_WORLD) == 0)</code>,
-                                // hardly a pretty solution.
-template <int dim>
-void BoussinesqFlowProblem<dim>::setup_dofs ()
-{
-  computing_timer.enter_section("Setup dof systems");
-  std::vector<unsigned int> stokes_sub_blocks (dim+1,0);
-  stokes_sub_blocks[dim] = 1;
-
-  GridTools::partition_triangulation (Utilities::System::
-                                     get_n_mpi_processes(MPI_COMM_WORLD),
-                                     triangulation);
-
-  {
-    stokes_dof_handler.distribute_dofs (stokes_fe);
-    DoFRenumbering::subdomain_wise (stokes_dof_handler);
-    DoFRenumbering::component_wise (stokes_dof_handler, stokes_sub_blocks);
-
-    stokes_constraints.clear ();
-    DoFTools::make_hanging_node_constraints (stokes_dof_handler,
-                                            stokes_constraints);
-
-    std::vector<bool> velocity_mask (dim+1, true);
-    velocity_mask[dim] = false;
-    VectorTools::interpolate_boundary_values (stokes_dof_handler,
-                                             0,
-                                             ZeroFunction<dim>(dim+1),
-                                             stokes_constraints,
-                                             velocity_mask);
-
-    std::set<unsigned char> no_normal_flux_boundaries;
-    no_normal_flux_boundaries.insert (1);
-    VectorTools::compute_no_normal_flux_constraints (stokes_dof_handler, 0,
-                                                    no_normal_flux_boundaries,
-                                                    stokes_constraints);
-    stokes_constraints.close ();
-  }
-  {
-    temperature_dof_handler.distribute_dofs (temperature_fe);
-    DoFRenumbering::subdomain_wise (temperature_dof_handler);
-
-    temperature_constraints.clear ();
-    VectorTools::interpolate_boundary_values (temperature_dof_handler,
-                                             0,
-                                             EquationData::TemperatureInitialValues<dim>(),
-                                             temperature_constraints);
-    VectorTools::interpolate_boundary_values (temperature_dof_handler,
-                                             1,
-                                             EquationData::TemperatureInitialValues<dim>(),
-                                             temperature_constraints);
-    DoFTools::make_hanging_node_constraints (temperature_dof_handler,
-                                            temperature_constraints);
-    temperature_constraints.close ();
-  }
-
-  std::vector<unsigned int> stokes_dofs_per_block (2);
-  DoFTools::count_dofs_per_block (stokes_dof_handler, stokes_dofs_per_block,
-                                 stokes_sub_blocks);
-
-  const unsigned int n_u = stokes_dofs_per_block[0],
-                     n_p = stokes_dofs_per_block[1],
-                    n_T = temperature_dof_handler.n_dofs();
-
-  pcout << "Number of active cells: "
-       << triangulation.n_active_cells()
-       << " (on "
-       << triangulation.n_levels()
-       << " levels)"
-       << std::endl
-       << "Number of degrees of freedom: "
-       << n_u + n_p + n_T
-       << " (" << n_u << '+' << n_p << '+'<< n_T <<')'
-       << std::endl
-       << std::endl;
-
-  std::vector<IndexSet> stokes_partitioning;
-  IndexSet temperature_partitioning (n_T);
-  {
-    const unsigned int my_id =
-      Utilities::System::get_this_mpi_process(MPI_COMM_WORLD);
-    IndexSet stokes_index_set =
-      DoFTools::dof_indices_with_subdomain_association(stokes_dof_handler,
-                                                      my_id);
-    stokes_partitioning.push_back(stokes_index_set.get_view(0,n_u));
-    stokes_partitioning.push_back(stokes_index_set.get_view(n_u,n_u+n_p));
-
-    temperature_partitioning =
-      DoFTools::dof_indices_with_subdomain_association(temperature_dof_handler,
-                                                      my_id);
-  }
-
-  if (Utilities::System::job_supports_mpi() == false)
-    {
-      Threads::TaskGroup<> tasks;
-      tasks += Threads::new_task (&BoussinesqFlowProblem<dim>::setup_stokes_matrix,
-                                 *this,
-                                 stokes_partitioning);
-      tasks += Threads::new_task (&BoussinesqFlowProblem<dim>::setup_stokes_preconditioner,
-                                 *this,
-                                 stokes_partitioning);
-      tasks += Threads::new_task (&BoussinesqFlowProblem<dim>::setup_temperature_matrices,
-                                 *this,
-                                 temperature_partitioning);
-      tasks.join_all ();
-    }
-  else
-    {
-      setup_stokes_matrix (stokes_partitioning);
-      setup_stokes_preconditioner (stokes_partitioning);
-      setup_temperature_matrices (temperature_partitioning);
-    }
-
-  stokes_rhs.reinit (stokes_partitioning, MPI_COMM_WORLD);
-  stokes_solution.reinit (stokes_rhs);
-  old_stokes_solution.reinit (stokes_solution);
-
-  temperature_rhs.reinit (temperature_partitioning, MPI_COMM_WORLD);
-  temperature_solution.reinit (temperature_rhs);
-  old_temperature_solution.reinit (temperature_solution);
-  old_old_temperature_solution.reinit (temperature_solution);
-
-  computing_timer.exit_section();
-}
-
-
-
-                                // @sect4{The BoussinesqFlowProblem assembly functions}
-                                //
-                                // Following the discussion in the
-                                // introduction and in the @ref threads
-                                // module, we split the assembly functions
-                                // into different parts:
-                                //
-                                // <ul>
-                                // <li> The local calculations of matrices
-                                // and right hand sides, given a certain cell
-                                // as input (these functions are named
-                                // <code>local_assemble_*</code> below). The
-                                // resulting function is, in other words,
-                                // essentially the body of the loop over all
-                                // cells in step-31. Note, however, that
-                                // these functions store the result from the
-                                // local calculations in variables of classes
-                                // from the CopyData namespace.
-                                //
-                                // <li>These objects are then given to the
-                                // second step which writes the local data
-                                // into the global data structures (these
-                                // functions are named
-                                // <code>copy_local_to_global_*</code>
-                                // below). These functions are pretty
-                                // trivial.
-                                //
-                                // <li>These two subfunctions are then used
-                                // in the respective assembly routine (called
-                                // <code>assemble_*</code> below), where a
-                                // WorkStream object is set up and runs over
-                                // all the cells that belong to the
-                                // processor's subdomain.
-                                // </ul>
-
-                                // @sect5{Stokes preconditioner assembly}
-                                //
-                                // Let us start with the functions that
-                                // builds the Stokes preconditioner. The
-                                // first two of these are pretty trivial,
-                                // given the discussion above. Note in
-                                // particular that the main point in using
-                                // the scratch data object is that we want to
-                                // avoid allocating any objects on the free
-                                // space each time we visit a new cell. As a
-                                // consequence, the assembly function below
-                                // only has automatic local variables, and
-                                // everything else is accessed through the
-                                // scratch data object, which is allocated
-                                // only once before we start the loop over
-                                // all cells:
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-local_assemble_stokes_preconditioner (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                     Assembly::Scratch::StokesPreconditioner<dim> &scratch,
-                                     Assembly::CopyData::StokesPreconditioner<dim> &data)
-{
-  const unsigned int   dofs_per_cell   = stokes_fe.dofs_per_cell;
-  const unsigned int   n_q_points      = scratch.stokes_fe_values.n_quadrature_points;
-
-  const FEValuesExtractors::Vector velocities (0);
-  const FEValuesExtractors::Scalar pressure (dim);
-
-  scratch.stokes_fe_values.reinit (cell);
-  cell->get_dof_indices (data.local_dof_indices);
-
-  data.local_matrix = 0;
-
-  for (unsigned int q=0; q<n_q_points; ++q)
-    {
-      for (unsigned int k=0; k<dofs_per_cell; ++k)
-       {
-         scratch.grad_phi_u[k] = scratch.stokes_fe_values[velocities].gradient(k,q);
-         scratch.phi_p[k]      = scratch.stokes_fe_values[pressure].value (k, q);
-       }
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       for (unsigned int j=0; j<dofs_per_cell; ++j)
-         data.local_matrix(i,j) += (EquationData::eta *
-                                    scalar_product (scratch.grad_phi_u[i],
-                                                    scratch.grad_phi_u[j])
-                                    +
-                                    (1./EquationData::eta) *
-                                    EquationData::pressure_scaling *
-                                    EquationData::pressure_scaling *
-                                    scratch.phi_p[i] * scratch.phi_p[j])
-                                   * scratch.stokes_fe_values.JxW(q);
-    }
-}
-
-
-
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-copy_local_to_global_stokes_preconditioner (const Assembly::CopyData::StokesPreconditioner<dim> &data)
-{
-  stokes_constraints.distribute_local_to_global (data.local_matrix,
-                                                data.local_dof_indices,
-                                                stokes_preconditioner_matrix);
-}
-
-
-
-                                // When we create the WorkStream, we modify
-                                // the start and end iterator into a
-                                // so-called <code>SubdomainFilter</code>
-                                // that tells the individual processes which
-                                // cells to work on. This is exactly the case
-                                // discussed in the introduction. Note how we
-                                // use the construct
-                                // <code>std_cxx1x::bind</code> to create a
-                                // function object that is compatible with
-                                // the WorkStream class. It uses placeholders
-                                // <code>_1, _2, _3</code> for the local
-                                // assembly function that specify cell,
-                                // scratch data, and copy data, as well as
-                                // the placeholder <code>_1</code> for the
-                                // copy function that expects the data to be
-                                // written into the global matrix. On the
-                                // other hand, the implicit zeroth argument
-                                // of member functions (namely the
-                                // <code>this</code> pointer of the object on
-                                // which that member function is to operate
-                                // on) is <i>bound</i> to the
-                                // <code>this</code> pointer of the current
-                                // function. The WorkStream class, as a
-                                // consequence, does not need to know
-                                // anything about the object these functions
-                                // work on.
-                                //
-                                // When the
-                                // WorkStream is executed, it will create
-                                // several local assembly routines of the
-                                // first kind for several cells and let
-                                // some available processors work on
-                                // them. The function that needs to be
-                                // synchronized, i.e., the write operation
-                                // into the global matrix, however, is
-                                // executed by only one thread at a time in
-                                // the prescribed order. Of course, this
-                                // only holds for the parallelization on a
-                                // single MPI process. Different MPI
-                                // processes will have their own WorkStream
-                                // objects and do that work completely
-                                // independently. In a distributed
-                                // calculation, some data will accumulate
-                                // at degrees of freedom that are not owned
-                                // by the respective processor. It would be
-                                // inefficient to send data around every
-                                // time we encounter such a dof. What
-                                // happens instead is that the Trilinos
-                                // sparse matrix will keep that data and
-                                // send it to the owner at the end of
-                                // assembly, by calling the
-                                // <code>compress()</code> command.
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::assemble_stokes_preconditioner ()
-{
-  stokes_preconditioner_matrix = 0;
-
-  const QGauss<dim> quadrature_formula(stokes_degree+2);
-
-  typedef
-    FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
-    SubdomainFilter;
-
-  WorkStream::
-    run (SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         stokes_dof_handler.begin_active()),
-        SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         stokes_dof_handler.end()),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         local_assemble_stokes_preconditioner,
-                         this,
-                         _1,
-                         _2,
-                         _3),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         copy_local_to_global_stokes_preconditioner,
-                         this,
-                         _1),
-        Assembly::Scratch::
-        StokesPreconditioner<dim> (stokes_fe, quadrature_formula,
-                                   update_JxW_values |
-                                   update_values |
-                                   update_gradients),
-        Assembly::CopyData::
-        StokesPreconditioner<dim> (stokes_fe));
-
-  stokes_preconditioner_matrix.compress();
-}
-
-
-
-                                // The final function in this block initiates
-                                // assemble of the Stokes preconditioner
-                                // matrix and then builds the Stokes
-                                // preconditioner. It is mostly the same as
-                                // in the serial case. The only difference to
-                                // step-31 is that we use an ILU
-                                // preconditioner for the pressure mass
-                                // matrix instead of IC, as discussed in the
-                                // introduction.
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::build_stokes_preconditioner ()
-{
-  if (rebuild_stokes_preconditioner == false)
-    return;
-
-  computing_timer.enter_section ("   Build Stokes preconditioner");
-  pcout << "   Rebuilding Stokes preconditioner..." << std::flush;
-
-  assemble_stokes_preconditioner ();
-
-  std::vector<std::vector<bool> > constant_modes;
-  std::vector<bool>  velocity_components (dim+1,true);
-  velocity_components[dim] = false;
-  DoFTools::extract_constant_modes (stokes_dof_handler, velocity_components,
-                                   constant_modes);
-
-  Mp_preconditioner  = std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionILU>
-                      (new TrilinosWrappers::PreconditionILU());
-  Amg_preconditioner = std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionAMG>
-                      (new TrilinosWrappers::PreconditionAMG());
-
-  TrilinosWrappers::PreconditionAMG::AdditionalData Amg_data;
-  Amg_data.constant_modes = constant_modes;
-  Amg_data.elliptic = true;
-  Amg_data.higher_order_elements = true;
-  Amg_data.smoother_sweeps = 2;
-  Amg_data.aggregation_threshold = 0.02;
-
-  Mp_preconditioner->initialize (stokes_preconditioner_matrix.block(1,1));
-  Amg_preconditioner->initialize (stokes_preconditioner_matrix.block(0,0),
-                                 Amg_data);
-
-  rebuild_stokes_preconditioner = false;
-
-  pcout << std::endl;
-  computing_timer.exit_section();
-}
-
-                                // @sect5{Stokes system assembly}
-
-                                // The next three functions implement the
-                                // assembly of the Stokes system, again
-                                // split up into a part performing local
-                                // calculations, one for writing the local
-                                // data into the global matrix and vector,
-                                // and one for actually running the loop
-                                // over all cells with the help of the
-                                // WorkStream class. Note that the assembly
-                                // of the Stokes matrix needs only to be
-                                // done in case we have changed the
-                                // mesh. Otherwise, just the
-                                // (temperature-dependent) right hand side
-                                // needs to be calculated here. Since we
-                                // are working with distributed matrices
-                                // and vectors, we have to call the
-                                // respective <code>compress()</code>
-                                // functions in the end of the assembly in
-                                // order to send non-local data to the
-                                // owner process.
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-local_assemble_stokes_system (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                             Assembly::Scratch::StokesSystem<dim> &scratch,
-                             Assembly::CopyData::StokesSystem<dim> &data)
-{
-  const unsigned int dofs_per_cell = scratch.stokes_fe_values.get_fe().dofs_per_cell;
-  const unsigned int n_q_points    = scratch.stokes_fe_values.n_quadrature_points;
-
-  const FEValuesExtractors::Vector velocities (0);
-  const FEValuesExtractors::Scalar pressure (dim);
-
-  scratch.stokes_fe_values.reinit (cell);
-
-  typename DoFHandler<dim>::active_cell_iterator
-    temperature_cell (&triangulation,
-                     cell->level(),
-                     cell->index(),
-                     &temperature_dof_handler);
-  scratch.temperature_fe_values.reinit (temperature_cell);
-
-  if (rebuild_stokes_matrix)
-    data.local_matrix = 0;
-  data.local_rhs = 0;
-
-  scratch.temperature_fe_values.get_function_values (old_temperature_solution,
-                                                    scratch.old_temperature_values);
-
-  for (unsigned int q=0; q<n_q_points; ++q)
-    {
-      const double old_temperature = scratch.old_temperature_values[q];
-
-      for (unsigned int k=0; k<dofs_per_cell; ++k)
-       {
-         scratch.phi_u[k] = scratch.stokes_fe_values[velocities].value (k,q);
-         if (rebuild_stokes_matrix)
-           {
-             scratch.grads_phi_u[k] = scratch.stokes_fe_values[velocities].symmetric_gradient(k,q);
-             scratch.div_phi_u[k]   = scratch.stokes_fe_values[velocities].divergence (k, q);
-             scratch.phi_p[k]       = scratch.stokes_fe_values[pressure].value (k, q);
-           }
-       }
-
-      if (rebuild_stokes_matrix)
-       for (unsigned int i=0; i<dofs_per_cell; ++i)
-         for (unsigned int j=0; j<dofs_per_cell; ++j)
-           data.local_matrix(i,j) += (EquationData::eta * 2 *
-                                      (scratch.grads_phi_u[i] * scratch.grads_phi_u[j])
-                                      - (EquationData::pressure_scaling *
-                                         scratch.div_phi_u[i] * scratch.phi_p[j])
-                                      - (EquationData::pressure_scaling *
-                                         scratch.phi_p[i] * scratch.div_phi_u[j]))
-                                     * scratch.stokes_fe_values.JxW(q);
-
-      const Tensor<1,dim>
-       gravity = EquationData::gravity_vector (scratch.stokes_fe_values
-                                               .quadrature_point(q));
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       data.local_rhs(i) += (EquationData::density(old_temperature) *
-                             gravity  *
-                             scratch.phi_u[i]) *
-                            scratch.stokes_fe_values.JxW(q);
-    }
-
-  cell->get_dof_indices (data.local_dof_indices);
-}
-
-
-
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-copy_local_to_global_stokes_system (const Assembly::CopyData::StokesSystem<dim> &data)
-{
-  if (rebuild_stokes_matrix == true)
-    stokes_constraints.distribute_local_to_global (data.local_matrix,
-                                                  data.local_rhs,
-                                                  data.local_dof_indices,
-                                                  stokes_matrix,
-                                                  stokes_rhs);
-  else
-    stokes_constraints.distribute_local_to_global (data.local_rhs,
-                                                  data.local_dof_indices,
-                                                  stokes_rhs);
-}
-
-
-
-template <int dim>
-void BoussinesqFlowProblem<dim>::assemble_stokes_system ()
-{
-  pcout << "   Assembling..." << std::flush;
-
-  computing_timer.enter_section ("   Assemble Stokes system");
-
-  if (rebuild_stokes_matrix == true)
-    stokes_matrix=0;
-
-  stokes_rhs=0;
-
-  const QGauss<dim> quadrature_formula(stokes_degree+2);
-
-  typedef
-    FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
-    SubdomainFilter;
-
-  WorkStream::
-    run (SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         stokes_dof_handler.begin_active()),
-        SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         stokes_dof_handler.end()),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         local_assemble_stokes_system,
-                         this,
-                         _1,
-                         _2,
-                         _3),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         copy_local_to_global_stokes_system,
-                         this,
-                         _1),
-        Assembly::Scratch::
-        StokesSystem<dim> (stokes_fe, quadrature_formula,
-                           (update_values    |
-                            update_quadrature_points  |
-                            update_JxW_values |
-                            (rebuild_stokes_matrix == true
-                             ?
-                             update_gradients
-                             :
-                             UpdateFlags(0))),
-                           temperature_fe,
-                           update_values),
-        Assembly::CopyData::
-        StokesSystem<dim> (stokes_fe));
-
-  stokes_matrix.compress();
-  stokes_rhs.compress();
-
-  rebuild_stokes_matrix = false;
-
-  pcout << std::endl;
-  computing_timer.exit_section();
-}
-
-
-                                // @sect5{Temperature matrix assembly}
-
-                                // The task to be performed by the next three
-                                // functions is to calculate a mass matrix
-                                // and a Laplace matrix on the temperature
-                                // system. These will be combined in order to
-                                // yield the semi-implicit time stepping
-                                // matrix that consists of the mass matrix
-                                // plus a time step weight times the Laplace
-                                // matrix. This function is again essentially
-                                // the body of the loop over all cells from
-                                // step-31.
-                                //
-                                // The two following functions perform
-                                // similar services as the ones above.
-template <int dim>
-void BoussinesqFlowProblem<dim>::
-local_assemble_temperature_matrix (const typename DoFHandler<dim>::active_cell_iterator &cell,
-                                  Assembly::Scratch::TemperatureMatrix<dim> &scratch,
-                                  Assembly::CopyData::TemperatureMatrix<dim> &data)
-{
-  const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
-  const unsigned int n_q_points    = scratch.temperature_fe_values.n_quadrature_points;
-
-  scratch.temperature_fe_values.reinit (cell);
-  cell->get_dof_indices (data.local_dof_indices);
-
-  data.local_mass_matrix = 0;
-  data.local_stiffness_matrix = 0;
-
-  for (unsigned int q=0; q<n_q_points; ++q)
-    {
-      for (unsigned int k=0; k<dofs_per_cell; ++k)
-       {
-         scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
-         scratch.phi_T[k]      = scratch.temperature_fe_values.shape_value (k, q);
-       }
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       for (unsigned int j=0; j<dofs_per_cell; ++j)
-         {
-           data.local_mass_matrix(i,j)
-             += (scratch.phi_T[i] * scratch.phi_T[j]
-                 *
-                 scratch.temperature_fe_values.JxW(q));
-           data.local_stiffness_matrix(i,j)
-             += (EquationData::kappa * scratch.grad_phi_T[i] * scratch.grad_phi_T[j]
-                 *
-                 scratch.temperature_fe_values.JxW(q));
-         }
-    }
-}
-
-
-
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-copy_local_to_global_temperature_matrix (const Assembly::CopyData::TemperatureMatrix<dim> &data)
-{
-  temperature_constraints.distribute_local_to_global (data.local_mass_matrix,
-                                                     data.local_dof_indices,
-                                                     temperature_mass_matrix);
-  temperature_constraints.distribute_local_to_global (data.local_stiffness_matrix,
-                                                     data.local_dof_indices,
-                                                     temperature_stiffness_matrix);
-}
-
-
-template <int dim>
-void BoussinesqFlowProblem<dim>::assemble_temperature_matrix ()
-{
-  if (rebuild_temperature_matrices == false)
-    return;
-
-  computing_timer.enter_section ("   Assemble temperature matrices");
-  temperature_mass_matrix = 0;
-  temperature_stiffness_matrix = 0;
-
-  const QGauss<dim> quadrature_formula(temperature_degree+2);
-
-  typedef
-    FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
-    SubdomainFilter;
-
-  WorkStream::
-    run (SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         temperature_dof_handler.begin_active()),
-        SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         temperature_dof_handler.end()),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         local_assemble_temperature_matrix,
-                         this,
-                         _1,
-                         _2,
-                         _3),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         copy_local_to_global_temperature_matrix,
-                         this,
-                         _1),
-        Assembly::Scratch::
-        TemperatureMatrix<dim> (temperature_fe, quadrature_formula),
-        Assembly::CopyData::
-        TemperatureMatrix<dim> (temperature_fe));
-
-  temperature_mass_matrix.compress();
-  temperature_stiffness_matrix.compress();
-
-  rebuild_temperature_matrices = false;
-  rebuild_temperature_preconditioner = true;
-
-  computing_timer.exit_section();
-}
-
-
-                                // @sect5{Temperature right hand side assembly}
-
-                                // This is the last assembly function. It
-                                // calculates the right hand side of the
-                                // temperature system, which includes the
-                                // convection and the stabilization
-                                // terms. It includes a lot of evaluations
-                                // of old solutions at the quadrature
-                                // points (which are necessary for
-                                // calculating the artificial viscosity of
-                                // stabilization), but is otherwise similar
-                                // to the other assembly functions. Notice,
-                                // once again, how we resolve the dilemma
-                                // of having inhomogeneous boundary
-                                // conditions, but just making a right hand
-                                // side at this point (compare the comments
-                                // for the project function): We create
-                                // some matrix columns with exactly the
-                                // values that would be entered for the
-                                // temperature stiffness matrix, in case we
-                                // have inhomogeneously constrained
-                                // dofs. That will account for the correct
-                                // balance of the right hand side vector
-                                // with the matrix system of temperature.
-template <int dim>
-void BoussinesqFlowProblem<dim>::
-local_assemble_temperature_rhs (const std::pair<double,double> global_T_range,
-                               const double                   global_max_velocity,
-                               const typename DoFHandler<dim>::active_cell_iterator &cell,
-                               Assembly::Scratch::TemperatureRHS<dim> &scratch,
-                               Assembly::CopyData::TemperatureRHS<dim> &data)
-{
-  const bool use_bdf2_scheme = (timestep_number != 0);
-
-  const unsigned int dofs_per_cell = scratch.temperature_fe_values.get_fe().dofs_per_cell;
-  const unsigned int n_q_points    = scratch.temperature_fe_values.n_quadrature_points;
-
-  EquationData::TemperatureRightHandSide<dim>  temperature_right_hand_side;
-
-  const FEValuesExtractors::Vector velocities (0);
-
-  data.local_rhs = 0;
-  data.matrix_for_bc = 0;
-  cell->get_dof_indices (data.local_dof_indices);
-
-  scratch.temperature_fe_values.reinit (cell);
-
-  typename DoFHandler<dim>::active_cell_iterator
-    stokes_cell (&triangulation,
-                cell->level(),
-                cell->index(),
-                &stokes_dof_handler);
-  scratch.stokes_fe_values.reinit (stokes_cell);
-
-  scratch.temperature_fe_values.get_function_values (old_temperature_solution,
-                                                    scratch.old_temperature_values);
-  scratch.temperature_fe_values.get_function_values (old_old_temperature_solution,
-                                                    scratch.old_old_temperature_values);
-
-  scratch.temperature_fe_values.get_function_gradients (old_temperature_solution,
-                                                       scratch.old_temperature_grads);
-  scratch.temperature_fe_values.get_function_gradients (old_old_temperature_solution,
-                                                       scratch.old_old_temperature_grads);
-
-  scratch.temperature_fe_values.get_function_laplacians (old_temperature_solution,
-                                                        scratch.old_temperature_laplacians);
-  scratch.temperature_fe_values.get_function_laplacians (old_old_temperature_solution,
-                                                        scratch.old_old_temperature_laplacians);
-
-  temperature_right_hand_side.value_list (scratch.temperature_fe_values.get_quadrature_points(),
-                                         scratch.gamma_values);
-
-  scratch.stokes_fe_values[velocities].get_function_values (stokes_solution,
-                                                           scratch.old_velocity_values);
-  scratch.stokes_fe_values[velocities].get_function_values (old_stokes_solution,
-                                                           scratch.old_old_velocity_values);
-
-  const double nu
-    = compute_viscosity (scratch.old_temperature_values,
-                        scratch.old_old_temperature_values,
-                        scratch.old_temperature_grads,
-                        scratch.old_old_temperature_grads,
-                        scratch.old_temperature_laplacians,
-                        scratch.old_old_temperature_laplacians,
-                        scratch.old_velocity_values,
-                        scratch.old_old_velocity_values,
-                        scratch.gamma_values,
-                        global_max_velocity,
-                        global_T_range.second - global_T_range.first,
-                        cell->diameter());
-
-  for (unsigned int q=0; q<n_q_points; ++q)
-    {
-      for (unsigned int k=0; k<dofs_per_cell; ++k)
-       {
-         scratch.phi_T[k]      = scratch.temperature_fe_values.shape_value (k, q);
-         scratch.grad_phi_T[k] = scratch.temperature_fe_values.shape_grad (k,q);
-       }
-
-
-      const double old_Ts
-       = (use_bdf2_scheme ?
-          (scratch.old_temperature_values[q] *
-           (time_step + old_time_step) / old_time_step
-           -
-           scratch.old_old_temperature_values[q] *
-           (time_step * time_step) /
-           (old_time_step * (time_step + old_time_step)))
-          :
-          scratch.old_temperature_values[q]);
-
-      const Tensor<1,dim> ext_grad_T
-       = (use_bdf2_scheme ?
-          (scratch.old_temperature_grads[q] *
-           (1+time_step/old_time_step)
-           -
-           scratch.old_old_temperature_grads[q] *
-           time_step / old_time_step)
-          :
-          scratch.old_temperature_grads[q]);
-
-      const Tensor<1,dim> extrapolated_u
-       = (use_bdf2_scheme ?
-          (scratch.old_velocity_values[q] * (1+time_step/old_time_step) -
-           scratch.old_old_velocity_values[q] * time_step/old_time_step)
-          :
-          scratch.old_velocity_values[q]);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       {
-         data.local_rhs(i) += (old_Ts * scratch.phi_T[i]
-                               -
-                               time_step *
-                               extrapolated_u * ext_grad_T * scratch.phi_T[i]
-                               -
-                               time_step *
-                               nu * ext_grad_T * scratch.grad_phi_T[i]
-                               +
-                               time_step *
-                               scratch.gamma_values[q] * scratch.phi_T[i])
-                              *
-                              scratch.temperature_fe_values.JxW(q);
-
-         if (temperature_constraints.is_inhomogeneously_constrained(data.local_dof_indices[i]))
-           {
-             for (unsigned int j=0; j<dofs_per_cell; ++j)
-               data.matrix_for_bc(j,i) += (scratch.phi_T[i] * scratch.phi_T[j] *
-                                           (use_bdf2_scheme ?
-                                            ((2*time_step + old_time_step) /
-                                             (time_step + old_time_step)) : 1.)
-                                           +
-                                           scratch.grad_phi_T[i] *
-                                           scratch.grad_phi_T[j] *
-                                           EquationData::kappa *
-                                           time_step)
-                                          *
-                                          scratch.temperature_fe_values.JxW(q);
-           }
-       }
-    }
-}
-
-
-template <int dim>
-void
-BoussinesqFlowProblem<dim>::
-copy_local_to_global_temperature_rhs (const Assembly::CopyData::TemperatureRHS<dim> &data)
-{
-  temperature_constraints.distribute_local_to_global (data.local_rhs,
-                                                     data.local_dof_indices,
-                                                     temperature_rhs,
-                                                     data.matrix_for_bc);
-}
-
-
-
-                                // In the function that runs the WorkStream
-                                // for actually calculating the right hand
-                                // side, we also generate the final
-                                // matrix. As mentioned above, it is a sum
-                                // of the mass matrix and the Laplace
-                                // matrix, times some time step
-                                // weight. This weight is specified by the
-                                // BDF-2 time integration scheme, see the
-                                // introduction in step-31. What is new in
-                                // this tutorial program (in addition to
-                                // the use of MPI parallelization and the
-                                // WorkStream class), is that we now
-                                // precompute the temperature
-                                // preconditioner as well. The reason is
-                                // that the setup of the IC preconditioner
-                                // takes a noticable time compared to the
-                                // solver because we usually only need
-                                // between 10 and 20 iterations for solving
-                                // the temperature system. Hence, it is
-                                // more efficient to precompute the
-                                // preconditioner, even though the matrix
-                                // entries may slightly change because the
-                                // time step might change. This is not
-                                // too big a problem because we remesh every
-                                // fifth time step (and regenerate the
-                                // preconditioner then).
-template <int dim>
-void BoussinesqFlowProblem<dim>::assemble_temperature_system (const double maximal_velocity)
-{
-  const bool use_bdf2_scheme = (timestep_number != 0);
-
-  if (use_bdf2_scheme == true)
-    {
-      temperature_matrix.copy_from (temperature_mass_matrix);
-      temperature_matrix *= (2*time_step + old_time_step) /
-                           (time_step + old_time_step);
-      temperature_matrix.add (time_step, temperature_stiffness_matrix);
-    }
-  else
-    {
-      temperature_matrix.copy_from (temperature_mass_matrix);
-      temperature_matrix.add (time_step, temperature_stiffness_matrix);
-    }
-  temperature_matrix.compress();
-
-  if (rebuild_temperature_preconditioner == true)
-    {
-      T_preconditioner =  std_cxx1x::shared_ptr<TrilinosWrappers::PreconditionIC>
-                         (new TrilinosWrappers::PreconditionIC());
-      T_preconditioner->initialize (temperature_matrix);
-      rebuild_temperature_preconditioner = false;
-    }
-
-  temperature_rhs = 0;
-
-  const QGauss<dim> quadrature_formula(temperature_degree+2);
-  const std::pair<double,double>
-    global_T_range = get_extrapolated_temperature_range();
-
-  typedef
-    FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
-    SubdomainFilter;
-
-  WorkStream::
-    run (SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         temperature_dof_handler.begin_active()),
-        SubdomainFilter (IteratorFilters::SubdomainEqualTo
-                         (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD)),
-                         temperature_dof_handler.end()),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         local_assemble_temperature_rhs,
-                         this,
-                         global_T_range,
-                         maximal_velocity,
-                         _1,
-                         _2,
-                         _3),
-        std_cxx1x::bind (&BoussinesqFlowProblem<dim>::
-                         copy_local_to_global_temperature_rhs,
-                         this,
-                         _1),
-        Assembly::Scratch::
-        TemperatureRHS<dim> (temperature_fe, stokes_fe, quadrature_formula),
-        Assembly::CopyData::
-        TemperatureRHS<dim> (temperature_fe));
-
-  temperature_rhs.compress();
-}
-
-
-
-
-                                // @sect4{BoussinesqFlowProblem::solve}
-
-                                // This function solves the linear systems
-                                // in each time step of the Boussinesq
-                                // problem. First, we
-                                // work on the Stokes system and then on
-                                // the temperature system. In essence, it
-                                // does the same things as the respective
-                                // function in step-31. However, there are
-                                // a few things that we need to pay some
-                                // attention to. The first thing is, as
-                                // mentioned in the introduction, the way
-                                // we store our solution: we keep the full
-                                // vector with all degrees of freedom on
-                                // each MPI node. When we enter a solver
-                                // which is supposed to perform
-                                // matrix-vector products with a
-                                // distributed matrix, this is not the
-                                // appropriate form, though. There, we will
-                                // want to have the solution vector to be
-                                // distributed in the same way as the
-                                // matrix. So what we do first (after
-                                // initializing the Schur-complement based
-                                // preconditioner) is to generate a
-                                // distributed vector called
-                                // <code>distributed_stokes_solution</code>
-                                // and put only the locally owned dofs into
-                                // that, which is neatly done by the
-                                // <code>operator=</code> of the Trilinos
-                                // vector. Next, we need to set the
-                                // pressure values at hanging nodes to
-                                // zero. This we also did in step-31 in
-                                // order not to disturb the Schur
-                                // complement by some vector entries that
-                                // actually are irrelevant during the solve
-                                // stage. As a difference to step-31, here
-                                // we do it only for the locally owned
-                                // pressure dofs. After solving for the
-                                // Stokes solution, each processor copies
-                                // distributed solution back into the solution
-                                // vector for which every element is locally
-                                // owned.
-                                //
-                                // Apart from these two changes, everything
-                                // is the same as in step-31, so we don't
-                                // need to further comment on it.
-template <int dim>
-void BoussinesqFlowProblem<dim>::solve ()
-{
-  computing_timer.enter_section ("   Solve Stokes system");
-  pcout << "   Solving..." << std::endl;
-
-  {
-    const LinearSolvers::BlockSchurPreconditioner<TrilinosWrappers::PreconditionAMG,
-                                                  TrilinosWrappers::PreconditionILU>
-      preconditioner (stokes_matrix, *Mp_preconditioner, *Amg_preconditioner);
-
-    TrilinosWrappers::MPI::BlockVector
-      distributed_stokes_solution (stokes_rhs);
-    distributed_stokes_solution = stokes_solution;
-
-    const unsigned int
-      start = (distributed_stokes_solution.block(0).size() +
-              distributed_stokes_solution.block(1).local_range().first),
-      end   = (distributed_stokes_solution.block(0).size() +
-              distributed_stokes_solution.block(1).local_range().second);
-    for (unsigned int i=start; i<end; ++i)
-      if (stokes_constraints.is_constrained (i))
-       distributed_stokes_solution(i) = 0;
-
-    SolverControl solver_control (stokes_matrix.m(), 1e-22*stokes_rhs.l2_norm());
-    SolverBicgstab<TrilinosWrappers::MPI::BlockVector>
-      bicgstab (solver_control, false);
-
-    bicgstab.solve(stokes_matrix, distributed_stokes_solution, stokes_rhs,
-                  preconditioner);
-
-    stokes_solution = distributed_stokes_solution;
-
-    pcout << "   "
-         << solver_control.last_step()
-         << " BiCGStab iterations for Stokes subsystem."
-         << std::endl;
-
-    stokes_constraints.distribute (stokes_solution);
-  }
-  computing_timer.exit_section();
-
-
-  computing_timer.enter_section ("   Assemble temperature rhs");
-
-  old_time_step = time_step;
-  const double maximal_velocity = get_maximal_velocity();
-  if (maximal_velocity > 1e-10)
-    time_step = 1./(1.6*dim*std::sqrt(1.*dim)) /
-               temperature_degree *
-               GridTools::minimal_cell_diameter(triangulation) /
-               maximal_velocity;
-  else
-    time_step = 1./(1.6*dim*std::sqrt(1.*dim)) /
-               temperature_degree *
-               GridTools::minimal_cell_diameter(triangulation) /
-               1e-10;
-
-  pcout << "   Maximal velocity: "
-       << maximal_velocity * EquationData::year_in_seconds * 100
-       << " cm/year"
-       << std::endl;
-  pcout << "   " << "Time step: "
-       << time_step/EquationData::year_in_seconds
-       << " years"
-       << std::endl;
-
-  temperature_solution = old_temperature_solution;
-
-  assemble_temperature_system (maximal_velocity);
-
-  computing_timer.exit_section ();
-
-  computing_timer.enter_section ("   Solve temperature system");
-
-  {
-    SolverControl solver_control (temperature_matrix.m(),
-                                 1e-12*temperature_rhs.l2_norm());
-    SolverCG<TrilinosWrappers::MPI::Vector>   cg (solver_control);
-
-    TrilinosWrappers::MPI::Vector
-      distributed_temperature_solution (temperature_rhs);
-    distributed_temperature_solution = temperature_solution;
-
-    cg.solve (temperature_matrix, distributed_temperature_solution,
-             temperature_rhs, *T_preconditioner);
-
-    temperature_solution = distributed_temperature_solution;
-    temperature_constraints.distribute (temperature_solution);
-
-    pcout << "   "
-         << solver_control.last_step()
-         << " CG iterations for temperature" << std::endl;
-    computing_timer.exit_section();
-
-    double min_temperature = temperature_solution(0),
-          max_temperature = temperature_solution(0);
-    for (unsigned int i=1; i<temperature_solution.size(); ++i)
-      {
-       min_temperature = std::min<double> (min_temperature,
-                                           temperature_solution(i));
-       max_temperature = std::max<double> (max_temperature,
-                                           temperature_solution(i));
-      }
-
-    pcout << "   Temperature range: "
-         << min_temperature << ' ' << max_temperature
-         << std::endl;
-  }
-}
-
-
-
-                                // @sect4{BoussinesqFlowProblem::output_results}
-
-                                // This function does mostly what the
-                                // corresponding one did in to
-                                // step-31, in particular merging
-                                // data from the two DoFHandler
-                                // objects (for the Stokes and the
-                                // temperature parts of the problem)
-                                // into one is the same. There are
-                                // three minor changes: we make sure
-                                // that only a single processor
-                                // actually does some work here; take
-                                // care of scaling variables in a
-                                // useful way; and in addition to the
-                                // Stokes and temperature parts in
-                                // the <code>joint_fe</code> finite
-                                // element, we also add a piecewise
-                                // constant field that denotes the
-                                // subdomain id a cell corresponds
-                                // to. This allows us to visualize
-                                // the partitioning of the domain. As
-                                // a consequence, we also have to
-                                // change the assertion about the
-                                // number of degrees of freedom in
-                                // the joint DoFHandler object (which
-                                // is now equal to the number of
-                                // Stokes degrees of freedom plus the
-                                // temperature degrees of freedom
-                                // plus the number of active cells as
-                                // that is the number of partition
-                                // variables we want to add), and
-                                // adjust the number of elements in
-                                // the arrays we use to name the
-                                // components of the joint solution
-                                // vector and to identify which of
-                                // these components are scalars or
-                                // parts of dim-dimensional vectors.
-                                //
-                                // As for scaling: as mentioned in
-                                // the introduction, to keep the
-                                // Stokes equations properly scaled
-                                // and symmetric, we introduced a new
-                                // pressure $\hat p =
-                                // \frac{L}{\eta}p$. What we really
-                                // wanted, however, was the original
-                                // pressure $p$, so while copying
-                                // data from the Stokes DoFHandler
-                                // into the joint one, we undo this
-                                // scaling. While we're at it messing
-                                // with the results of the
-                                // simulation, we do two more things:
-                                // First, the pressure is only
-                                // defined up to a constant. To make
-                                // it more easily comparable, we
-                                // compute the minimal value of the
-                                // pressure computed and shift all
-                                // values up by that amount -- in
-                                // essence making all pressure
-                                // variables positive or
-                                // zero. Secondly, let's also take
-                                // care of the awkward units we use
-                                // for the velocity: it is computed
-                                // in SI units of meters per second,
-                                // which of course is a very small
-                                // number in the earth mantle. We
-                                // therefore rescale things into
-                                // centimeters per year, the unit
-                                // commonly used in geophysics.
-template <int dim>
-void BoussinesqFlowProblem<dim>::output_results ()
-{
-  if (timestep_number % 25 != 0)
-    return;
-
-  computing_timer.enter_section ("Postprocessing");
-
-  if (Utilities::System::get_this_mpi_process(MPI_COMM_WORLD) == 0)
-    {
-
-      const FESystem<dim> joint_fe (stokes_fe, 1,
-                                   temperature_fe, 1,
-                                   FE_DGQ<dim>(0), 1);
-      DoFHandler<dim> joint_dof_handler (triangulation);
-      joint_dof_handler.distribute_dofs (joint_fe);
-      Assert (joint_dof_handler.n_dofs() ==
-             stokes_dof_handler.n_dofs() +
-             temperature_dof_handler.n_dofs() +
-             triangulation.n_active_cells(),
-             ExcInternalError());
-
-      Vector<double> joint_solution (joint_dof_handler.n_dofs());
-
-      {
-       double minimal_pressure = stokes_solution.block(1)(0);
-       for (unsigned int i=0; i<stokes_solution.block(1).size(); ++i)
-         minimal_pressure = std::min<double> (stokes_solution.block(1)(i),
-                                              minimal_pressure);
-
-       std::vector<unsigned int> local_joint_dof_indices (joint_fe.dofs_per_cell);
-       std::vector<unsigned int> local_stokes_dof_indices (stokes_fe.dofs_per_cell);
-       std::vector<unsigned int> local_temperature_dof_indices (temperature_fe.dofs_per_cell);
-
-       typename DoFHandler<dim>::active_cell_iterator
-         joint_cell       = joint_dof_handler.begin_active(),
-         joint_endc       = joint_dof_handler.end(),
-         stokes_cell      = stokes_dof_handler.begin_active(),
-         temperature_cell = temperature_dof_handler.begin_active();
-       for (; joint_cell!=joint_endc; ++joint_cell, ++stokes_cell, ++temperature_cell)
-         {
-           joint_cell->get_dof_indices (local_joint_dof_indices);
-           stokes_cell->get_dof_indices (local_stokes_dof_indices);
-           temperature_cell->get_dof_indices (local_temperature_dof_indices);
-
-           for (unsigned int i=0; i<joint_fe.dofs_per_cell; ++i)
-             if (joint_fe.system_to_base_index(i).first.first == 0)
-               {
-                 Assert (joint_fe.system_to_base_index(i).second
-                         <
-                         local_stokes_dof_indices.size(),
-                         ExcInternalError());
-
-                 const unsigned int index_in_stokes_fe
-                   = joint_fe.system_to_base_index(i).second;
-
-                 if (stokes_fe.system_to_component_index(index_in_stokes_fe).first
-                     < dim)
-                   {
-                     joint_solution(local_joint_dof_indices[i])
-                       = (stokes_solution(local_stokes_dof_indices
-                                          [joint_fe.system_to_base_index(i).second])
-                          *
-                          EquationData::year_in_seconds
-                          *
-                          100);
-                   }
-                 else
-                   {
-                     Assert (stokes_fe.system_to_component_index(index_in_stokes_fe).first
-                             ==
-                             dim,
-                             ExcInternalError());
-
-                     joint_solution(local_joint_dof_indices[i])
-                       = ((stokes_solution(local_stokes_dof_indices
-                                           [joint_fe.system_to_base_index(i).second])
-                           -
-                           minimal_pressure)
-                          *
-                          EquationData::pressure_scaling);
-                   }
-               }
-             else if (joint_fe.system_to_base_index(i).first.first == 1)
-               {
-                 Assert (joint_fe.system_to_base_index(i).second
-                         <
-                         local_temperature_dof_indices.size(),
-                         ExcInternalError());
-                 joint_solution(local_joint_dof_indices[i])
-                   = temperature_solution(local_temperature_dof_indices
-                                          [joint_fe.system_to_base_index(i).second]);
-               }
-             else
-               {
-                 Assert (joint_fe.system_to_base_index(i).first.first == 2,
-                         ExcInternalError());
-                 Assert (joint_fe.system_to_base_index(i).second
-                         == 0,
-                         ExcInternalError());
-                 joint_solution(local_joint_dof_indices[i])
-                   = joint_cell->subdomain_id();
-               }
-         }
-      }
-
-      std::vector<std::string> joint_solution_names (dim, "velocity");
-      joint_solution_names.push_back ("p");
-      joint_solution_names.push_back ("T");
-      joint_solution_names.push_back ("partition");
-
-      DataOut<dim> data_out;
-
-      data_out.attach_dof_handler (joint_dof_handler);
-
-      std::vector<DataComponentInterpretation::DataComponentInterpretation>
-       data_component_interpretation
-       (dim+3, DataComponentInterpretation::component_is_scalar);
-      for (unsigned int i=0; i<dim; ++i)
-       data_component_interpretation[i]
-         = DataComponentInterpretation::component_is_part_of_vector;
-
-      data_out.add_data_vector (joint_solution, joint_solution_names,
-                               DataOut<dim>::type_dof_data,
-                               data_component_interpretation);
-      data_out.build_patches (std::min(stokes_degree, temperature_degree));
-
-      std::ostringstream filename;
-      filename << "solution-" << Utilities::int_to_string(timestep_number, 5)
-              << ".vtk";
-
-      std::ofstream output (filename.str().c_str());
-      data_out.write_vtk (output);
-    }
-
-  computing_timer.exit_section ();
-}
-
-
-
-                                // @sect4{BoussinesqFlowProblem::refine_mesh}
-
-                                // This function isn't really new
-                                // either. Since the
-                                // <code>setup_dofs</code> function
-                                // that we call in the middle has its
-                                // own timer section, we split timing
-                                // this function into two
-                                // sections. It will also allow us to
-                                // easily identify which of the two
-                                // is more expensive.
-                                //
-                                // One thing of note, however, is
-                                // that we don't want to compute all
-                                // error indicators on all cells, of
-                                // course. Rather, it would be nice
-                                // if each processor could only
-                                // compute the error indicators for
-                                // those cells it actually
-                                // owns. However, in order for mesh
-                                // refinement to proceed in the same
-                                // way on all processors, all
-                                // processors would have to exchange
-                                // their refinement indicators. We do
-                                // so in two steps: first, we call
-                                // the KellyErrorEstimator::estimate
-                                // function with an argument (usually
-                                // defaulted, but explicitly given
-                                // here) thatindicates the subdomain
-                                // id of all those cells that we want
-                                // to work on; note that this means
-                                // that we also have to specify
-                                // values for all those default
-                                // arguments that lie before the one
-                                // we want to give.
-                                //
-                                // Secondly, we need to exchange the
-                                // data. To do this, we could add up
-                                // the refinement indicators from all
-                                // processors, since they all only
-                                // worked on a disjoint subset of the
-                                // elements of the vector that holds
-                                // these indicators. We could set up
-                                // a distributed Trilinos vector for
-                                // this, but that appears
-                                // unnecessarily complicated because
-                                // we would have to specify a
-                                // partition of this vector, and none
-                                // appears immediately
-                                // obvious. Rather, we want to use
-                                // the Trilinos communicator class to
-                                // this for us, taking the local
-                                // indicators as a collection of
-                                // floating point values rather than
-                                // a linear algebra
-                                // vector. Unfortunately, the
-                                // Trilinos communicator class
-                                // doesn't appear to have function
-                                // that wraps around the MPI add
-                                // function; it has one that computes
-                                // the maximum of a bunch of values,
-                                // though, which in our case is
-                                // equally good -- maybe even better,
-                                // in case two processors should
-                                // compute values for the same cell
-                                // (which they shouldn't of course,
-                                // unless we have made a mistake in
-                                // specifying the arguments to the
-                                // estimate function below). There is
-                                // little snag again, however, that
-                                // makes this a bit awkward: the
-                                // Trilinos communicator class can
-                                // take the maximum over all
-                                // processors for each element of a
-                                // vector, but only if the vector
-                                // contains doubles. The vector
-                                // returned by the
-                                // KellyErrorEstimator::estimate
-                                // function, on the other hand, has
-                                // floats as its data type. An ugly,
-                                // if workable way, is therefore to
-                                // compute the indicators as floats,
-                                // convert the vector to doubles, and
-                                // form the maximum of that.
-                                //
-                                // At the end of this chain of
-                                // events, every processors has the
-                                // complete set of refinement
-                                // indicators, and the rest of the
-                                // function proceeds as before.
-template <int dim>
-void BoussinesqFlowProblem<dim>::refine_mesh (const unsigned int max_grid_level)
-{
-  computing_timer.enter_section ("Refine mesh structure, part 1");
-  Vector<float> local_estimated_error_per_cell (triangulation.n_active_cells());
-
-  KellyErrorEstimator<dim>::estimate (temperature_dof_handler,
-                                     QGauss<dim-1>(temperature_degree+1),
-                                     typename FunctionMap<dim>::type(),
-                                     temperature_solution,
-                                     local_estimated_error_per_cell,
-                                     std::vector<bool>(),
-                                     0,
-                                     0,
-                                     Utilities::System::get_this_mpi_process(MPI_COMM_WORLD));
-
-  Vector<double> x_local_estimated_error_per_cell (triangulation.n_active_cells());
-  x_local_estimated_error_per_cell = local_estimated_error_per_cell;
-  Vector<double> estimated_error_per_cell (triangulation.n_active_cells());
-#ifdef DEAL_II_COMPILER_SUPPORTS_MPI
-  MPI_Allreduce (&x_local_estimated_error_per_cell(0),
-                &estimated_error_per_cell(0),
-                triangulation.n_active_cells(), MPI_DOUBLE, MPI_MAX, MPI_COMM_WORLD);
-#else
-  estimated_error_per_cell = x_local_estimated_error_per_cell;
-#endif
-
-  GridRefinement::refine_and_coarsen_fixed_fraction (triangulation,
-                                                    estimated_error_per_cell,
-                                                    0.6, 0.2);
-  if (triangulation.n_levels() > max_grid_level)
-    for (typename Triangulation<dim>::active_cell_iterator
-          cell = triangulation.begin_active(max_grid_level);
-        cell != triangulation.end(); ++cell)
-      cell->clear_refine_flag ();
-
-  std::vector<TrilinosWrappers::Vector> x_temperature (2);
-  x_temperature[0] = temperature_solution;
-  x_temperature[1] = old_temperature_solution;
-  TrilinosWrappers::BlockVector x_stokes = stokes_solution;
-
-  SolutionTransfer<dim,TrilinosWrappers::Vector>
-    temperature_trans(temperature_dof_handler);
-  SolutionTransfer<dim,TrilinosWrappers::BlockVector>
-    stokes_trans(stokes_dof_handler);
-
-  triangulation.prepare_coarsening_and_refinement();
-  temperature_trans.prepare_for_coarsening_and_refinement(x_temperature);
-  stokes_trans.prepare_for_coarsening_and_refinement(x_stokes);
-
-  triangulation.execute_coarsening_and_refinement ();
-  computing_timer.exit_section();
-
-  setup_dofs ();
-
-  computing_timer.enter_section ("Refine mesh structure, part 2");
-
-  std::vector<TrilinosWrappers::Vector> tmp (2);
-  tmp[0].reinit (temperature_solution);
-  tmp[1].reinit (temperature_solution);
-  temperature_trans.interpolate(x_temperature, tmp);
-
-  temperature_solution = tmp[0];
-  old_temperature_solution = tmp[1];
-  temperature_constraints.distribute(temperature_solution);
-  temperature_constraints.distribute(old_temperature_solution);
-
-  TrilinosWrappers::BlockVector x_stokes_new = stokes_solution;
-  stokes_trans.interpolate (x_stokes, x_stokes_new);
-  stokes_solution = x_stokes_new;
-
-  rebuild_stokes_matrix              = true;
-  rebuild_stokes_preconditioner      = true;
-  rebuild_temperature_matrices       = true;
-  rebuild_temperature_preconditioner = true;
-
-  computing_timer.exit_section();
-}
-
-
-
-                                // @sect4{BoussinesqFlowProblem::run}
-
-                                // This is the final function in this
-                                // class. It actually runs the program. It
-                                // is, once more, very similar to
-                                // step-31. The only thing that really
-                                // changed is that we use the
-                                // <code>project_temperature_field()</code>
-                                // function instead of the library function
-                                // <code>VectorTools::project</code>, the
-                                // rest is as before.
-template <int dim>
-void BoussinesqFlowProblem<dim>::run ()
-{
-  const unsigned int initial_refinement = (dim == 2 ? 5 : 2);
-  const unsigned int n_pre_refinement_steps = (dim == 2 ? 2 : 2);
-
-  GridGenerator::hyper_shell (triangulation,
-                             Point<dim>(),
-                             EquationData::R0,
-                             EquationData::R1,
-                             12,
-                             true);
-
-  static HyperShellBoundary<dim> boundary;
-  triangulation.set_boundary (0, boundary);
-  triangulation.set_boundary (1, boundary);
-
-  global_Omega_diameter = GridTools::diameter (triangulation);
-
-  triangulation.refine_global (initial_refinement);
-
-  setup_dofs();
-
-  unsigned int pre_refinement_step = 0;
-
-  start_time_iteration:
-
-  project_temperature_field ();
-
-  timestep_number           = 0;
-  time_step = old_time_step = 0;
-
-  double time = 0;
-
-  do
-    {
-      pcout << "Timestep " << timestep_number
-           << ":  t=" << time/EquationData::year_in_seconds
-           << " years"
-           << std::endl;
-
-      assemble_stokes_system ();
-      build_stokes_preconditioner ();
-      assemble_temperature_matrix ();
-
-      solve ();
-
-      pcout << std::endl;
-
-      if ((timestep_number == 0) &&
-         (pre_refinement_step < n_pre_refinement_steps))
-       {
-         refine_mesh (initial_refinement + n_pre_refinement_steps);
-         ++pre_refinement_step;
-         goto start_time_iteration;
-       }
-      else
-       if ((timestep_number > 0) && (timestep_number % 10 == 0))
-         refine_mesh (initial_refinement + n_pre_refinement_steps);
-
-      output_results ();
-
-      time += time_step;
-      ++timestep_number;
-
-      old_stokes_solution          = stokes_solution;
-      old_old_temperature_solution = old_temperature_solution;
-      old_temperature_solution     = temperature_solution;
-    }
-  while (time <= EquationData::end_time);
-}
-
-
-
-                                // @sect3{The <code>main</code> function}
-
-                                // This is copied verbatim from step-31:
-int main (int argc, char *argv[])
-{
-  try
-    {
-      deallog.depth_console (0);
-
-      Utilities::System::MPI_InitFinalize mpi_initialization(argc, argv);
-
-      BoussinesqFlowProblem<2> flow_problem;
-      flow_problem.run ();
-    }
-  catch (std::exception &exc)
-    {
-      std::cerr << std::endl << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      std::cerr << "Exception on processing: " << std::endl
-                << exc.what() << std::endl
-                << "Aborting!" << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-
-      return 1;
-    }
-  catch (...)
-    {
-      std::cerr << std::endl << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      std::cerr << "Unknown exception!" << std::endl
-                << "Aborting!" << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      return 1;
-    }
-
-  return 0;
-}
diff --git a/deal.II/examples/step-37/Makefile b/deal.II/examples/step-37/Makefile
deleted file mode 100644 (file)
index ebd829a..0000000
+++ /dev/null
@@ -1,157 +0,0 @@
-# $Id$
-
-# For the small projects Makefile, you basically need to fill in only
-# four fields.
-#
-# The first is the name of the application. It is assumed that the
-# application name is the same as the base file name of the single C++
-# file from which the application is generated.
-target = $(basename $(shell echo step-*.cc))
-
-# The second field determines whether you want to run your program in
-# debug or optimized mode. The latter is significantly faster, but no
-# run-time checking of parameters and internal states is performed, so
-# you should set this value to `on' while you develop your program,
-# and to `off' when running production computations.
-debug-mode = on
-
-
-# As third field, we need to give the path to the top-level deal.II
-# directory. You need to adjust this to your needs. Since this path is
-# probably the most often needed one in the Makefile internals, it is
-# designated by a single-character variable, since that can be
-# reference using $D only, i.e. without the parentheses that are
-# required for most other parameters, as e.g. in $(target).
-D = ../../
-
-
-# The last field specifies the names of data and other files that
-# shall be deleted when calling `make clean'. Object and backup files,
-# executables and the like are removed anyway. Here, we give a list of
-# files in the various output formats that deal.II supports.
-clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk
-
-
-
-
-#
-#
-# Usually, you will not need to change anything beyond this point.
-#
-#
-# The next statement tell the `make' program where to find the
-# deal.II top level directory and to include the file with the global
-# settings
-include $D/common/Make.global_options
-
-
-
-
-# Since the whole project consists of only one file, we need not
-# consider difficult dependencies. We only have to declare the
-# libraries which we want to link to the object file, and there need
-# to be two sets of libraries: one for the debug mode version of the
-# application and one for the optimized mode. Here we have selected
-# the versions for 2d. Note that the order in which the libraries are
-# given here is important and that your applications won't link
-# properly if they are given in another order.
-#
-# You may need to augment the lists of libraries when compiling your
-# program for other dimensions, or when using third party libraries
-libs.g   = $(lib-deal2-2d.g) \
-          $(lib-deal2-3d.g) \
-          $(lib-lac.g)      \
-           $(lib-base.g)
-libs.o   = $(lib-deal2-2d.o) \
-          $(lib-deal2-3d.o) \
-          $(lib-lac.o)      \
-           $(lib-base.o)
-
-
-# We now use the variable defined above which switch between debug and
-# optimized mode to select the set of libraries to link with. Included
-# in the list of libraries is the name of the object file which we
-# will produce from the single C++ file. Note that by default we use
-# the extension .g.o for object files compiled in debug mode and .o for
-# object files in optimized mode (or whatever the local default on your
-# system is instead of .o).
-ifeq ($(debug-mode),on)
-  libraries = $(target).g.$(OBJEXT) $(libs.g)
-else
-  libraries = $(target).$(OBJEXT) $(libs.o)
-endif
-
-
-# Now comes the first production rule: how to link the single object
-# file produced from the single C++ file into the executable. Since
-# this is the first rule in the Makefile, it is the one `make' selects
-# if you call it without arguments.
-$(target) : $(libraries)
-       @echo ============================ Linking $@
-       @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS)
-
-
-# To make running the application somewhat independent of the actual
-# program name, we usually declare a rule `run' which simply runs the
-# program. You can then run it by typing `make run'. This is also
-# useful if you want to call the executable with arguments which do
-# not change frequently. You may then want to add them to the
-# following rule:
-run: $(target)
-       @echo ============================ Running $<
-       @./$(target)$(EXEEXT)
-
-
-# As a last rule to the `make' program, we define what to do when
-# cleaning up a directory. This usually involves deleting object files
-# and other automatically created files such as the executable itself,
-# backup files, and data files. Since the latter are not usually quite
-# diverse, you needed to declare them at the top of this file.
-clean:
-       -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files)
-
-
-# Since we have not yet stated how to make an object file from a C++
-# file, we should do so now. Since the many flags passed to the
-# compiler are usually not of much interest, we suppress the actual
-# command line using the `at' sign in the first column of the rules
-# and write the string indicating what we do instead.
-./%.g.$(OBJEXT) :
-       @echo ==============debug========= $(<F)
-       @$(CXX) $(CXXFLAGS.g) -c $< -o $@
-./%.$(OBJEXT) :
-       @echo ==============optimized===== $(<F)
-       @$(CXX) $(CXXFLAGS.o) -c $< -o $@
-
-
-# The following statement tells make that the rules `run' and `clean'
-# are not expected to produce files of the same name as Makefile rules
-# usually do.
-.PHONY: run clean
-
-
-# Finally there is a rule which you normally need not care much about:
-# since the executable depends on some include files from the library,
-# besides the C++ application file of course, it is necessary to
-# re-generate the executable when one of the files it depends on has
-# changed. The following rule to created a dependency file
-# `Makefile.dep', which `make' uses to determine when to regenerate
-# the executable. This file is automagically remade whenever needed,
-# i.e. whenever one of the cc-/h-files changed. Make detects whether
-# to remake this file upon inclusion at the bottom of this file.
-#
-# If the creation of Makefile.dep fails, blow it away and fail
-Makefile.dep: $(target).cc Makefile \
-              $(shell echo $D/*/include/*/*.h)
-       @echo ============================ Remaking $@
-       @$D/common/scripts/make_dependencies  $(INCLUDE) -B. $(target).cc \
-               > $@ \
-         || (rm -f $@ ; false)
-       @if test -s $@ ; then : else rm $@ ; fi
-
-
-# To make the dependencies known to `make', we finally have to include
-# them:
-include Makefile.dep
-
-
diff --git a/deal.II/examples/step-37/doc/builds-on b/deal.II/examples/step-37/doc/builds-on
deleted file mode 100644 (file)
index 42c2846..0000000
+++ /dev/null
@@ -1 +0,0 @@
-step-16
diff --git a/deal.II/examples/step-37/doc/intro.dox b/deal.II/examples/step-37/doc/intro.dox
deleted file mode 100644 (file)
index 9d488ec..0000000
+++ /dev/null
@@ -1,406 +0,0 @@
-<br>
-
-<i>
-This program was contributed by Katharina Kormann and Martin
-Kronbichler.
-
-This program is currently under construction.
-
-The algorithm for the matrix-vector product is built upon the report &quot;MPI
-parallelization of a cell-based matrix-vector product for finite elements. An
-application from quantum dynamics&quot; by Katharina Kormann, Uppsala
-University, June 2009.  
-</i>
-
-
-<a name="Intro"></a>
-<h1>Introduction</h1>
-
-This example shows how to implement a matrix-free method, that is, a method
-that does not explicitly store the matrix elements, for a
-second-order Poisson equation with variable coefficients on a fairly 
-unstructured mesh representing a circle.
-
-<h3>Matrix-vector product implementation</h3>
-
-In order to find out how we can write a code that performs a matrix-vector
-product, but does not need to store the matrix elements, let us start at
-looking how some finite-element related matrix <i>A</i> is assembled:
-@f{eqnarray*}
-A = \sum_{\mathrm{cell}=1}^{\mathrm{n\_cells}} P_\mathrm{cell,{loc-glob}}^T A_\mathrm{cell}
-P_\mathrm{cell,{loc-glob}}.
-@f}
-In this formula, the matrix <i>P</i><sub>cell,loc-glob</sub> is a rectangular
-matrix that defines the index mapping from local degrees of freedom in the
-current cell to the global degrees of freedom. The information from which this
-operator can be built is usually encoded in the <code>local_dof_indices</code>
-variable we have always used in the assembly of matrices.
-
-If we are to perform a matrix-vector product, we can hence use that
-@f{eqnarray*}
-y &=& A\cdot x = \left(\sum_{\text{cell}=1}^{\mathrm{n\_cells}} P_\mathrm{cell,{loc-glob}}^T
-A_\mathrm{cell} P_\mathrm{cell,{loc-glob}}\right) \cdot x
-\\
-&=& \sum_{\mathrm{cell}=1}^{\mathrm{n\_cells}} P_\mathrm{cell,{loc-glob}}^T
-A_\mathrm{cell} x_\mathrm{cell}
-\\
-&=& \sum_{\mathrm{cell}=1}^{\mathrm{n\_cells}} P_\mathrm{cell,{loc-glob}}^T
-y_\mathrm{cell},
-@f}
-where <i>x</i><sub>cell</sub> are the values of <i>x</i> at the degrees of freedom
-of the respective cell, and  <i>x</i><sub>cell</sub> correspondingly for the result.
-A naive attempt to implement the local action of the Laplacian would hence be
-to use the following code:
-@code
-MatrixFree::vmult (Vector<double>       &dst,
-                  const Vector<double> &src) const
-{
-  dst = 0;
-
-  QGauss<dim>  quadrature_formula(fe.degree+1);
-  FEValues<dim> fe_values (fe, quadrature_formula,
-                           update_gradients | update_JxW_values);
-
-  const unsigned int   dofs_per_cell = fe.dofs_per_cell;
-  const unsigned int   n_q_points    = quadrature_formula.size();
-
-  FullMatrix<double>   cell_matrix (dofs_per_cell, dofs_per_cell);
-  Vector<double>       cell_src (dofs_per_cell),
-                      cell_dst (dofs_per_cell);
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-
-  typename DoFHandler<dim>::active_cell_iterator
-    cell = dof_handler.begin_active(),
-    endc = dof_handler.end();
-  for (; cell!=endc; ++cell)
-    {
-      cell_matrix = 0;
-      fe_values.reinit (cell);
-
-      for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-        for (unsigned int i=0; i<dofs_per_cell; ++i)
-         for (unsigned int j=0; j<dofs_per_cell; ++j)
-            cell_matrix(i,j) += (fe_values.shape_grad(i,q_point) *
-                                 fe_values.shape_grad(j,q_point) *
-                                 fe_values.JxW(q_point));
-
-      cell->get_dof_indices (local_dof_indices);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        cell_src(i) = src(local_dof_indices(i));
-
-      cell_matrix.vmult (cell_dst, cell_src);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        dst(local_dof_indices(i)) += cell_dst;
-    }
-}
-@endcode
-
-Here we neglected boundary conditions as well as any hanging nodes we may
-have, though neither would be very difficult to include using the
-ConstraintMatrix class. Note how we first generate the local matrix in the
-usual way. To form the actual product as expressed in the above formula, we
-read in the values of <code>src</code> of the cell-related degrees of freedom
-(the action of <i>P</i><sub>cell,loc-glob</sub>), multiply by the local matrix
-(the action of <i>A</i><sub>cell</sub>), and finally add the result to the
-destination vector <code>dst</code> (the action of
-<i>P</i><sub>cell,loc-glob</sub><sup>T</sup>, added over all the elements). It
-is not more difficult than that, in principle.
-
-While this code is completely correct, it is very slow. For every cell, we
-generate a local matrix, which takes three nested loops with as many
-elements as there are degrees of freedom on the actual cell to compute. The
-multiplication itself is then done by two nested loops, which means that it
-is much cheaper.
-
-One way to improve this is to realize that conceptually the local
-matrix can be thought of as the product of three matrices,
-@f{eqnarray*}
-A_\mathrm{cell} = B_\mathrm{cell}^T D_\mathrm{cell} B_\mathrm{cell},
-@f}
-where for the example of the Laplace operator the (<i>q</i>*dim+<i>d,i</i>)-th
-element of <i>B</i><sub>cell</sub> is given by
-<code>fe_values.shape_grad(i,q)[d]</code>. The matrix consists of
-<code>dim*n_q_points</code> rows and @p dofs_per_cell columns). The matrix
-<i>D</i><sub>cell</sub> is diagonal and contains the values
-<code>fe_values.JxW(q)</code> (or, rather, @p dim copies of it).
-
-Every numerical analyst learns in one of her first classes that for
-forming a product of the form
-@f{eqnarray*}
-A_\mathrm{cell}\cdot x_\mathrm{cell} = B_\mathrm{cell} D_\mathrm{cell} 
-                    B_\mathrm{cell}^T \cdot x_\mathrm{cell},
-@f}
-one should never form the matrix-matrix products, but rather multiply with the
-vector from right to left so that only three successive matrix-vector products
-are formed. To put this into code, we can write:
-@code
-...
-  for (; cell!=endc; ++cell)
-    {
-      fe_values.reinit (cell);
-
-      cell->get_dof_indices (local_dof_indices);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        cell_src(i) = src(local_dof_indices(i));
-
-      temp_vector = 0;
-      for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-        for (unsigned int d=0; d<dim; ++d)
-          for (unsigned int i=0; i<dofs_per_cell; ++i)
-           temp_vector(q_point*dim+d) += fe_values.shape_grad(i,q_point)[d] *
-                                         cell_src(i);
-
-      for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-        for (unsigned int d=0; d<dim; ++d)
-         temp_vector(q_point*dim+d) *= fe_values.JxW(q_point);
-
-      cell_dst = 0;
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-          for (unsigned int d=0; d<dim; ++d)
-           cell_dst(i) += fe_values.shape_grad(i,q_point)[d] *
-                          temp_vector(q_point*dim+d);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        dst(local_dof_indices(i)) += cell_dst(i);
-    }
-}
-@endcode
-
-This removed the three nested loops in the calculation of the local matrix
-(here the loop over <i>d</i> is a not really a loop, rather two or three
-operations). What happens is as follows: We first transform the vector of
-values on the local dofs to a vector of gradients on the quadrature
-points. In the second loop, we multiply these gradients by the integration
-weight. The third loop applies the second gradient (in transposed form), so
-that we get back to a vector of (Laplacian) values on the cell dofs.
-
-This improves the situation a lot and reduced the complexity of the product
-from something like $\mathcal {O}(\mathrm{dofs\_per\_cell}^3)$ to $\mathcal
-{O}(\mathrm{dofs\_per\_cell}^2)$. In fact, all the remainder is just to make
-a slightly more clever use of data in order to gain some extra speed. It does
-not change the code structure, though.
-
-The bottleneck in the above code is the operations done by the call
-<code>fe_values.reinit(cell)</code>, which take about as much time as the
-other steps together (at least if the mesh is unstructured; deal.II can
-recognize that the gradients are often unchanged on structured meshes). That
-is certainly not ideal and we would like to do better than this. What the
-reinit function does is to calculate the gradient in real space by
-transforming the gradient on the reference cell using the Jacobian of the
-transformation from real to reference cell. This is done for each basis
-function on the cell, for each quadrature point. The Jacobian does not depend on
-the basis function, but it is different on different quadrature points in
-general. The trick is now to factor out the Jacobian transformation and first
-apply the operation that leads us to <code>temp_vector</code> only with the
-gradient on the reference cell. That transforms the vector of values on the
-local dofs to a vector of gradients on the quadrature points. There, we first
-apply the Jacobian that we factored out from the gradient, then we apply the
-weights of the quadrature, and we apply with the transposed Jacobian for
-preparing the third loop which again uses the gradients on the unit cell.
-
-Let us again write this in terms of matrices. Let the matrix
-<i>B</i><sub>cell</sub> denote the cell-related gradient matrix, with each row
-containing the values of the quadrature points. It is constructed by a
-matrix-matrix product as
-@f{eqnarray*}
-B_\mathrm{cell} = J_\mathrm{cell} B_\mathrm{ref\_cell},
-@f}
-where <i>B</i><sub>ref_cell</sub> denotes the gradient on the reference cell
-and <i>J</i><sub>cell</sub> denotes the Jacobian
-transformation. <i>J</i><sub>cell</sub> is block-diagonal, and the blocks size
-is equal to the dimension of the problem. Each diagonal block is the Jacobian
-transformation that goes from the reference cell to the real cell.
-
-Putting things together, we find that
-@f{eqnarray*}
-A_\mathrm{cell} = B_\mathrm{cell}^T D B_\mathrm{cell} 
-               = B_\mathrm{ref\_cell}^T J_\mathrm{cell}^T 
-                 D_\mathrm{cell} 
-                 J_\mathrm{cell} B_\mathrm{ref\_cell},
-@f}
-so we calculate the product (starting the local product from the right)
-@f{eqnarray*}
-y_\mathrm{cell} = B_\mathrm{ref\_cell}^T J_\mathrm{cell}^T D J_\mathrm{cell} 
-B_\mathrm{ref\_cell} x_\mathrm{cell}, \quad
-y = \sum_{\mathrm{cell}=1}^{\mathrm{n\_cells}} P_\mathrm{cell,{loc-glob}}^T
-y_\mathrm{cell}.
-@f}
-@code
-...
-  FEValues<dim> fe_values_reference (fe, quadrature_formula,
-                                    update_gradients);
-  Triangulation<dim> reference_cell;
-  GridGenerator::hyper_cube(reference_cell, 0., 1.);
-  fe_values_reference.reinit (reference_cell.begin());
-
-  FEValues<dim> fe_values (fe, quadrature_formula,
-                           update_inverse_jacobians | update_JxW_values);
-
-  for (; cell!=endc; ++cell)
-    {
-      fe_values.reinit (cell);
-
-      cell->get_dof_indices (local_dof_indices);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        cell_src(i) = src(local_dof_indices(i));
-
-      temp_vector = 0;
-      for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-        for (unsigned int d=0; d<dim; ++d)
-          for (unsigned int i=0; i<dofs_per_cell; ++i)
-           temp_vector(q_point*dim+d) +=
-             fe_values_reference.shape_grad(i,q_point)[d] * cell_src(i);
-
-      for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-        {
-          // apply the Jacobian of the mapping from unit to real cell
-         Tensor<1,dim> temp;
-         for (unsigned int d=0; d<dim; ++d)
-           temp[d] = temp_vector(q_point*dim+d);
-         for (unsigned int d=0; d<dim; ++d)
-           {
-             double sum = 0;
-             for (unsigned int e=0; e<dim; ++e)
-               sum += fe_values.inverse_jacobian(q_point)[d][e] *
-                      temp[e];
-             temp_vector(q_point*dim+d) = sum;
-           }
-
-          // multiply with integration weight
-         for (unsigned int d=0; d<dim; ++d)
-           temp_vector(q_point*dim+d) *= fe_values.JxW(q_point);
-
-          // apply the transpose of the Jacobian of the mapping from unit
-         // to real cell
-         for (unsigned int d=0; d<dim; ++d)
-           temp[d] = temp_vector(q_point*dim+d);
-         for (unsigned int d=0; d<dim; ++d)
-           {
-             double sum = 0;
-             for (unsigned int e=0; e<dim; ++e)
-               sum += fe_values.inverse_jacobian(q_point)[e][d] *
-                      temp[e];
-             temp_vector(q_point*dim+d) = sum;
-           }
-        }
-
-      cell_dst = 0;
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
-          for (unsigned int d=0; d<dim; ++d)
-           cell_dst(i) += fe_values_reference.shape_grad(i,q_point)[d] *
-                          temp_vector(q_point*dim+d);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-        dst(local_dof_indices(i)) += cell_dst(i);
-    }
-}
-@endcode
-
-Note how we create an additional FEValues object for the reference cell
-gradients and how we initialize it to the reference cell. The actual
-derivative data is then applied by the Jacobians (deal.II calls the Jacobian
-matrix from unit to real cell inverse_jacobian, because the transformation
-direction in deal.II is from real to unit cell).
-
-To sum up, we want to look at the additional costs introduced by this
-written-out matrix-matrix-vector product compared to a sparse matrix. To first
-approximation, we have increased the number of operations for the local
-matrix-vector product by a factor of 4 in 2D and 6 in 3D, i.e., two
-matrix-vector products with matrices that have <code>dim</code> as many
-columns than before (here we assume that the number of quadrature points is
-the same as the number of degrees of freedom per cell, which is usual for
-scalar problems). Then, we also need to keep in mind that we touch some
-degrees of freedom several times because they belong to several cells. This
-also increases computational costs. A realistic value compared to a sparse
-matrix is that we now have to perform about 10 times as many operations (a bit
-less in 2D, a bit more in 3D).
-
-The above is, in essence, what happens in the code below and if you have
-difficulties in understanding the implementation, you should try to first
-understand what happens in the code above. In the actual implementation
-there are a few more points done to be even more efficient, namely:
-<ul>
-  <li>We pre-compute the inverse of the Jacobian of the transformation and
-      store it in an extra array. This allows us to fuse the three
-      operations <i>J</i><sub>cell</sub><sup>T</sup><i> D</i><sub>cell</sub>
-      <i>J</i><sub>cell</sub> (apply Jacobian, multiply by 
-      weights, apply transposed
-      Jacobian) into one second-rank tensor that is also symmetric (so we
-      only need to store half the tensor).
-  <li>We work on several cells at once when we apply the gradients of the
-      unit cell (it is always the same matrix with the reference cell
-      data). This allows us to replace the matrix-vector product by a
-      matrix-matrix product (several vectors of cell-data form a matrix),
-      which enables a faster implementation. Obviously, we need some adapted
-      data structures for that, but it isn't too hard to provide that. What
-      is nice is that dense matrix-matrix products are close to today's
-      processors' peak performance if the matrices are neither too small nor
-      too large &mdash; and these operations are the most expensive part in
-      the implementation shown here. 
-</ul>
-
-The implementation of the matrix-free matrix-vector product shown in this
-tutorial is slower than a matrix-vector product using a sparse matrix for
-linear and quadratic elements, but on par with third order elements and faster
-for even higher order elements. An additional gain with this implementation is
-that we do not have to build the sparse matrix itself, which can also be quite
-expensive depending on the underlying differential equation.
-
-
-<h3>Combination with multigrid</h3>
-
-Above, we have gone to significant lengths to implement a matrix-vector
-product that does not actually store the matrix elements. In many user codes,
-however, one wants more than just performing some uncertain number of
-matrix-vector products &mdash; one wants to do as little of these operations
-as possible when solving linear equation systems. In theory, we could use the
-CG method without preconditioning; however, that would not be very
-efficient. Rather, one uses preconditioners for improving speed. On the other
-hand, most of the more frequently used preconditioners such as SSOR, ILU or
-algebraic multigrid (AMG) can now no longer be used here because their
-implementation requires knowledge of the elements of the system matrix.
-
-One solution is to use multigrid methods as shown in
-step-16. They are known to be very fast, and they are suitable for our
-purpose since they can be designed based purely on matrix-vector products. All
-one needs to do is to find a smoother that works with matrix-vector products
-only (our choice requires knowledge of the diagonal entries of the matrix,
-though). One such candidate would be a damped Jacobi iteration, but that is
-often not sufficiently good in damping high-frequency errors.
-A Chebyshev preconditioner, eventually, is what we use here. It can be
-seen as an extension of the Jacobi method by using Chebyshev polynomials. With
-degree zero, the Jacobi method with optimal damping parameter is retrieved,
-whereas higher order corrections improve the smoothing properties if some
-parameters are suitably chosen. The effectiveness of Chebyshev smoothing in
-multigrid has been demonstrated, e.g., in the article <i>M. Adams, M. Brezina,
-J. Hu, R. Tuminaro. Parallel multigrid smoothers: polynomial versus
-Gauss&ndash;Seidel, J. Comput. Phys. 188:593&ndash;610, 2003</i>. This
-publication also identifies one more advantage of Chebyshev smoothers that we
-exploit here, namely that they are easy to parallelize, whereas
-SOR/Gauss&ndash;Seidel smoothing relies on substitutions, which can often only
-be parallelized by working on diagonal sub-blocks of the matrix, which
-decreases efficiency.
-
-The implementation into the multigrid framework is then straightforward. This
-program is based on an earlier version of step-16 that demonstrated multigrid
-on uniformly refined grids. However, the present matrix-free techniques would
-obviously also apply to the adaptive meshes the current step-16 uses.
-
-
-<h3>The test case</h3>
-
-In order to demonstrate the capabilities of the method, we work on a rather
-general Poisson problem, based on a more or less unstructured mesh (where
-the Jacobians are different from cell to cell), higher order mappings to a
-curved boundary, and a non-constant coefficient in the equation. If we
-worked on a constant-coefficient case with structured mesh, we could
-decrease the operation count by a factor of 4 in 2D and 6 in 3D by building
-a local matrix (which is then the same for all cells), and doing the
-products as in the first developing step of the above code pieces.
diff --git a/deal.II/examples/step-37/doc/kind b/deal.II/examples/step-37/doc/kind
deleted file mode 100644 (file)
index c1d9154..0000000
+++ /dev/null
@@ -1 +0,0 @@
-techniques
diff --git a/deal.II/examples/step-37/doc/results.dox b/deal.II/examples/step-37/doc/results.dox
deleted file mode 100644 (file)
index 6e00f0b..0000000
+++ /dev/null
@@ -1,311 +0,0 @@
-<h1>Results</h1>
-
-<h3>Program output</h3>
-
-Since this example solves the same problem as step-5 (except for
-a different coefficient), we refer to the graphical output there. Here, we
-evaluate some aspects of the multigrid solver.
-
-When we run this program in 2D for quadratic ($Q_2$) elements, we get the
-following output:
-@code
-Cycle 0
-Number of degrees of freedom: 337
-System matrix memory consumption: 0.02573 MiB.
-Multigrid objects memory consumption: 0.05083 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 1
-Number of degrees of freedom: 1313
-System matrix memory consumption: 0.09257 MiB.
-Multigrid objects memory consumption: 0.1794 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 2
-Number of degrees of freedom: 5185
-System matrix memory consumption: 0.3553 MiB.
-Multigrid objects memory consumption: 0.6779 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 3
-Number of degrees of freedom: 20609
-System matrix memory consumption: 1.397 MiB.
-Multigrid objects memory consumption: 2.645 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 4
-Number of degrees of freedom: 82177
-System matrix memory consumption: 5.546 MiB.
-Multigrid objects memory consumption: 10.46 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 5
-Number of degrees of freedom: 328193
-System matrix memory consumption: 22.11 MiB.
-Multigrid objects memory consumption: 41.65 MiB.
-Convergence in 10 CG iterations.
-@endcode
-
-As in step-16, we see that the number of CG iterations remains constant with
-increasing number of degrees of freedom. We can also see that the various
-objects we have to store for the multigrid method on the individual levels of
-our mesh together make up about twice as much as the matrix on the finest
-level.
-
-Not much changes if we run the
-program in three spatial dimensions, with the exception that the multilevel
-objects now take up comparatively less space (because in 3d, each level has
-only one eighth the number of cells of the next finer one, whereas in 2d this
-factor if one quarter):
-
-@code
-Cycle 0
-Number of degrees of freedom: 517
-System matrix memory consumption: 0.1001 MiB.
-Multigrid objects memory consumption: 0.1463 MiB.
-Convergence in 9 CG iterations.
-
-Cycle 1
-Number of degrees of freedom: 3817
-System matrix memory consumption: 0.6613 MiB.
-Multigrid objects memory consumption: 0.8896 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 2
-Number of degrees of freedom: 29521
-System matrix memory consumption: 5.1 MiB.
-Multigrid objects memory consumption: 6.653 MiB.
-Convergence in 10 CG iterations.
-
-Cycle 3
-Number of degrees of freedom: 232609
-System matrix memory consumption: 40.4 MiB.
-Multigrid objects memory consumption: 52.24 MiB.
-Convergence in 11 CG iterations.
-
-Cycle 4
-Number of degrees of freedom: 1847617
-System matrix memory consumption: 322 MiB.
-Multigrid objects memory consumption: 415.1 MiB.
-Convergence in 11 CG iterations.
-@endcode
-
-
-<h3>Comparison with a sparse matrix</h3>
-
-In order to understand the capabilities of this class, we compare the memory
-consumption and execution (wallclock) time for assembly and 50 matrix-vector
-products (MV) on a <b>3D problem with one million unknowns</b> the classical
-sparse matrix implementation (SpM) and the MatrixFree implementation shown
-here (M-F). Both matrices are based on @p double %numbers. The program is run
-on a 2.8 GHz Opteron processor with the <a href="http://www.amd.com/acml">ACML
-BLAS</a>. We present results running on one core core and four cores,
-respectively. Moreover, we measure the time it takes to construct the
-individual matrices and filling them with data (@p setup and @p assemble
-functions). The sparse matrix is initialized using a
-CompressedSimpleSparsityPattern for calling the
-DoFTools::make_sparsity_pattern function, and then copied to a SparsityPattern
-object. The boundary nodes are eliminated using the ConstraintMatrix class, so
-that only elements that are actually nonzero are stored in the matrix.
-
-<table align="center" border="1">
-  <tr>
-    <th>&nbsp;</th>
-    <th colspan="2">Memory consumption</th>
-    <th colspan="2">Time assembly</th>
-    <th colspan="2">Time 50 MV, 1 CPU</th>
-    <th colspan="2">Time 50 MV, 4 CPUs</th>
-  </tr>
-  <tr>
-    <th align="center">element order</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-  </tr>
-  <tr>
-    <td align="center">1</td>
-    <td align="center">299 MiB</td>
-    <td align="center">394 MiB</td>
-    <td align="center">8.09 s</td>
-    <td align="center">3.43 s</td>
-    <td align="center">5.50 s</td>
-    <td align="center">22.4 s</td>
-    <td align="center">4.30 s</td>
-    <td align="center">11.0 s</td>
-  </tr>
-  <tr>
-    <td align="center">2</td>
-    <td align="center">698 MiB</td>
-    <td align="center">177 MiB</td>
-    <td align="center">12.43 s</td>
-    <td align="center">1.32 s</td>
-    <td align="center">12.0 s</td>
-    <td align="center">18.6 s</td>
-    <td align="center">9.10 s</td>
-    <td align="center">6.31 s</td>
-  </tr>
-  <tr>
-    <td align="center">3</td>
-    <td align="center">1295 MiB</td>
-    <td align="center">124 MiB</td>
-    <td align="center">41.1 s</td>
-    <td align="center">1.31 s</td>
-    <td align="center">21.2 s</td>
-    <td align="center">23.7 s</td>
-    <td align="center">16.0 s</td>
-    <td align="center">7.43 s</td>
-  </tr>
-  <tr>
-    <td align="center">4</td>
-    <td align="center">2282 MiB</td>
-    <td align="center">107 MiB</td>
-    <td align="center">117 s</td>
-    <td align="center">1.97 s</td>
-    <td align="center">40.8 s</td>
-    <td align="center">36.3 s</td>
-    <td align="center">19.7 s</td>
-    <td align="center">10.9 s</td>
-  </tr>
-  <tr>
-    <td align="center">5</td>
-    <td align="center">3597 MiB</td>
-    <td align="center">96.4 MiB</td>
-    <td align="center">510 s</td>
-    <td align="center">5.52 s</td>
-    <td align="center">75.7 s</td>
-    <td align="center">53.9 s</td>
-    <td align="center">29.3 s</td>
-    <td align="center">15.7 s</td>
-  </tr>
-  <tr>
-    <td align="center">6</td>
-    <td align="center">5679 MiB</td>
-    <td align="center">96.3 MiB</td>
-    <td align="center">2389 s</td>
-    <td align="center">26.1 s</td>
-    <td align="center">135 s</td>
-    <td align="center">79.1 s</td>
-    <td align="center">45.8 s</td>
-    <td align="center">24.3 s</td>
-  </tr>
-</table>
-
-There are a few interesting things with the %numbers in this table. 
-
-Firstly, we see the disappointing fact that for <b>linear elements</b> the
-MatrixFree class does actually consume more memory than a SparseMatrix with
-its SparsityPattern, despite the efforts made in this program. As mentioned
-earlier, this is mostly because the Transformation data is stored for every
-quadrature point. For each quadrature point, the transformation consists of
-six doubles, and there are about eight times as many quadrature points as
-there are degrees of freedom. In first approximation, this means that the
-matrix consumes 384 (= @p sizeof(double) * 6 * 8) bytes for each degree of
-freedom. On the other hand, the sparse matrix has a bandwidth of 27 or less,
-so each dof gives rise to at most 324 (= 27 * 12) bytes. A more clever
-implementation would try to compress the Jacobian transformation data, by
-exploiting similarities between the mappings within the cells, as well as from
-one cell to the next. This could dramatically reduce the memory requirements,
-and hence, increase the speed for lower-order implementations.
-
-Secondly, we observe that the memory requirements for a SparseMatrix grow
-quickly as the order of the elements increases. This is because there are
-increasingly many entries in each row, which exist due to more degrees of
-freedom that couple to each other. The matrix-free implementation does not
-suffer from this drawback. Here, the memory consumption decreases instead,
-since there are less DoFs that are shared among elements, which decreases the
-relative amount of quadrature points. Regarding the execution speed, we see
-that the matrix-free variant gets more competitive with higher order, and it
-does scale better when run on multiple processors (3.5 speedup with four
-processors compared to the serial case, compared to 2-2.5 speedup for the
-SparseMatrix). The advantage in %parallel scaling was expected, because the
-matrix-free variant is less memory-bound for higher order implementations, so
-that the additional computing power from many cores can better be exploited.
-
-A third thing, which is unrelated to this tutorial program, is the fact that
-standard matrix assembly gets really slow for high order elements. The %numbers
-shown here are based on the usual routines that many other tutorial programs
-make use of. A closer analysis of this shows that the cell data does not fit
-into cache anymore. One could circumvent this problem by writing the assembly
-as a matrix-matrix product, and using (cache-aware) BLAS implementations.
-
-For completeness, here comes a similar table for a <b>2D problem with 5.7
-million unknowns</b>. Since the excess in work for the matrix-free
-implementation is less compared to 3D, the implementation is more competitive
-for lower-order elements.
-
-<table align="center" border="1">
-  <tr>
-    <th>&nbsp;</th>
-    <th colspan="2">Memory consumption</th>
-    <th colspan="2">Time assembly</th>
-    <th colspan="2">Time 50 MV, 4 CPUs</th>
-  </tr>
-  <tr>
-    <th align="center">element order</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-    <th align="center">SpM</th>
-    <th align="center">M-F</th>
-  </tr>
-  <tr>
-    <td align="center">1</td>
-    <td align="center">659 MiB</td>
-    <td align="center">661 MiB</td>
-    <td align="center">18.8 s</td>
-    <td align="center">6.45 s</td>
-    <td align="center">11.0 s</td>
-    <td align="center">28.8 s</td>
-  </tr>
-  <tr>
-    <td align="center">2</td>
-    <td align="center">1119 MiB</td>
-    <td align="center">391 MiB</td>
-    <td align="center">15.6 s</td>
-    <td align="center">2.46 s</td>
-    <td align="center">17.1 s</td>
-    <td align="center">16.2 s</td>
-  </tr>
-  <tr>
-    <td align="center">3</td>
-    <td align="center">1711 MiB</td>
-    <td align="center">318 MiB</td>
-    <td align="center">17.4 s</td>
-    <td align="center">1.82 s</td>
-    <td align="center">23.1 s</td>
-    <td align="center">13.7 s</td>
-  </tr>
-  <tr>
-    <td align="center">4</td>
-    <td align="center">2434 MiB</td>
-    <td align="center">285 MiB</td>
-    <td align="center">24.2 s</td>
-    <td align="center">1.34 s</td>
-    <td align="center">31.1 s</td>
-    <td align="center">14.6 s</td>
-  </tr>
-  <tr>
-    <td align="center">5</td>
-    <td align="center">3289 MiB</td>
-    <td align="center">266 MiB</td>
-    <td align="center">35.9 s</td>
-    <td align="center">1.26 s</td>
-    <td align="center">29.6 s</td>
-    <td align="center">16.7 s</td>
-  </tr>
-  <tr>
-    <td align="center">6</td>
-    <td align="center">4274 MiB</td>
-    <td align="center">254 MiB</td>
-    <td align="center">58.0 s</td>
-    <td align="center">1.12 s</td>
-    <td align="center">35.9 s</td>
-    <td align="center">19.4 s</td>
-  </tr>
-</table>
diff --git a/deal.II/examples/step-37/doc/tooltip b/deal.II/examples/step-37/doc/tooltip
deleted file mode 100644 (file)
index d469776..0000000
+++ /dev/null
@@ -1 +0,0 @@
-Matrix-free methods. Multigrid.
diff --git a/deal.II/examples/step-37/step-37.cc b/deal.II/examples/step-37/step-37.cc
deleted file mode 100644 (file)
index 7b5a100..0000000
+++ /dev/null
@@ -1,1629 +0,0 @@
-/* $Id$ */
-/* Author: Katharina Kormann, Martin Kronbichler, Uppsala University, 2009 */
-
-/*    $Id$       */
-/*                                                                */
-/*    Copyright (C) 2009 by the deal.II authors                   */
-/*                                                                */
-/*    This file is subject to QPL and may not be  distributed     */
-/*    without copyright and license information. Please refer     */
-/*    to the file deal.II/doc/license.html for the  text  and     */
-/*    further information on this license.                        */
-
-
-                                // To start with the include files are more
-                                // or less the same as in step-16:
-#include <base/quadrature_lib.h>
-#include <base/function.h>
-#include <base/logstream.h>
-#include <base/work_stream.h>
-
-#include <lac/vector.h>
-#include <lac/full_matrix.h>
-#include <lac/solver_cg.h>
-#include <lac/precondition.h>
-
-#include <fe/fe_q.h>
-#include <fe/fe_values.h>
-
-#include <grid/tria.h>
-#include <grid/tria_accessor.h>
-#include <grid/tria_iterator.h>
-#include <grid/tria_boundary_lib.h>
-#include <grid/grid_generator.h>
-
-#include <multigrid/multigrid.h>
-#include <multigrid/mg_dof_handler.h>
-#include <multigrid/mg_dof_accessor.h>
-#include <multigrid/mg_transfer.h>
-#include <multigrid/mg_tools.h>
-#include <multigrid/mg_coarse.h>
-#include <multigrid/mg_smoother.h>
-#include <multigrid/mg_matrix.h>
-
-#include <numerics/data_out.h>
-#include <numerics/vectors.h>
-
-#include <fstream>
-#include <sstream>
-
-using namespace dealii;
-
-
-
-                                // @sect3{Equation data}
-
-                                // We define a variable coefficient function
-                                // for the Poisson problem. It is similar to
-                                // the function in step-5 but we use the form
-                                // $a(\mathbf x)=\frac{1}{0.1 + \|\bf x\|^2}$
-                                // instead of a discontinuous one. It is
-                                // merely to demonstrate the possibilities of
-                                // this implementation, rather than making
-                                // much sense physically.
-template <int dim>
-class Coefficient : public Function<dim>
-{
-  public:
-    Coefficient ()  : Function<dim>() {}
-
-    virtual double value (const Point<dim>   &p,
-                         const unsigned int  component = 0) const;
-
-    virtual void value_list (const std::vector<Point<dim> > &points,
-                            std::vector<double>            &values,
-                            const unsigned int              component = 0) const;
-};
-
-
-
-template <int dim>
-double Coefficient<dim>::value (const Point<dim> &p,
-                               const unsigned int /*component*/) const
-{
-  return 1./(0.1+p.square());
-}
-
-
-
-template <int dim>
-void Coefficient<dim>::value_list (const std::vector<Point<dim> > &points,
-                                  std::vector<double>            &values,
-                                  const unsigned int              component) const
-{
-  Assert (values.size() == points.size(),
-         ExcDimensionMismatch (values.size(), points.size()));
-  Assert (component == 0,
-         ExcIndexRange (component, 0, 1));
-
-  const unsigned int n_points = points.size();
-
-  for (unsigned int i=0; i<n_points; ++i)
-    values[i] = 1./(0.1+points[i].square());
-}
-
-
-
-                                // @sect3{Matrix-free implementation}
-
-                                // In this program, we want to make
-                                // use of the ability of deal.II to
-                                // runs things in %parallel if compute
-                                // resources are available. We will
-                                // follow the general framework laid
-                                // out in the @ref threads module and
-                                // use the WorkStream class to do
-                                // operations on the range of all
-                                // cells.
-                                //
-                                // To this end, we first have to have
-                                // a few declarations that we use for
-                                // defining the %parallel layout of
-                                // the vector multiplication function
-                                // with the WorkStream concept in the
-                                // Matrix-free class. These comprise
-                                // so-called scratch data that we use
-                                // for calculating cell-related
-                                // information, and copy data that is
-                                // eventually used in a separate
-                                // function for writing local data
-                                // into the global vector. The reason
-                                // for this split-up definition is
-                                // that many threads at a time can
-                                // execute the local multiplications
-                                // (and filling up the copy data),
-                                // but than that copy data needs to
-                                // be worked on by one process at a
-                                // time.
-namespace WorkStreamData
-{
-  template <typename number>
-  struct ScratchData
-  {
-      ScratchData ();
-      ScratchData (const ScratchData &scratch);
-      FullMatrix<number> solutions;
-  };
-
-  template<typename number>
-  ScratchData<number>::ScratchData ()
-                 :
-                 solutions ()
-  {}
-
-  template<typename number>
-  ScratchData<number>::ScratchData (const ScratchData &)
-                 :
-                 solutions ()
-  {}
-
-  template <typename number>
-  struct CopyData : public ScratchData<number>
-  {
-      CopyData ();
-      CopyData (const CopyData &scratch);
-      unsigned int first_cell;
-      unsigned int n_dofs;
-  };
-
-  template <typename number>
-  CopyData<number>::CopyData ()
-                 :
-                 ScratchData<number> ()
-  {}
-
-  template <typename number>
-  CopyData<number>::CopyData (const CopyData &)
-                 :
-                 ScratchData<number> ()
-  {}
-
-}
-
-
-
-                                // Next comes the implementation of the
-                                // matrix-free class. It provides some
-                                // standard information we expect for
-                                // matrices (like returning the dimensions
-                                // of the matrix), it implements
-                                // matrix-vector multiplications in several
-                                // forms, and it provides functions for
-                                // filling the matrix with data.
-                                //
-                                // We choose to make this class generic,
-                                // i.e., we do not implement the actual
-                                // differential operator (here: Laplace
-                                // operator) directly in this class.  We
-                                // instead let the actual transformation
-                                // (which happens on the level of quadrature
-                                // points, see the discussion in the
-                                // introduction) be a template parameter that
-                                // is implemented by another class. We then
-                                // only have to store a list of these objects
-                                // for each quadrature point on each cell in
-                                // a big list &ndash; we choose a
-                                // <code>Table<2,Transformation></code> data
-                                // format) &ndash; and call a transform
-                                // command of the @p Transformation
-                                // class. This template magic makes it easy
-                                // to reuse this MatrixFree class for other
-                                // problems that are based on a symmetric
-                                // operation without the need for substantial
-                                // changes.
-template <typename number, class Transformation>
-class MatrixFree : public Subscriptor
-{
-  public:
-    MatrixFree ();
-
-    void reinit (const unsigned int        n_dofs,
-                const unsigned int        n_cells,
-                const FullMatrix<double> &cell_matrix,
-                const unsigned int        n_points_per_cell);
-    void clear();
-
-    unsigned int m () const;
-    unsigned int n () const;
-    ConstraintMatrix & get_constraints ();
-
-    void set_local_dof_indices (const unsigned int               cell_no,
-                               const std::vector<unsigned int> &local_dof_indices);
-    void set_derivative_data (const unsigned int    cell_no,
-                             const unsigned int    quad_point,
-                             const Transformation &trans_in);
-
-    template <typename number2>
-    void vmult (Vector<number2> &dst,
-               const Vector<number2> &src) const;
-    template <typename number2>
-    void Tvmult (Vector<number2> &dst,
-                const Vector<number2> &src) const;
-    template <typename number2>
-    void vmult_add (Vector<number2> &dst,
-                   const Vector<number2> &src) const;
-    template <typename number2>
-    void Tvmult_add (Vector<number2> &dst,
-                    const Vector<number2> &src) const;
-
-    number el (const unsigned int row,
-              const unsigned int col) const;
-    void calculate_diagonal () const;
-
-    std::size_t memory_consumption () const;
-
-                                    // The private member variables of the
-                                    // @p MatrixFree class are a
-                                    // small matrix that does the
-                                    // transformation from solution values to
-                                    // quadrature points, a list with the
-                                    // mapping between local degrees of freedom
-                                    // and global degrees of freedom for each
-                                    // cell (stored as a two-dimensional array,
-                                    // where each row corresponds to one
-                                    // cell, and the columns within individual
-                                    // cells are the local degrees of freedom),
-                                    // the transformation variable for
-                                    // implementing derivatives, a constraint
-                                    // matrix for handling boundary conditions
-                                    // as well as a few other variables that
-                                    // store matrix properties.
-  private:
-    typedef std::vector<std::pair<unsigned int,unsigned int> >::const_iterator
-    CellChunkIterator;
-    template <typename number2>
-    void local_vmult (CellChunkIterator                    cell_range,
-                     WorkStreamData::ScratchData<number> &scratch,
-                     WorkStreamData::CopyData<number>    &copy,
-                     const Vector<number2>               &src) const;
-
-    template <typename number2>
-    void
-    copy_local_to_global (const WorkStreamData::CopyData<number> &copy,
-                         Vector<number2>                        &dst) const;
-
-    FullMatrix<number>      B_ref_cell;
-    Table<2,unsigned int>   indices_local_to_global;
-    Table<2,Transformation> derivatives;
-
-    ConstraintMatrix        constraints;
-
-    mutable Vector<number>  diagonal_values;
-    mutable bool            diagonal_is_calculated;
-
-    struct MatrixSizes
-    {
-       unsigned int n_dofs, n_cells;
-       unsigned int m, n;
-       unsigned int n_points, n_comp;
-       std::vector<std::pair<unsigned int,unsigned int> > chunks;
-    }  matrix_sizes;
-};
-
-
-
-                                // This is the constructor of the @p
-                                // MatrixFree class. All it does is to
-                                // subscribe to the general deal.II @p
-                                // Subscriptor scheme that makes sure that we
-                                // do not delete an object of this class as
-                                // long as it used somewhere else, e.g. in a
-                                // preconditioner.
-template <typename number, class Transformation>
-MatrixFree<number,Transformation>::MatrixFree ()
-               :
-               Subscriptor()
-{}
-
-
-
-                                // The next functions return the
-                                // number of rows and columns of the
-                                // global matrix (i.e. the dimensions
-                                // of the operator this class
-                                // represents, the point of this
-                                // tutorial program was, after all,
-                                // that we don't actually store the
-                                // elements of the rows and columns
-                                // of this operator). Since the
-                                // matrix is square, the returned
-                                // numbers are the same.
-template <typename number, class Transformation>
-unsigned int
-MatrixFree<number,Transformation>::m () const
-{
-  return matrix_sizes.n_dofs;
-}
-
-
-
-template <typename number, class Transformation>
-unsigned int
-MatrixFree<number,Transformation>::n () const
-{
-  return matrix_sizes.n_dofs;
-}
-
-
-
-                                // One more function that just returns an
-                                // %internal variable. Note that the user
-                                // will need to change this variable, so it
-                                // returns a non-constant reference to the
-                                // ConstraintMatrix.
-template <typename number, class Transformation>
-ConstraintMatrix &
-MatrixFree<number,Transformation>::get_constraints ()
-{
-  return constraints;
-}
-
-
-
-                                // The following function takes a vector of
-                                // local dof indices on cell level and writes
-                                // the data into the
-                                // @p indices_local_to_global field
-                                // in order to have fast access to it. It
-                                // performs a few sanity checks like whether
-                                // the sizes in the matrix are set
-                                // correctly. One tiny thing: Whenever we
-                                // enter this function, we probably make some
-                                // modification to the matrix. This means
-                                // that the diagonal of the matrix, which we
-                                // might have computed to have fast access to
-                                // those elements, is invalidated. We set the
-                                // respective flag to @p false.
-template <typename number, class Transformation>
-void MatrixFree<number,Transformation>::
-set_local_dof_indices (const unsigned int               cell_no,
-                      const std::vector<unsigned int> &local_dof_indices)
-{
-  Assert (local_dof_indices.size() == matrix_sizes.m,
-         ExcDimensionMismatch(local_dof_indices.size(),
-                              matrix_sizes.m));
-  for (unsigned int i=0; i<matrix_sizes.m; ++i)
-    {
-      Assert (local_dof_indices[i] < matrix_sizes.n_dofs, ExcInternalError());
-      indices_local_to_global(cell_no,i) = local_dof_indices[i];
-    }
-  diagonal_is_calculated = false;
-}
-
-
-
-                                // Next a function that writes the derivative
-                                // data on a certain cell and a certain
-                                // quadrature point to the array that keeps
-                                // the data around. Even though the array @p
-                                // derivatives stands for the majority of the
-                                // matrix memory consumption, it still pays
-                                // off to have that data around since it
-                                // would be quite expensive to manually
-                                // compute it every time we make a
-                                // matrix-vector product.
-template <typename number, class Transformation>
-void MatrixFree<number,Transformation>::
-set_derivative_data (const unsigned int cell_no,
-                    const unsigned int quad_point,
-                    const Transformation &trans_in)
-{
-  Assert (quad_point < matrix_sizes.n_points, ExcInternalError());
-  derivatives(cell_no,quad_point) = trans_in;
-  diagonal_is_calculated = false;
-}
-
-
-
-                                // Now finally to the central function of the
-                                // matrix-free class, implementing the
-                                // multiplication of the matrix with a
-                                // vector. This function does not actually
-                                // work on all cells of a mesh, but only the
-                                // subset of cells specified by the first
-                                // argument @p cell_range. Since this
-                                // function operates similarly irrespective
-                                // on which cell chunk we are sitting, we can
-                                // call it simultaneously on many processors,
-                                // but with different cell range data.
-                                //
-                                // The goal of this function is to provide
-                                // the multiplication of a vector with the
-                                // local contributions of a set of cells. As
-                                // mentioned in the introduction, if we were
-                                // to deal with a single cell, this would
-                                // amount to performing the product 
-                                // @f{eqnarray*}
-                                // P^T_\mathrm{cell,local-global} A_\mathrm{cell} 
-                                // P_\mathrm{cell,local-global} x
-                                // @f}
-                                // where 
-                                // @f{eqnarray*}
-                                // A_\mathrm{cell} = 
-                                // B_\mathrm{ref\_cell}^T J_\mathrm{cell}^T 
-                                // D_\mathrm{cell} 
-                                // J_\mathrm{cell} B_\mathrm{ref\_cell}
-                                // @f}
-                                // and <i>P</i><sub>cell,local-global</sub>
-                                // is the transformation from local to global
-                                // indices.
-                                //
-                                // To do this, we would have to do the
-                                // following steps:
-                                // <ol>
-                                //   <li> Form $x_\mathrm{cell} =
-                                //   P_\mathrm{cell,local-global} x$. This is
-                                //   done by using the command
-                                //   ConstraintMatrix::get_dof_values.
-                                //   <li> Form $x_1 = B_\mathrm{ref\_cell}
-                                //   x_\mathrm{cell}$. The vector
-                                //   <i>x</i><sub>1</sub> contains the
-                                //   reference cell gradient to the local
-                                //   cell vector.
-                                //   <li> Form $x_2 = J_\mathrm{cell}^T
-                                //   D_\mathrm{cell} J_\mathrm{cell}
-                                //   x_1$. This is a block-diagonal
-                                //   operation, with the block size equal to
-                                //   @p dim. The blocks just
-                                //   correspond to the individual quadrature
-                                //   points. The operation on each quadrature
-                                //   point is implemented by the
-                                //   Transformation class object that this
-                                //   class is equipped with. Compared to the
-                                //   introduction, the matrix
-                                //   <i>D</i><sub>cell</sub> now contains the
-                                //   @p JxW values and the
-                                //   inhomogeneous coefficient.
-                                //   <li> Form $y_\mathrm{cell} =
-                                //   B_\mathrm{ref\_cell}^T x_2$. This gives
-                                //   the local result of the matrix-vector
-                                //   product.
-                                //   <li> Form $y \leftarrow y +
-                                //   P_\mathrm{cell,local-global}^T
-                                //   y_\mathrm{cell}$. This adds the local
-                                //   result to the global vector, which is
-                                //   realized using the method
-                                //   ConstraintMatrix::distribute_local_to_global.
-                                //   Note that we do this in an extra
-                                //   function called
-                                //   @p copy_local_to_global
-                                //   because that operation must not be done
-                                //   in %parallel, in order to avoid two or
-                                //   more processes trying to add to the same
-                                //   positions in the result vector <i>y</i>.
-                                //   </ol>
-                                // The steps 1 to 4 can be done in %parallel
-                                // by multiple processes.
-
-                                // Now, it turns out that the most expensive
-                                // part of the above is the multiplication
-                                // <i>B</i><sub>ref_cell</sub>
-                                // <i>x</i><sub>cell</sub> in the second step
-                                // and the transpose operation in step
-                                // 4. Note that the matrix
-                                // <i>J</i><sup>T</sup><i> D J</i> is
-                                // block-diagonal, and hence, its application
-                                // is cheaper. Since the matrix
-                                // <i>B</i><sub>ref_cell</sub> is the same
-                                // for all cells, all that changes is the
-                                // vector <i>x</i><sub>cell</sub>. Hence,
-                                // nothing prevents us from collecting
-                                // several cell vectors to a (rectangular)
-                                // matrix, and then perform a matrix-matrix
-                                // product. These matrices are both full, but
-                                // not very large, having of the order @p
-                                // dofs_per_cell rows and columns. This is an
-                                // operation that can be much better
-                                // optimized than matrix-vector products. The
-                                // functions @p FullMatrix<number>::mmult and
-                                // @p FullMatrix<number>::mTmult use the BLAS
-                                // dgemm function (as long as BLAS has been
-                                // detected in deal.II configuration), which
-                                // provides optimized kernels for doing this
-                                // product. In our case, a matrix-matrix
-                                // product is between three and five times
-                                // faster than doing the matrix-vector
-                                // product on one cell after the other. The
-                                // variables that hold the solution on the
-                                // respective cell's support points and the
-                                // quadrature points are thus full matrices,
-                                // which we set to the correct size as a
-                                // first action in this function. The number
-                                // of rows in the two matrices @p
-                                // scratch.solutions and @p copy.solutions is
-                                // given by the number of cells they work on,
-                                // and the number of columns is the number of
-                                // degrees of freedom per cell for the first
-                                // and the number of quadrature points times
-                                // the number of components per point for the
-                                // latter.
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::
-local_vmult (CellChunkIterator                    cell_range,
-            WorkStreamData::ScratchData<number> &scratch,
-            WorkStreamData::CopyData<number>    &copy,
-            const Vector<number2>               &src) const
-{
-  const unsigned int chunk_size = cell_range->second - cell_range->first;
-
-  scratch.solutions.reinit (chunk_size, matrix_sizes.n, true);
-  copy.solutions.reinit    (chunk_size, matrix_sizes.m, true);
-  copy.first_cell         = cell_range->first;
-  copy.n_dofs             = chunk_size*matrix_sizes.m;
-
-  constraints.get_dof_values(src, &indices_local_to_global(copy.first_cell,0),
-                            &copy.solutions(0,0),
-                            &copy.solutions(0,0)+copy.n_dofs);
-
-  copy.solutions.mmult (scratch.solutions, B_ref_cell);
-
-  for (unsigned int i=0, k = copy.first_cell; i<chunk_size; ++i, ++k)
-    for (unsigned int j=0; j<matrix_sizes.n_points; ++j)
-      derivatives(k,j).transform(&scratch.solutions(i, j*matrix_sizes.n_comp));
-
-  scratch.solutions.mTmult (copy.solutions, B_ref_cell);
-}
-
-
-
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::
-copy_local_to_global (const WorkStreamData::CopyData<number> &copy,
-                     Vector<number2>                        &dst) const
-{
-  constraints.distribute_local_to_global (&copy.solutions(0,0),
-                                         &copy.solutions(0,0)+copy.n_dofs,
-                                         &indices_local_to_global(copy.first_cell,0),
-                                         dst);
-}
-
-
-
-                                // Now to the @p vmult function that is
-                                // called externally: In addition to what we
-                                // do in a @p vmult_add function, we set the
-                                // destination to zero first.
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::vmult (Vector<number2>       &dst,
-                                         const Vector<number2> &src) const
-{
-  dst = 0;
-  vmult_add (dst, src);
-}
-
-
-
-                                // Transposed matrix-vector products (needed
-                                // for the multigrid operations to be
-                                // well-defined): do the same. Since we
-                                // implement a symmetric operation, we can
-                                // refer to the @p vmult_add operation.
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::Tvmult (Vector<number2>       &dst,
-                                          const Vector<number2> &src) const
-{
-  dst = 0;
-  Tvmult_add (dst,src);
-}
-
-
-
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::Tvmult_add (Vector<number2>       &dst,
-                                              const Vector<number2> &src) const
-{
-  vmult_add (dst,src);
-}
-
-
-
-                                // This is the @p vmult_add function that
-                                // multiplies the matrix with vector @p src
-                                // and adds the result to vector @p dst.  We
-                                // include a few sanity checks to make sure
-                                // that the size of the vectors is the same
-                                // as the dimension of the matrix. We call a
-                                // %parallel function that applies the
-                                // multiplication on a chunk of cells at once
-                                // using the WorkStream module (cf. also the
-                                // @ref threads module). The subdivision into
-                                // chunks will be performed in the reinit
-                                // function and is stored in the field @p
-                                // matrix_sizes.chunks. What the rather
-                                // cryptic command to @p std_cxx1x::bind does
-                                // is to transform a function that has
-                                // several arguments (source vector, chunk
-                                // information) into a function which has
-                                // three arguments (in the first case) or one
-                                // argument (in the second), which is what
-                                // the WorkStream::run function expects. The
-                                // placeholders <code>_1, _2, _3</code> in
-                                // the local vmult specify variable input
-                                // values, given by the chunk information,
-                                // scratch data and copy data that the
-                                // WorkStream::run function will provide,
-                                // whereas the other arguments to the @p
-                                // local_vmult function are bound: to @p this
-                                // and a constant reference to the @p src in
-                                // the first case, and @p this and a
-                                // reference to the output vector in the
-                                // second. Similarly, the placeholder
-                                // @p _1 argument in the
-                                // @p copy_local_to_global function
-                                // sets the first explicit argument of that
-                                // function, which is of class
-                                // @p CopyData. We need to
-                                // abstractly specify these arguments because
-                                // the tasks defined by different cell chunks
-                                // will be scheduled by the WorkStream class,
-                                // and we will reuse available scratch and
-                                // copy data.
-template <typename number, class Transformation>
-template <typename number2>
-void
-MatrixFree<number,Transformation>::vmult_add (Vector<number2>       &dst,
-                                             const Vector<number2> &src) const
-{
-  Assert (src.size() == n(), ExcDimensionMismatch(src.size(), n()));
-  Assert (dst.size() == m(), ExcDimensionMismatch(dst.size(), m()));
-
-  WorkStream::run (matrix_sizes.chunks.begin(), matrix_sizes.chunks.end(),
-                  std_cxx1x::bind(&MatrixFree<number,Transformation>::
-                                  template local_vmult<number2>,
-                                  this, _1, _2, _3, boost::cref(src)),
-                  std_cxx1x::bind(&MatrixFree<number,Transformation>::
-                                  template copy_local_to_global<number2>,
-                                  this, _1, boost::ref(dst)),
-                  WorkStreamData::ScratchData<number>(),
-                  WorkStreamData::CopyData<number>(),
-                  2*multithread_info.n_default_threads,1);
-
-                                  // One thing to be cautious about:
-                                  // The deal.II classes expect that
-                                  // the matrix still contains a
-                                  // diagonal entry for constrained
-                                  // dofs (otherwise, the matrix
-                                  // would be singular, which is not
-                                  // what we want). Since the
-                                  // <code>distribute_local_to_global</code>
-                                  // command of the constraint matrix
-                                  // which we used for adding the
-                                  // local elements into the global
-                                  // vector does not do anything with
-                                  // constrained elements, we have to
-                                  // circumvent that problem by
-                                  // artificially setting the
-                                  // diagonal to some non-zero value
-                                  // and adding the source values. We
-                                  // simply set it to one, which
-                                  // corresponds to copying the
-                                  // respective elements of the
-                                  // source vector into the matching
-                                  // entry of the destination vector.
-  for (unsigned int i=0; i<matrix_sizes.n_dofs; ++i)
-    if (constraints.is_constrained(i) == true)
-      dst(i) += 1.0 * src(i);
-}
-
-
-
-                                // The next function initializes the
-                                // structures of the matrix. It writes the
-                                // number of total degrees of freedom in the
-                                // problem as well as the number of cells to
-                                // the MatrixSizes struct and copies the
-                                // small matrix that transforms the solution
-                                // from support points to quadrature
-                                // points. It uses the small matrix for
-                                // determining the number of degrees of
-                                // freedom per cell (number of rows in @p
-                                // B_ref_cell). The number of quadrature
-                                // points needs to be passed through the last
-                                // variable @p n_points_per_cell, since the
-                                // number of columns in the small matrix is
-                                // @p dim*n_points_per_cell for the Laplace
-                                // problem (the Laplacian is a tensor and has
-                                // @p dim components). In this function, we
-                                // also give the fields containing the
-                                // derivative information and the local dof
-                                // indices the correct sizes. They will be
-                                // filled by calling the respective set
-                                // function defined above.
-template <typename number, class Transformation>
-void MatrixFree<number,Transformation>::
-reinit (const unsigned int        n_dofs_in,
-       const unsigned int        n_cells_in,
-       const FullMatrix<double> &B_ref_cell_in,
-       const unsigned int        n_points_per_cell)
-{
-  B_ref_cell = B_ref_cell_in;
-
-  derivatives.reinit (n_cells_in, n_points_per_cell);
-  indices_local_to_global.reinit (n_cells_in, B_ref_cell.m());
-
-  diagonal_is_calculated = false;
-
-  matrix_sizes.n_dofs = n_dofs_in;
-  matrix_sizes.n_cells = n_cells_in;
-  matrix_sizes.m = B_ref_cell.m();
-  matrix_sizes.n = B_ref_cell.n();
-  matrix_sizes.n_points = n_points_per_cell;
-  matrix_sizes.n_comp   = B_ref_cell.n()/matrix_sizes.n_points;
-  Assert(matrix_sizes.n_comp * n_points_per_cell == B_ref_cell.n(),
-        ExcInternalError());
-
-                                  // One thing to make the matrix-vector
-                                  // product with this class efficient is to
-                                  // decide how many cells should be combined
-                                  // to one chunk, which will determine the
-                                  // size of the full matrix that we work
-                                  // on. If we choose too few cells, then the
-                                  // gains from using the matrix-matrix
-                                  // product will not be fully utilized
-                                  // (dgemm tends to provide more efficiency
-                                  // the larger the matrix dimensions get),
-                                  // so we choose at least 60 cells for one
-                                  // chunk (except when there are very few
-                                  // cells, like on the coarse levels of the
-                                  // multigrid scheme). If we choose too
-                                  // many, we will degrade parallelization
-                                  // (we need to have sufficiently
-                                  // independent tasks). We need to also
-                                  // think about the fact that most high
-                                  // performance BLAS implementations
-                                  // internally work with square
-                                  // sub-matrices. Choosing as many cells in
-                                  // a chunk as there are degrees of freedom
-                                  // on each cell (coded in @p
-                                  // matrix_sizes.m) respects the BLAS GEMM
-                                  // design, whenever we exceed 60. Clearly,
-                                  // the chunk size is an
-                                  // architecture-dependent value and the
-                                  // interested user can squeeze out some
-                                  // extra performance by hand-tuning this
-                                  // parameter. Once we have chosen the
-                                  // number of cells we collect in one chunk,
-                                  // we determine how many chunks we have on
-                                  // the given cell range and recalculate the
-                                  // actual chunk size in order to evenly
-                                  // distribute the chunks.
-  const unsigned int divisor = std::max(60U, matrix_sizes.m);
-  const unsigned int n_chunks = std::max (matrix_sizes.n_cells/divisor + 1,
-                                         2*multithread_info.n_default_threads);
-
-  const unsigned int chunk_size = (matrix_sizes.n_cells/n_chunks +
-                                  (matrix_sizes.n_cells%n_chunks>0));
-
-  std::pair<unsigned int, unsigned int> chunk;
-  for (unsigned int i=0; i<n_chunks; ++i)
-    {
-      chunk.first = i*chunk_size;
-      if ((i+1)*chunk_size > matrix_sizes.n_cells)
-       chunk.second = matrix_sizes.n_cells;
-      else
-       chunk.second = (i+1)*chunk_size;
-
-      if (chunk.second > chunk.first)
-       matrix_sizes.chunks.push_back(chunk);
-      else
-       break;
-    }
-}
-
-
-
-                                // Then we need a function if we want to
-                                // delete the content of the matrix,
-                                // e.g. when we are finished with one grid
-                                // level and continue to the next one. Just
-                                // set all the field sizes to 0.
-template <typename number, class Transformation>
-void
-MatrixFree<number,Transformation>::clear ()
-{
-  B_ref_cell.reinit(0,0);
-  derivatives.reinit (0,0);
-  indices_local_to_global.reinit(0,0);
-
-  constraints.clear();
-
-  diagonal_values.reinit (0);
-  diagonal_is_calculated = false;
-
-  matrix_sizes.n_dofs = 0;
-  matrix_sizes.n_cells = 0;
-  matrix_sizes.chunks.clear();
-}
-
-
-
-                                // The next function returns the entries of the
-                                // matrix. Since this class is intended not
-                                // to store the matrix entries, it would make
-                                // no sense to provide all those
-                                // elements. However, diagonal entries are
-                                // explicitly needed for the implementation
-                                // of the Chebyshev smoother that we intend
-                                // to use in the multigrid
-                                // preconditioner. This matrix is equipped
-                                // with a vector that stores the diagonal,
-                                // and we compute it when this function is
-                                // called for the first time.
-template <typename number, class Transformation>
-number
-MatrixFree<number,Transformation>::el (const unsigned int row,
-                                      const unsigned int col) const
-{
-  Assert (row == col, ExcNotImplemented());
-  if (diagonal_is_calculated == false)
-    calculate_diagonal();
-
-  return diagonal_values(row);
-}
-
-
-
-                                // Regarding the calculation of the diagonal,
-                                // remember that this is as simple (or
-                                // complicated) as assembling a right hand
-                                // side in deal.II. Well, it is a bit easier
-                                // to do this within this class since we have
-                                // all the derivative information
-                                // available. What we do is to go through all
-                                // the cells (now in serial, since this
-                                // function should not be called very often
-                                // anyway), then all the degrees of
-                                // freedom. At this place, we first copy the
-                                // first basis functions in all the
-                                // quadrature points to a temporary array,
-                                // apply the derivatives from the Jacobian
-                                // matrix, and finally multiply with the
-                                // second basis function. This is exactly the
-                                // value that would be written into the
-                                // diagonal of a sparse matrix. Note that we
-                                // need to condense hanging node constraints
-                                // and set the constrained diagonals to one.
-template <typename number, class Transformation>
-void
-MatrixFree<number,Transformation>::calculate_diagonal() const
-{
-  diagonal_values.reinit (matrix_sizes.n_dofs);
-  std::vector<number> calculation (matrix_sizes.n);
-  for (unsigned int cell=0; cell<matrix_sizes.n_cells; ++cell)
-    for (unsigned int dof=0; dof<matrix_sizes.m; ++dof)
-      {
-       memcpy (&calculation[0],&B_ref_cell(dof,0),
-               matrix_sizes.n*sizeof(number));
-       for (unsigned int q=0; q<matrix_sizes.n_points; ++q)
-         derivatives(cell,q).transform(&calculation[q*matrix_sizes.n_comp]);
-       double diag_value = 0;
-       for (unsigned int q=0; q<matrix_sizes.n; ++q)
-         diag_value += calculation[q] * B_ref_cell(dof,q);
-       diagonal_values(indices_local_to_global(cell,dof)) += diag_value;
-      }
-  constraints.condense (diagonal_values);
-  for (unsigned int i=0; i<matrix_sizes.n_dofs; ++i)
-    if (constraints.is_constrained(i) == true)
-      diagonal_values(i) = 1.0;
-
-  diagonal_is_calculated = true;
-}
-
-
-
-                                // Eventually, we provide a function that
-                                // calculates how much memory this class
-                                // uses. We just need to sum up the memory
-                                // consumption of the arrays, the
-                                // constraints, the small matrix and of the
-                                // local variables. Just as a remark: In 2D
-                                // and with data type @p double,
-                                // about 80 per cent of the memory
-                                // consumption is due to the
-                                // @p derivatives array, while in 3D
-                                // this number is even 85 per cent.
-template <typename number, class Transformation>
-std::size_t MatrixFree<number,Transformation>::memory_consumption () const
-{
-  std::size_t glob_size = derivatives.memory_consumption() +
-                         indices_local_to_global.memory_consumption() +
-                         constraints.memory_consumption() +
-                         B_ref_cell.memory_consumption() +
-                         diagonal_values.memory_consumption() +
-                         matrix_sizes.chunks.size()*2*sizeof(unsigned int) +
-                         sizeof(*this);
-  return glob_size;
-}
-
-
-
-                                // @sect3{Laplace operator implementation}
-
-                                // This class implements the local action of
-                                // a Laplace operator on a quadrature
-                                // point. This is a very basic class
-                                // implementation, providing functions for
-                                // initialization with a Tensor of rank 2 and
-                                // implementing the @p transform operation
-                                // needed by the @p MatrixFree class. There
-                                // is one point worth noting: The
-                                // quadrature-point related action of the
-                                // Laplace operator is a tensor of rank
-                                // two. It is symmetric since it is the
-                                // product of the inverse Jacobian
-                                // transformation between unit and real cell
-                                // with its transpose (times quadrature
-                                // weights and a coefficient, which are
-                                // scalar), so we can just save the diagonal
-                                // and upper diagonal part. We could use the
-                                // SymmetricTensor<2,dim> class for doing
-                                // this, however, that class is only based on
-                                // @p double %numbers. Since we also want to
-                                // use @p float %numbers for the multigrid
-                                // preconditioner (in order to save memory
-                                // and computing time), we manually implement
-                                // this operator. Note that @p dim is a
-                                // template argument and hence known at
-                                // compile-time, so the compiler knows that
-                                // this symmetric rank-2 tensor has 3 entries
-                                // if used in 2D and 6 entries if used in 3D.
-template <int dim,typename number>
-class LaplaceOperator
-{
-  public:
-    LaplaceOperator ();
-
-    LaplaceOperator (const Tensor<2,dim> &tensor);
-
-    void transform (number * result) const;
-
-    LaplaceOperator<dim,number>&
-    operator = (const Tensor<2,dim> &tensor);
-
-  private:
-    number transformation[dim*(dim+1)/2];
-};
-
-template<int dim,typename number>
-LaplaceOperator<dim,number>::LaplaceOperator()
-{}
-
-
-template<int dim,typename number>
-LaplaceOperator<dim,number>::LaplaceOperator(const Tensor<2,dim> &tensor)
-{
-  *this = tensor;
-}
-
-                                // Now implement the transformation, which is
-                                // just a so-called contraction
-                                // operation between a tensor of rank two and a
-                                // tensor of rank one. Unfortunately, we
-                                // need to implement this by hand, since we
-                                // chose not to use the
-                                // SymmetricTensor<2,dim> class (note that
-                                // the resulting values are entries in a full
-                                // matrix that consists of doubles or
-                                // floats). It feels a bit unsafe to operate
-                                // on a pointer to the data, but that is the
-                                // only possibility if we do not want to copy
-                                // data back and forth, which is expensive
-                                // since this is the innermost position of
-                                // the loop in the @p vmult
-                                // operation of the MatrixFree class. We need
-                                // to pay attention to the fact that we only
-                                // saved half of the (symmetric) rank-two
-                                // tensor.
-                                //
-                                // At first sight, it seems inefficient that
-                                // we have an @p if clause at this position
-                                // in the code at the innermost loop, but
-                                // note once again that @p dim is known when
-                                // this piece of code is compiled, so the
-                                // compiler can optimize away the @p if
-                                // statement (and actually even inline these
-                                // few lines of code into the @p MatrixFree
-                                // class).
-template <int dim, typename number>
-void LaplaceOperator<dim,number>::transform (number* result) const
-{
-  if (dim == 2)
-    {
-      const number temp = result[0];
-      result[0] = transformation[0] * temp + transformation[1] * result[1];
-      result[1] = transformation[1] * temp + transformation[2] * result[1];
-    }
-  else if (dim == 3)
-    {
-      const number temp1 = result[0];
-      const number temp2 = result[1];
-      result[0] = transformation[0] * temp1 + transformation[1] * temp2 +
-                 transformation[2] * result[2];
-      result[1] = transformation[1] * temp1 + transformation[3] * temp2 +
-                 transformation[4] * result[2];
-      result[2] = transformation[2] * temp1 + transformation[4] * temp2 +
-                 transformation[5] * result[2];
-    }
-  else
-    ExcNotImplemented();
-}
-
-                                // The final function in this group
-                                // takes the content of a rank-2
-                                // tensor and writes it to the field
-                                // @p transformation of
-                                // this class. We save the upper part
-                                // of the symmetric tensor row-wise:
-                                // we first take the (0,0)-entry,
-                                // then the (0,1)-entry, and so
-                                // on. We only implement this for
-                                // dimensions two and three, which
-                                // for the moment should do just
-                                // fine:
-template <int dim, typename number>
-LaplaceOperator<dim,number>&
-LaplaceOperator<dim,number>::operator=(const Tensor<2,dim> &tensor)
-{
-  if (dim == 2)
-    {
-      transformation[0] = tensor[0][0];
-      transformation[1] = tensor[0][1];
-      transformation[2] = tensor[1][1];
-      Assert (std::fabs(tensor[1][0]-tensor[0][1])<1e-15,
-             ExcInternalError());
-    }
-  else if (dim == 3)
-    {
-      transformation[0] = tensor[0][0];
-      transformation[1] = tensor[0][1];
-      transformation[2] = tensor[0][2];
-      transformation[3] = tensor[1][1];
-      transformation[4] = tensor[1][2];
-      transformation[5] = tensor[2][2];
-      Assert (std::fabs(tensor[1][0]-tensor[0][1])<1e-15,
-             ExcInternalError());
-      Assert (std::fabs(tensor[2][0]-tensor[0][2])<1e-15,
-             ExcInternalError());
-      Assert (std::fabs(tensor[2][1]-tensor[1][2])<1e-15,
-             ExcInternalError());
-    }
-  else
-    ExcNotImplemented();
-  return *this;
-}
-
-
-
-                                // @sect3{LaplaceProblem class}
-
-                                // This class is based on the same
-                                // class in step-16. However, we
-                                // replaced the SparseMatrix<double>
-                                // class by our matrix-free
-                                // implementation, which means that
-                                // we can also skip the sparsity
-                                // patterns.
-template <int dim>
-class LaplaceProblem
-{
-  public:
-    LaplaceProblem (const unsigned int degree);
-    void run ();
-
-  private:
-    void setup_system ();
-    void assemble_system ();
-    void assemble_multigrid ();
-    void solve ();
-    void output_results (const unsigned int cycle) const;
-
-    Triangulation<dim>   triangulation;
-    FE_Q<dim>            fe;
-    MGDoFHandler<dim>    mg_dof_handler;
-
-    MatrixFree<double,LaplaceOperator<dim,double> > system_matrix;
-    typedef MatrixFree<float,LaplaceOperator<dim,float> > MatrixFreeType;
-    MGLevelObject<MatrixFreeType> mg_matrices;
-    FullMatrix<float>             coarse_matrix;
-
-    Vector<double>       solution;
-    Vector<double>       system_rhs;
-};
-
-
-
-template <int dim>
-LaplaceProblem<dim>::LaplaceProblem (const unsigned int degree)
-               :
-                fe (degree),
-               mg_dof_handler (triangulation)
-{}
-
-
-
-                                // @sect4{LaplaceProblem::setup_system}
-
-                                // This is the function of step-16 with
-                                // relevant changes due to the MatrixFree
-                                // class. What we need to do is to somehow
-                                // create a local gradient matrix that does
-                                // not contain any cell-related data
-                                // (gradient on the reference cell). The
-                                // way to get to this matrix is to create
-                                // an FEValues object with gradient
-                                // information on a cell that corresponds
-                                // to the reference cell, which is a cube
-                                // with side length 1. So we create a
-                                // pseudo triangulation, initialize the
-                                // FEValues to the only cell of that
-                                // triangulation, and read off the
-                                // gradients (which we put in a
-                                // FullMatrix). That full matrix is then
-                                // passed to the reinit function of the
-                                // MatrixFree class used as a system matrix
-                                // and, further down, as multigrid matrices
-                                // on the individual levels. We need to
-                                // implement Dirichlet boundary conditions
-                                // here, which is done with the
-                                // ConstraintMatrix function as shown,
-                                // e.g., in step-22.
-template <int dim>
-void LaplaceProblem<dim>::setup_system ()
-{
-  system_matrix.clear();
-  mg_matrices.clear();
-
-  mg_dof_handler.distribute_dofs (fe);
-
-  std::cout << "Number of degrees of freedom: "
-           << mg_dof_handler.n_dofs()
-           << std::endl;
-
-  const unsigned int nlevels = triangulation.n_levels();
-  mg_matrices.resize(0, nlevels-1);
-
-  QGauss<dim>  quadrature_formula(fe.degree+1);
-  FEValues<dim> fe_values_reference (fe, quadrature_formula,
-                                    update_gradients);
-  Triangulation<dim> reference_cell;
-  GridGenerator::hyper_cube (reference_cell, 0, 1);
-  fe_values_reference.reinit (reference_cell.begin());
-  FullMatrix<double> ref_cell_gradients (fe.dofs_per_cell,
-                                        quadrature_formula.size()*dim);
-  for (unsigned int i=0; i<fe.dofs_per_cell; ++i)
-    {
-      for (unsigned int j=0; j<quadrature_formula.size(); ++j)
-       {
-         for (unsigned int d=0; d<dim; ++d)
-           ref_cell_gradients(i,j*dim+d) = fe_values_reference.shape_grad(i,j)[d];
-       }
-    }
-  system_matrix.reinit (mg_dof_handler.n_dofs(), triangulation.n_active_cells(),
-                       ref_cell_gradients, quadrature_formula.size());
-  VectorTools::interpolate_boundary_values (mg_dof_handler,
-                                           0,
-                                           ZeroFunction<dim>(),
-                                           system_matrix.get_constraints());
-  system_matrix.get_constraints().close();
-  std::cout.precision(4);
-  std::cout << "System matrix memory consumption: "
-           << system_matrix.memory_consumption()/double(1<<20)
-           << " MiB."
-           << std::endl;
-
-  solution.reinit (mg_dof_handler.n_dofs());
-  system_rhs.reinit (mg_dof_handler.n_dofs());
-
-                                  // Next, initialize the matrices for the
-                                  // multigrid method on all the
-                                  // levels. Unfortunately, the function
-                                  // MGTools::make_boundary_list cannot write
-                                  // Dirichlet boundary conditions into a
-                                  // ConstraintMatrix object directly, so we
-                                  // first have to make the boundary list and
-                                  // then manually fill the boundary
-                                  // conditions using the command
-                                  // ConstraintMatrix::add_line. Once this is
-                                  // done, we close the ConstraintMatrix so
-                                  // it can be used for matrix-vector
-                                  // products.
-  typename FunctionMap<dim>::type dirichlet_boundary;
-  ZeroFunction<dim>               homogeneous_dirichlet_bc (1);
-  dirichlet_boundary[0] = &homogeneous_dirichlet_bc;
-  std::vector<std::set<unsigned int> > boundary_indices(triangulation.n_levels());
-  MGTools::make_boundary_list (mg_dof_handler,
-                              dirichlet_boundary,
-                              boundary_indices);
-  for (unsigned int level=0;level<nlevels;++level)
-    {
-      mg_matrices[level].reinit(mg_dof_handler.n_dofs(level),
-                               triangulation.n_cells(level),
-                               ref_cell_gradients,
-                               quadrature_formula.size());
-      std::set<unsigned int>::iterator bc_it = boundary_indices[level].begin();
-      for ( ; bc_it != boundary_indices[level].end(); ++bc_it)
-       mg_matrices[level].get_constraints().add_line(*bc_it);
-      mg_matrices[level].get_constraints().close();
-    }
-  coarse_matrix.reinit (mg_dof_handler.n_dofs(0),
-                       mg_dof_handler.n_dofs(0));
-}
-
-
-
-                                // @sect4{LaplaceProblem::assemble_system}
-
-                                // The assemble function is significantly
-                                // reduced compared to step-16. All we need
-                                // to do is to assemble the right hand side
-                                // and to calculate the cell-dependent part
-                                // of the Laplace operator. The first task is
-                                // standard. The second is also not too hard
-                                // given the discussion in the introduction:
-                                // We need to take the inverse of the
-                                // Jacobian of the transformation from unit
-                                // to real cell, multiply it with its
-                                // transpose and multiply the resulting
-                                // rank-2 tensor with the quadrature weights
-                                // and the coefficient values at the
-                                // quadrature points. To make this work, we
-                                // add the update flag @p
-                                // update_inverse_jacobians to the FEValues
-                                // constructor, and query the inverse of the
-                                // Jacobian in a loop over the quadrature
-                                // points (note that the Jacobian is not
-                                // related to any kind of degrees of freedom
-                                // directly). In the end, we condense the
-                                // constraints from Dirichlet boundary
-                                // conditions away from the right hand side.
-template <int dim>
-void LaplaceProblem<dim>::assemble_system ()
-{
-  QGauss<dim>  quadrature_formula(fe.degree+1);
-  MappingQ<dim> mapping (fe.degree);
-  FEValues<dim> fe_values (mapping, fe, quadrature_formula,
-                          update_values   | update_inverse_jacobians |
-                           update_quadrature_points | update_JxW_values);
-
-  const unsigned int   dofs_per_cell = fe.dofs_per_cell;
-  const unsigned int   n_q_points    = quadrature_formula.size();
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-  const Coefficient<dim> coefficient;
-  std::vector<double>    coefficient_values (n_q_points);
-
-  unsigned int cell_no = 0;
-
-  typename DoFHandler<dim>::active_cell_iterator cell = mg_dof_handler.begin_active(),
-                                                endc = mg_dof_handler.end();
-  for (; cell!=endc; ++cell, ++cell_no)
-    {
-      cell->get_dof_indices (local_dof_indices);
-      fe_values.reinit (cell);
-      coefficient.value_list (fe_values.get_quadrature_points(),
-                             coefficient_values);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       {
-         double rhs_val = 0;
-         for (unsigned int q=0; q<n_q_points; ++q)
-           rhs_val += (fe_values.shape_value(i,q) * 1.0 *
-                       fe_values.JxW(q));
-         system_rhs(local_dof_indices[i]) += rhs_val;
-       }
-
-      system_matrix.set_local_dof_indices (cell_no, local_dof_indices);
-      for (unsigned int q=0; q<n_q_points; ++q)
-       system_matrix.set_derivative_data (cell_no, q,
-                                          (transpose
-                                           (fe_values.inverse_jacobian(q)) *
-                                           fe_values.inverse_jacobian(q)) *
-                                          fe_values.JxW(q) *
-                                          coefficient_values[q]);
-    }
-  system_matrix.get_constraints().condense(system_rhs);
-}
-
-
-                                // @sect4{LaplaceProblem::assemble_multigrid}
-
-                                // Here is another assemble
-                                // function. The integration core is
-                                // the same as above. Only the loop
-                                // goes over all existing cells now
-                                // and the results must be entered
-                                // into the correct matrix.
-
-                                // Since we only do multilevel
-                                // preconditioning, no right-hand side is
-                                // assembled here. Compared to step-16, there
-                                // is one new thing here: we manually
-                                // calculate the matrix on the coarsest
-                                // level. In step-16, we could simply copy
-                                // the entries from the respective sparse
-                                // matrix, but this is obviously not possible
-                                // here. We could have integrated this into the
-                                // MatrixFree class as well, but it is simple
-                                // enough, so calculate it here instead.
-template <int dim>
-void LaplaceProblem<dim>::assemble_multigrid ()
-{
-  coarse_matrix = 0;
-  QGauss<dim>  quadrature_formula(fe.degree+1);
-  MappingQ<dim> mapping (fe.degree);
-  FEValues<dim> fe_values (mapping, fe, quadrature_formula,
-                          update_gradients  | update_inverse_jacobians |
-                           update_quadrature_points | update_JxW_values);
-
-  const unsigned int   dofs_per_cell = fe.dofs_per_cell;
-  const unsigned int   n_q_points    = quadrature_formula.size();
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-  const Coefficient<dim> coefficient;
-  std::vector<double>    coefficient_values (n_q_points);
-
-  std::vector<unsigned int> cell_no(triangulation.n_levels());
-  typename MGDoFHandler<dim>::cell_iterator cell = mg_dof_handler.begin(),
-                                           endc = mg_dof_handler.end();
-  for (; cell!=endc; ++cell)
-    {
-      const unsigned int level = cell->level();
-      cell->get_mg_dof_indices (local_dof_indices);
-      fe_values.reinit (cell);
-      coefficient.value_list (fe_values.get_quadrature_points(),
-                             coefficient_values);
-
-      mg_matrices[level].set_local_dof_indices (cell_no[level],
-                                               local_dof_indices);
-      for (unsigned int q=0; q<n_q_points; ++q)
-       mg_matrices[level].set_derivative_data (cell_no[level], q,
-                                               (transpose
-                                                (fe_values.inverse_jacobian(q)) *
-                                                fe_values.inverse_jacobian(q)) *
-                                               fe_values.JxW(q) *
-                                               coefficient_values[q]);
-
-      ++cell_no[level];
-      if (level == 0)
-       {
-         for (unsigned int i=0; i<dofs_per_cell; ++i)
-           for (unsigned int j=0; j<dofs_per_cell; ++j)
-             {
-               double add_value = 0;
-               for (unsigned int q=0; q<n_q_points; ++q)
-                 add_value += (fe_values.shape_grad(i,q) *
-                               fe_values.shape_grad(j,q) *
-                               coefficient_values[q] *
-                               fe_values.JxW(q));
-               coarse_matrix(local_dof_indices[i],
-                             local_dof_indices[j]) += add_value;
-             }
-       }
-    }
-
-                                  // In a final step, we need to
-                                  // condense the boundary conditions
-                                  // on the coarse matrix. There is
-                                  // no built-in function for doing
-                                  // this on a full matrix, so
-                                  // manually delete the rows and
-                                  // columns of the matrix that are
-                                  // constrained.
-  for (unsigned int i=0; i<coarse_matrix.m(); ++i)
-    if (mg_matrices[0].get_constraints().is_constrained(i))
-      for (unsigned int j=0; j<coarse_matrix.n(); ++j)
-       if (i!=j)
-         {
-           coarse_matrix(i,j) = 0;
-           coarse_matrix(j,i) = 0;
-         }
-}
-
-
-
-                                // @sect4{LaplaceProblem::solve}
-
-                                // The solution process again looks like
-                                // step-16. We now use a Chebyshev smoother
-                                // instead of SOR (SOR would be very
-                                // difficult to implement because we do not
-                                // have the matrix elements available
-                                // explicitly, and it is difficult to make it
-                                // work efficiently in %parallel). The
-                                // multigrid classes provide a simple
-                                // interface for using the Chebyshev smoother
-                                // which is defined in a preconditioner
-                                // class: MGSmootherPrecondition.
-template <int dim>
-void LaplaceProblem<dim>::solve ()
-{
-  GrowingVectorMemory<>   vector_memory;
-
-  MGTransferPrebuilt<Vector<double> > mg_transfer;
-  mg_transfer.build_matrices(mg_dof_handler);
-
-  MGCoarseGridHouseholder<float, Vector<double> > mg_coarse;
-  mg_coarse.initialize(coarse_matrix);
-
-  typedef PreconditionChebyshev<MatrixFreeType,Vector<double> > SMOOTHER;
-  MGSmootherPrecondition<MatrixFreeType, SMOOTHER, Vector<double> >
-    mg_smoother(vector_memory);
-
-                                  // Then, we initialize the smoother
-                                  // with our level matrices and the
-                                  // required, additional data for
-                                  // the Chebyshev smoother. In
-                                  // particular, we use a higher
-                                  // polynomial degree for higher
-                                  // order elements, since smoothing
-                                  // gets more difficult for
-                                  // these. Smooth out a range of
-                                  // $[\lambda_{\max}/10,\lambda_{\max}]$. In
-                                  // order to compute the maximum
-                                  // eigenvalue of the corresponding
-                                  // matrix, the Chebyshev
-                                  // initializations performs a few
-                                  // steps of a CG algorithm. Since
-                                  // all we need is a rough estimate,
-                                  // we choose some eight iterations
-                                  // (more if the finite element
-                                  // polynomial degree is larger,
-                                  // less if it is smaller than
-                                  // quadratic).
-  typename SMOOTHER::AdditionalData smoother_data;
-  smoother_data.smoothing_range = 10.;
-  smoother_data.degree = fe.degree;
-  smoother_data.eig_cg_n_iterations = 4+2*fe.degree;
-  mg_smoother.initialize(mg_matrices, smoother_data);
-
-  MGMatrix<MatrixFreeType, Vector<double> >
-    mg_matrix(&mg_matrices);
-
-  Multigrid<Vector<double> > mg(mg_dof_handler,
-                               mg_matrix,
-                               mg_coarse,
-                               mg_transfer,
-                               mg_smoother,
-                               mg_smoother);
-  PreconditionMG<dim, Vector<double>,
-    MGTransferPrebuilt<Vector<double> > >
-  preconditioner(mg_dof_handler, mg, mg_transfer);
-
-                                // Finally, write out the memory
-                                // consumption of the Multigrid object
-                                // (or rather, of its most significant
-                                // components, since there is no built-in
-                                // function for the total multigrid
-                                // object), then create the solver object
-                                // and solve the system. This is very
-                                // easy, and we didn't even see any
-                                // difference in the solve process
-                                // compared to step-16. The magic is all
-                                // hidden behind the implementation of
-                                // the MatrixFree::vmult operation.
-  const unsigned int multigrid_memory
-    = (mg_matrices.memory_consumption() +
-       mg_transfer.memory_consumption() +
-       coarse_matrix.memory_consumption());
-  std::cout << "Multigrid objects memory consumption: "
-           << multigrid_memory/double(1<<20)
-           << " MiB."
-           << std::endl;
-
-  SolverControl           solver_control (1000, 1e-12);
-  SolverCG<>              cg (solver_control);
-
-  cg.solve (system_matrix, solution, system_rhs,
-           preconditioner);
-
-  std::cout << "Convergence in " << solver_control.last_step()
-           << " CG iterations." << std::endl;
-}
-
-
-
-                                // @sect4{LaplaceProblem::output_results}
-
-                                // Here is the data output, which is a
-                                // simplified version of step-5. We use the
-                                // standard VTK output for each grid
-                                // produced in the refinement process.
-template <int dim>
-void LaplaceProblem<dim>::output_results (const unsigned int cycle) const
-{
-  DataOut<dim> data_out;
-
-  data_out.attach_dof_handler (mg_dof_handler);
-  data_out.add_data_vector (solution, "solution");
-  data_out.build_patches ();
-
-  std::ostringstream filename;
-  filename << "solution-"
-          << cycle
-          << ".vtk";
-
-  std::ofstream output (filename.str().c_str());
-  data_out.write_vtk (output);
-}
-
-
-
-                                // @sect4{LaplaceProblem::run}
-
-                                // The function that runs the program is
-                                // very similar to the one in step-16. We
-                                // make less refinement steps in 3D
-                                // compared to 2D, but that's it.
-template <int dim>
-void LaplaceProblem<dim>::run ()
-{
-  for (unsigned int cycle=0; cycle<8-dim; ++cycle)
-    {
-      std::cout << "Cycle " << cycle << std::endl;
-
-      if (cycle == 0)
-       {
-         GridGenerator::hyper_ball(triangulation);
-         static const HyperBallBoundary<dim> boundary;
-         triangulation.set_boundary (0, boundary);
-         triangulation.refine_global (3-dim);
-       }
-      triangulation.refine_global (1);
-      setup_system ();
-      assemble_system ();
-      assemble_multigrid ();
-      solve ();
-      output_results (cycle);
-      std::cout << std::endl;
-    };
-}
-
-
-
-                                // @sect3{The <code>main</code> function}
-
-                                // This is as in all other programs:
-int main ()
-{
-  try
-    {
-      deallog.depth_console (0);
-      LaplaceProblem<2> laplace_problem (2);
-      laplace_problem.run ();
-    }
-  catch (std::exception &exc)
-    {
-      std::cerr << std::endl << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      std::cerr << "Exception on processing: " << std::endl
-               << exc.what() << std::endl
-               << "Aborting!" << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      return 1;
-    }
-  catch (...)
-    {
-      std::cerr << std::endl << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      std::cerr << "Unknown exception!" << std::endl
-               << "Aborting!" << std::endl
-               << "----------------------------------------------------"
-               << std::endl;
-      return 1;
-    }
-
-  return 0;
-}
diff --git a/deal.II/examples/step-42/Makefile b/deal.II/examples/step-42/Makefile
deleted file mode 100644 (file)
index a443603..0000000
+++ /dev/null
@@ -1,156 +0,0 @@
-# $Id: Makefile,v 1.4 2006/02/10 17:53:05 wolf Exp $
-
-
-# For the small projects Makefile, you basically need to fill in only
-# four fields.
-#
-# The first is the name of the application. It is assumed that the
-# application name is the same as the base file name of the single C++
-# file from which the application is generated.
-target = $(basename $(shell echo step-*.cc))
-
-# The second field determines whether you want to run your program in
-# debug or optimized mode. The latter is significantly faster, but no
-# run-time checking of parameters and internal states is performed, so
-# you should set this value to `on' while you develop your program,
-# and to `off' when running production computations.
-debug-mode = on
-
-
-# As third field, we need to give the path to the top-level deal.II
-# directory. You need to adjust this to your needs. Since this path is
-# probably the most often needed one in the Makefile internals, it is
-# designated by a single-character variable, since that can be
-# reference using $D only, i.e. without the parentheses that are
-# required for most other parameters, as e.g. in $(target).
-D = ../../
-
-
-# The last field specifies the names of data and other files that
-# shall be deleted when calling `make clean'. Object and backup files,
-# executables and the like are removed anyway. Here, we give a list of
-# files in the various output formats that deal.II supports.
-clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk
-
-
-
-
-#
-#
-# Usually, you will not need to change anything beyond this point.
-#
-#
-# The next statement tell the `make' program where to find the
-# deal.II top level directory and to include the file with the global
-# settings
-include $D/common/Make.global_options
-
-
-# Since the whole project consists of only one file, we need not
-# consider difficult dependencies. We only have to declare the
-# libraries which we want to link to the object file, and there need
-# to be two sets of libraries: one for the debug mode version of the
-# application and one for the optimized mode. Here we have selected
-# the versions for 2d. Note that the order in which the libraries are
-# given here is important and that your applications won't link
-# properly if they are given in another order.
-#
-# You may need to augment the lists of libraries when compiling your
-# program for other dimensions, or when using third party libraries
-libs.g   = $(lib-deal2-2d.g) \
-          $(lib-deal2-3d.g) \
-          $(lib-lac.g)      \
-           $(lib-base.g)
-libs.o   = $(lib-deal2-2d.o) \
-          $(lib-deal2-3d.o) \
-          $(lib-lac.o)      \
-           $(lib-base.o)
-
-
-# We now use the variable defined above which switch between debug and
-# optimized mode to select the set of libraries to link with. Included
-# in the list of libraries is the name of the object file which we
-# will produce from the single C++ file. Note that by default we use
-# the extension .g.o for object files compiled in debug mode and .o for
-# object files in optimized mode (or whatever the local default on your
-# system is instead of .o).
-ifeq ($(debug-mode),on)
-  libraries = $(target).g.$(OBJEXT) $(libs.g)
-else
-  libraries = $(target).$(OBJEXT) $(libs.o)
-endif
-
-
-# Now comes the first production rule: how to link the single object
-# file produced from the single C++ file into the executable. Since
-# this is the first rule in the Makefile, it is the one `make' selects
-# if you call it without arguments.
-$(target) : $(libraries)
-       @echo ============================ Linking $@
-       @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS)
-
-
-# To make running the application somewhat independent of the actual
-# program name, we usually declare a rule `run' which simply runs the
-# program. You can then run it by typing `make run'. This is also
-# useful if you want to call the executable with arguments which do
-# not change frequently. You may then want to add them to the
-# following rule:
-run: $(target)
-       @echo ============================ Running $<
-       @./$(target)$(EXEEXT)
-
-
-# As a last rule to the `make' program, we define what to do when
-# cleaning up a directory. This usually involves deleting object files
-# and other automatically created files such as the executable itself,
-# backup files, and data files. Since the latter are not usually quite
-# diverse, you needed to declare them at the top of this file.
-clean:
-       -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files)
-
-
-# Since we have not yet stated how to make an object file from a C++
-# file, we should do so now. Since the many flags passed to the
-# compiler are usually not of much interest, we suppress the actual
-# command line using the `at' sign in the first column of the rules
-# and write the string indicating what we do instead.
-./%.g.$(OBJEXT) :
-       @echo ==============debug========= $(<F)
-       @$(CXX) $(CXXFLAGS.g) -c $< -o $@
-./%.$(OBJEXT) :
-       @echo ==============optimized===== $(<F)
-       @$(CXX) $(CXXFLAGS.o) -c $< -o $@
-
-
-# The following statement tells make that the rules `run' and `clean'
-# are not expected to produce files of the same name as Makefile rules
-# usually do.
-.PHONY: run clean
-
-
-# Finally there is a rule which you normally need not care much about:
-# since the executable depends on some include files from the library,
-# besides the C++ application file of course, it is necessary to
-# re-generate the executable when one of the files it depends on has
-# changed. The following rule to created a dependency file
-# `Makefile.dep', which `make' uses to determine when to regenerate
-# the executable. This file is automagically remade whenever needed,
-# i.e. whenever one of the cc-/h-files changed. Make detects whether
-# to remake this file upon inclusion at the bottom of this file.
-#
-# If the creation of Makefile.dep fails, blow it away and fail
-Makefile.dep: $(target).cc Makefile \
-              $(shell echo $D/*/include/*/*.h)
-       @echo ============================ Remaking $@
-       @$D/common/scripts/make_dependencies  $(INCLUDE) -B. $(target).cc \
-               > $@ \
-         || (rm -f $@ ; false)
-       @if test -s $@ ; then : else rm $@ ; fi
-
-
-# To make the dependencies known to `make', we finally have to include
-# them:
-include Makefile.dep
-
-
diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox
deleted file mode 100644 (file)
index c59cfdf..0000000
+++ /dev/null
@@ -1,2 +0,0 @@
-<a name="Intro"></a>
-<h1>Introduction</h1>
diff --git a/deal.II/examples/step-42/doc/results.dox b/deal.II/examples/step-42/doc/results.dox
deleted file mode 100644 (file)
index 747bdbb..0000000
+++ /dev/null
@@ -1,3 +0,0 @@
-<a name="Results"></a>
-<h1>Results</h1>
-
diff --git a/deal.II/examples/step-42/step-42.cc b/deal.II/examples/step-42/step-42.cc
deleted file mode 100644 (file)
index d5bcacb..0000000
+++ /dev/null
@@ -1,1187 +0,0 @@
-/* $Id$ */
-/* Author: Wolfgang Bangerth, Texas A&M University, 2008 */
-
-/*    $Id$       */
-/*                                                                */
-/*    Copyright (C) 2008, 2009, 2010 by the deal.II authors */
-/*                                                                */
-/*    This file is subject to QPL and may not be  distributed     */
-/*    without copyright and license information. Please refer     */
-/*    to the file deal.II/doc/license.html for the  text  and     */
-/*    further information on this license.                        */
-
-
-                                // @sect3{Include files}
-
-                                // As usual, we start by including
-                                // some well-known files:
-#include <base/quadrature_lib.h>
-#include <base/logstream.h>
-#include <base/function.h>
-#include <base/utilities.h>
-
-#include <lac/block_vector.h>
-#include <lac/full_matrix.h>
-#include <lac/block_sparse_matrix.h>
-#include <lac/solver_cg.h>
-#include <lac/solver_gmres.h>
-#include <lac/precondition.h>
-#include <lac/constraint_matrix.h>
-
-#include <grid/tria.h>
-#include <grid/grid_generator.h>
-#include <grid/tria_accessor.h>
-#include <grid/tria_iterator.h>
-#include <grid/tria_boundary_lib.h>
-#include <grid/grid_tools.h>
-#include <grid/grid_out.h>
-#include <grid/grid_refinement.h>
-
-#include <dofs/dof_handler.h>
-#include <dofs/dof_renumbering.h>
-#include <dofs/dof_accessor.h>
-#include <dofs/dof_tools.h>
-
-#include <fe/fe_q.h>
-#include <fe/fe_system.h>
-#include <fe/fe_values.h>
-#include <fe/mapping_q1.h>
-
-#include <numerics/vectors.h>
-#include <numerics/matrices.h>
-#include <numerics/data_out.h>
-#include <numerics/error_estimator.h>
-
-
-#include <lac/sparse_direct.h>
-
-#include <lac/sparse_ilu.h>
-
-#include <multigrid/multigrid.h>
-#include <multigrid/mg_dof_handler.h>
-#include <multigrid/mg_dof_accessor.h>
-#include <multigrid/mg_transfer.h>
-#include <multigrid/mg_tools.h>
-#include <multigrid/mg_coarse.h>
-#include <multigrid/mg_smoother.h>
-#include <multigrid/mg_matrix.h>
-
-#include <fstream>
-#include <sstream>
-
-
-using namespace dealii;
-
-
-template <int dim>
-struct InnerPreconditioner;
-
-
-template <>
-struct InnerPreconditioner<2>
-{
-    typedef SparseDirectUMFPACK type;
-};
-
-
-template <>
-struct InnerPreconditioner<3>
-{
-    typedef SparseILU<double> type;
-};
-
-template <typename MATRIX>
-void copy(const MATRIX &matrix,
-    FullMatrix<double> &full_matrix)
-{
-  const unsigned int m = matrix.m();
-  const unsigned int n = matrix.n();
-  full_matrix.reinit(n,m);
-
-  Vector<double> unit (n);
-  Vector<double> result (m);
-  for(unsigned int i=0; i<n; ++i)
-  {
-    unit(i) = 1;
-    for(unsigned int j=0; j<m; ++j)
-    {
-      matrix.vmult(result,unit);
-      full_matrix(i,j) = result(j);
-    }
-    unit(i) = 0;
-  }
-}
-
-template <int dim>
-class StokesProblem
-{
-  public:
-    StokesProblem (const unsigned int degree);
-    void run ();
-
-  private:
-    void setup_dofs ();
-    void assemble_system ();
-    void assemble_multigrid ();
-    void solve ();
-    void solve_block ();
-
-    void find_dofs_on_lower_level (std::vector<std::vector<bool> > &lower_dofs,
-        std::vector<std::vector<bool> > &boundary_dofs);
-
-    void output_results (const unsigned int refinement_cycle) const;
-    void refine_mesh ();
-
-    const unsigned int   degree;
-
-    Triangulation<dim>   triangulation;
-    FESystem<dim>        fe;
-    MGDoFHandler<dim>    dof_handler;
-
-    ConstraintMatrix     constraints;
-
-    BlockSparsityPattern      sparsity_pattern;
-    BlockSparseMatrix<double> system_matrix;
-
-    BlockVector<double> solution;
-    BlockVector<double> system_rhs;
-
-    MGLevelObject<ConstraintMatrix>           mg_constraints;
-    MGLevelObject<BlockSparsityPattern>       mg_sparsity;
-    MGLevelObject<BlockSparseMatrix<double> > mg_matrices;
-
-    MGLevelObject<BlockSparseMatrix<double> > mg_interface_matrices;
-    std::vector<std::vector<unsigned int> >   mg_dofs_per_component;
-
-    std::vector<std_cxx1x::shared_ptr<typename InnerPreconditioner<dim>::type> > mg_A_preconditioner;
-    std_cxx1x::shared_ptr<typename InnerPreconditioner<dim>::type> A_preconditioner;
-};
-
-
-template <int dim>
-class BoundaryValues : public Function<dim>
-{
-  public:
-    BoundaryValues () : Function<dim>(dim+1) {}
-
-    virtual double value (const Point<dim>   &p,
-                          const unsigned int  component = 0) const;
-
-    virtual void vector_value (const Point<dim> &p,
-                               Vector<double>   &value) const;
-};
-
-
-template <int dim>
-double
-BoundaryValues<dim>::value (const Point<dim>  &p,
-                           const unsigned int component) const
-{
-  Assert (component < this->n_components,
-         ExcIndexRange (component, 0, this->n_components));
-
-   if (component == 0 && p[0] == 0)
-    return (dim == 2 ? - p[1]*(p[1]-1.) : p[1]*(p[1]-1.) * p[2]*(p[2]-1.));
-  return 0;
-}
-
-
-template <int dim>
-void
-BoundaryValues<dim>::vector_value (const Point<dim> &p,
-                                  Vector<double>   &values) const
-{
-  for (unsigned int c=0; c<this->n_components; ++c)
-    values(c) = BoundaryValues<dim>::value (p, c);
-}
-
-
-
-
-template <int dim>
-class RightHandSide : public Function<dim>
-{
-  public:
-    RightHandSide () : Function<dim>(dim+1) {}
-
-    virtual double value (const Point<dim>   &p,
-                          const unsigned int  component = 0) const;
-
-    virtual void vector_value (const Point<dim> &p,
-                               Vector<double>   &value) const;
-
-};
-
-
-template <int dim>
-double
-RightHandSide<dim>::value (const Point<dim>  &/*p*/,
-                           const unsigned int component) const
-{
-  return (component == 1 ? 1 : 0);
-}
-
-
-template <int dim>
-void
-RightHandSide<dim>::vector_value (const Point<dim> &p,
-                                  Vector<double>   &values) const
-{
-  for (unsigned int c=0; c<this->n_components; ++c)
-    values(c) = RightHandSide<dim>::value (p, c);
-}
-
-
-
-
-template <class Matrix, class Preconditioner>
-class InverseMatrix : public Subscriptor
-{
-  public:
-    InverseMatrix (const Matrix         &m,
-                   const Preconditioner &preconditioner);
-
-    void vmult (Vector<double>       &dst,
-                const Vector<double> &src) const;
-
-    mutable std::string name;
-  private:
-    const SmartPointer<const Matrix> matrix;
-    const SmartPointer<const Preconditioner> preconditioner;
-};
-
-
-template <class Matrix, class Preconditioner>
-InverseMatrix<Matrix,Preconditioner>::InverseMatrix (const Matrix &m,
-                                                    const Preconditioner &preconditioner)
-               :
-               matrix (&m),
-               preconditioner (&preconditioner)
-{}
-
-
-template <class Matrix, class Preconditioner>
-void InverseMatrix<Matrix,Preconditioner>::vmult (Vector<double>       &dst,
-                                                 const Vector<double> &src) const
-{
-  SolverControl solver_control (src.size(), 1.0e-12*src.l2_norm());
-  SolverCG<>    cg (solver_control);
-
-  dst = 0;
-
-  try
-    {
-      cg.solve (*matrix, dst, src, *preconditioner);
-    }
-  catch (...)
-    {
-      std::cout << "Failure in " << __PRETTY_FUNCTION__ << std::endl;
-      abort ();
-    }
-
-#ifdef STEP_42_TEST
-  if (name == "in schur")
-    std::cout << "      " << solver_control.last_step()
-             << " inner CG steps inside the Schur complement ";
-  else if (name == "top left")
-    std::cout << "    " << solver_control.last_step()
-             << " CG steps on the top left block ";
-  else if (name == "rhs")
-    std::cout << "    " << solver_control.last_step()
-             << " CG steps for computing the r.h.s. ";
-  else
-    abort ();
-
-  std::cout << solver_control.initial_value() << "->" << solver_control.last_value()
-           << std::endl;
-#endif
-}
-
-
-template <class PreconditionerA, class PreconditionerMp>
-class BlockSchurPreconditioner : public Subscriptor
-{
-  public:
-    BlockSchurPreconditioner (const BlockSparseMatrix<double>         &S,
-          const InverseMatrix<SparseMatrix<double>,PreconditionerMp>  &Mpinv,
-          const PreconditionerA &Apreconditioner);
-
-  void vmult (BlockVector<double>       &dst,
-              const BlockVector<double> &src) const;
-
-  private:
-    const SmartPointer<const BlockSparseMatrix<double> > system_matrix;
-    const SmartPointer<const InverseMatrix<SparseMatrix<double>, 
-                       PreconditionerMp > > m_inverse;
-    const PreconditionerA &a_preconditioner;
-    
-    mutable Vector<double> tmp;
-
-};
-
-template <class PreconditionerA, class PreconditionerMp>
-BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::BlockSchurPreconditioner(
-          const BlockSparseMatrix<double>                            &S,
-          const InverseMatrix<SparseMatrix<double>,PreconditionerMp> &Mpinv,
-          const PreconditionerA &Apreconditioner
-          )
-                :
-                system_matrix           (&S),
-                m_inverse               (&Mpinv),
-                a_preconditioner        (Apreconditioner),
-                tmp                     (S.block(1,1).m())
-{}
-
-        // Now the interesting function, the multiplication of
-        // the preconditioner with a BlockVector. 
-template <class PreconditionerA, class PreconditionerMp>
-void BlockSchurPreconditioner<PreconditionerA, PreconditionerMp>::vmult (
-                                     BlockVector<double>       &dst,
-                                     const BlockVector<double> &src) const
-{
-        // Form u_new = A^{-1} u
-  a_preconditioner.vmult (dst.block(0), src.block(0));
-        // Form tmp = - B u_new + p 
-        // (<code>SparseMatrix::residual</code>
-        // does precisely this)
-  system_matrix->block(1,0).residual(tmp, dst.block(0), src.block(1));
-        // Change sign in tmp
-  tmp *= -1;
-        // Multiply by approximate Schur complement 
-        // (i.e. a pressure mass matrix)
-  m_inverse->vmult (dst.block(1), tmp);
-}
-
-template <class Preconditioner>
-class SchurComplement : public Subscriptor
-{
-  public:
-    SchurComplement (const BlockSparseMatrix<double> &system_matrix,
-                    const InverseMatrix<SparseMatrix<double>, Preconditioner> &A_inverse);
-
-    void vmult (Vector<double>       &dst,
-               const Vector<double> &src) const;
-
-    unsigned int m() const
-      {
-       return system_matrix->block(1,1).m();
-      }
-
-    unsigned int n() const
-      {
-       return system_matrix->block(1,1).n();
-      }
-
-  private:
-    const SmartPointer<const BlockSparseMatrix<double> > system_matrix;
-    const SmartPointer<const InverseMatrix<SparseMatrix<double>, Preconditioner> > A_inverse;
-
-    mutable Vector<double> tmp1, tmp2;
-};
-
-
-
-template <class Preconditioner>
-SchurComplement<Preconditioner>::
-SchurComplement (const BlockSparseMatrix<double> &system_matrix,
-                const InverseMatrix<SparseMatrix<double>,Preconditioner> &A_inverse)
-               :
-               system_matrix (&system_matrix),
-               A_inverse (&A_inverse),
-               tmp1 (system_matrix.block(0,0).m()),
-               tmp2 (system_matrix.block(0,0).m())
-{}
-
-
-template <class Preconditioner>
-void SchurComplement<Preconditioner>::vmult (Vector<double>       &dst,
-                                            const Vector<double> &src) const
-{
-  system_matrix->block(0,1).vmult (tmp1, src);
-  A_inverse->name = "in schur";
-  A_inverse->vmult (tmp2, tmp1);
-  system_matrix->block(1,0).vmult (dst, tmp2);
-  dst *= -1;
-  system_matrix->block(1,1).vmult_add (dst, src);
-  dst *= -1;
-}
-
-
-
-template <int dim>
-StokesProblem<dim>::StokesProblem (const unsigned int degree)
-                :
-                degree (degree),
-                triangulation (Triangulation<dim>::limit_level_difference_at_vertices),
-                fe (FE_Q<dim>(degree+1), dim,
-                    FE_Q<dim>(degree), 1),
-                dof_handler (triangulation)
-{}
-
-
-
-
-
-template <int dim>
-void StokesProblem<dim>::setup_dofs ()
-{
-  A_preconditioner.reset ();
-  mg_A_preconditioner.resize (0);
-  system_matrix.clear ();
-
-  dof_handler.distribute_dofs (fe);
-//  DoFRenumbering::Cuthill_McKee (dof_handler);
-
-  std::vector<unsigned int> block_component (dim+1,0);
-  block_component[dim] = 1;
-  DoFRenumbering::component_wise (dof_handler, block_component);
-
-
-  {
-    constraints.clear ();
-    std::vector<bool> component_mask (dim+1, true);
-    component_mask[dim] = false;
-    VectorTools::interpolate_boundary_values (dof_handler,
-                                             0,
-                                             ZeroFunction<dim>(dim+1), //TODO: go back to BoundaryValues
-                                             constraints,
-                                             component_mask);
-    DoFTools::make_hanging_node_constraints (dof_handler,
-                                            constraints);
-  }
-
-  constraints.close ();
-
-
-  std::vector<unsigned int> dofs_per_block (2);
-  DoFTools::count_dofs_per_block (dof_handler, dofs_per_block,
-                                 block_component);
-  const unsigned int n_u = dofs_per_block[0],
-                     n_p = dofs_per_block[1];
-
-  std::cout << "   Number of active cells: "
-            << triangulation.n_active_cells()
-            << std::endl
-            << "   Number of degrees of freedom: "
-            << dof_handler.n_dofs()
-            << " (" << n_u << '+' << n_p << ')'
-            << std::endl;
-
-  {
-    BlockCompressedSimpleSparsityPattern csp (2,2);
-
-    csp.block(0,0).reinit (n_u, n_u);
-    csp.block(1,0).reinit (n_p, n_u);
-    csp.block(0,1).reinit (n_u, n_p);
-    csp.block(1,1).reinit (n_p, n_p);
-
-    csp.collect_sizes();
-
-    DoFTools::make_sparsity_pattern (
-        static_cast<const DoFHandler<dim>&>(dof_handler),
-        csp, constraints, false);
-    sparsity_pattern.copy_from (csp);
-  }
-
-
-  system_matrix.reinit (sparsity_pattern);
-
-  solution.reinit (2);
-  solution.block(0).reinit (n_u);
-  solution.block(1).reinit (n_p);
-  solution.collect_sizes ();
-
-  system_rhs.reinit (2);
-  system_rhs.block(0).reinit (n_u);
-  system_rhs.block(1).reinit (n_p);
-  system_rhs.collect_sizes ();
-
-  //now setup stuff for mg
-  const unsigned int nlevels = triangulation.n_levels();
-
-  mg_matrices.resize(0, nlevels-1);
-  mg_matrices.clear ();
-  mg_interface_matrices.resize(0, nlevels-1);
-  mg_interface_matrices.clear ();
-  mg_sparsity.resize(0, nlevels-1);
-
-  mg_dofs_per_component.resize (nlevels);
-  for (unsigned int level=0; level<nlevels; ++level)
-    mg_dofs_per_component[level].resize (2);
-
-  MGTools::count_dofs_per_block (dof_handler, mg_dofs_per_component,
-                                block_component);
-  for (unsigned int level=0; level<nlevels; ++level)
-    std::cout << "                        Level " << level << ": "
-             << dof_handler.n_dofs (level) << " ("
-             << mg_dofs_per_component[level][0] << '+'
-             << mg_dofs_per_component[level][1] << ')'
-             << std::endl;
-
-  for (unsigned int level=0; level<nlevels; ++level)
-  {
-    DoFRenumbering::component_wise (dof_handler, level, block_component);
-
-    BlockCompressedSparsityPattern bcsp (mg_dofs_per_component[level],
-        mg_dofs_per_component[level]);
-    MGTools::make_sparsity_pattern(dof_handler, bcsp, level);
-    mg_sparsity[level].copy_from (bcsp);
-    mg_matrices[level].reinit (mg_sparsity[level]);
-    mg_interface_matrices[level].reinit (mg_sparsity[level]);
-  }
-}
-
-
-
-template <int dim>
-void StokesProblem<dim>::assemble_system ()
-{
-  system_matrix=0;
-  system_rhs=0;
-
-  QGauss<dim>   quadrature_formula(degree+2);
-
-  FEValues<dim> fe_values (fe, quadrature_formula,
-                           update_values    |
-                           update_quadrature_points  |
-                           update_JxW_values |
-                           update_gradients);
-
-  const unsigned int   dofs_per_cell   = fe.dofs_per_cell;
-
-  const unsigned int   n_q_points      = quadrature_formula.size();
-
-  FullMatrix<double>   local_matrix (dofs_per_cell, dofs_per_cell);
-  Vector<double>       local_rhs (dofs_per_cell);
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-
-  const RightHandSide<dim>          right_hand_side;
-  std::vector<Vector<double> >      rhs_values (n_q_points,
-                                                Vector<double>(dim+1));
-
-
-  const FEValuesExtractors::Vector velocities (0);
-  const FEValuesExtractors::Scalar pressure (dim);
-
-
-
-  std::vector<Tensor<2,dim> >          phi_grads_u (dofs_per_cell);
-  std::vector<double>                  div_phi_u   (dofs_per_cell);
-  std::vector<double>                  phi_p       (dofs_per_cell);
-
-  typename MGDoFHandler<dim>::active_cell_iterator
-    cell = dof_handler.begin_active(),
-    endc = dof_handler.end();
-  for (; cell!=endc; ++cell)
-    {
-      fe_values.reinit (cell);
-      local_matrix = 0;
-      local_rhs = 0;
-
-      right_hand_side.vector_value_list(fe_values.get_quadrature_points(),
-                                        rhs_values);
-
-      for (unsigned int q=0; q<n_q_points; ++q)
-       {
-         for (unsigned int k=0; k<dofs_per_cell; ++k)
-           {
-             phi_grads_u[k] = fe_values[velocities].gradient (k, q);
-             div_phi_u[k]   = fe_values[velocities].divergence (k, q);
-             phi_p[k]       = fe_values[pressure].value (k, q);
-           }
-
-         for (unsigned int i=0; i<dofs_per_cell; ++i)
-           {
-             for (unsigned int j=0; j<dofs_per_cell; ++j)
-               {
-                 local_matrix(i,j) += (scalar_product(phi_grads_u[i], phi_grads_u[j])
-                                       - div_phi_u[i] * phi_p[j]
-                                       - phi_p[i] * div_phi_u[j]
-                                        - phi_p[i] * phi_p[j]
-                                        )
-                                      * fe_values.JxW(q);
-               }
-
-             const unsigned int component_i =
-               fe.system_to_component_index(i).first;
-             local_rhs(i) += fe_values.shape_value(i,q) *
-                             rhs_values[q](component_i) *
-                             fe_values.JxW(q);
-           }
-       }
-
-
-
-
-      cell->get_dof_indices (local_dof_indices);
-      constraints.distribute_local_to_global (local_matrix, local_rhs,
-                                             local_dof_indices,
-                                             system_matrix, system_rhs);
-    }
-}
-
-
-template <int dim>
-void StokesProblem<dim>::assemble_multigrid ()
-{
-  QGauss<dim>   quadrature_formula(degree+2);
-  FEValues<dim> fe_values (fe, quadrature_formula,
-                           update_values    |
-                           update_quadrature_points  |
-                           update_JxW_values |
-                           update_gradients);
-
-  const unsigned int   dofs_per_cell   = fe.dofs_per_cell;
-  const unsigned int   n_q_points      = quadrature_formula.size();
-
-  FullMatrix<double>   local_matrix (dofs_per_cell, dofs_per_cell);
-
-  std::vector<unsigned int> local_dof_indices (dofs_per_cell);
-
-  const FEValuesExtractors::Vector velocities (0);
-  const FEValuesExtractors::Scalar pressure (dim);
-
-
-  std::vector<Tensor<2,dim> >          phi_grads_u (dofs_per_cell);
-  std::vector<double>                  div_phi_u   (dofs_per_cell);
-  std::vector<double>                  phi_p       (dofs_per_cell);
-
-  std::vector<std::vector<bool> > interface_dofs;
-  std::vector<std::vector<bool> > boundary_interface_dofs;
-  for (unsigned int level = 0; level<triangulation.n_levels(); ++level)
-    {
-      std::vector<bool> tmp (dof_handler.n_dofs(level));
-      interface_dofs.push_back (tmp);
-      boundary_interface_dofs.push_back (tmp);
-    }
-  MGTools::extract_inner_interface_dofs (dof_handler,
-                                        interface_dofs, boundary_interface_dofs);
-
-  typename FunctionMap<dim>::type      dirichlet_boundary;
-  BoundaryValues<dim>                  dirichlet_bc;
-  dirichlet_boundary[0] =             &dirichlet_bc;
-
-  std::vector<std::set<unsigned int> > boundary_indices(triangulation.n_levels());
-  std::vector<bool> component_mask (dim+1, true);
-  component_mask[dim] = false;
-  MGTools::make_boundary_list (dof_handler, dirichlet_boundary,
-                              boundary_indices, component_mask);
-
-  std::vector<ConstraintMatrix> boundary_constraints (triangulation.n_levels());
-  std::vector<ConstraintMatrix> boundary_interface_constraints (triangulation.n_levels());
-  for (unsigned int level=0; level<triangulation.n_levels(); ++level)
-    {
-      boundary_constraints[level].add_lines (interface_dofs[level]);
-      boundary_constraints[level].add_lines (boundary_indices[level]);
-      boundary_constraints[level].close ();
-
-      boundary_interface_constraints[level]
-       .add_lines (boundary_interface_dofs[level]);
-      boundary_interface_constraints[level].close ();
-    }
-
-  typename MGDoFHandler<dim>::cell_iterator
-    cell = dof_handler.begin(),
-    endc = dof_handler.end();
-  for (; cell!=endc; ++cell)
-    {
-                                      // Remember the level of the
-                                      // current cell.
-      const unsigned int level = cell->level();
-                                      // Compute the values specified
-                                      // by update flags above.
-      fe_values.reinit (cell);
-      local_matrix = 0;
-
-      for (unsigned int q=0; q<n_q_points; ++q)
-      {
-         for (unsigned int k=0; k<dofs_per_cell; ++k)
-           {
-             phi_grads_u[k] = fe_values[velocities].gradient (k, q);
-             div_phi_u[k]   = fe_values[velocities].divergence (k, q);
-             phi_p[k]       = fe_values[pressure].value (k, q);
-           }
-
-         for (unsigned int i=0; i<dofs_per_cell; ++i)
-             for (unsigned int j=0; j<dofs_per_cell; ++j)
-                 local_matrix(i,j) += (
-                      scalar_product(phi_grads_u[i], phi_grads_u[j])
-                                       - div_phi_u[i] * phi_p[j]
-                                       - phi_p[i] * div_phi_u[j]
-//                                        - phi_p[i] * phi_p[j]
-                                        )
-                                      * fe_values.JxW(q);
-      }
-
-      cell->get_mg_dof_indices (local_dof_indices);
-      boundary_constraints[level]
-       .distribute_local_to_global (local_matrix,
-                                    local_dof_indices,
-                                    mg_matrices[level]);
-
-      for (unsigned int i=0; i<dofs_per_cell; ++i)
-       for (unsigned int j=0; j<dofs_per_cell; ++j)
-         if( !(interface_dofs[level][local_dof_indices[i]]==true &&
-               interface_dofs[level][local_dof_indices[j]]==false))
-           local_matrix(i,j) = 0;
-
-      boundary_interface_constraints[level]
-       .distribute_local_to_global (local_matrix,
-                                    local_dof_indices,
-                                    mg_interface_matrices[level]);
-    }
-
-  mg_A_preconditioner.resize (triangulation.n_levels());
-  for (unsigned int level=0; level<triangulation.n_levels(); ++level)
-    {
-      mg_A_preconditioner[level]
-       = std_cxx1x::shared_ptr<typename InnerPreconditioner<dim>::type>(new typename InnerPreconditioner<dim>::type());
-      mg_A_preconditioner[level]
-       ->initialize (mg_matrices[level].block(0,0),
-                     typename InnerPreconditioner<dim>::type::AdditionalData());
-    }
-}
-
-
-template <typename InnerPreconditioner>
-class SchurComplementSmoother
-{
-  public:
-    struct AdditionalData
-    {
-       const InnerPreconditioner *A_preconditioner;
-    };
-
-    void initialize (const BlockSparseMatrix<double> &system_matrix,
-                    const AdditionalData            &data);
-
-    void vmult (BlockVector<double> &dst,
-               const BlockVector<double> &src) const;
-
-    void Tvmult (BlockVector<double> &dst,
-                const BlockVector<double> &src) const;
-
-    void clear ();
-
-  private:
-    SmartPointer<const BlockSparseMatrix<double> > system_matrix;
-    SmartPointer<const InnerPreconditioner>        A_preconditioner;
-};
-
-
-template <typename InnerPreconditioner>
-void
-SchurComplementSmoother<InnerPreconditioner>::
-initialize (const BlockSparseMatrix<double> &system_matrix,
-           const AdditionalData            &data)
-{
-  this->system_matrix    = &system_matrix;
-  this->A_preconditioner = data.A_preconditioner;
-}
-
-
-
-
-template <typename InnerPreconditioner>
-void
-SchurComplementSmoother<InnerPreconditioner>::
-vmult (BlockVector<double> &dst,
-       const BlockVector<double> &src) const
-{
-#ifdef STEP_42_TEST
-  std::cout << "Entering smoother with " << dst.size() << " unknowns" << std::endl;
-#endif
-
-  SparseDirectUMFPACK direct_solver;
-  direct_solver.initialize(*system_matrix);
-  Vector<double> solution, rhs;
-  solution = dst;
-  rhs = src;
-  direct_solver.vmult(solution, rhs);
-  dst = solution;
-/*
-  const InverseMatrix<SparseMatrix<double>,InnerPreconditioner>
-    A_inverse (system_matrix->block(0,0), *A_preconditioner);
-  Vector<double> tmp (dst.block(0).size());
-
-
-  {
-    Vector<double> schur_rhs (dst.block(1).size());
-    A_inverse.name = "rhs";
-    A_inverse.vmult (tmp, src.block(0));
-//    std::cout << " TMP " << tmp.l2_norm() << std::endl;
-    system_matrix->block(1,0).vmult (schur_rhs, tmp);
-    schur_rhs -= src.block(1);
-//    std::cout << " BLOCK 1 " << src.block(1).l2_norm() << std::endl;
-//    std::cout << " SCHUR RHS " << schur_rhs.l2_norm() << std::endl;
-
-    SchurComplement<InnerPreconditioner>
-      schur_complement (*system_matrix, A_inverse);
-
-                                    // The usual control structures for
-                                    // the solver call are created...
-    SolverControl solver_control (dst.block(1).size(),
-                                 1e-1*schur_rhs.l2_norm());
-    SolverGMRES<>    cg (solver_control);
-
-#ifdef STEP_42_TEST
-    std::cout << "    Starting Schur complement solver -- "
-             << schur_complement.m() << " unknowns"
-             << std::endl;
-#endif
-    try
-      {
-       cg.solve (schur_complement, dst.block(1), schur_rhs,
-                 PreconditionIdentity());
-      }
-    catch (...)
-      {
-       std::cout << "Failure in " << __PRETTY_FUNCTION__ << std::endl;
-       std::cout << schur_rhs.l2_norm () << std::endl;
-       abort ();
-      }
-
-// no constraints to be taken care of here
-#ifdef STEP_42_TEST
-    std::cout << "    "
-             << solver_control.last_step()
-             << " CG Schur complement iterations in smoother "
-             << solver_control.initial_value() << "->" << solver_control.last_value()
-             << std::endl;
-#endif
-  }
-
-
-  {
-    system_matrix->block(0,1).vmult (tmp, dst.block(1));
-    tmp *= -1;
-    tmp += src.block(0);
-
-    A_inverse.name = "top left";
-    A_inverse.vmult (dst.block(0), tmp);
-// no constraints here either
-  }
-#ifdef STEP_42_TEST
-  std::cout << "Exiting smoother with " << dst.size() << " unknowns" << std::endl;
-#endif
-*/
-}
-
-
-
-template <typename InnerPreconditioner>
-void
-SchurComplementSmoother<InnerPreconditioner>::clear ()
-{}
-
-
-
-template <typename InnerPreconditioner>
-void
-SchurComplementSmoother<InnerPreconditioner>::
-Tvmult (BlockVector<double> &,
-       const BlockVector<double> &) const
-{
-  Assert (false, ExcNotImplemented());
-}
-
-
-
-template <int dim>
-void StokesProblem<dim>::solve ()
-{
-  system_matrix.block(1,1) = 0;
-  assemble_multigrid ();
-  typedef PreconditionMG<dim, BlockVector<double>, MGTransferPrebuilt<BlockVector<double> > >
-    MGPREC;
-
-  GrowingVectorMemory<BlockVector<double> >  mg_vector_memory;
-
-  MGTransferPrebuilt<BlockVector<double> > mg_transfer(constraints);
-  std::vector<unsigned int> block_component (dim+1,0);
-  block_component[dim] = 1;
-  mg_transfer.set_component_to_block_map (block_component);
-  mg_transfer.build_matrices(dof_handler);
-
-  FullMatrix<float> mg_coarse_matrix;
-  mg_coarse_matrix.copy_from (mg_matrices[0]);
-  MGCoarseGridHouseholder<float, BlockVector<double> > mg_coarse;
-  mg_coarse.initialize(mg_coarse_matrix);
-
-  MGMatrix<BlockSparseMatrix<double>, BlockVector<double> >
-    mg_matrix(&mg_matrices);
-  MGMatrix<BlockSparseMatrix<double>, BlockVector<double> >
-    mg_interface_up(&mg_interface_matrices);
-  MGMatrix<BlockSparseMatrix<double>, BlockVector<double> >
-    mg_interface_down(&mg_interface_matrices);
-
-  typedef
-    SchurComplementSmoother<typename InnerPreconditioner<dim>::type>
-    Smoother;
-
-  MGSmootherPrecondition<BlockSparseMatrix<double>,
-    Smoother,
-    BlockVector<double> >
-  mg_smoother(mg_vector_memory);
-
-  MGLevelObject<typename Smoother::AdditionalData>
-    smoother_data (0, triangulation.n_levels()-1);
-
-  for (unsigned int level=0; level<triangulation.n_levels(); ++level)
-    smoother_data[level].A_preconditioner = mg_A_preconditioner[level].get();
-
-  mg_smoother.initialize(mg_matrices, smoother_data);
-  mg_smoother.set_steps(2);
-
-  Multigrid<BlockVector<double> > mg(dof_handler,
-      mg_matrix,
-      mg_coarse,
-      mg_transfer,
-      mg_smoother,
-      mg_smoother);
-  mg.set_debug(3);
-  mg.set_edge_matrices(mg_interface_down, mg_interface_up);
-
-  MGPREC  preconditioner(dof_handler, mg, mg_transfer);
-
-  SolverControl solver_control (system_matrix.m(),
-      1e-6*system_rhs.l2_norm());
-  GrowingVectorMemory<BlockVector<double> > vector_memory;
-  SolverGMRES<BlockVector<double> >::AdditionalData gmres_data;
-  gmres_data.max_n_tmp_vectors = 100;
-
-  SolverGMRES<BlockVector<double> > gmres(solver_control, vector_memory,
-      gmres_data);
-
-//  PreconditionIdentity precondition_identity;
-#ifdef STEP_42_TEST
-  std::cout << "Starting outer GMRES complement solver" << std::endl;
-#endif
-  try
-    {
-      gmres.solve(system_matrix, solution, system_rhs,
-                 preconditioner);
-    }
-  catch (...)
-    {
-      std::cout << "Failure in " << __PRETTY_FUNCTION__ << std::endl;
-      abort ();
-    }
-
-  constraints.distribute (solution);
-
-  std::cout << solver_control.last_step()
-           << " outer GMRES iterations ";
-}
-
-
-  template <int dim>
-void StokesProblem<dim>::solve_block () 
-{
-  std::cout << "   Computing preconditioner..." << std::endl << std::flush;
-      
-  A_preconditioner
-    = std_cxx1x::shared_ptr<typename InnerPreconditioner<dim>::type>(new typename InnerPreconditioner<dim>::type());
-  A_preconditioner->initialize (system_matrix.block(0,0),
-                               typename InnerPreconditioner<dim>::type::AdditionalData());
-
-  SparseMatrix<double> pressure_mass_matrix;
-  pressure_mass_matrix.reinit(sparsity_pattern.block(1,1));
-  pressure_mass_matrix.copy_from(system_matrix.block(1,1));
-  system_matrix.block(1,1) = 0;
-
-  SparseILU<double> pmass_preconditioner;
-  pmass_preconditioner.initialize (pressure_mass_matrix, 
-      SparseILU<double>::AdditionalData());
-
-  InverseMatrix<SparseMatrix<double>,SparseILU<double> >
-    m_inverse (pressure_mass_matrix, pmass_preconditioner);
-
-  BlockSchurPreconditioner<typename InnerPreconditioner<dim>::type,
-    SparseILU<double> > 
-      preconditioner (system_matrix, m_inverse, *A_preconditioner);
-
-  SolverControl solver_control (system_matrix.m(),
-      1e-6*system_rhs.l2_norm());
-  GrowingVectorMemory<BlockVector<double> > vector_memory;
-  SolverGMRES<BlockVector<double> >::AdditionalData gmres_data;
-  gmres_data.max_n_tmp_vectors = 100;
-
-  SolverGMRES<BlockVector<double> > gmres(solver_control, vector_memory,
-      gmres_data);
-
-  gmres.solve(system_matrix, solution, system_rhs,
-      preconditioner);
-
-  constraints.distribute (solution);
-
-  std::cout << " "
-    << solver_control.last_step()
-    << " block GMRES iterations ";
-}
-
-
-template <int dim>
-void
-StokesProblem<dim>::output_results (const unsigned int refinement_cycle)  const
-{
-  std::vector<std::string> solution_names (dim, "velocity");
-  solution_names.push_back ("pressure");
-
-  std::vector<DataComponentInterpretation::DataComponentInterpretation>
-    data_component_interpretation
-    (dim, DataComponentInterpretation::component_is_part_of_vector);
-  data_component_interpretation
-    .push_back (DataComponentInterpretation::component_is_scalar);
-
-  DataOut<dim> data_out;
-  data_out.attach_dof_handler (dof_handler);
-  data_out.add_data_vector (solution, solution_names,
-                           DataOut<dim>::type_dof_data,
-                           data_component_interpretation);
-  data_out.build_patches ();
-
-  std::ostringstream filename;
-  filename << "solution-"
-           << Utilities::int_to_string (refinement_cycle, 2)
-           << ".vtk";
-
-  std::ofstream output (filename.str().c_str());
-  data_out.write_vtk (output);
-}
-
-
-
-template <int dim>
-void
-StokesProblem<dim>::refine_mesh ()
-{
-  Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
-
-  std::vector<bool> component_mask (dim+1, false);
-  component_mask[dim] = true;
-  KellyErrorEstimator<dim>::estimate (static_cast<const DoFHandler<dim>&>(dof_handler),
-                                      QGauss<dim-1>(degree+1),
-                                      typename FunctionMap<dim>::type(),
-                                      solution,
-                                      estimated_error_per_cell,
-                                      component_mask);
-
-  GridRefinement::refine_and_coarsen_fixed_number (triangulation,
-                                                   estimated_error_per_cell,
-                                                   0.3, 0.0);
-  triangulation.execute_coarsening_and_refinement ();
-}
-
-
-
-template <int dim>
-void StokesProblem<dim>::run ()
-{
-  {
-    std::vector<unsigned int> subdivisions (dim, 1);
-    subdivisions[0] = 1;
-
-    const Point<dim> bottom_left = (dim == 2 ?
-                                   Point<dim>(0,0) :
-                                   Point<dim>(0,0,0));
-    const Point<dim> top_right   = (dim == 2 ?
-                                   Point<dim>(1,1) :
-                                   Point<dim>(1,1,1));
-
-    GridGenerator::subdivided_hyper_rectangle (triangulation,
-                                              subdivisions,
-                                              bottom_left,
-                                              top_right);
-  }
-
-
-  for (typename Triangulation<dim>::active_cell_iterator
-        cell = triangulation.begin_active();
-       cell != triangulation.end(); ++cell)
-    for (unsigned int f=0; f<GeometryInfo<dim>::faces_per_cell; ++f)
-      if (cell->face(f)->center()[0] == 1)
-       cell->face(f)->set_all_boundary_indicators(1);
-
-
-
-  triangulation.refine_global (1);
-
-
-  for (unsigned int refinement_cycle = 0; refinement_cycle<10;
-       ++refinement_cycle)
-    {
-      std::cout << "Refinement cycle " << refinement_cycle << std::endl;
-
-      if (refinement_cycle > 0)
-        refine_mesh ();
-
-      std::ostringstream out_filename;
-      out_filename << "gitter"
-        << refinement_cycle
-        << ".eps";
-
-      std::ofstream grid_output (out_filename.str().c_str());
-      GridOut grid_out;
-      grid_out.write_eps (triangulation, grid_output);
-
-      setup_dofs ();
-
-      std::cout << "   Assembling..." << std::endl << std::flush;
-      assemble_system ();
-
-      std::cout << "   Solving..." << std::flush;
-
-      solve_block ();
-      output_results (refinement_cycle);
-      system ("mv solution-* block");
-
-      solution = 0;
-
-      solve ();
-      output_results (refinement_cycle);
-      system ("mv solution-* mg");
-
-      std::cout << std::endl;
-    }
-}
-
-
-
-int main ()
-{
-  try
-    {
-      deallog.depth_console (0);
-
-      StokesProblem<2> flow_problem(1);
-      flow_problem.run ();
-    }
-  catch (std::exception &exc)
-    {
-      std::cerr << std::endl << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      std::cerr << "Exception on processing: " << std::endl
-                << exc.what() << std::endl
-                << "Aborting!" << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-
-      return 1;
-    }
-  catch (...)
-    {
-      std::cerr << std::endl << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      std::cerr << "Unknown exception!" << std::endl
-                << "Aborting!" << std::endl
-                << "----------------------------------------------------"
-                << std::endl;
-      return 1;
-    }
-
-  return 0;
-}

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