]> https://gitweb.dealii.org/ - dealii.git/commitdiff
reserve step-63
authorTimo Heister <timo.heister@gmail.com>
Mon, 11 Feb 2019 23:11:08 +0000 (16:11 -0700)
committerTimo Heister <timo.heister@gmail.com>
Tue, 12 Feb 2019 16:04:49 +0000 (09:04 -0700)
examples/step-63/CMakeLists.txt [new file with mode: 0644]
examples/step-63/doc/builds-on [new file with mode: 0644]
examples/step-63/doc/intro.dox [new file with mode: 0644]
examples/step-63/doc/kind [new file with mode: 0644]
examples/step-63/doc/results.dox [new file with mode: 0644]
examples/step-63/doc/tooltip [new file with mode: 0644]
examples/step-63/step-63.cc [new file with mode: 0644]

diff --git a/examples/step-63/CMakeLists.txt b/examples/step-63/CMakeLists.txt
new file mode 100644 (file)
index 0000000..476f55c
--- /dev/null
@@ -0,0 +1,39 @@
+##
+#  CMake script for the step-63 tutorial program:
+##
+
+# Set the name of the project and target:
+SET(TARGET "step-63")
+
+# 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
+  )
+
+# 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 ${deal.II_FOUND})
+  MESSAGE(FATAL_ERROR "\n"
+    "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n"
+    "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n"
+    "or set an environment variable \"DEAL_II_DIR\" that contains this path."
+    )
+ENDIF()
+
+DEAL_II_INITIALIZE_CACHED_VARIABLES()
+PROJECT(${TARGET})
+DEAL_II_INVOKE_AUTOPILOT()
diff --git a/examples/step-63/doc/builds-on b/examples/step-63/doc/builds-on
new file mode 100644 (file)
index 0000000..42c2846
--- /dev/null
@@ -0,0 +1 @@
+step-16
diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox
new file mode 100644 (file)
index 0000000..b8535c1
--- /dev/null
@@ -0,0 +1,9 @@
+<br>
+
+<i>This program was contributed by Thomas Clevenger and Timo Heister.</i>
+
+<a name="Intro"></a>
+<h1>Introduction</h1>
+
+Please note: This is work in progress and will be an example for block
+smoothers in geometric multigrid. For now, this is just step-16.
diff --git a/examples/step-63/doc/kind b/examples/step-63/doc/kind
new file mode 100644 (file)
index 0000000..6816e90
--- /dev/null
@@ -0,0 +1 @@
+unfinished
diff --git a/examples/step-63/doc/results.dox b/examples/step-63/doc/results.dox
new file mode 100644 (file)
index 0000000..b5eaba9
--- /dev/null
@@ -0,0 +1,2 @@
+<h1>Results</h1>
+
diff --git a/examples/step-63/doc/tooltip b/examples/step-63/doc/tooltip
new file mode 100644 (file)
index 0000000..9aad4b3
--- /dev/null
@@ -0,0 +1 @@
+Block smoothers for Geometric Multigrid.
diff --git a/examples/step-63/step-63.cc b/examples/step-63/step-63.cc
new file mode 100644 (file)
index 0000000..86234e2
--- /dev/null
@@ -0,0 +1,715 @@
+/* ---------------------------------------------------------------------
+ *
+ * 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.
+ *
+ * ---------------------------------------------------------------------
+ *
+ * Authors: Thomas Clevenger, Clemson University
+ *          Timo Heister, University of Utah
+ */
+
+// @note: This is work in progress and will be an example for block smoothers
+// in geometric multigrid. For now, this is just step-16.
+
+// @sect3{Include files}
+
+// Again, the first few include files are already known, so we won't comment
+// on them:
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/function.h>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/utilities.h>
+
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/vector.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/precondition.h>
+
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/grid_refinement.h>
+
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_values.h>
+
+#include <deal.II/numerics/vector_tools.h>
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/error_estimator.h>
+
+// These, now, are the include necessary for the multilevel methods. The first
+// one declares how to handle Dirichlet boundary conditions on each of the
+// levels of the multigrid method. For the actual description of the degrees
+// of freedom, we do not need any new include file because DoFHandler already
+// has all necessary methods implemented. We will only need to distribute the
+// DoFs for the levels further down.
+//
+// The rest of the include files deals with the mechanics of multigrid as a
+// linear operator (solver or preconditioner).
+#include <deal.II/multigrid/mg_constrained_dofs.h>
+#include <deal.II/multigrid/multigrid.h>
+#include <deal.II/multigrid/mg_transfer.h>
+#include <deal.II/multigrid/mg_tools.h>
+#include <deal.II/multigrid/mg_coarse.h>
+#include <deal.II/multigrid/mg_smoother.h>
+#include <deal.II/multigrid/mg_matrix.h>
+
+// We will be using MeshWorker::mesh_loop to loop over the cells, so include it
+// here:
+#include <deal.II/meshworker/mesh_loop.h>
+
+
+// This is C++:
+#include <iostream>
+#include <fstream>
+
+using namespace dealii;
+
+namespace Step16
+{
+  // @sect3{The Scratch and Copy objects}
+  //
+  // We use MeshWorker::mesh_loop() to assemble our matrices. For this, we
+  // need a ScratchData object to store temporary data on each cell (this is
+  // just the FEValues object) and a CopyData object that will contain the
+  // output of each cell assembly. For more details about the usage of scratch
+  // and copy objects, see the WorkStream namespace.
+  template <int dim>
+  struct ScratchData
+  {
+    ScratchData(const Mapping<dim> &      mapping,
+                const FiniteElement<dim> &fe,
+                const unsigned int        quadrature_degree,
+                const UpdateFlags         update_flags)
+      : fe_values(mapping, fe, QGauss<dim>(quadrature_degree), update_flags)
+    {}
+
+    ScratchData(const ScratchData<dim> &scratch_data)
+      : fe_values(scratch_data.fe_values.get_mapping(),
+                  scratch_data.fe_values.get_fe(),
+                  scratch_data.fe_values.get_quadrature(),
+                  scratch_data.fe_values.get_update_flags())
+    {}
+
+    FEValues<dim> fe_values;
+  };
+
+  struct CopyData
+  {
+    unsigned int                         level;
+    FullMatrix<double>                   cell_matrix;
+    Vector<double>                       cell_rhs;
+    std::vector<types::global_dof_index> local_dof_indices;
+
+    template <class Iterator>
+    void reinit(const Iterator &cell, unsigned int dofs_per_cell)
+    {
+      cell_matrix.reinit(dofs_per_cell, dofs_per_cell);
+      cell_rhs.reinit(dofs_per_cell);
+
+      local_dof_indices.resize(dofs_per_cell);
+      cell->get_active_or_mg_dof_indices(local_dof_indices);
+      level = cell->level();
+    }
+  };
+
+  // @sect3{The <code>LaplaceProblem</code> class template}
+
+  // This main class is similar to the same class in step-6. As far as
+  // member functions is concerned, the only additions are:
+  // - The <code>assemble_multigrid</code> function that assembles the matrices
+  // that correspond to the discrete operators on intermediate levels.
+  // - The <code>cell_worker</code> function that assembles our PDE on a single
+  // cell.
+  template <int dim>
+  class LaplaceProblem
+  {
+  public:
+    LaplaceProblem(const unsigned int degree);
+    void run();
+
+  private:
+    template <class Iterator>
+    void cell_worker(const Iterator &  cell,
+                     ScratchData<dim> &scratch_data,
+                     CopyData &        copy_data);
+
+    void setup_system();
+    void assemble_system();
+    void assemble_multigrid();
+    void solve();
+    void refine_grid();
+    void output_results(const unsigned int cycle) const;
+
+    Triangulation<dim> triangulation;
+    FE_Q<dim>          fe;
+    DoFHandler<dim>    dof_handler;
+
+    SparsityPattern      sparsity_pattern;
+    SparseMatrix<double> system_matrix;
+
+    AffineConstraints<double> constraints;
+
+    Vector<double> solution;
+    Vector<double> system_rhs;
+
+    const unsigned int degree;
+
+    // The following members are the essential data structures for the multigrid
+    // method. The first four represent the sparsity patterns and the matrices
+    // on individual levels of the multilevel hierarchy, very much like the
+    // objects for the global mesh above.
+    //
+    // Then we have two new matrices only needed for multigrid methods with
+    // local smoothing on adaptive meshes. They convey data between the interior
+    // part of the refined region and the refinement edge, as outlined in detail
+    // in the @ref mg_paper "multigrid paper".
+    //
+    // The last object stores information about the boundary indices on each
+    // level and information about indices lying on a refinement edge between
+    // two different refinement levels. It thus serves a similar purpose as
+    // AffineConstraints, but on each level.
+    MGLevelObject<SparsityPattern> mg_sparsity_patterns;
+    MGLevelObject<SparsityPattern> mg_interface_sparsity_patterns;
+
+    MGLevelObject<SparseMatrix<double>> mg_matrices;
+    MGLevelObject<SparseMatrix<double>> mg_interface_matrices;
+    MGConstrainedDoFs                   mg_constrained_dofs;
+  };
+
+
+  // @sect3{The <code>LaplaceProblem</code> class implementation}
+
+  // Just one short remark about the constructor of the Triangulation:
+  // 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 <int dim>
+  LaplaceProblem<dim>::LaplaceProblem(const unsigned int degree)
+    : triangulation(Triangulation<dim>::limit_level_difference_at_vertices)
+    , fe(degree)
+    , dof_handler(triangulation)
+    , degree(degree)
+  {}
+
+
+
+  // @sect4{LaplaceProblem::setup_system}
+
+  // In addition to just distributing the degrees of freedom in
+  // the DoFHandler, we do the same on each level. Then, we follow the
+  // same procedure as before to set up the system on the leaf mesh.
+  template <int dim>
+  void LaplaceProblem<dim>::setup_system()
+  {
+    dof_handler.distribute_dofs(fe);
+    dof_handler.distribute_mg_dofs();
+
+    std::cout << "   Number of degrees of freedom: " << dof_handler.n_dofs()
+              << " (by level: ";
+    for (unsigned int level = 0; level < triangulation.n_levels(); ++level)
+      std::cout << dof_handler.n_dofs(level)
+                << (level == triangulation.n_levels() - 1 ? ")" : ", ");
+    std::cout << std::endl;
+
+
+    solution.reinit(dof_handler.n_dofs());
+    system_rhs.reinit(dof_handler.n_dofs());
+
+    constraints.clear();
+    DoFTools::make_hanging_node_constraints(dof_handler, constraints);
+
+    std::set<types::boundary_id> dirichlet_boundary_ids = {0};
+    Functions::ZeroFunction<dim> homogeneous_dirichlet_bc;
+    const std::map<types::boundary_id, const Function<dim> *>
+      dirichlet_boundary_functions = {
+        {types::boundary_id(0), &homogeneous_dirichlet_bc}};
+    VectorTools::interpolate_boundary_values(dof_handler,
+                                             dirichlet_boundary_functions,
+                                             constraints);
+    constraints.close();
+
+    {
+      DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+      DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints);
+      sparsity_pattern.copy_from(dsp);
+    }
+    system_matrix.reinit(sparsity_pattern);
+
+    // The multigrid constraints have to be initialized. They need to know
+    // where Dirichlet boundary conditions are prescribed.
+    mg_constrained_dofs.clear();
+    mg_constrained_dofs.initialize(dof_handler);
+    mg_constrained_dofs.make_zero_boundary_constraints(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 <code>n_levels-1</code>. 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_levels();
+
+    mg_interface_matrices.resize(0, n_levels - 1);
+    mg_matrices.resize(0, n_levels - 1);
+    mg_sparsity_patterns.resize(0, n_levels - 1);
+    mg_interface_sparsity_patterns.resize(0, n_levels - 1);
+
+    // Now, we have to provide a matrix on each level. To this end, we first use
+    // the MGTools::make_sparsity_pattern function to 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 the interface matrices with
+    // the fitting sparsity pattern.
+    //
+    // 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. Therefore, we use a function specifically build for this
+    // purpose to generate it.
+    for (unsigned int level = 0; level < n_levels; ++level)
+      {
+        {
+          DynamicSparsityPattern dsp(dof_handler.n_dofs(level),
+                                     dof_handler.n_dofs(level));
+          MGTools::make_sparsity_pattern(dof_handler, dsp, level);
+
+          mg_sparsity_patterns[level].copy_from(dsp);
+          mg_matrices[level].reinit(mg_sparsity_patterns[level]);
+        }
+        {
+          DynamicSparsityPattern dsp(dof_handler.n_dofs(level),
+                                     dof_handler.n_dofs(level));
+          MGTools::make_interface_sparsity_pattern(dof_handler,
+                                                   mg_constrained_dofs,
+                                                   dsp,
+                                                   level);
+          mg_interface_sparsity_patterns[level].copy_from(dsp);
+          mg_interface_matrices[level].reinit(
+            mg_interface_sparsity_patterns[level]);
+        }
+      }
+  }
+
+
+  // @sect4{LaplaceProblem::cell_worker}
+
+  // The cell_worker function is used to assemble the matrix and right-hand side
+  // on the given cell. This function is used for the active cells to generate
+  // the system_matrix and on each level to build the level matrices.
+  //
+  // Note that we also assemble a right-hand side when called from
+  // assemble_multigrid() even though it is not used.
+  template <int dim>
+  template <class Iterator>
+  void LaplaceProblem<dim>::cell_worker(const Iterator &  cell,
+                                        ScratchData<dim> &scratch_data,
+                                        CopyData &        copy_data)
+  {
+    FEValues<dim> &fe_values = scratch_data.fe_values;
+    fe_values.reinit(cell);
+
+    const unsigned int dofs_per_cell = fe_values.get_fe().dofs_per_cell;
+    const unsigned int n_q_points    = fe_values.get_quadrature().size();
+
+    copy_data.reinit(cell, dofs_per_cell);
+
+    const std::vector<double> &JxW = fe_values.get_JxW_values();
+
+    for (unsigned int q = 0; q < n_q_points; ++q)
+      {
+        const double coefficient =
+          (fe_values.get_quadrature_points()[q][0] < 0.0) ? 1.0 : 0.1;
+        //(cell->center().square() < 0.5 * 0.5) ? 10.0:1.0;
+
+        for (unsigned int i = 0; i < dofs_per_cell; ++i)
+          {
+            for (unsigned int j = 0; j < dofs_per_cell; ++j)
+              {
+                copy_data.cell_matrix(i, j) +=
+                  coefficient *
+                  (fe_values.shape_grad(i, q) * fe_values.shape_grad(j, q)) *
+                  JxW[q];
+              }
+            copy_data.cell_rhs(i) += 1.0 * fe_values.shape_value(i, q) * JxW[q];
+          }
+      }
+  }
+
+
+
+  // @sect4{LaplaceProblem::assemble_system}
+
+  // The following function assembles the linear system on the active cells of
+  // the mesh. For this, we pass two lambda functions to the mesh_loop()
+  // function. The cell_worker function redirects to the class member function
+  // of the same name, while the copier is specific to this function and copies
+  // local matrix and vector to the corresponding global ones using the
+  // constraints.
+  template <int dim>
+  void LaplaceProblem<dim>::assemble_system()
+  {
+    MappingQ1<dim> mapping;
+
+    auto cell_worker =
+      [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
+          ScratchData<dim> &                                    scratch_data,
+          CopyData &                                            copy_data) {
+        this->cell_worker(cell, scratch_data, copy_data);
+      };
+
+    auto copier = [&](const CopyData &cd) {
+      this->constraints.distribute_local_to_global(cd.cell_matrix,
+                                                   cd.cell_rhs,
+                                                   cd.local_dof_indices,
+                                                   system_matrix,
+                                                   system_rhs);
+    };
+
+    const unsigned int n_gauss_points = degree + 1;
+
+    ScratchData<dim> scratch_data(mapping,
+                                  fe,
+                                  n_gauss_points,
+                                  update_values | update_gradients |
+                                    update_JxW_values |
+                                    update_quadrature_points);
+
+    MeshWorker::mesh_loop(dof_handler.begin_active(),
+                          dof_handler.end(),
+                          cell_worker,
+                          copier,
+                          scratch_data,
+                          CopyData(),
+                          MeshWorker::assemble_own_cells);
+  }
+
+
+  // @sect4{LaplaceProblem::assemble_multigrid}
+
+  // The next function is the one that builds the 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 level matrices. Fortunately, MeshWorker hides most of that from
+  // us, and thus the difference between this function and the previous lies
+  // only in the setup of the assembler and the different iterators in the loop.
+  //
+  // We generate an AffineConstraints<> object for each level containing the
+  // boundary and interface dofs as constrained entries. The corresponding
+  // object is then used to generate the level matrices.
+  template <int dim>
+  void LaplaceProblem<dim>::assemble_multigrid()
+  {
+    MappingQ1<dim>     mapping;
+    const unsigned int n_levels = triangulation.n_levels();
+
+    std::vector<AffineConstraints<>> boundary_constraints(n_levels);
+    for (unsigned int level = 0; level < n_levels; ++level)
+      {
+        IndexSet dofset;
+        DoFTools::extract_locally_relevant_level_dofs(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();
+      }
+
+    auto cell_worker =
+      [&](const typename DoFHandler<dim>::level_cell_iterator &cell,
+          ScratchData<dim> &                                   scratch_data,
+          CopyData &                                           copy_data) {
+        this->cell_worker(cell, scratch_data, copy_data);
+      };
+
+    auto copier = [&](const CopyData &cd) {
+      boundary_constraints[cd.level].distribute_local_to_global(
+        cd.cell_matrix, cd.local_dof_indices, mg_matrices[cd.level]);
+
+      const unsigned int dofs_per_cell = cd.local_dof_indices.size();
+
+      // TODO EXPLAIN:
+
+      for (unsigned int i = 0; i < dofs_per_cell; ++i)
+        for (unsigned int j = 0; j < dofs_per_cell; ++j)
+          if (mg_constrained_dofs.is_interface_matrix_entry(
+                cd.level, cd.local_dof_indices[i], cd.local_dof_indices[j]))
+            {
+              mg_interface_matrices[cd.level].add(cd.local_dof_indices[i],
+                                                  cd.local_dof_indices[j],
+                                                  cd.cell_matrix(i, j));
+            }
+    };
+
+    const unsigned int n_gauss_points = degree + 1;
+
+    ScratchData<dim> scratch_data(mapping,
+                                  fe,
+                                  n_gauss_points,
+                                  update_values | update_gradients |
+                                    update_JxW_values |
+                                    update_quadrature_points);
+
+    MeshWorker::mesh_loop(dof_handler.begin_mg(),
+                          dof_handler.end_mg(),
+                          cell_worker,
+                          copier,
+                          scratch_data,
+                          CopyData(),
+                          MeshWorker::assemble_own_cells);
+  }
+
+
+
+  // @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
+  // of the final linear system and the MGConstrainedDoFs object that knows
+  // about the boundary conditions on the each level and the degrees of
+  // freedom on interfaces between different refinement level can build the
+  // matrices for those transfer operations from a DoFHandler object with
+  // level degrees of freedom.
+  //
+  // 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 <int dim>
+  void LaplaceProblem<dim>::solve()
+  {
+    MGTransferPrebuilt<Vector<double>> mg_transfer(mg_constrained_dofs);
+    mg_transfer.build_matrices(dof_handler);
+
+    FullMatrix<double> coarse_matrix;
+    coarse_matrix.copy_from(mg_matrices[0]);
+    MGCoarseGridHouseholder<> coarse_grid_solver;
+    coarse_grid_solver.initialize(coarse_matrix);
+
+    // 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) or a small number of iterations of a solver method (such as CG or
+    // GMRES). The mg::SmootherRelaxation and MGSmootherPrecondition classes
+    // provide support for these two kinds of smoothers. 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
+    // <code>initialize()</code> 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 <code>set_steps()</code> 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 <code>mg_smoother</code> 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 = PreconditionSOR<SparseMatrix<double>>;
+    mg::SmootherRelaxation<Smoother, Vector<double>> mg_smoother;
+    mg_smoother.initialize(mg_matrices);
+    mg_smoother.set_steps(2);
+    mg_smoother.set_symmetric(true);
+
+    // 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<Vector<double>> mg_matrix(mg_matrices);
+    mg::Matrix<Vector<double>> mg_interface_up(mg_interface_matrices);
+    mg::Matrix<Vector<double>> mg_interface_down(mg_interface_matrices);
+
+    // Now, we are ready to set up the V-cycle operator and the multilevel
+    // preconditioner.
+    Multigrid<Vector<double>> mg(
+      mg_matrix, coarse_grid_solver, mg_transfer, mg_smoother, mg_smoother);
+    mg.set_edge_matrices(mg_interface_down, mg_interface_up);
+
+    PreconditionMG<dim, Vector<double>, MGTransferPrebuilt<Vector<double>>>
+      preconditioner(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(1000, 1e-12);
+    SolverCG<>    solver(solver_control);
+
+    solution = 0;
+
+    solver.solve(system_matrix, solution, system_rhs, preconditioner);
+    std::cout << "   Number of CG iterations: " << solver_control.last_step()
+              << "\n"
+              << 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 functions are almost unchanged from those in step-6.
+  template <int dim>
+  void LaplaceProblem<dim>::refine_grid()
+  {
+    Vector<float> estimated_error_per_cell(triangulation.n_active_cells());
+
+    KellyErrorEstimator<dim>::estimate(
+      dof_handler,
+      QGauss<dim - 1>(degree + 2),
+      std::map<types::boundary_id, const Function<dim> *>(),
+      solution,
+      estimated_error_per_cell);
+    GridRefinement::refine_and_coarsen_fixed_number(triangulation,
+                                                    estimated_error_per_cell,
+                                                    0.3,
+                                                    0.03);
+    triangulation.execute_coarsening_and_refinement();
+  }
+
+
+
+  template <int dim>
+  void LaplaceProblem<dim>::output_results(const unsigned int cycle) 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("solution-" + std::to_string(cycle) + ".vtk");
+    data_out.write_vtk(output);
+  }
+
+
+  // @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
+  // <code>assemble_multigrid</code> that takes care of forming the matrices
+  // on every level that we need in the multigrid method.
+  template <int dim>
+  void LaplaceProblem<dim>::run()
+  {
+    for (unsigned int cycle = 0; cycle < 8; ++cycle)
+      {
+        std::cout << "Cycle " << cycle << std::endl;
+
+        if (cycle == 0)
+          {
+            GridGenerator::hyper_ball(triangulation);
+            triangulation.refine_global(2);
+          }
+        else
+          refine_grid();
+
+        std::cout << "   Number of active cells:       "
+                  << triangulation.n_active_cells() << std::endl;
+
+        setup_system();
+
+        assemble_system();
+        assemble_multigrid();
+
+        solve();
+        output_results(cycle);
+      }
+  }
+} // namespace Step16
+
+
+// @sect3{The main() function}
+//
+// This is again the same function as in step-6:
+int main()
+{
+  try
+    {
+      using namespace Step16;
+
+      LaplaceProblem<2> laplace_problem(1);
+      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;
+}

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