From: Matthias Maier Date: Tue, 14 May 2019 14:18:00 +0000 (-0500) Subject: removed unfinished steps (release tasks step 8). X-Git-Tag: v9.1.0-rc1~1^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9f714ea428778f58a032737c3926b770ec62490d;p=dealii.git removed unfinished steps (release tasks step 8). --- diff --git a/examples/step-50/CMakeLists.txt b/examples/step-50/CMakeLists.txt deleted file mode 100644 index 58b0d6532a..0000000000 --- a/examples/step-50/CMakeLists.txt +++ /dev/null @@ -1,59 +0,0 @@ -## -# CMake script for the step-50 tutorial program: -## - -# Set the name of the project and target: -SET(TARGET "step-50") - -# Declare all source files the target consists of. Here, this is only -# the one step-X.cc file, but as you expand your project you may wish -# to add other source files as well. If your project becomes much larger, -# you may want to either replace the following statement by something like -# FILE(GLOB_RECURSE TARGET_SRC "source/*.cc") -# FILE(GLOB_RECURSE TARGET_INC "include/*.h") -# SET(TARGET_SRC ${TARGET_SRC} ${TARGET_INC}) -# or switch altogether to the large project CMakeLists.txt file discussed -# in the "CMake in user projects" page accessible from the "User info" -# page of the documentation. -SET(TARGET_SRC - ${TARGET}.cc - ) - -# Define the output that should be cleaned: -SET(CLEAN_UP_FILES *.vtu *.pvtu *.visit) - -# Usually, you will not need to modify anything beyond this point... - -CMAKE_MINIMUM_REQUIRED(VERSION 2.8.12) - -FIND_PACKAGE(deal.II 9.1.0 QUIET - HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR} - ) -IF(NOT ${deal.II_FOUND}) - MESSAGE(FATAL_ERROR "\n" - "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n" - "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n" - "or set an environment variable \"DEAL_II_DIR\" that contains this path." - ) -ENDIF() - -# -# Are all dependencies fulfilled? -# -IF(NOT DEAL_II_WITH_MPI OR NOT DEAL_II_WITH_P4EST OR NOT DEAL_II_WITH_TRILINOS) # keep in one line - MESSAGE(FATAL_ERROR " -Error! This tutorial requires a deal.II library that was configured with the following options: - DEAL_II_WITH_MPI = ON - DEAL_II_WITH_P4EST = ON - DEAL_II_WITH_TRILINOS = ON -However, the deal.II library found at ${DEAL_II_PATH} was configured with these options - DEAL_II_WITH_MPI = ${DEAL_II_WITH_MPI} - DEAL_II_WITH_P4EST = ${DEAL_II_WITH_P4EST} - DEAL_II_WITH_TRILINOS = ${DEAL_II_WITH_TRILINOS} -which conflict with the requirements." - ) -ENDIF() - -DEAL_II_INITIALIZE_CACHED_VARIABLES() -PROJECT(${TARGET}) -DEAL_II_INVOKE_AUTOPILOT() diff --git a/examples/step-50/doc/builds-on b/examples/step-50/doc/builds-on deleted file mode 100644 index 79df0eea06..0000000000 --- a/examples/step-50/doc/builds-on +++ /dev/null @@ -1 +0,0 @@ -step-16 step-40 diff --git a/examples/step-50/doc/intro.dox b/examples/step-50/doc/intro.dox deleted file mode 100644 index e597798731..0000000000 --- a/examples/step-50/doc/intro.dox +++ /dev/null @@ -1,90 +0,0 @@ -
- -This program has evolved from a version originally written by Guido -Kanschat in 2003. It has undergone significant revisions by Bärbel -Janssen, Guido Kanschat and Wolfgang Bangerth in 2009 and 2010 to demonstrate -multigrid algorithms on adaptively refined meshes. - - - - -

Introduction

- - -This example shows the basic usage of the multilevel functions in -deal.II. It solves the same problem as used in step-6, -but demonstrating the things one has to provide when using multigrid -as a preconditioner. In particular, this requires that we define a -hierarchy of levels, provide transfer operators from one level to the -next and back, and provide representations of the Laplace operator on -each level. - -In order to allow sufficient flexibility in conjunction with systems of -differential equations and block preconditioners, quite a few different objects -have to be created before starting the multilevel method, although -most of what needs to be done is provided by deal.II itself. These are - -Most of these objects will only be needed inside the function that -actually solves the linear system. There, these objects are combined -in an object of type Multigrid, containing the implementation of the -V-cycle, which is in turn used by the preconditioner PreconditionMG, -ready for plug-in into a linear solver of the LAC library. - -The multilevel method in deal.II follows in many respects the outlines -of the various publications by James Bramble, Joseph Pasciak and -Jinchao Xu (i.e. the "BPX" framework). In order to understand many of -the options, a rough familiarity with their work is quite helpful. - -However, in comparison to this framework, the implementation in -deal.II has to take into account the fact that we want to solve linear -systems on adaptively refined meshes. This leads to the complication -that it isn't quite as clear any more what exactly a "level" in a -multilevel hierarchy of a mesh is. The following image shows what we -consider to be a "level": - -

- @image html "hanging_nodes.png" "" -

- -In other words, the fine level in this mesh consists only of the -degrees of freedom that are defined on the refined cells, but does not -extend to that part of the domain that is not refined. While this -guarantees that the overall effort grows as ${\cal O}(N)$ as necessary -for optimal multigrid complexity, it leads to problems when defining -where to smooth and what boundary conditions to pose for the operators -defined on individual levels if the level boundary is not an external -boundary. These questions are discussed in detail in the -@ref mg_paper "Multigrid paper by Janssen and Kanschat" that describes -the implementation in deal.II. - - - -

The testcase

- -The problem we solve here is exactly the same as in -step-6, the only difference being the solver we use -here. You may want to look there for a definition of what we solve, -right hand side and boundary conditions. Obviously, the program would -also work if we changed the geometry and other pieces of data that -defines this particular problem. - -The things that are new are all those parts that concern the -multigrid. In particular, this includes the following members of the -main class: -- LaplaceProblem::mg_dof_handler -- LaplaceProblem::mg_sparsity -- LaplaceProblem::mg_matrices -- LaplaceProblem::mg_interface_matrices_up -- LaplaceProblem::assemble_multigrid () -- LaplaceProblem::solve () -Take a look at these functions. diff --git a/examples/step-50/doc/kind b/examples/step-50/doc/kind deleted file mode 100644 index 6816e9090f..0000000000 --- a/examples/step-50/doc/kind +++ /dev/null @@ -1 +0,0 @@ -unfinished diff --git a/examples/step-50/doc/results.dox b/examples/step-50/doc/results.dox deleted file mode 100644 index 2d5fe0deb0..0000000000 --- a/examples/step-50/doc/results.dox +++ /dev/null @@ -1,98 +0,0 @@ -

Results

- -The output that this program generates is, of course, the same as that -of step-6, so you may see there for more results. On the -other hand, since no tutorial program is a good one unless it has at -least one colorful picture, here is, again, the solution: - - -When run, the output of this program is -
-Cycle 0:
-   Number of active cells:       20
-   Number of degrees of freedom: 25 (by level: 8, 25)
-   7 CG iterations needed to obtain convergence.
-Cycle 1:
-   Number of active cells:       44
-   Number of degrees of freedom: 57 (by level: 8, 25, 48)
-   8 CG iterations needed to obtain convergence.
-Cycle 2:
-   Number of active cells:       92
-   Number of degrees of freedom: 117 (by level: 8, 25, 80, 60)
-   9 CG iterations needed to obtain convergence.
-Cycle 3:
-   Number of active cells:       188
-   Number of degrees of freedom: 221 (by level: 8, 25, 80, 200)
-   12 CG iterations needed to obtain convergence.
-Cycle 4:
-   Number of active cells:       416
-   Number of degrees of freedom: 485 (by level: 8, 25, 89, 288, 280)
-   13 CG iterations needed to obtain convergence.
-Cycle 5:
-   Number of active cells:       800
-   Number of degrees of freedom: 925 (by level: 8, 25, 89, 288, 784, 132)
-   14 CG iterations needed to obtain convergence.
-Cycle 6:
-   Number of active cells:       1628
-   Number of degrees of freedom: 1865 (by level: 8, 25, 89, 304, 1000, 1164, 72)
-   14 CG iterations needed to obtain convergence.
-Cycle 7:
-   Number of active cells:       3194
-   Number of degrees of freedom: 3603 (by level: 8, 25, 89, 328, 1032, 2200, 1392)
-   16 CG iterations needed to obtain convergence.
-
-That's not perfect — we would have hoped for a constant number -of iterations rather than one that increases as we get more and more -degrees of freedom — but it is also not far away. The reason for -this is easy enough to understand, however: since we have a strongly -varying coefficient, the operators that we assembly by quadrature on -the lower levels become worse and worse approximations of the operator -on the finest level. Consequently, even if we had perfect solvers on -the coarser levels, they would not be good preconditioners on the -finest level. This theory is easily tested by comparing results when -we use a constant coefficient: in that case, the number of iterations -remains constant at 9 after the first three or four refinement steps. - -We can also compare what this program produces with how @ref step_5 -"step-5" performed. To solve the same problem as in step-5, the only -two changes that are necessary are (i) to replace the body of the -function LaplaceProblem::refine_grid by a call to -triangulation.refine_global(1), and (ii) to use the same -SolverControl object and tolerance as in step-5 — the rest of the -program remains unchanged. In that case, here is how the solvers used -in step-5 and the multigrid solver used in the current program -compare: - - - - - - - - -
cellsstep-5step-16
20 13 6
80 17 7
320 29 9
1280 51 10
5120 94 11
2048018013
-This isn't only fewer iterations than in step-5 (each of which -is, however, much more expensive) but more importantly, the number of -iterations also grows much more slowly under mesh refinement (again, -it would be almost constant if the coefficient was constant rather -than strongly varying as chosen here). This justifies the common -observation that, whenever possible, multigrid methods should be used -for second order problems. - - -

Possible extensions

- -A close inspection of this program's performance shows that it is mostly -dominated by matrix-vector operations. step-37 shows one way -how this can be avoided by working with matrix-free methods. - -Another avenue would be to use algebraic multigrid methods. The -geometric multigrid method used here can at times be a bit awkward to -implement because it needs all those additional data structures, and -it becomes even more difficult if the program is to run in %parallel on -machines coupled through MPI, for example. In that case, it would be -simpler if one could use a black-box preconditioner that uses some -sort of multigrid hierarchy for good performance but can figure out -level matrices and similar things out by itself. Algebraic multigrid -methods do exactly this, and we will use them in -step-31 for the solution of a Stokes problem. diff --git a/examples/step-50/doc/tooltip b/examples/step-50/doc/tooltip deleted file mode 100644 index 2fd65590a0..0000000000 --- a/examples/step-50/doc/tooltip +++ /dev/null @@ -1 +0,0 @@ -Multigrid on adaptive meshes. diff --git a/examples/step-50/step-50.cc b/examples/step-50/step-50.cc deleted file mode 100644 index 14d782b080..0000000000 --- a/examples/step-50/step-50.cc +++ /dev/null @@ -1,1006 +0,0 @@ -/* --------------------------------------------------------------------- - * - * Copyright (C) 2003 - 2018 by the deal.II authors - * - * This file is part of the deal.II library. - * - * The deal.II library is free software; you can use it, redistribute - * it, and/or modify it under the terms of the GNU Lesser General - * Public License as published by the Free Software Foundation; either - * version 2.1 of the License, or (at your option) any later version. - * The full text of the license can be found in the file LICENSE.md at - * the top level directory of deal.II. - * - * --------------------------------------------------------------------- - - * - * Author: Guido Kanschat and Timo Heister - */ - - -// @note: This a work in progress example of parallel geometric -// multigrid. Some parts are still in heavy development. - -// This program is a parallel version of step-16 with a slightly different -// problem setup. - -// @sect3{Include files} - -// Again, the first few include files -// are already known, so we won't -// comment on them: -#include -#include -#include -#include -#include - -#include -#include -#include -#include -#include -#include -#include - -#include -#include -#include -#include -#include -#include - -#include -#include - -#include -#include - -#include -#include -#include - -#include -#include -#include - -#include -#include -#include -#include -#include -#include -#include - - -#include - -// #define USE_PETSC_LA PETSc is not quite supported yet - -namespace LA -{ -#ifdef USE_PETSC_LA - using namespace dealii::LinearAlgebraPETSc; -#else - using namespace dealii::LinearAlgebraTrilinos; -#endif -} // namespace LA - -// This is C++: -#include -#include - -// The last step is as in all -// previous programs: -namespace Step50 -{ - using namespace dealii; - - - // @sect3{The LaplaceProblem class template} - - // This main class is very similar to step-16, except that we are storing a - // parallel Triangulation and parallel versions of matrices and vectors. - template - class LaplaceProblem - { - public: - LaplaceProblem(const unsigned int deg); - void run(); - - private: - void setup_system(); - void assemble_system(); - void assemble_multigrid(); - void solve(); - void refine_grid(); - void output_results(const unsigned int cycle) const; - - ConditionalOStream pcout; - - parallel::distributed::Triangulation triangulation; - FE_Q fe; - DoFHandler mg_dof_handler; - - using matrix_t = LA::MPI::SparseMatrix; - using vector_t = LA::MPI::Vector; - - matrix_t system_matrix; - - IndexSet locally_relevant_set; - - AffineConstraints constraints; - - vector_t solution; - vector_t system_rhs; - - const unsigned int degree; - - // Finally we are storing the various parallel multigrid matrices. Our - // problem is self-adjoint, so the interface matrices are the transpose - // of each other, so we only need to compute/store them once. - MGLevelObject mg_matrices; - MGLevelObject mg_interface_matrices; - // - MGConstrainedDoFs mg_constrained_dofs; - }; - - - - // @sect3{Nonconstant coefficients} - - // The implementation of nonconstant - // coefficients is copied verbatim - // from step-5 and step-6: - - template - class Coefficient : public Function - { - public: - Coefficient() - : Function() - {} - - virtual double value(const Point & p, - const unsigned int component = 0) const override; - - virtual void value_list(const std::vector> &points, - std::vector & values, - const unsigned int component = 0) const override; - }; - - - - template - double Coefficient::value(const Point &p, const unsigned int) const - { - if (p.square() < 0.5 * 0.5) - return 5; - else - return 1; - } - - - - template - void Coefficient::value_list(const std::vector> &points, - std::vector & values, - const unsigned int component) const - { - (void)component; - const unsigned int n_points = points.size(); - - Assert(values.size() == n_points, - ExcDimensionMismatch(values.size(), n_points)); - - Assert(component == 0, ExcIndexRange(component, 0, 1)); - - for (unsigned int i = 0; i < n_points; ++i) - values[i] = Coefficient::value(points[i]); - } - - - // @sect3{The LaplaceProblem class implementation} - - // @sect4{LaplaceProblem::LaplaceProblem} - - // The constructor is left mostly - // unchanged. We take the polynomial degree - // of the finite elements to be used as a - // constructor argument and store it in a - // member variable. - // - // By convention, all adaptively refined - // triangulations in deal.II never change by - // more than one level across a face between - // cells. For our multigrid algorithms, - // however, we need a slightly stricter - // guarantee, namely that the mesh also does - // not change by more than refinement level - // across vertices that might connect two - // cells. In other words, we must prevent the - // following situation: - // - // @image html limit_level_difference_at_vertices.png "" - // - // This is achieved by passing the - // Triangulation::limit_level_difference_at_vertices - // flag to the constructor of the - // triangulation class. - template - LaplaceProblem::LaplaceProblem(const unsigned int degree) - : pcout(std::cout, (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)) - , triangulation(MPI_COMM_WORLD, - Triangulation::limit_level_difference_at_vertices, - parallel::distributed::Triangulation< - dim>::construct_multigrid_hierarchy) - , fe(degree) - , mg_dof_handler(triangulation) - , degree(degree) - {} - - - // @sect4{LaplaceProblem::setup_system} - - // The following function extends what the - // corresponding one in step-6 did. The top - // part, apart from the additional output, - // does the same: - template - void LaplaceProblem::setup_system() - { - mg_dof_handler.distribute_dofs(fe); - mg_dof_handler.distribute_mg_dofs(); - - DoFTools::extract_locally_relevant_dofs(mg_dof_handler, - locally_relevant_set); - - solution.reinit(mg_dof_handler.locally_owned_dofs(), MPI_COMM_WORLD); - system_rhs.reinit(mg_dof_handler.locally_owned_dofs(), MPI_COMM_WORLD); - - // But it starts to be a wee bit different - // here, although this still doesn't have - // anything to do with multigrid - // methods. step-6 took care of boundary - // values and hanging nodes in a separate - // step after assembling the global matrix - // from local contributions. This works, - // but the same can be done in a slightly - // simpler way if we already take care of - // these constraints at the time of copying - // local contributions into the global - // matrix. To this end, we here do not just - // compute the constraints do to hanging - // nodes, but also due to zero boundary - // conditions. We will - // use this set of constraints later on to - // help us copy local contributions - // correctly into the global linear system - // right away, without the need for a later - // clean-up stage: - constraints.reinit(locally_relevant_set); - DoFTools::make_hanging_node_constraints(mg_dof_handler, constraints); - - std::set dirichlet_boundary_ids; - std::map *> dirichlet_boundary; - Functions::ConstantFunction homogeneous_dirichlet_bc(1.0); - dirichlet_boundary_ids.insert(0); - dirichlet_boundary[0] = &homogeneous_dirichlet_bc; - VectorTools::interpolate_boundary_values(mg_dof_handler, - dirichlet_boundary, - constraints); - constraints.close(); - - DynamicSparsityPattern dsp(mg_dof_handler.n_dofs(), - mg_dof_handler.n_dofs()); - DoFTools::make_sparsity_pattern(mg_dof_handler, dsp, constraints); - system_matrix.reinit(mg_dof_handler.locally_owned_dofs(), - dsp, - MPI_COMM_WORLD, - true); - - - // The multigrid constraints have to be - // initialized. They need to know about - // the boundary values as well, so we - // pass the dirichlet_boundary - // here as well. - mg_constrained_dofs.clear(); - mg_constrained_dofs.initialize(mg_dof_handler); - mg_constrained_dofs.make_zero_boundary_constraints(mg_dof_handler, - dirichlet_boundary_ids); - - - // Now for the things that concern the - // multigrid data structures. First, we - // resize the multilevel objects to hold - // matrices and sparsity patterns for every - // level. The coarse level is zero (this is - // mandatory right now but may change in a - // future revision). Note that these - // functions take a complete, inclusive - // range here (not a starting index and - // size), so the finest level is - // n_levels-1. We first have - // to resize the container holding the - // SparseMatrix classes, since they have to - // release their SparsityPattern before the - // can be destroyed upon resizing. - const unsigned int n_levels = triangulation.n_global_levels(); - - mg_interface_matrices.resize(0, n_levels - 1); - mg_interface_matrices.clear_elements(); - mg_matrices.resize(0, n_levels - 1); - mg_matrices.clear_elements(); - - // Now, we have to provide a matrix on each - // level. To this end, we first use the - // MGTools::make_sparsity_pattern function - // to first generate a preliminary - // compressed sparsity pattern on each - // level (see the @ref Sparsity module for - // more information on this topic) and then - // copy it over to the one we really - // want. The next step is to initialize - // both kinds of level matrices with these - // sparsity patterns. - // - // It may be worth pointing out that the - // interface matrices only have entries for - // degrees of freedom that sit at or next - // to the interface between coarser and - // finer levels of the mesh. They are - // therefore even sparser than the matrices - // on the individual levels of our - // multigrid hierarchy. If we were more - // concerned about memory usage (and - // possibly the speed with which we can - // multiply with these matrices), we should - // use separate and different sparsity - // patterns for these two kinds of - // matrices. - for (unsigned int level = 0; level < n_levels; ++level) - { - DynamicSparsityPattern dsp(mg_dof_handler.n_dofs(level), - mg_dof_handler.n_dofs(level)); - MGTools::make_sparsity_pattern(mg_dof_handler, dsp, level); - - mg_matrices[level].reinit(mg_dof_handler.locally_owned_mg_dofs(level), - mg_dof_handler.locally_owned_mg_dofs(level), - dsp, - MPI_COMM_WORLD, - true); - - mg_interface_matrices[level].reinit( - mg_dof_handler.locally_owned_mg_dofs(level), - mg_dof_handler.locally_owned_mg_dofs(level), - dsp, - MPI_COMM_WORLD, - true); - } - } - - - // @sect4{LaplaceProblem::assemble_system} - - // The following function assembles the - // linear system on the finest level of the - // mesh. It is almost exactly the same as in - // step-6, with the exception that we don't - // eliminate hanging nodes and boundary - // values after assembling, but while copying - // local contributions into the global - // matrix. This is not only simpler but also - // more efficient for large problems. - // - // This latter trick is something that only - // found its way into deal.II over time and - // wasn't used in the initial version of this - // tutorial program. There is, however, a - // discussion of this function in the - // introduction of step-27. - template - void LaplaceProblem::assemble_system() - { - const QGauss quadrature_formula(degree + 1); - - FEValues fe_values(fe, - quadrature_formula, - update_values | update_gradients | - 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(); - - FullMatrix cell_matrix(dofs_per_cell, dofs_per_cell); - Vector cell_rhs(dofs_per_cell); - - std::vector local_dof_indices(dofs_per_cell); - - const Coefficient coefficient; - std::vector coefficient_values(n_q_points); - - typename DoFHandler::active_cell_iterator cell = mg_dof_handler - .begin_active(), - endc = mg_dof_handler.end(); - for (; cell != endc; ++cell) - if (cell->is_locally_owned()) - { - cell_matrix = 0; - cell_rhs = 0; - - fe_values.reinit(cell); - - coefficient.value_list(fe_values.get_quadrature_points(), - coefficient_values); - - 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) += - (coefficient_values[q_point] * - fe_values.shape_grad(i, q_point) * - fe_values.shape_grad(j, q_point) * fe_values.JxW(q_point)); - - cell_rhs(i) += (fe_values.shape_value(i, q_point) * 10.0 * - fe_values.JxW(q_point)); - } - - cell->get_dof_indices(local_dof_indices); - constraints.distribute_local_to_global(cell_matrix, - cell_rhs, - local_dof_indices, - system_matrix, - system_rhs); - } - - system_matrix.compress(VectorOperation::add); - system_rhs.compress(VectorOperation::add); - } - - - // @sect4{LaplaceProblem::assemble_multigrid} - - // The next function is the one that builds - // the linear operators (matrices) that - // define the multigrid method on each level - // of the mesh. The integration core is the - // same as above, but the loop below will go - // over all existing cells instead of just - // the active ones, and the results must be - // entered into the correct matrix. Note also - // that since we only do multilevel - // preconditioning, no right-hand side needs - // to be assembled here. - // - // Before we go there, however, we have to - // take care of a significant amount of book - // keeping: - template - void LaplaceProblem::assemble_multigrid() - { - QGauss quadrature_formula(1 + degree); - - FEValues fe_values(fe, - quadrature_formula, - update_values | update_gradients | - 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(); - - FullMatrix cell_matrix(dofs_per_cell, dofs_per_cell); - - std::vector local_dof_indices(dofs_per_cell); - - const Coefficient coefficient; - std::vector coefficient_values(n_q_points); - - // Next a few things that are specific to building the multigrid - // data structures (since we only need them in the current - // function, rather than also elsewhere, we build them here - // instead of the setup_system function). Some of the - // following may be a bit obscure if you're not familiar with the - // algorithm actually implemented in deal.II to support multilevel - // algorithms on adaptive meshes; if some of the things below seem - // strange, take a look at the @ref mg_paper. - // - // Our first job is to identify those degrees of freedom on each level - // that are located on interfaces between adaptively refined levels, and - // those that lie on the interface but also on the exterior boundary of - // the domain. The MGConstrainedDoFs already computed the - // information for us when we called initialize in - - // setup_system(). - // of type IndexSet on each level (get_refinement_edge_indices(), - - // The indices just identified will later be used to decide where - // the assembled value has to be added into on each level. On the - // other hand, we also have to impose zero boundary conditions on - // the external boundary of each level. But this the - // MGConstrainedDoFs knows it. So we simply ask for them - // by calling get_boundary_indices (). The third - // step is to construct constraints on all those degrees of - // freedom: their value should be zero after each application of - // the level operators. To this end, we construct AffineConstraints - // objects for each level, and add to each of these constraints - // for each degree of freedom. Due to the way the AffineConstraints class - // stores its data, the function to add a constraint on a single - // degree of freedom and force it to be zero is called - // AffineConstraints::add_line(); doing so for several degrees of - // freedom at once can be done using - // AffineConstraints::add_lines(): - std::vector> boundary_constraints( - triangulation.n_global_levels()); - AffineConstraints empty_constraints; - for (unsigned int level = 0; level < triangulation.n_global_levels(); - ++level) - { - IndexSet dofset; - DoFTools::extract_locally_relevant_level_dofs(mg_dof_handler, - level, - dofset); - boundary_constraints[level].reinit(dofset); - boundary_constraints[level].add_lines( - mg_constrained_dofs.get_refinement_edge_indices(level)); - boundary_constraints[level].add_lines( - mg_constrained_dofs.get_boundary_indices(level)); - - boundary_constraints[level].close(); - } - - // Now that we're done with most of our preliminaries, let's start - // the integration loop. It looks mostly like the loop in - // assemble_system, with two exceptions: (i) we don't - // need a right hand side, and more significantly (ii) we don't - // just loop over all active cells, but in fact all cells, active - // or not. Consequently, the correct iterator to use is - // DoFHandler::cell_iterator rather than - // DoFHandler::active_cell_iterator. Let's go about it: - typename DoFHandler::cell_iterator cell = mg_dof_handler.begin(), - endc = mg_dof_handler.end(); - - for (; cell != endc; ++cell) - if (cell->level_subdomain_id() == triangulation.locally_owned_subdomain()) - { - cell_matrix = 0; - fe_values.reinit(cell); - - coefficient.value_list(fe_values.get_quadrature_points(), - coefficient_values); - - 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) += - (coefficient_values[q_point] * - fe_values.shape_grad(i, q_point) * - fe_values.shape_grad(j, q_point) * fe_values.JxW(q_point)); - - // The rest of the assembly is again slightly - // different. This starts with a gotcha that is easily - // forgotten: The indices of global degrees of freedom we - // want here are the ones for current level, not for the - // global matrix. We therefore need the function - // MGDoFAccessorLLget_mg_dof_indices, not - // MGDoFAccessor::get_dof_indices as used in the assembly of - // the global system: - cell->get_mg_dof_indices(local_dof_indices); - - // Next, we need to copy local contributions into the level - // objects. We can do this in the same way as in the global - // assembly, using a constraint object that takes care of - // constrained degrees (which here are only boundary nodes, - // as the individual levels have no hanging node - // constraints). Note that the - // boundary_constraints object makes sure that - // the level matrices contains no contributions from degrees - // of freedom at the interface between cells of different - // refinement level. - boundary_constraints[cell->level()].distribute_local_to_global( - cell_matrix, local_dof_indices, mg_matrices[cell->level()]); - - // The next step is again slightly more obscure (but - // explained in the @ref mg_paper): We need the remainder of - // the operator that we just copied into the - // mg_matrices object, namely the part on the - // interface between cells at the current level and cells - // one level coarser. This matrix exists in two directions: - // for interior DoFs (index $i$) of the current level to - // those sitting on the interface (index $j$), and the other - // way around. Of course, since we have a symmetric - // operator, one of these matrices is the transpose of the - // other. - // - // The way we assemble these matrices is as follows: since - // the are formed from parts of the local contributions, we - // first delete all those parts of the local contributions - // that we are not interested in, namely all those elements - // of the local matrix for which not $i$ is an interface DoF - // and $j$ is not. The result is one of the two matrices - // that we are interested in, and we then copy it into the - // mg_interface_matrices object. The - // boundary_interface_constraints object at the - // same time makes sure that we delete contributions from - // all degrees of freedom that are not only on the interface - // but also on the external boundary of the domain. - // - // The last part to remember is how to get the other - // matrix. Since it is only the transpose, we will later (in - // the solve() function) be able to just pass - // the transpose matrix where necessary. - - const IndexSet &interface_dofs_on_level = - mg_constrained_dofs.get_refinement_edge_indices(cell->level()); - const unsigned int lvl = cell->level(); - - for (unsigned int i = 0; i < dofs_per_cell; ++i) - for (unsigned int j = 0; j < dofs_per_cell; ++j) - if (interface_dofs_on_level.is_element( - local_dof_indices[i]) // at_refinement_edge(i) - && !interface_dofs_on_level.is_element( - local_dof_indices[j]) // !at_refinement_edge(j) - && - ((!mg_constrained_dofs.is_boundary_index( - lvl, local_dof_indices[i]) && - !mg_constrained_dofs.is_boundary_index( - lvl, - local_dof_indices[j])) // ( !boundary(i) && !boundary(j) ) - || (mg_constrained_dofs.is_boundary_index( - lvl, local_dof_indices[i]) && - local_dof_indices[i] == - local_dof_indices[j]) // ( boundary(i) && boundary(j) - // && i==j ) - )) - { - // do nothing, so add entries to interface matrix - } - else - { - cell_matrix(i, j) = 0; - } - - - empty_constraints.distribute_local_to_global( - cell_matrix, - local_dof_indices, - mg_interface_matrices[cell->level()]); - } - - for (unsigned int i = 0; i < triangulation.n_global_levels(); ++i) - { - mg_matrices[i].compress(VectorOperation::add); - mg_interface_matrices[i].compress(VectorOperation::add); - } - } - - - - // @sect4{LaplaceProblem::solve} - - // This is the other function that is significantly different in - // support of the multigrid solver (or, in fact, the preconditioner - // for which we use the multigrid method). - // - // Let us start out by setting up two of the components of - // multilevel methods: transfer operators between levels, and a - // solver on the coarsest level. In finite element methods, the - // transfer operators are derived from the finite element function - // spaces involved and can often be computed in a generic way - // independent of the problem under consideration. In that case, we - // can use the MGTransferPrebuilt class that, given the constraints - // on the global level and an DoFHandler object computes the - // matrices corresponding to these transfer operators. - // - // The second part of the following lines deals with the coarse grid - // solver. Since our coarse grid is very coarse indeed, we decide - // for a direct solver (a Householder decomposition of the coarsest - // level matrix), even if its implementation is not particularly - // sophisticated. If our coarse mesh had many more cells than the - // five we have here, something better suited would obviously be - // necessary here. - template - void LaplaceProblem::solve() - { - // Create the object that deals with the transfer between - // different refinement levels. - MGTransferPrebuilt mg_transfer(mg_constrained_dofs); - // Now the prolongation matrix has to be built. - mg_transfer.build_matrices(mg_dof_handler); - - matrix_t &coarse_matrix = mg_matrices[0]; - - SolverControl coarse_solver_control(1000, 1e-10, false, false); - SolverCG coarse_solver(coarse_solver_control); - PreconditionIdentity id; - MGCoarseGridIterativeSolver, - matrix_t, - PreconditionIdentity> - coarse_grid_solver(coarse_solver, coarse_matrix, id); - - // The next component of a multilevel solver or preconditioner is - // that we need a smoother on each level. A common choice for this - // is to use the application of a relaxation method (such as the - // SOR, Jacobi or Richardson method). The MGSmootherPrecondition - // class provides support for this kind of smoother. Here, we opt - // for the application of a single SOR iteration. To this end, we - // define an appropriate alias and then setup a smoother object. - // - // The last step is to initialize the smoother object with our - // level matrices and to set some smoothing parameters. The - // initialize() function can optionally take - // additional arguments that will be passed to the smoother object - // on each level. In the current case for the SOR smoother, this - // could, for example, include a relaxation parameter. However, we - // here leave these at their default values. The call to - // set_steps() indicates that we will use two pre- - // and two post-smoothing steps on each level; to use a variable - // number of smoother steps on different levels, more options can - // be set in the constructor call to the mg_smoother - // object. - // - // The last step results from the fact that - // we use the SOR method as a smoother - - // which is not symmetric - but we use the - // conjugate gradient iteration (which - // requires a symmetric preconditioner) - // below, we need to let the multilevel - // preconditioner make sure that we get a - // symmetric operator even for nonsymmetric - // smoothers: - using Smoother = LA::MPI::PreconditionJacobi; - MGSmootherPrecondition mg_smoother; - mg_smoother.initialize(mg_matrices, Smoother::AdditionalData(0.5)); - mg_smoother.set_steps(2); - // mg_smoother.set_symmetric(false); - - // The next preparatory step is that we - // must wrap our level and interface - // matrices in an object having the - // required multiplication functions. We - // will create two objects for the - // interface objects going from coarse to - // fine and the other way around; the - // multigrid algorithm will later use the - // transpose operator for the latter - // operation, allowing us to initialize - // both up and down versions of the - // operator with the matrices we already - // built: - mg::Matrix mg_matrix(mg_matrices); - mg::Matrix mg_interface_up(mg_interface_matrices); - mg::Matrix mg_interface_down(mg_interface_matrices); - - // Now, we are ready to set up the - // V-cycle operator and the - // multilevel preconditioner. - Multigrid mg( - mg_matrix, coarse_grid_solver, mg_transfer, mg_smoother, mg_smoother); - // mg.set_debug(6); - mg.set_edge_matrices(mg_interface_down, mg_interface_up); - - PreconditionMG> preconditioner( - mg_dof_handler, mg, mg_transfer); - - - // With all this together, we can finally - // get about solving the linear system in - // the usual way: - SolverControl solver_control(500, 1e-8 * system_rhs.l2_norm(), false); - SolverCG solver(solver_control); - - if (false) - { - /* - // code to optionally compare to Trilinos ML - TrilinosWrappers::PreconditionAMG prec; - - 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; - // Amg_data.symmetric = true; - - prec.initialize (system_matrix, - Amg_data); - solver.solve (system_matrix, solution, system_rhs, prec); - */ - } - else - { - solver.solve(system_matrix, solution, system_rhs, preconditioner); - } - pcout << " CG converged in " << solver_control.last_step() - << " iterations." << std::endl; - - constraints.distribute(solution); - } - - - - // @sect4{Postprocessing} - - // The following two functions postprocess a solution once it is - // computed. In particular, the first one refines the mesh at the beginning - // of each cycle while the second one outputs results at the end of each - // such cycle. The refine_grid() method is almost unchanged - // from step-6: the only substantial difference is that this method uses a - // distributed grid refinement function instead of a serial one. The - // output_results() method is quite different since each - // processor writes only part of the overall graphical output. - template - void LaplaceProblem::refine_grid() - { - Vector estimated_error_per_cell(triangulation.n_active_cells()); - - LA::MPI::Vector temp_solution; - temp_solution.reinit(locally_relevant_set, MPI_COMM_WORLD); - temp_solution = solution; - - KellyErrorEstimator::estimate( - mg_dof_handler, - QGauss(degree + 1), - std::map *>(), - temp_solution, - estimated_error_per_cell); - - parallel::distributed::GridRefinement::refine_and_coarsen_fixed_fraction( - triangulation, estimated_error_per_cell, 0.3, 0.0); - - triangulation.execute_coarsening_and_refinement(); - } - - - - template - void LaplaceProblem::output_results(const unsigned int cycle) const - { - DataOut data_out; - - LA::MPI::Vector temp_solution; - temp_solution.reinit(locally_relevant_set, MPI_COMM_WORLD); - temp_solution = solution; - - - LA::MPI::Vector temp = solution; - system_matrix.residual(temp, solution, system_rhs); - LA::MPI::Vector res_ghosted = temp_solution; - res_ghosted = temp; - - data_out.attach_dof_handler(mg_dof_handler); - data_out.add_data_vector(temp_solution, "solution"); - data_out.add_data_vector(res_ghosted, "res"); - Vector subdomain(triangulation.n_active_cells()); - for (unsigned int i = 0; i < subdomain.size(); ++i) - subdomain(i) = triangulation.locally_owned_subdomain(); - data_out.add_data_vector(subdomain, "subdomain"); - - data_out.build_patches(0); - - const std::string filename = - ("solution-" + Utilities::int_to_string(cycle, 5) + "." + - Utilities::int_to_string(triangulation.locally_owned_subdomain(), 4) + - ".vtu"); - std::ofstream output(filename); - data_out.write_vtu(output); - - if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0) - { - std::vector filenames; - for (unsigned int i = 0; - i < Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD); - ++i) - filenames.push_back(std::string("solution-") + - Utilities::int_to_string(cycle, 5) + "." + - Utilities::int_to_string(i, 4) + ".vtu"); - const std::string pvtu_master_filename = - ("solution-" + Utilities::int_to_string(cycle, 5) + ".pvtu"); - std::ofstream pvtu_master(pvtu_master_filename); - data_out.write_pvtu_record(pvtu_master, filenames); - - const std::string visit_master_filename = - ("solution-" + Utilities::int_to_string(cycle, 5) + ".visit"); - std::ofstream visit_master(visit_master_filename); - DataOutBase::write_visit_record(visit_master, filenames); - - std::cout << " wrote " << pvtu_master_filename << std::endl; - } - } - - - // @sect4{LaplaceProblem::run} - - // Like several of the functions above, this - // is almost exactly a copy of the - // corresponding function in step-6. The only - // difference is the call to - // assemble_multigrid that takes - // care of forming the matrices on every - // level that we need in the multigrid - // method. - template - void LaplaceProblem::run() - { - for (unsigned int cycle = 0; cycle < 15; ++cycle) - { - pcout << "Cycle " << cycle << ':' << std::endl; - - if (cycle == 0) - { - GridGenerator::hyper_cube(triangulation); - - triangulation.refine_global(4); - } - else - refine_grid(); - - pcout << " Number of active cells: " - << triangulation.n_global_active_cells() << std::endl; - - setup_system(); - - pcout << " Number of degrees of freedom: " << mg_dof_handler.n_dofs() - << " (by level: "; - for (unsigned int level = 0; level < triangulation.n_global_levels(); - ++level) - pcout << mg_dof_handler.n_dofs(level) - << (level == triangulation.n_global_levels() - 1 ? ")" : ", "); - pcout << std::endl; - - assemble_system(); - assemble_multigrid(); - - solve(); - output_results(cycle); - } - } -} // namespace Step50 - - -// @sect3{The main() function} -// -// This is again the same function as -// in step-6: -int main(int argc, char *argv[]) -{ - try - { - using namespace dealii; - using namespace Step50; - - Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1); - - LaplaceProblem<2> laplace_problem(1 /*degree*/); - 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; - } - catch (...) - { - std::cerr << std::endl - << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Unknown exception!" << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - throw; - } - - return 0; -}