step-50.h

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,
 *                          const unsigned int /*component*/ = 0) const override
 *     {
 *       return 1.0;
 *     }
 *   
 *   
 *     template <typename number>
 *     VectorizedArray<number>
 *     value(const Point<dim, VectorizedArray<number>> & /*p*/,
 *           const unsigned int /*component*/ = 0) const
 *     {
 *       return VectorizedArray<number>(1.0);
 *     }
 *   };
 *   
 *   
 * @endcode
 * 
 * This next class represents the diffusion coefficient. We use a variable
 * coefficient which is 100.0 at any point where at least one coordinate is
 * less than -0.5, and 1.0 at all other points. As above, a separate value()
 * returning a VectorizedArray is used for the matrix-free code. An @p
 * average() function computes the arithmetic average for a set of points.
 * 
 * @code
 *   template <int dim>
 *   class Coefficient : public Function<dim>
 *   {
 *   public:
 *     virtual double value(const Point<dim> &p,
 *                          const unsigned int /*component*/ = 0) const override;
 *   
 *     template <typename number>
 *     VectorizedArray<number> value(const Point<dim, VectorizedArray<number>> &p,
 *                                   const unsigned int /*component*/ = 0) const;
 *   
 *     template <typename number>
 *     number average_value(const std::vector<Point<dim, number>> &points) const;
 *   
 * @endcode
 * 
 * When using a coefficient in the MatrixFree framework, we also
 * need a function that creates a Table of coefficient values for a
 * set of cells provided by the MatrixFree operator argument here.
 * 
 * @code
 *     template <typename number>
 *     std::shared_ptr<Table<2, VectorizedArray<number>>> make_coefficient_table(
 *       const MatrixFree<dim, number, VectorizedArray<number>> &mf_storage) const;
 *   };
 *   
 *   
 *   
 *   template <int dim>
 *   double Coefficient<dim>::value(const Point<dim> &p, const unsigned int) const
 *   {
 *     for (int d = 0; d < dim; ++d)
 *       {
 *         if (p[d] < -0.5)
 *           return 100.0;
 *       }
 *     return 1.0;
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   template <typename number>
 *   VectorizedArray<number>
 *   Coefficient<dim>::value(const Point<dim, VectorizedArray<number>> &p,
 *                           const unsigned int) const
 *   {
 *     VectorizedArray<number> return_value = VectorizedArray<number>(1.0);
 *     for (unsigned int i = 0; i < VectorizedArray<number>::size(); ++i)
 *       {
 *         for (int d = 0; d < dim; ++d)
 *           if (p[d][i] < -0.5)
 *             {
 *               return_value[i] = 100.0;
 *               break;
 *             }
 *       }
 *   
 *     return return_value;
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   template <typename number>
 *   number Coefficient<dim>::average_value(
 *     const std::vector<Point<dim, number>> &points) const
 *   {
 *     number average(0);
 *     for (unsigned int i = 0; i < points.size(); ++i)
 *       average += value(points[i]);
 *     average /= points.size();
 *   
 *     return average;
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   template <typename number>
 *   std::shared_ptr<Table<2, VectorizedArray<number>>>
 *   Coefficient<dim>::make_coefficient_table(
 *     const MatrixFree<dim, number, VectorizedArray<number>> &mf_storage) const
 *   {
 *     auto coefficient_table =
 *       std::make_shared<Table<2, VectorizedArray<number>>>();
 *   
 *     FEEvaluation<dim, -1, 0, 1, number> fe_eval(mf_storage);
 *   
 *     const unsigned int n_cells    = mf_storage.n_cell_batches();
 *     const unsigned int n_q_points = fe_eval.n_q_points;
 *   
 *     coefficient_table->reinit(n_cells, 1);
 *   
 *     for (unsigned int cell = 0; cell < n_cells; ++cell)
 *       {
 *         fe_eval.reinit(cell);
 *   
 *         VectorizedArray<number> average_value = 0.;
 *         for (unsigned int q = 0; q < n_q_points; ++q)
 *           average_value += value(fe_eval.quadrature_point(q));
 *         average_value /= n_q_points;
 *   
 *         (*coefficient_table)(cell, 0) = average_value;
 *       }
 *   
 *     return coefficient_table;
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Runtimeparameters"></a> 
 * <h3>Run time parameters</h3>
 * 
 
 * 
 * We will use ParameterHandler to pass in parameters at runtime.  The
 * structure @p Settings parses and stores these parameters to be queried
 * throughout the program.
 * 
 * @code
 *   struct Settings
 *   {
 *     bool try_parse(const std::string &prm_filename);
 *   
 *     enum SolverType
 *     {
 *       gmg_mb,
 *       gmg_mf,
 *       amg
 *     };
 *   
 *     SolverType solver;
 *   
 *     int          dimension;
 *     double       smoother_dampen;
 *     unsigned int smoother_steps;
 *     unsigned int n_steps;
 *     bool         output;
 *   };
 *   
 *   
 *   
 *   bool Settings::try_parse(const std::string &prm_filename)
 *   {
 *     ParameterHandler prm;
 *     prm.declare_entry("dim", "2", Patterns::Integer(), "The problem dimension.");
 *     prm.declare_entry("n_steps",
 *                       "10",
 *                       Patterns::Integer(0),
 *                       "Number of adaptive refinement steps.");
 *     prm.declare_entry("smoother dampen",
 *                       "1.0",
 *                       Patterns::Double(0.0),
 *                       "Dampen factor for the smoother.");
 *     prm.declare_entry("smoother steps",
 *                       "1",
 *                       Patterns::Integer(1),
 *                       "Number of smoother steps.");
 *     prm.declare_entry("solver",
 *                       "MF",
 *                       Patterns::Selection("MF|MB|AMG"),
 *                       "Switch between matrix-free GMG, "
 *                       "matrix-based GMG, and AMG.");
 *     prm.declare_entry("output",
 *                       "false",
 *                       Patterns::Bool(),
 *                       "Output graphical results.");
 *   
 *     if (prm_filename.size() == 0)
 *       {
 *         std::cout << "****  Error: No input file provided!\n"
 *                   << "****  Error: Call this program as './step-50 input.prm\n"
 *                   << '\n'
 *                   << "****  You may want to use one of the input files in this\n"
 *                   << "****  directory, or use the following default values\n"
 *                   << "****  to create an input file:\n";
 *         if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
 *           prm.print_parameters(std::cout, ParameterHandler::Text);
 *         return false;
 *       }
 *   
 *     try
 *       {
 *         prm.parse_input(prm_filename);
 *       }
 *     catch (std::exception &e)
 *       {
 *         if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
 *           std::cerr << e.what() << std::endl;
 *         return false;
 *       }
 *   
 *     if (prm.get("solver") == "MF")
 *       this->solver = gmg_mf;
 *     else if (prm.get("solver") == "MB")
 *       this->solver = gmg_mb;
 *     else if (prm.get("solver") == "AMG")
 *       this->solver = amg;
 *     else
 *       AssertThrow(false, ExcNotImplemented());
 *   
 *     this->dimension       = prm.get_integer("dim");
 *     this->n_steps         = prm.get_integer("n_steps");
 *     this->smoother_dampen = prm.get_double("smoother dampen");
 *     this->smoother_steps  = prm.get_integer("smoother steps");
 *     this->output          = prm.get_bool("output");
 *   
 *     return true;
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemclass"></a> 
 * <h3>LaplaceProblem class</h3>
 * 
 
 * 
 * This is the main class of the program. It looks very similar to
 * @ref step_16 "step-16", @ref step_37 "step-37", and @ref step_40 "step-40". For the MatrixFree setup, we use the
 * MatrixFreeOperators::LaplaceOperator class which defines `local_apply()`,
 * `compute_diagonal()`, and `set_coefficient()` functions internally. Note that
 * the polynomial degree is a template parameter of this class. This is
 * necessary for the matrix-free code.
 * 
 * @code
 *   template <int dim, int degree>
 *   class LaplaceProblem
 *   {
 *   public:
 *     LaplaceProblem(const Settings &settings);
 *     void run();
 *   
 *   private:
 * @endcode
 * 
 * We will use the following types throughout the program. First the
 * matrix-based types, after that the matrix-free classes. For the
 * matrix-free implementation, we use @p float for the level operators.
 * 
 * @code
 *     using MatrixType      = LA::MPI::SparseMatrix;
 *     using VectorType      = LA::MPI::Vector;
 *     using PreconditionAMG = LA::MPI::PreconditionAMG;
 *   
 *     using MatrixFreeLevelMatrix = MatrixFreeOperators::LaplaceOperator<
 *       dim,
 *       degree,
 *       degree + 1,
 *       1,
 *       LinearAlgebra::distributed::Vector<float>>;
 *     using MatrixFreeActiveMatrix = MatrixFreeOperators::LaplaceOperator<
 *       dim,
 *       degree,
 *       degree + 1,
 *       1,
 *       LinearAlgebra::distributed::Vector<double>>;
 *   
 *     using MatrixFreeLevelVector  = LinearAlgebra::distributed::Vector<float>;
 *     using MatrixFreeActiveVector = LinearAlgebra::distributed::Vector<double>;
 *   
 *     void setup_system();
 *     void setup_multigrid();
 *     void assemble_system();
 *     void assemble_multigrid();
 *     void assemble_rhs();
 *     void solve();
 *     void estimate();
 *     void refine_grid();
 *     void output_results(const unsigned int cycle);
 *   
 *     Settings settings;
 *   
 *     MPI_Comm           mpi_communicator;
 *     ConditionalOStream pcout;
 *   
 *     parallel::distributed::Triangulation<dim> triangulation;
 *     const MappingQ1<dim>                      mapping;
 *     FE_Q<dim>                                 fe;
 *   
 *     DoFHandler<dim> dof_handler;
 *   
 *     IndexSet                  locally_owned_dofs;
 *     IndexSet                  locally_relevant_dofs;
 *     AffineConstraints<double> constraints;
 *   
 *     MatrixType             system_matrix;
 *     MatrixFreeActiveMatrix mf_system_matrix;
 *     VectorType             solution;
 *     VectorType             right_hand_side;
 *     Vector<double>         estimated_error_square_per_cell;
 *   
 *     MGLevelObject<MatrixType> mg_matrix;
 *     MGLevelObject<MatrixType> mg_interface_in;
 *     MGConstrainedDoFs         mg_constrained_dofs;
 *   
 *     MGLevelObject<MatrixFreeLevelMatrix> mf_mg_matrix;
 *   
 *     TimerOutput computing_timer;
 *   };
 *   
 *   
 * @endcode
 * 
 * The only interesting part about the constructor is that we construct the
 * multigrid hierarchy unless we use AMG. For that, we need to parse the
 * run time parameters before this constructor completes.
 * 
 * @code
 *   template <int dim, int degree>
 *   LaplaceProblem<dim, degree>::LaplaceProblem(const Settings &settings)
 *     : settings(settings)
 *     , mpi_communicator(MPI_COMM_WORLD)
 *     , pcout(std::cout, (Utilities::MPI::this_mpi_process(mpi_communicator) == 0))
 *     , triangulation(mpi_communicator,
 *                     Triangulation<dim>::limit_level_difference_at_vertices,
 *                     (settings.solver == Settings::amg) ?
 *                       parallel::distributed::Triangulation<dim>::default_setting :
 *                       parallel::distributed::Triangulation<
 *                         dim>::construct_multigrid_hierarchy)
 *     , mapping()
 *     , fe(degree)
 *     , dof_handler(triangulation)
 *     , computing_timer(pcout, TimerOutput::never, TimerOutput::wall_times)
 *   {
 *     GridGenerator::hyper_L(triangulation, -1., 1., /*colorize*/ false);
 *     triangulation.refine_global(1);
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemsetup_system"></a> 
 * <h4>LaplaceProblem::setup_system()</h4>
 * 
 
 * 
 * Unlike @ref step_16 "step-16" and @ref step_37 "step-37", we split the set up into two parts,
 * setup_system() and setup_multigrid(). Here is the typical setup_system()
 * function for the active mesh found in most tutorials. For matrix-free, the
 * active mesh set up is similar to @ref step_37 "step-37"; for matrix-based (GMG and AMG
 * solvers), the setup is similar to @ref step_40 "step-40".
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::setup_system()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Setup");
 *   
 *     dof_handler.distribute_dofs(fe);
 *   
 *     locally_relevant_dofs = DoFTools::extract_locally_relevant_dofs(dof_handler);
 *     locally_owned_dofs    = dof_handler.locally_owned_dofs();
 *   
 *     solution.reinit(locally_owned_dofs, mpi_communicator);
 *     right_hand_side.reinit(locally_owned_dofs, mpi_communicator);
 *     constraints.reinit(locally_relevant_dofs);
 *     DoFTools::make_hanging_node_constraints(dof_handler, constraints);
 *   
 *     VectorTools::interpolate_boundary_values(
 *       mapping, dof_handler, 0, Functions::ZeroFunction<dim>(), constraints);
 *     constraints.close();
 *   
 *     switch (settings.solver)
 *       {
 *         case Settings::gmg_mf:
 *           {
 *             typename MatrixFree<dim, double>::AdditionalData additional_data;
 *             additional_data.tasks_parallel_scheme =
 *               MatrixFree<dim, double>::AdditionalData::none;
 *             additional_data.mapping_update_flags =
 *               (update_gradients | update_JxW_values | update_quadrature_points);
 *             std::shared_ptr<MatrixFree<dim, double>> mf_storage =
 *               std::make_shared<MatrixFree<dim, double>>();
 *             mf_storage->reinit(mapping,
 *                                dof_handler,
 *                                constraints,
 *                                QGauss<1>(degree + 1),
 *                                additional_data);
 *   
 *             mf_system_matrix.initialize(mf_storage);
 *   
 *             const Coefficient<dim> coefficient;
 *             mf_system_matrix.set_coefficient(
 *               coefficient.make_coefficient_table(*mf_storage));
 *   
 *             break;
 *           }
 *   
 *         case Settings::gmg_mb:
 *         case Settings::amg:
 *           {
 *   #ifdef USE_PETSC_LA
 *             DynamicSparsityPattern dsp(locally_relevant_dofs);
 *             DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints);
 *   
 *             SparsityTools::distribute_sparsity_pattern(dsp,
 *                                                        locally_owned_dofs,
 *                                                        mpi_communicator,
 *                                                        locally_relevant_dofs);
 *   
 *             system_matrix.reinit(locally_owned_dofs,
 *                                  locally_owned_dofs,
 *                                  dsp,
 *                                  mpi_communicator);
 *   #else
 *             TrilinosWrappers::SparsityPattern dsp(locally_owned_dofs,
 *                                                   locally_owned_dofs,
 *                                                   locally_relevant_dofs,
 *                                                   mpi_communicator);
 *             DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints);
 *             dsp.compress();
 *             system_matrix.reinit(dsp);
 *   #endif
 *   
 *             break;
 *           }
 *   
 *         default:
 *           Assert(false, ExcNotImplemented());
 *       }
 *   }
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemsetup_multigrid"></a> 
 * <h4>LaplaceProblem::setup_multigrid()</h4>
 * 
 
 * 
 * This function does the multilevel setup for both matrix-free and
 * matrix-based GMG. The matrix-free setup is similar to that of @ref step_37 "step-37", and
 * the matrix-based is similar to @ref step_16 "step-16", except we must use appropriate
 * distributed sparsity patterns.
 * 
 
 * 
 * The function is not called for the AMG approach, but to err on the
 * safe side, the main `switch` statement of this function
 * nevertheless makes sure that the function only operates on known
 * multigrid settings by throwing an assertion if the function were
 * called for anything other than the two geometric multigrid methods.
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::setup_multigrid()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Setup multigrid");
 *   
 *     dof_handler.distribute_mg_dofs();
 *   
 *     mg_constrained_dofs.clear();
 *     mg_constrained_dofs.initialize(dof_handler);
 *   
 *     const std::set<types::boundary_id> boundary_ids = {types::boundary_id(0)};
 *     mg_constrained_dofs.make_zero_boundary_constraints(dof_handler, boundary_ids);
 *   
 *     const unsigned int n_levels = triangulation.n_global_levels();
 *   
 *     switch (settings.solver)
 *       {
 *         case Settings::gmg_mf:
 *           {
 *             mf_mg_matrix.resize(0, n_levels - 1);
 *   
 *             for (unsigned int level = 0; level < n_levels; ++level)
 *               {
 *                 const IndexSet relevant_dofs =
 *                   DoFTools::extract_locally_relevant_level_dofs(dof_handler,
 *                                                                 level);
 *                 AffineConstraints<double> level_constraints;
 *                 level_constraints.reinit(relevant_dofs);
 *                 level_constraints.add_lines(
 *                   mg_constrained_dofs.get_boundary_indices(level));
 *                 level_constraints.close();
 *   
 *                 typename MatrixFree<dim, float>::AdditionalData additional_data;
 *                 additional_data.tasks_parallel_scheme =
 *                   MatrixFree<dim, float>::AdditionalData::none;
 *                 additional_data.mapping_update_flags =
 *                   (update_gradients | update_JxW_values |
 *                    update_quadrature_points);
 *                 additional_data.mg_level = level;
 *                 std::shared_ptr<MatrixFree<dim, float>> mf_storage_level(
 *                   new MatrixFree<dim, float>());
 *                 mf_storage_level->reinit(mapping,
 *                                          dof_handler,
 *                                          level_constraints,
 *                                          QGauss<1>(degree + 1),
 *                                          additional_data);
 *   
 *                 mf_mg_matrix[level].initialize(mf_storage_level,
 *                                                mg_constrained_dofs,
 *                                                level);
 *   
 *                 const Coefficient<dim> coefficient;
 *                 mf_mg_matrix[level].set_coefficient(
 *                   coefficient.make_coefficient_table(*mf_storage_level));
 *   
 *                 mf_mg_matrix[level].compute_diagonal();
 *               }
 *   
 *             break;
 *           }
 *   
 *         case Settings::gmg_mb:
 *           {
 *             mg_matrix.resize(0, n_levels - 1);
 *             mg_matrix.clear_elements();
 *             mg_interface_in.resize(0, n_levels - 1);
 *             mg_interface_in.clear_elements();
 *   
 *             for (unsigned int level = 0; level < n_levels; ++level)
 *               {
 *                 const IndexSet dof_set =
 *                   DoFTools::extract_locally_relevant_level_dofs(dof_handler,
 *                                                                 level);
 *   
 *                 {
 *   #ifdef USE_PETSC_LA
 *                   DynamicSparsityPattern dsp(dof_set);
 *                   MGTools::make_sparsity_pattern(dof_handler, dsp, level);
 *                   dsp.compress();
 *                   SparsityTools::distribute_sparsity_pattern(
 *                     dsp,
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     mpi_communicator,
 *                     dof_set);
 *   
 *                   mg_matrix[level].reinit(
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dsp,
 *                     mpi_communicator);
 *   #else
 *                   TrilinosWrappers::SparsityPattern dsp(
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_set,
 *                     mpi_communicator);
 *                   MGTools::make_sparsity_pattern(dof_handler, dsp, level);
 *   
 *                   dsp.compress();
 *                   mg_matrix[level].reinit(dsp);
 *   #endif
 *                 }
 *   
 *                 {
 *   #ifdef USE_PETSC_LA
 *                   DynamicSparsityPattern dsp(dof_set);
 *                   MGTools::make_interface_sparsity_pattern(dof_handler,
 *                                                            mg_constrained_dofs,
 *                                                            dsp,
 *                                                            level);
 *                   dsp.compress();
 *                   SparsityTools::distribute_sparsity_pattern(
 *                     dsp,
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     mpi_communicator,
 *                     dof_set);
 *   
 *                   mg_interface_in[level].reinit(
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dsp,
 *                     mpi_communicator);
 *   #else
 *                   TrilinosWrappers::SparsityPattern dsp(
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_handler.locally_owned_mg_dofs(level),
 *                     dof_set,
 *                     mpi_communicator);
 *   
 *                   MGTools::make_interface_sparsity_pattern(dof_handler,
 *                                                            mg_constrained_dofs,
 *                                                            dsp,
 *                                                            level);
 *                   dsp.compress();
 *                   mg_interface_in[level].reinit(dsp);
 *   #endif
 *                 }
 *               }
 *             break;
 *           }
 *   
 *         default:
 *           Assert(false, ExcNotImplemented());
 *       }
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemassemble_system"></a> 
 * <h4>LaplaceProblem::assemble_system()</h4>
 * 
 
 * 
 * The assembly is split into three parts: `assemble_system()`,
 * `assemble_multigrid()`, and `assemble_rhs()`. The
 * `assemble_system()` function here assembles and stores the (global)
 * system matrix and the right-hand side for the matrix-based
 * methods. It is similar to the assembly in @ref step_40 "step-40".
 * 
 
 * 
 * Note that the matrix-free method does not execute this function as it does
 * not need to assemble a matrix, and it will instead assemble the right-hand
 * side in assemble_rhs().
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::assemble_system()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Assemble");
 *   
 *     const QGauss<dim> quadrature_formula(degree + 1);
 *   
 *     FEValues<dim> fe_values(fe,
 *                             quadrature_formula,
 *                             update_values | update_gradients |
 *                               update_quadrature_points | update_JxW_values);
 *   
 *     const unsigned int dofs_per_cell = fe.n_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_rhs(dofs_per_cell);
 *   
 *     std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *   
 *     const Coefficient<dim> coefficient;
 *     RightHandSide<dim>     rhs;
 *     std::vector<double>    rhs_values(n_q_points);
 *   
 *     for (const auto &cell : dof_handler.active_cell_iterators())
 *       if (cell->is_locally_owned())
 *         {
 *           cell_matrix = 0;
 *           cell_rhs    = 0;
 *   
 *           fe_values.reinit(cell);
 *   
 *           const double coefficient_value =
 *             coefficient.average_value(fe_values.get_quadrature_points());
 *           rhs.value_list(fe_values.get_quadrature_points(), rhs_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_value *                // epsilon(x)
 *                     fe_values.shape_grad(i, q_point) * // * grad phi_i(x)
 *                     fe_values.shape_grad(j, q_point) * // * grad phi_j(x)
 *                     fe_values.JxW(q_point);            // * dx
 *   
 *                 cell_rhs(i) +=
 *                   fe_values.shape_value(i, q_point) * // grad phi_i(x)
 *                   rhs_values[q_point] *               // * f(x)
 *                   fe_values.JxW(q_point);             // * dx
 *               }
 *   
 *           cell->get_dof_indices(local_dof_indices);
 *           constraints.distribute_local_to_global(cell_matrix,
 *                                                  cell_rhs,
 *                                                  local_dof_indices,
 *                                                  system_matrix,
 *                                                  right_hand_side);
 *         }
 *   
 *     system_matrix.compress(VectorOperation::add);
 *     right_hand_side.compress(VectorOperation::add);
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemassemble_multigrid"></a> 
 * <h4>LaplaceProblem::assemble_multigrid()</h4>
 * 
 
 * 
 * The following function assembles and stores the multilevel matrices for the
 * matrix-based GMG method. This function is similar to the one found in
 * @ref step_16 "step-16", only here it works for distributed meshes. This difference amounts
 * to adding a condition that we only assemble on locally owned level cells and
 * a call to compress() for each matrix that is built.
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::assemble_multigrid()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Assemble multigrid");
 *   
 *     QGauss<dim> quadrature_formula(degree + 1);
 *   
 *     FEValues<dim> fe_values(fe,
 *                             quadrature_formula,
 *                             update_values | update_gradients |
 *                               update_quadrature_points | update_JxW_values);
 *   
 *     const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
 *     const unsigned int n_q_points    = quadrature_formula.size();
 *   
 *     FullMatrix<double> cell_matrix(dofs_per_cell, dofs_per_cell);
 *   
 *     std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *   
 *     const Coefficient<dim> coefficient;
 *   
 *     std::vector<AffineConstraints<double>> boundary_constraints(
 *       triangulation.n_global_levels());
 *     for (unsigned int level = 0; level < triangulation.n_global_levels(); ++level)
 *       {
 *         const IndexSet dof_set =
 *           DoFTools::extract_locally_relevant_level_dofs(dof_handler, level);
 *         boundary_constraints[level].reinit(dof_set);
 *         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();
 *       }
 *   
 *     for (const auto &cell : dof_handler.cell_iterators())
 *       if (cell->level_subdomain_id() == triangulation.locally_owned_subdomain())
 *         {
 *           cell_matrix = 0;
 *           fe_values.reinit(cell);
 *   
 *           const double coefficient_value =
 *             coefficient.average_value(fe_values.get_quadrature_points());
 *   
 *           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_value * fe_values.shape_grad(i, q_point) *
 *                   fe_values.shape_grad(j, q_point) * fe_values.JxW(q_point);
 *   
 *           cell->get_mg_dof_indices(local_dof_indices);
 *   
 *           boundary_constraints[cell->level()].distribute_local_to_global(
 *             cell_matrix, local_dof_indices, mg_matrix[cell->level()]);
 *   
 *           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(
 *                     cell->level(), local_dof_indices[i], local_dof_indices[j]))
 *                 mg_interface_in[cell->level()].add(local_dof_indices[i],
 *                                                    local_dof_indices[j],
 *                                                    cell_matrix(i, j));
 *         }
 *   
 *     for (unsigned int i = 0; i < triangulation.n_global_levels(); ++i)
 *       {
 *         mg_matrix[i].compress(VectorOperation::add);
 *         mg_interface_in[i].compress(VectorOperation::add);
 *       }
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemassemble_rhs"></a> 
 * <h4>LaplaceProblem::assemble_rhs()</h4>
 * 
 
 * 
 * The final function in this triptych assembles the right-hand side
 * vector for the matrix-free method -- because in the matrix-free
 * framework, we don't have to assemble the matrix and can get away
 * with only assembling the right hand side. We could do this by extracting the
 * code from the `assemble_system()` function above that deals with the right
 * hand side, but we decide instead to go all in on the matrix-free approach and
 * do the assembly using that way as well.
 * 
 
 * 
 * The result is a function that is similar
 * to the one found in the "Use FEEvaluation::read_dof_values_plain()
 * to avoid resolving constraints" subsection in the "Possibilities
 * for extensions" section of @ref step_37 "step-37".
 * 
 
 * 
 * The reason for this function is that the MatrixFree operators do not take
 * into account non-homogeneous Dirichlet constraints, instead treating all
 * Dirichlet constraints as homogeneous. To account for this, the right-hand
 * side here is assembled as the residual @f$r_0 = f-Au_0@f$, where @f$u_0@f$ is a
 * zero vector except in the Dirichlet values. Then when solving, we have that
 * the solution is @f$u = u_0 + A^{-1}r_0@f$. This can be seen as a Newton
 * iteration on a linear system with initial guess @f$u_0@f$. The CG solve in the
 * `solve()` function below computes @f$A^{-1}r_0@f$ and the call to
 * `constraints.distribute()` (which directly follows) adds the @f$u_0@f$.
 * 
 
 * 
 * Obviously, since we are considering a problem with zero Dirichlet boundary,
 * we could have taken a similar approach to @ref step_37 "step-37" `assemble_rhs()`, but this
 * additional work allows us to change the problem declaration if we so
 * choose.
 * 
 
 * 
 * This function has two parts in the integration loop: applying the negative
 * of matrix @f$A@f$ to @f$u_0@f$ by submitting the negative of the gradient, and adding
 * the right-hand side contribution by submitting the value @f$f@f$. We must be sure
 * to use `read_dof_values_plain()` for evaluating @f$u_0@f$ as `read_dof_values()`
 * would set all Dirichlet values to zero.
 * 
 
 * 
 * Finally, the system_rhs vector is of type LA::MPI::Vector, but the
 * MatrixFree class only work for
 * LinearAlgebra::distributed::Vector.  Therefore we must
 * compute the right-hand side using MatrixFree functionality and then
 * use the functions in the `ChangeVectorType` namespace to copy it to
 * the correct type.
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::assemble_rhs()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Assemble right-hand side");
 *   
 *     MatrixFreeActiveVector solution_copy;
 *     MatrixFreeActiveVector right_hand_side_copy;
 *     mf_system_matrix.initialize_dof_vector(solution_copy);
 *     mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
 *   
 *     solution_copy = 0.;
 *     constraints.distribute(solution_copy);
 *     solution_copy.update_ghost_values();
 *     right_hand_side_copy = 0;
 *     const Table<2, VectorizedArray<double>> &coefficient =
 *       *(mf_system_matrix.get_coefficient());
 *   
 *     RightHandSide<dim> right_hand_side_function;
 *   
 *     FEEvaluation<dim, degree, degree + 1, 1, double> phi(
 *       *mf_system_matrix.get_matrix_free());
 *   
 *     for (unsigned int cell = 0;
 *          cell < mf_system_matrix.get_matrix_free()->n_cell_batches();
 *          ++cell)
 *       {
 *         phi.reinit(cell);
 *         phi.read_dof_values_plain(solution_copy);
 *         phi.evaluate(EvaluationFlags::gradients);
 *   
 *         for (unsigned int q = 0; q < phi.n_q_points; ++q)
 *           {
 *             phi.submit_gradient(-1.0 *
 *                                   (coefficient(cell, 0) * phi.get_gradient(q)),
 *                                 q);
 *             phi.submit_value(
 *               right_hand_side_function.value(phi.quadrature_point(q)), q);
 *           }
 *   
 *         phi.integrate_scatter(EvaluationFlags::values |
 *                                 EvaluationFlags::gradients,
 *                               right_hand_side_copy);
 *       }
 *   
 *     right_hand_side_copy.compress(VectorOperation::add);
 *   
 *     ChangeVectorTypes::copy(right_hand_side, right_hand_side_copy);
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemsolve"></a> 
 * <h4>LaplaceProblem::solve()</h4>
 * 
 
 * 
 * Here we set up the multigrid preconditioner, test the timing of a single
 * V-cycle, and solve the linear system. Unsurprisingly, this is one of the
 * places where the three methods differ the most.
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::solve()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Solve");
 *   
 *     SolverControl solver_control(1000, 1.e-10 * right_hand_side.l2_norm());
 *     solver_control.enable_history_data();
 *   
 *     solution = 0.;
 *   
 * @endcode
 * 
 * The solver for the matrix-free GMG method is similar to @ref step_37 "step-37", apart
 * from adding some interface matrices in complete analogy to @ref step_16 "step-16".
 * 
 * @code
 *     switch (settings.solver)
 *       {
 *         case Settings::gmg_mf:
 *           {
 *             computing_timer.enter_subsection("Solve: Preconditioner setup");
 *   
 *             MGTransferMatrixFree<dim, float> mg_transfer(mg_constrained_dofs);
 *             mg_transfer.build(dof_handler);
 *   
 *             SolverControl coarse_solver_control(1000, 1e-12, false, false);
 *             SolverCG<MatrixFreeLevelVector> coarse_solver(coarse_solver_control);
 *             PreconditionIdentity            identity;
 *             MGCoarseGridIterativeSolver<MatrixFreeLevelVector,
 *                                         SolverCG<MatrixFreeLevelVector>,
 *                                         MatrixFreeLevelMatrix,
 *                                         PreconditionIdentity>
 *               coarse_grid_solver(coarse_solver, mf_mg_matrix[0], identity);
 *   
 *             using Smoother = PreconditionJacobi<MatrixFreeLevelMatrix>;
 *             MGSmootherPrecondition<MatrixFreeLevelMatrix,
 *                                    Smoother,
 *                                    MatrixFreeLevelVector>
 *               smoother;
 *             smoother.initialize(mf_mg_matrix,
 *                                 typename Smoother::AdditionalData(
 *                                   settings.smoother_dampen));
 *             smoother.set_steps(settings.smoother_steps);
 *   
 *             mg::Matrix<MatrixFreeLevelVector> mg_m(mf_mg_matrix);
 *   
 *             MGLevelObject<
 *               MatrixFreeOperators::MGInterfaceOperator<MatrixFreeLevelMatrix>>
 *               mg_interface_matrices;
 *             mg_interface_matrices.resize(0, triangulation.n_global_levels() - 1);
 *             for (unsigned int level = 0; level < triangulation.n_global_levels();
 *                  ++level)
 *               mg_interface_matrices[level].initialize(mf_mg_matrix[level]);
 *             mg::Matrix<MatrixFreeLevelVector> mg_interface(mg_interface_matrices);
 *   
 *             Multigrid<MatrixFreeLevelVector> mg(
 *               mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
 *             mg.set_edge_matrices(mg_interface, mg_interface);
 *   
 *             PreconditionMG<dim,
 *                            MatrixFreeLevelVector,
 *                            MGTransferMatrixFree<dim, float>>
 *               preconditioner(dof_handler, mg, mg_transfer);
 *   
 * @endcode
 * 
 * Copy the solution vector and right-hand side from LA::MPI::Vector
 * to LinearAlgebra::distributed::Vector so that we can solve.
 * 
 * @code
 *             MatrixFreeActiveVector solution_copy;
 *             MatrixFreeActiveVector right_hand_side_copy;
 *             mf_system_matrix.initialize_dof_vector(solution_copy);
 *             mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
 *   
 *             ChangeVectorTypes::copy(solution_copy, solution);
 *             ChangeVectorTypes::copy(right_hand_side_copy, right_hand_side);
 *             computing_timer.leave_subsection("Solve: Preconditioner setup");
 *   
 * @endcode
 * 
 * Timing for 1 V-cycle.
 * 
 * @code
 *             {
 *               TimerOutput::Scope timing(computing_timer,
 *                                         "Solve: 1 multigrid V-cycle");
 *               preconditioner.vmult(solution_copy, right_hand_side_copy);
 *             }
 *             solution_copy = 0.;
 *   
 * @endcode
 * 
 * Solve the linear system, update the ghost values of the solution,
 * copy back to LA::MPI::Vector and distribute constraints.
 * 
 * @code
 *             {
 *               SolverCG<MatrixFreeActiveVector> solver(solver_control);
 *   
 *               TimerOutput::Scope timing(computing_timer, "Solve: CG");
 *               solver.solve(mf_system_matrix,
 *                            solution_copy,
 *                            right_hand_side_copy,
 *                            preconditioner);
 *             }
 *   
 *             solution_copy.update_ghost_values();
 *             ChangeVectorTypes::copy(solution, solution_copy);
 *             constraints.distribute(solution);
 *   
 *             break;
 *           }
 *   
 * @endcode
 * 
 * Solver for the matrix-based GMG method, similar to @ref step_16 "step-16", only
 * using a Jacobi smoother instead of a SOR smoother (which is not
 * implemented in parallel).
 * 
 * @code
 *         case Settings::gmg_mb:
 *           {
 *             computing_timer.enter_subsection("Solve: Preconditioner setup");
 *   
 *             MGTransferPrebuilt<VectorType> mg_transfer(mg_constrained_dofs);
 *             mg_transfer.build(dof_handler);
 *   
 *             SolverControl        coarse_solver_control(1000, 1e-12, false, false);
 *             SolverCG<VectorType> coarse_solver(coarse_solver_control);
 *             PreconditionIdentity identity;
 *             MGCoarseGridIterativeSolver<VectorType,
 *                                         SolverCG<VectorType>,
 *                                         MatrixType,
 *                                         PreconditionIdentity>
 *               coarse_grid_solver(coarse_solver, mg_matrix[0], identity);
 *   
 *             using Smoother = LA::MPI::PreconditionJacobi;
 *             MGSmootherPrecondition<MatrixType, Smoother, VectorType> smoother;
 *   
 *   #ifdef USE_PETSC_LA
 *             smoother.initialize(mg_matrix);
 *             Assert(
 *               settings.smoother_dampen == 1.0,
 *               ExcNotImplemented(
 *                 "PETSc's PreconditionJacobi has no support for a damping parameter."));
 *   #else
 *             smoother.initialize(mg_matrix, settings.smoother_dampen);
 *   #endif
 *   
 *             smoother.set_steps(settings.smoother_steps);
 *   
 *             mg::Matrix<VectorType> mg_m(mg_matrix);
 *             mg::Matrix<VectorType> mg_in(mg_interface_in);
 *             mg::Matrix<VectorType> mg_out(mg_interface_in);
 *   
 *             Multigrid<VectorType> mg(
 *               mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
 *             mg.set_edge_matrices(mg_out, mg_in);
 *   
 *   
 *             PreconditionMG<dim, VectorType, MGTransferPrebuilt<VectorType>>
 *               preconditioner(dof_handler, mg, mg_transfer);
 *   
 *             computing_timer.leave_subsection("Solve: Preconditioner setup");
 *   
 * @endcode
 * 
 * Timing for 1 V-cycle.
 * 
 * @code
 *             {
 *               TimerOutput::Scope timing(computing_timer,
 *                                         "Solve: 1 multigrid V-cycle");
 *               preconditioner.vmult(solution, right_hand_side);
 *             }
 *             solution = 0.;
 *   
 * @endcode
 * 
 * Solve the linear system and distribute constraints.
 * 
 * @code
 *             {
 *               SolverCG<VectorType> solver(solver_control);
 *   
 *               TimerOutput::Scope timing(computing_timer, "Solve: CG");
 *               solver.solve(system_matrix,
 *                            solution,
 *                            right_hand_side,
 *                            preconditioner);
 *             }
 *   
 *             constraints.distribute(solution);
 *   
 *             break;
 *           }
 *   
 * @endcode
 * 
 * Solver for the AMG method, similar to @ref step_40 "step-40".
 * 
 * @code
 *         case Settings::amg:
 *           {
 *             computing_timer.enter_subsection("Solve: Preconditioner setup");
 *   
 *             PreconditionAMG                 preconditioner;
 *             PreconditionAMG::AdditionalData Amg_data;
 *   
 *   #ifdef USE_PETSC_LA
 *             Amg_data.symmetric_operator = true;
 *   #else
 *             Amg_data.elliptic              = true;
 *             Amg_data.smoother_type         = "Jacobi";
 *             Amg_data.higher_order_elements = true;
 *             Amg_data.smoother_sweeps       = settings.smoother_steps;
 *             Amg_data.aggregation_threshold = 0.02;
 *   #endif
 *   
 *             Amg_data.output_details = false;
 *   
 *             preconditioner.initialize(system_matrix, Amg_data);
 *             computing_timer.leave_subsection("Solve: Preconditioner setup");
 *   
 * @endcode
 * 
 * Timing for 1 V-cycle.
 * 
 * @code
 *             {
 *               TimerOutput::Scope timing(computing_timer,
 *                                         "Solve: 1 multigrid V-cycle");
 *               preconditioner.vmult(solution, right_hand_side);
 *             }
 *             solution = 0.;
 *   
 * @endcode
 * 
 * Solve the linear system and distribute constraints.
 * 
 * @code
 *             {
 *               SolverCG<VectorType> solver(solver_control);
 *   
 *               TimerOutput::Scope timing(computing_timer, "Solve: CG");
 *               solver.solve(system_matrix,
 *                            solution,
 *                            right_hand_side,
 *                            preconditioner);
 *             }
 *             constraints.distribute(solution);
 *   
 *             break;
 *           }
 *   
 *         default:
 *           Assert(false, ExcInternalError());
 *       }
 *   
 *     pcout << "   Number of CG iterations:      " << solver_control.last_step()
 *           << std::endl;
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Theerrorestimator"></a> 
 * <h3>The error estimator</h3>
 * 
 
 * 
 * We use the FEInterfaceValues class to assemble an error estimator to decide
 * which cells to refine. See the exact definition of the cell and face
 * integrals in the introduction. To use the method, we define Scratch and
 * Copy objects for the MeshWorker::mesh_loop() with much of the following code
 * being in essence as was set up in @ref step_12 "step-12" already (or at least similar in
 * spirit).
 * 
 * @code
 *   template <int dim>
 *   struct ScratchData
 *   {
 *     ScratchData(const Mapping<dim> &      mapping,
 *                 const FiniteElement<dim> &fe,
 *                 const unsigned int        quadrature_degree,
 *                 const UpdateFlags         update_flags,
 *                 const UpdateFlags         interface_update_flags)
 *       : fe_values(mapping, fe, QGauss<dim>(quadrature_degree), update_flags)
 *       , fe_interface_values(mapping,
 *                             fe,
 *                             QGauss<dim - 1>(quadrature_degree),
 *                             interface_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())
 *       , fe_interface_values(scratch_data.fe_values.get_mapping(),
 *                             scratch_data.fe_values.get_fe(),
 *                             scratch_data.fe_interface_values.get_quadrature(),
 *                             scratch_data.fe_interface_values.get_update_flags())
 *     {}
 *   
 *     FEValues<dim>          fe_values;
 *     FEInterfaceValues<dim> fe_interface_values;
 *   };
 *   
 *   
 *   
 *   struct CopyData
 *   {
 *     CopyData()
 *       : cell_index(numbers::invalid_unsigned_int)
 *       , value(0.)
 *     {}
 *   
 *     struct FaceData
 *     {
 *       unsigned int cell_indices[2];
 *       double       values[2];
 *     };
 *   
 *     unsigned int          cell_index;
 *     double                value;
 *     std::vector<FaceData> face_data;
 *   };
 *   
 *   
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::estimate()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Estimate");
 *   
 *     VectorType temp_solution;
 *     temp_solution.reinit(locally_owned_dofs,
 *                          locally_relevant_dofs,
 *                          mpi_communicator);
 *     temp_solution = solution;
 *   
 *     const Coefficient<dim> coefficient;
 *   
 *     estimated_error_square_per_cell.reinit(triangulation.n_active_cells());
 *   
 *     using Iterator = typename DoFHandler<dim>::active_cell_iterator;
 *   
 * @endcode
 * 
 * Assembler for cell residual @f$h^2 \| f + \epsilon \triangle u \|_K^2@f$
 * 
 * @code
 *     auto cell_worker = [&](const Iterator &  cell,
 *                            ScratchData<dim> &scratch_data,
 *                            CopyData &        copy_data) {
 *       FEValues<dim> &fe_values = scratch_data.fe_values;
 *       fe_values.reinit(cell);
 *   
 *       RightHandSide<dim> rhs;
 *       const double       rhs_value = rhs.value(cell->center());
 *   
 *       const double nu = coefficient.value(cell->center());
 *   
 *       std::vector<Tensor<2, dim>> hessians(fe_values.n_quadrature_points);
 *       fe_values.get_function_hessians(temp_solution, hessians);
 *   
 *       copy_data.cell_index = cell->active_cell_index();
 *   
 *       double residual_norm_square = 0.;
 *       for (unsigned k = 0; k < fe_values.n_quadrature_points; ++k)
 *         {
 *           const double residual = (rhs_value + nu * trace(hessians[k]));
 *           residual_norm_square += residual * residual * fe_values.JxW(k);
 *         }
 *   
 *       copy_data.value =
 *         cell->diameter() * cell->diameter() * residual_norm_square;
 *     };
 *   
 * @endcode
 * 
 * Assembler for face term @f$\sum_F h_F \| \jump{\epsilon \nabla u \cdot n}
 * \|_F^2@f$
 * 
 * @code
 *     auto face_worker = [&](const Iterator &    cell,
 *                            const unsigned int &f,
 *                            const unsigned int &sf,
 *                            const Iterator &    ncell,
 *                            const unsigned int &nf,
 *                            const unsigned int &nsf,
 *                            ScratchData<dim> &  scratch_data,
 *                            CopyData &          copy_data) {
 *       FEInterfaceValues<dim> &fe_interface_values =
 *         scratch_data.fe_interface_values;
 *       fe_interface_values.reinit(cell, f, sf, ncell, nf, nsf);
 *   
 *       copy_data.face_data.emplace_back();
 *       CopyData::FaceData &copy_data_face = copy_data.face_data.back();
 *   
 *       copy_data_face.cell_indices[0] = cell->active_cell_index();
 *       copy_data_face.cell_indices[1] = ncell->active_cell_index();
 *   
 *       const double coeff1 = coefficient.value(cell->center());
 *       const double coeff2 = coefficient.value(ncell->center());
 *   
 *       std::vector<Tensor<1, dim>> grad_u[2];
 *   
 *       for (unsigned int i = 0; i < 2; ++i)
 *         {
 *           grad_u[i].resize(fe_interface_values.n_quadrature_points);
 *           fe_interface_values.get_fe_face_values(i).get_function_gradients(
 *             temp_solution, grad_u[i]);
 *         }
 *   
 *       double jump_norm_square = 0.;
 *   
 *       for (unsigned int qpoint = 0;
 *            qpoint < fe_interface_values.n_quadrature_points;
 *            ++qpoint)
 *         {
 *           const double jump =
 *             coeff1 * grad_u[0][qpoint] * fe_interface_values.normal(qpoint) -
 *             coeff2 * grad_u[1][qpoint] * fe_interface_values.normal(qpoint);
 *   
 *           jump_norm_square += jump * jump * fe_interface_values.JxW(qpoint);
 *         }
 *   
 *       const double h           = cell->face(f)->measure();
 *       copy_data_face.values[0] = 0.5 * h * jump_norm_square;
 *       copy_data_face.values[1] = copy_data_face.values[0];
 *     };
 *   
 *     auto copier = [&](const CopyData &copy_data) {
 *       if (copy_data.cell_index != numbers::invalid_unsigned_int)
 *         estimated_error_square_per_cell[copy_data.cell_index] += copy_data.value;
 *   
 *       for (auto &cdf : copy_data.face_data)
 *         for (unsigned int j = 0; j < 2; ++j)
 *           estimated_error_square_per_cell[cdf.cell_indices[j]] += cdf.values[j];
 *     };
 *   
 *     const unsigned int n_gauss_points = degree + 1;
 *     ScratchData<dim>   scratch_data(mapping,
 *                                   fe,
 *                                   n_gauss_points,
 *                                   update_hessians | update_quadrature_points |
 *                                     update_JxW_values,
 *                                   update_values | update_gradients |
 *                                     update_JxW_values | update_normal_vectors);
 *     CopyData           copy_data;
 *   
 * @endcode
 * 
 * We need to assemble each interior face once but we need to make sure that
 * both processes assemble the face term between a locally owned and a ghost
 * cell. This is achieved by setting the
 * MeshWorker::assemble_ghost_faces_both flag. We need to do this, because
 * we do not communicate the error estimator contributions here.
 * 
 * @code
 *     MeshWorker::mesh_loop(dof_handler.begin_active(),
 *                           dof_handler.end(),
 *                           cell_worker,
 *                           copier,
 *                           scratch_data,
 *                           copy_data,
 *                           MeshWorker::assemble_own_cells |
 *                             MeshWorker::assemble_ghost_faces_both |
 *                             MeshWorker::assemble_own_interior_faces_once,
 *                           /*boundary_worker=*/nullptr,
 *                           face_worker);
 *   
 *     const double global_error_estimate =
 *       std::sqrt(Utilities::MPI::sum(estimated_error_square_per_cell.l1_norm(),
 *                                     mpi_communicator));
 *     pcout << "   Global error estimate:        " << global_error_estimate
 *           << std::endl;
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemrefine_grid"></a> 
 * <h4>LaplaceProblem::refine_grid()</h4>
 * 
 
 * 
 * We use the cell-wise estimator stored in the vector @p estimate_vector and
 * refine a fixed number of cells (chosen here to roughly double the number of
 * DoFs in each step).
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::refine_grid()
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Refine grid");
 *   
 *     const double refinement_fraction = 1. / (std::pow(2.0, dim) - 1.);
 *     parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
 *       triangulation, estimated_error_square_per_cell, refinement_fraction, 0.0);
 *   
 *     triangulation.execute_coarsening_and_refinement();
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemoutput_results"></a> 
 * <h4>LaplaceProblem::output_results()</h4>
 * 
 
 * 
 * The output_results() function is similar to the ones found in many of the
 * tutorials (see @ref step_40 "step-40" for example).
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::output_results(const unsigned int cycle)
 *   {
 *     TimerOutput::Scope timing(computing_timer, "Output results");
 *   
 *     VectorType temp_solution;
 *     temp_solution.reinit(locally_owned_dofs,
 *                          locally_relevant_dofs,
 *                          mpi_communicator);
 *     temp_solution = solution;
 *   
 *     DataOut<dim> data_out;
 *     data_out.attach_dof_handler(dof_handler);
 *     data_out.add_data_vector(temp_solution, "solution");
 *   
 *     Vector<float> 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");
 *   
 *     Vector<float> level(triangulation.n_active_cells());
 *     for (const auto &cell : triangulation.active_cell_iterators())
 *       level(cell->active_cell_index()) = cell->level();
 *     data_out.add_data_vector(level, "level");
 *   
 *     if (estimated_error_square_per_cell.size() > 0)
 *       data_out.add_data_vector(estimated_error_square_per_cell,
 *                                "estimated_error_square_per_cell");
 *   
 *     data_out.build_patches();
 *   
 *     const std::string pvtu_filename = data_out.write_vtu_with_pvtu_record(
 *       "", "solution", cycle, mpi_communicator, 2 /*n_digits*/, 1 /*n_groups*/);
 *   
 *     pcout << "   Wrote " << pvtu_filename << std::endl;
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="LaplaceProblemrun"></a> 
 * <h4>LaplaceProblem::run()</h4>
 * 
 
 * 
 * As in most tutorials, this function calls the various functions defined
 * above to set up, assemble, solve, and output the results.
 * 
 * @code
 *   template <int dim, int degree>
 *   void LaplaceProblem<dim, degree>::run()
 *   {
 *     for (unsigned int cycle = 0; cycle < settings.n_steps; ++cycle)
 *       {
 *         pcout << "Cycle " << cycle << ':' << std::endl;
 *         if (cycle > 0)
 *           refine_grid();
 *   
 *         pcout << "   Number of active cells:       "
 *               << triangulation.n_global_active_cells();
 *   
 * @endcode
 * 
 * We only output level cell data for the GMG methods (same with DoF
 * data below). Note that the partition efficiency is irrelevant for AMG
 * since the level hierarchy is not distributed or used during the
 * computation.
 * 
 * @code
 *         if (settings.solver == Settings::gmg_mf ||
 *             settings.solver == Settings::gmg_mb)
 *           pcout << " (" << triangulation.n_global_levels() << " global levels)"
 *                 << std::endl
 *                 << "   Partition efficiency:         "
 *                 << 1.0 / MGTools::workload_imbalance(triangulation);
 *         pcout << std::endl;
 *   
 *         setup_system();
 *   
 * @endcode
 * 
 * Only set up the multilevel hierarchy for GMG.
 * 
 * @code
 *         if (settings.solver == Settings::gmg_mf ||
 *             settings.solver == Settings::gmg_mb)
 *           setup_multigrid();
 *   
 *         pcout << "   Number of degrees of freedom: " << dof_handler.n_dofs();
 *         if (settings.solver == Settings::gmg_mf ||
 *             settings.solver == Settings::gmg_mb)
 *           {
 *             pcout << " (by level: ";
 *             for (unsigned int level = 0; level < triangulation.n_global_levels();
 *                  ++level)
 *               pcout << dof_handler.n_dofs(level)
 *                     << (level == triangulation.n_global_levels() - 1 ? ")" :
 *                                                                        ", ");
 *           }
 *         pcout << std::endl;
 *   
 * @endcode
 * 
 * For the matrix-free method, we only assemble the right-hand side.
 * For both matrix-based methods, we assemble both active matrix and
 * right-hand side, and only assemble the multigrid matrices for
 * matrix-based GMG.
 * 
 * @code
 *         if (settings.solver == Settings::gmg_mf)
 *           assemble_rhs();
 *         else /*gmg_mb or amg*/
 *           {
 *             assemble_system();
 *             if (settings.solver == Settings::gmg_mb)
 *               assemble_multigrid();
 *           }
 *   
 *         solve();
 *         estimate();
 *   
 *         if (settings.output)
 *           output_results(cycle);
 *   
 *         computing_timer.print_summary();
 *         computing_timer.reset();
 *       }
 *   }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Themainfunction"></a> 
 * <h3>The main() function</h3>
 * 
 
 * 
 * This is a similar main function to @ref step_40 "step-40", with the exception that
 * we require the user to pass a .prm file as a sole command line
 * argument (see @ref step_29 "step-29" and the documentation of the ParameterHandler
 * class for a complete discussion of parameter files).
 * 
 * @code
 *   int main(int argc, char *argv[])
 *   {
 *     using namespace dealii;
 *     Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
 *   
 *     Settings settings;
 *     if (!settings.try_parse((argc > 1) ? (argv[1]) : ""))
 *       return 0;
 *   
 *     try
 *       {
 *         constexpr unsigned int fe_degree = 2;
 *   
 *         switch (settings.dimension)
 *           {
 *             case 2:
 *               {
 *                 LaplaceProblem<2, fe_degree> test(settings);
 *                 test.run();
 *   
 *                 break;
 *               }
 *   
 *             case 3:
 *               {
 *                 LaplaceProblem<3, fe_degree> test(settings);
 *                 test.run();
 *   
 *                 break;
 *               }
 *   
 *             default:
 *               Assert(false, ExcMessage("This program only works in 2d and 3d."));
 *           }
 *       }
 *     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;
 *         MPI_Abort(MPI_COMM_WORLD, 1);
 *         return 1;
 *       }
 *     catch (...)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Unknown exception!" << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         MPI_Abort(MPI_COMM_WORLD, 2);
 *         return 1;
 *       }
 *   
 *     return 0;
 *   }
 * @endcode
<a name="Results"></a><h1>Results</h1>
 
 
When you run the program using the following command
@code
mpirun -np 16 ./step-50  gmg_mf_2d.prm
@endcode
the screen output should look like the following:
@code
Cycle 0:
   Number of active cells:       12 (2 global levels)
   Partition efficiency:         0.1875
   Number of degrees of freedom: 65 (by level: 21, 65)
   Number of CG iterations:      10
   Global error estimate:        0.355373
   Wrote solution_00.pvtu
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |    0.0163s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| Assemble right-hand side        |         1 |  0.000374s |       2.3% |
| Estimate                        |         1 |  0.000724s |       4.4% |
| Output results                  |         1 |   0.00277s |        17% |
| Setup                           |         1 |   0.00225s |        14% |
| Setup multigrid                 |         1 |   0.00181s |        11% |
| Solve                           |         1 |   0.00364s |        22% |
| Solve: 1 multigrid V-cycle      |         1 |  0.000354s |       2.2% |
| Solve: CG                       |         1 |   0.00151s |       9.3% |
| Solve: Preconditioner setup     |         1 |   0.00125s |       7.7% |
+---------------------------------+-----------+------------+------------+
 
Cycle 1:
   Number of active cells:       24 (3 global levels)
   Partition efficiency:         0.276786
   Number of degrees of freedom: 139 (by level: 21, 65, 99)
   Number of CG iterations:      10
   Global error estimate:        0.216726
   Wrote solution_01.pvtu
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |    0.0169s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| Assemble right-hand side        |         1 |  0.000309s |       1.8% |
| Estimate                        |         1 |   0.00156s |       9.2% |
| Output results                  |         1 |   0.00222s |        13% |
| Refine grid                     |         1 |   0.00278s |        16% |
| Setup                           |         1 |   0.00196s |        12% |
| Setup multigrid                 |         1 |    0.0023s |        14% |
| Solve                           |         1 |   0.00565s |        33% |
| Solve: 1 multigrid V-cycle      |         1 |  0.000349s |       2.1% |
| Solve: CG                       |         1 |   0.00285s |        17% |
| Solve: Preconditioner setup     |         1 |   0.00195s |        12% |
+---------------------------------+-----------+------------+------------+
 
Cycle 2:
   Number of active cells:       51 (4 global levels)
   Partition efficiency:         0.41875
   Number of degrees of freedom: 245 (by level: 21, 65, 225, 25)
   Number of CG iterations:      11
   Global error estimate:        0.112098
   Wrote solution_02.pvtu
 
 
+---------------------------------------------+------------+------------+
| Total wallclock time elapsed since start    |    0.0183s |            |
|                                             |            |            |
| Section                         | no. calls |  wall time | % of total |
+---------------------------------+-----------+------------+------------+
| Assemble right-hand side        |         1 |  0.000274s |       1.5% |
| Estimate                        |         1 |   0.00127s |       6.9% |
| Output results                  |         1 |   0.00227s |        12% |
| Refine grid                     |         1 |    0.0024s |        13% |
| Setup                           |         1 |   0.00191s |        10% |
| Setup multigrid                 |         1 |   0.00295s |        16% |
| Solve                           |         1 |   0.00702s |        38% |
| Solve: 1 multigrid V-cycle      |         1 |  0.000398s |       2.2% |
| Solve: CG                       |         1 |   0.00376s |        21% |
| Solve: Preconditioner setup     |         1 |   0.00238s |        13% |
+---------------------------------+-----------+------------+------------+
.
.
.
@endcode
Here, the timing of the `solve()` function is split up in 3 parts: setting
up the multigrid preconditioner, execution of a single multigrid V-cycle, and
the CG solver. The V-cycle that is timed is unnecessary for the overall solve
and only meant to give an insight at the different costs for AMG and GMG.
Also it should be noted that when using the AMG solver, "Workload imbalance"
is not included in the output since the hierarchy of coarse meshes is not
required.
 
All results in this section are gathered on Intel Xeon Platinum 8280 (Cascade
Lake) nodes which have 56 cores and 192GB per node and support AVX-512 instructions,
allowing for vectorization over 8 doubles (vectorization used only in the matrix-free
computations). The code is compiled using gcc 7.1.0 with intel-mpi 17.0.3. Trilinos
12.10.1 is used for the matrix-based GMG/AMG computations.
 
We can then gather a variety of information by calling the program
with the input files that are provided in the directory in which
@ref step_50 "step-50" is located. Using these, and adjusting the number of mesh
refinement steps, we can produce information about how well the
program scales.
 
The following table gives weak scaling timings for this program on up to 256M DoFs
and 7,168 processors. (Recall that weak scaling keeps the number of
degrees of freedom per processor constant while increasing the number of
processors; i.e., it considers larger and larger problems.)
Here, @f$\mathbb{E}@f$ is the partition efficiency from the
 introduction (also equal to 1.0/workload imbalance), "Setup" is a combination
of setup, setup multigrid, assemble, and assemble multigrid from the timing blocks,
and "Prec" is the preconditioner setup. Ideally all times would stay constant
over each problem size for the individual solvers, but since the partition
efficiency decreases from 0.371 to 0.161 from largest to smallest problem size,
we expect to see an approximately @f$0.371/0.161=2.3@f$ times increase in timings
for GMG. This is, in fact, pretty close to what we really get:
 
<table align="center" class="doxtable">
<tr>
  <th colspan="4"></th>
  <th></th>
  <th colspan="4">MF-GMG</th>
  <th></th>
  <th colspan="4">MB-GMG</th>
  <th></th>
  <th colspan="4">AMG</th>
</tr>
<tr>
  <th align="right">Procs</th>
  <th align="right">Cycle</th>
  <th align="right">DoFs</th>
  <th align="right">@f$\mathbb{E}@f$</th>
  <th></th>
  <th align="right">Setup</th>
  <th align="right">Prec</th>
  <th align="right">Solve</th>
  <th align="right">Total</th>
  <th></th>
  <th align="right">Setup</th>
  <th align="right">Prec</th>
  <th align="right">Solve</th>
  <th align="right">Total</th>
  <th></th>
  <th align="right">Setup</th>
  <th align="right">Prec</th>
  <th align="right">Solve</th>
  <th align="right">Total</th>
</tr>
<tr>
  <td align="right">112</th>
  <td align="right">13</th>
  <td align="right">4M</th>
  <td align="right">0.37</th>
  <td></td>
  <td align="right">0.742</th>
  <td align="right">0.393</th>
  <td align="right">0.200</th>
  <td align="right">1.335</th>
  <td></td>
  <td align="right">1.714</th>
  <td align="right">2.934</th>
  <td align="right">0.716</th>
  <td align="right">5.364</th>
  <td></td>
  <td align="right">1.544</th>
  <td align="right">0.456</th>
  <td align="right">1.150</th>
  <td align="right">3.150</th>
</tr>
<tr>
  <td align="right">448</th>
  <td align="right">15</th>
  <td align="right">16M</th>
  <td align="right">0.29</th>
  <td></td>
  <td align="right">0.884</th>
  <td align="right">0.535</th>
  <td align="right">0.253</th>
  <td align="right">1.672</th>
  <td></td>
  <td align="right">1.927</th>
  <td align="right">3.776</th>
  <td align="right">1.190</th>
  <td align="right">6.893</th>
  <td></td>
  <td align="right">1.544</th>
  <td align="right">0.456</th>
  <td align="right">1.150</th>
  <td align="right">3.150</th>
</tr>
<tr>
  <td align="right">1,792</th>
  <td align="right">17</th>
  <td align="right">65M</th>
  <td align="right">0.22</th>
  <td></td>
  <td align="right">1.122</th>
  <td align="right">0.686</th>
  <td align="right">0.309</th>
  <td align="right">2.117</th>
  <td></td>
  <td align="right">2.171</th>
  <td align="right">4.862</th>
  <td align="right">1.660</th>
  <td align="right">8.693</th>
  <td></td>
  <td align="right">1.654</th>
  <td align="right">0.546</th>
  <td align="right">1.460</th>
  <td align="right">3.660</th>
</tr>
<tr>
  <td align="right">7,168</th>
  <td align="right">19</th>
  <td align="right">256M</th>
  <td align="right">0.16</th>
  <td></td>
  <td align="right">1.214</th>
  <td align="right">0.893</th>
  <td align="right">0.521</th>
  <td align="right">2.628</th>
  <td></td>
  <td align="right">2.386</th>
  <td align="right">7.260</th>
  <td align="right">2.560</th>
  <td align="right">12.206</th>
  <td></td>
  <td align="right">1.844</th>
  <td align="right">1.010</th>
  <td align="right">1.890</th>
  <td align="right">4.744</th>
</tr>
</table>
 
On the other hand, the algebraic multigrid in the last set of columns
is relatively unaffected by the increasing imbalance of the mesh
hierarchy (because it doesn't use the mesh hierarchy) and the growth
in time is rather driven by other factors that are well documented in
the literature (most notably that the algorithmic complexity of
some parts of algebraic multigrid methods appears to be @f${\cal O}(N
\log N)@f$ instead of @f${\cal O}(N)@f$ for geometric multigrid).
 
The upshort of the table above is that the matrix-free geometric multigrid
method appears to be the fastest approach to solving this equation if
not by a huge margin. Matrix-based methods, on the other hand, are
consistently the worst.
 
The following figure provides strong scaling results for each method, i.e.,
we solve the same problem on more and more processors. Specifically,
we consider the problems after 16 mesh refinement cycles
(32M DoFs) and 19 cycles (256M DoFs), on between 56 to 28,672 processors:
 
<img width="600px" src="https://www.dealii.org/images/steps/developer/step-50-strong-scaling.png" alt="">
 
While the matrix-based GMG solver and AMG scale similarly and have a
similar time to solution (at least as long as there is a substantial
number of unknowns per processor -- say, several 10,000), the
matrix-free GMG solver scales much better and solves the finer problem
in roughly the same time as the AMG solver for the coarser mesh with
only an eighth of the number of processors. Conversely, it can solve the
same problem on the same number of processors in about one eighth the
time.
 
 
<a name="Possibilitiesforextensions"></a><h3> Possibilities for extensions </h3>
 
 
<a name="Testingconvergenceandhigherorderelements"></a><h4> Testing convergence and higher order elements </h4>
 
 
The finite element degree is currently hard-coded as 2, see the template
arguments of the main class. It is easy to change. To test, it would be
interesting to switch to a test problem with a reference solution. This way,
you can compare error rates.
 
<a name="Coarsesolver"></a><h4> Coarse solver </h4>
 
 
A more interesting example would involve a more complicated coarse mesh (see
@ref step_49 "step-49" for inspiration). The issue in that case is that the coarsest
level of the mesh hierarchy is actually quite large, and one would
have to think about ways to solve the coarse level problem
efficiently. (This is not an issue for algebraic multigrid methods
because they would just continue to build coarser and coarser levels
of the matrix, regardless of their geometric origin.)
 
In the program here, we simply solve the coarse level problem with a
Conjugate Gradient method without any preconditioner. That is acceptable
if the coarse problem is really small -- for example, if the coarse
mesh had a single cell, then the coarse mesh problems has a @f$9\times 9@f$
matrix in 2d, and a @f$27\times 27@f$ matrix in 3d; for the coarse mesh we
use on the @f$L@f$-shaped domain of the current program, these sizes are
@f$21\times 21@f$ in 2d and @f$117\times 117@f$ in 3d. But if the coarse mesh
consists of hundreds or thousands of cells, this approach will no
longer work and might start to dominate the overall run-time of each V-cycle.
A common approach is then to solve the coarse mesh problem using an
algebraic multigrid preconditioner; this would then, however, require
assembling the coarse matrix (even for the matrix-free version) as
input to the AMG implementation.
 *
 *
<a name="PlainProg"></a>
<h1> The plain program</h1>
@include "step-50.cc"
*/