Heat_Eqn_Parallel.h

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 *   #include <deal.II/base/function.h>
 *   #include <deal.II/base/quadrature_lib.h>
 *   #include <deal.II/base/timer.h>
 *   #include <deal.II/lac/generic_linear_algebra.h>
 *   #define FORCE_USE_OF_TRILINOS
 *   namespace LA {
 *   #if defined(DEAL_II_WITH_PETSC) && !defined(DEAL_II_PETSC_WITH_COMPLEX) &&     \
 *       !(defined(DEAL_II_WITH_TRILINOS) && defined(FORCE_USE_OF_TRILINOS))
 *   using namespace dealii::LinearAlgebraPETSc;
 *   #define USE_PETSC_LA
 *   #elif defined(DEAL_II_WITH_TRILINOS)
 *   using namespace dealii::LinearAlgebraTrilinos;
 *   #else
 *   #error DEAL_II_WITH_PETSC or DEAL_II_WITH_TRILINOS required
 *   #endif
 *   } // namespace LA
 *   
 *   #include <deal.II/base/conditional_ostream.h>
 *   #include <deal.II/base/index_set.h>
 *   #include <deal.II/base/utilities.h>
 *   #include <deal.II/distributed/grid_refinement.h>
 *   #include <deal.II/distributed/tria.h>
 *   #include <deal.II/dofs/dof_handler.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/grid/grid_generator.h>
 *   #include <deal.II/lac/affine_constraints.h>
 *   #include <deal.II/lac/dynamic_sparsity_pattern.h>
 *   #include <deal.II/lac/full_matrix.h>
 *   #include <deal.II/lac/solver_cg.h>
 *   #include <deal.II/lac/sparsity_tools.h>
 *   #include <deal.II/lac/vector.h>
 *   #include <deal.II/numerics/data_out.h>
 *   #include <deal.II/numerics/error_estimator.h>
 *   #include <deal.II/numerics/matrix_creator.h>
 *   #include <deal.II/numerics/matrix_tools.h>
 *   #include <deal.II/numerics/solution_transfer.h>
 *   #include <deal.II/numerics/vector_tools.h>
 *   
 *   #include <fstream>
 *   #include <iostream>
 *   
 *   namespace MPIHeatEquation {
 *   using namespace dealii;
 *   
 *   template <int dim> class HeatEquation {
 *   public:
 *     HeatEquation();
 *     void run();
 *   
 *   private:
 *     void setup_system();
 *     void assemble_system(const double &time);
 *     void solve_time_step();
 *     void output_results() const;
 *     void refine_mesh(const unsigned int min_grid_level,
 *                      const unsigned int max_grid_level);
 *   
 *     MPI_Comm mpi_communicator;
 *     ConditionalOStream pcout;
 *     TimerOutput computing_timer;
 *   
 *     parallel::distributed::Triangulation<dim> triangulation;
 *     FE_Q<dim> fe;
 *     DoFHandler<dim> dof_handler;
 *   
 *     IndexSet locally_owned_dofs;
 *     IndexSet locally_relevant_dofs;
 *   
 *     AffineConstraints<double> constraints;
 *   
 *     LA::MPI::SparseMatrix system_matrix;
 *   
 *     LA::MPI::Vector locally_relevant_solution;
 *     LA::MPI::Vector old_locally_relevant_solution;
 *     LA::MPI::Vector system_rhs;
 *   
 *     double time;
 *     double time_step;
 *     unsigned int timestep_number;
 *     const double theta;
 *   };
 *   
 *   template <int dim> class RightHandSide : public Function<dim> {
 *   public:
 *     RightHandSide() : Function<dim>(), period(0.2) {}
 *   
 *     virtual double value(const Point<dim> &p,
 *                          const unsigned int component = 0) const override;
 *   
 *   private:
 *     const double period;
 *   };
 *   
 *   template <int dim>
 *   double RightHandSide<dim>::value(const Point<dim> &p,
 *                                    const unsigned int component) const {
 *     (void)component;
 *     AssertIndexRange(component, 1);
 *     Assert(dim == 2, ExcNotImplemented());
 *   
 *     const double time = this->get_time();
 *     const double point_within_period =
 *         (time / period - std::floor(time / period));
 *   
 *     if ((point_within_period >= 0.0) && (point_within_period <= 0.2)) {
 *       if ((p[0] > 0.5) && (p[1] > -0.5))
 *         return 1;
 *       else
 *         return 0;
 *     } else if ((point_within_period >= 0.5) && (point_within_period <= 0.7)) {
 *       if ((p[0] > -0.5) && (p[1] > 0.5))
 *         return 1;
 *       else
 *         return 0;
 *     } else
 *       return 0;
 *   }
 *   
 *   template <int dim> class BoundaryValues : public Function<dim> {
 *   public:
 *     virtual double value(const Point<dim> &p,
 *                          const unsigned int component = 0) const override;
 *   };
 *   
 *   template <int dim>
 *   double BoundaryValues<dim>::value(const Point<dim> & /*p*/,
 *                                     const unsigned int component) const {
 *     (void)component;
 *     Assert(component == 0, ExcIndexRange(component, 0, 1));
 *     return 0;
 *   }
 *   
 *   template <int dim>
 *   
 *   HeatEquation<dim>::HeatEquation()
 *   
 *       : mpi_communicator(MPI_COMM_WORLD),
 *         pcout(std::cout,
 *               (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)),
 *         computing_timer(mpi_communicator, pcout, TimerOutput::never,
 *                         TimerOutput::wall_times),
 *         triangulation(mpi_communicator), fe(1), dof_handler(triangulation),
 *         time_step(1. / 500), theta(0.5) {}
 *   
 *   template <int dim> void HeatEquation<dim>::setup_system() {
 *     TimerOutput::Scope t(computing_timer, "setup");
 *   
 *     dof_handler.distribute_dofs(fe);
 *   
 *     pcout << std::endl
 *           << "===========================================" << std::endl
 *           << "   Number of active cells:       "
 *           << triangulation.n_global_active_cells() << std::endl
 *           << "   Number of degrees of freedom: " << dof_handler.n_dofs()
 *           << std::endl;
 *   
 *     locally_owned_dofs = dof_handler.locally_owned_dofs();
 *     locally_relevant_dofs = DoFTools::extract_locally_relevant_dofs(dof_handler);
 *     locally_relevant_solution.reinit(locally_owned_dofs, locally_relevant_dofs,
 *                                      mpi_communicator);
 *     old_locally_relevant_solution.reinit(locally_owned_dofs,
 *                                          locally_relevant_dofs, mpi_communicator);
 *     system_rhs.reinit(locally_owned_dofs, mpi_communicator);
 *   
 *     constraints.clear();
 *     constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
 *     DoFTools::make_hanging_node_constraints(dof_handler, constraints);
 *     VectorTools::interpolate_boundary_values(
 *         dof_handler, 0, Functions::ZeroFunction<dim>(), constraints);
 *     constraints.close();
 *   
 *     DynamicSparsityPattern dsp(locally_relevant_dofs);
 *     DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, true);
 *     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);
 *   }
 *   
 *   template <int dim> void HeatEquation<dim>::assemble_system(const double &time) {
 *     TimerOutput::Scope t(computing_timer, "assemble_system");
 *   
 *     const QGauss<dim> quadrature_formula(fe.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);
 *     FullMatrix<double> cell_mass_matrix(dofs_per_cell, dofs_per_cell);
 *     FullMatrix<double> cell_laplace_matrix(dofs_per_cell, dofs_per_cell);
 *   
 *     Vector<double> cell_rhs(dofs_per_cell);
 *     Vector<double> cell_tmp(dofs_per_cell);
 *     Vector<double> cell_forcing_terms(dofs_per_cell);
 *     std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
 *     std::vector<double> rhs_values(n_q_points);
 *     std::vector<double> rhs_values_old(n_q_points);
 *     for (const auto &cell : dof_handler.active_cell_iterators())
 *       if (cell->is_locally_owned())
 *   
 *       {
 *         cell_matrix = 0;
 *         cell_mass_matrix = 0;
 *         cell_laplace_matrix = 0;
 *         cell_rhs = 0;
 *         cell_forcing_terms = 0;
 *         cell_tmp = 0;
 *   
 *         fe_values.reinit(cell);
 *   
 *         for (const unsigned int q_point : fe_values.quadrature_point_indices()) {
 *           for (const unsigned int i : fe_values.dof_indices()) {
 *   
 *             for (const unsigned int j : fe_values.dof_indices()) {
 *   
 *               cell_mass_matrix(i, j) += fe_values.shape_value(i, q_point) *
 *                                         fe_values.shape_value(j, q_point) *
 *                                         fe_values.JxW(q_point);
 *               cell_laplace_matrix(i, j) += fe_values.shape_grad(i, q_point) *
 *                                            fe_values.shape_grad(j, q_point) *
 *                                            fe_values.JxW(q_point);
 *   
 *               cell_matrix(i, j) +=
 *                   ((fe_values.shape_value(i, q_point) *
 *                     fe_values.shape_value(j, q_point)) +
 *                    (theta * time_step * fe_values.shape_grad(i, q_point) *
 *                     fe_values.shape_grad(j, q_point))) *
 *                   fe_values.JxW(q_point);
 *             }
 *           }
 *         }
 *   
 * @endcode
 * 
 * First compute M*U_old -(1-theta)*k*A*U_old
 * 
 * @code
 *         cell->get_dof_indices(local_dof_indices);
 *         Vector<double> old_cell_solution(dofs_per_cell);
 *         cell->get_dof_values(old_locally_relevant_solution, old_cell_solution);
 *         cell_mass_matrix.vmult(cell_rhs, old_cell_solution);
 *         cell_laplace_matrix.vmult(cell_tmp, old_cell_solution);
 *         cell_rhs.add(-(1 - theta) * time_step, cell_tmp);
 *   
 * @endcode
 * 
 * Add the rhs terms
 * 
 * @code
 *         RightHandSide<dim> rhs_function;
 * @endcode
 * 
 * k*theta*F^n
 * 
 * @code
 *         rhs_function.set_time(time);
 *         rhs_function.value_list(fe_values.get_quadrature_points(), rhs_values);
 *         for (const unsigned int q_point : fe_values.quadrature_point_indices())
 *           for (const unsigned int i : fe_values.dof_indices())
 *             cell_rhs(i) += time_step * theta *
 *                            (fe_values.shape_value(i, q_point) *
 *                             rhs_values[q_point] * fe_values.JxW(q_point));
 *   
 * @endcode
 * 
 * Adding:k*(1-theta)*F^{n-1}
 * 
 * @code
 *         rhs_function.set_time(time - time_step);
 *         rhs_function.value_list(fe_values.get_quadrature_points(),
 *                                 rhs_values_old);
 *         for (const unsigned int q_point : fe_values.quadrature_point_indices())
 *           for (const unsigned int i : fe_values.dof_indices())
 *             cell_rhs(i) += time_step * (1 - theta) *
 *                            (fe_values.shape_value(i, q_point) *
 *                             rhs_values_old[q_point] * fe_values.JxW(q_point));
 *   
 *         constraints.distribute_local_to_global(
 *             cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
 *       }
 *   
 *     system_matrix.compress(VectorOperation::add);
 *     system_rhs.compress(VectorOperation::add);
 *   }
 *   
 *   template <int dim> void HeatEquation<dim>::solve_time_step() {
 *     TimerOutput::Scope t(computing_timer, "solve");
 *   
 *     LA::MPI::Vector completely_distributed_solution(locally_owned_dofs,
 *                                                     mpi_communicator);
 *   
 *     SolverControl solver_control(1000, 1e-8 * system_rhs.l2_norm());
 *   
 *     SolverCG<LA::MPI::Vector> solver(solver_control);
 *   
 *     LA::MPI::PreconditionAMG::AdditionalData data;
 *   #ifdef USE_PETSC_LA
 *     data.symmetric_operator = true;
 *   #else
 *     /* Trilinos defaults are good */
 *   #endif
 *     LA::MPI::PreconditionAMG preconditioner;
 *     preconditioner.initialize(system_matrix, data);
 *   
 *     solver.solve(system_matrix, completely_distributed_solution, system_rhs,
 *                  preconditioner);
 *   
 *     pcout << "   Solved in " << solver_control.last_step() << " iterations."
 *           << std::endl;
 *   
 *     constraints.distribute(completely_distributed_solution);
 *   
 *     locally_relevant_solution = completely_distributed_solution;
 *   }
 *   
 *   template <int dim> void HeatEquation<dim>::output_results() const {
 *     DataOut<dim> data_out;
 *     data_out.attach_dof_handler(dof_handler);
 *     data_out.add_data_vector(locally_relevant_solution, "T");
 *   
 *     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");
 *   
 *     data_out.build_patches();
 *     data_out.set_flags(DataOutBase::VtkFlags(time, timestep_number));
 *   
 *     data_out.write_vtu_with_pvtu_record("./", "solution", timestep_number,
 *                                         mpi_communicator, 2, 8);
 *   }
 *   
 *   template <int dim>
 *   void HeatEquation<dim>::refine_mesh(const unsigned int min_grid_level,
 *                                       const unsigned int max_grid_level)
 *   
 *   {
 *     TimerOutput::Scope t(computing_timer, "refine");
 *     Vector<float> estimated_error_per_cell(
 *         triangulation.n_locally_owned_active_cells());
 *   
 *     KellyErrorEstimator<dim>::estimate(
 *         dof_handler, QGauss<dim - 1>(fe.degree + 1),
 *         std::map<types::boundary_id, const Function<dim> *>(),
 *         locally_relevant_solution, estimated_error_per_cell);
 *   
 *     parallel::distributed::GridRefinement::refine_and_coarsen_fixed_fraction(
 *         triangulation, estimated_error_per_cell, 0.6, 0.4);
 *   
 *     if (triangulation.n_levels() > max_grid_level) {
 *       for (const auto &cell :
 *            triangulation.active_cell_iterators_on_level(max_grid_level))
 *         if (cell->is_locally_owned())
 *           cell->clear_refine_flag();
 *     }
 *     for (const auto &cell :
 *          triangulation.active_cell_iterators_on_level(min_grid_level)) {
 *       if (cell->is_locally_owned())
 *         cell->clear_coarsen_flag();
 *     }
 *   
 *     SolutionTransfer<dim, LA::MPI::Vector> solution_trans(dof_handler);
 *   
 *     LA::MPI::Vector previous_locally_relevant_solution(
 *         locally_owned_dofs, locally_relevant_dofs, mpi_communicator);
 *     previous_locally_relevant_solution = locally_relevant_solution;
 *     triangulation.prepare_coarsening_and_refinement();
 *     solution_trans.prepare_for_coarsening_and_refinement(
 *         previous_locally_relevant_solution);
 *     triangulation.execute_coarsening_and_refinement();
 *     setup_system();
 *     LA::MPI::Vector completely_distributed_solution(locally_owned_dofs,
 *                                                     mpi_communicator);
 *   
 *     solution_trans.interpolate(completely_distributed_solution);
 *     constraints.distribute(completely_distributed_solution);
 *     locally_relevant_solution = completely_distributed_solution;
 *   }
 *   
 *   template <int dim> void HeatEquation<dim>::run() {
 *     pcout << "Running with "
 *   #ifdef USE_PETSC_LA
 *           << "PETSc"
 *   #else
 *           << "Trilinos"
 *   #endif
 *           << " on " << Utilities::MPI::n_mpi_processes(mpi_communicator)
 *           << " MPI rank(s)..." << std::endl;
 *   
 *     const unsigned int initial_global_refinement = 2;//12;
 *     const unsigned int n_adaptive_pre_refinement_steps = 4;//0;
 *     GridGenerator::hyper_L(triangulation);
 *     triangulation.refine_global(initial_global_refinement);
 *   
 *     setup_system();
 *   
 *     unsigned int pre_refinement_step = 0;
 *   
 *   start_time_iteration:
 *   
 *     time = 0.0;
 *     timestep_number = 0;
 *   
 *     LA::MPI::Vector completely_distributed_solution(locally_owned_dofs,
 *                                                     mpi_communicator);
 *     VectorTools::interpolate(dof_handler, Functions::ZeroFunction<dim>(),
 *                              completely_distributed_solution);
 *     old_locally_relevant_solution = completely_distributed_solution;
 *   
 *     locally_relevant_solution = completely_distributed_solution;
 *   
 *     {
 *       TimerOutput::Scope t(computing_timer, "output");
 *       output_results();
 *     }
 *   
 *     const double end_time = 0.5;//1./500;
 *     while (time < end_time) {
 * @endcode
 * 
 * while (time <= end_time) {
 * 
 * @code
 *       time += time_step;
 *       ++timestep_number;
 *   
 *       pcout << "Time step " << timestep_number << " at t=" << time << std::endl;
 *   
 *       system_rhs.reinit(locally_owned_dofs, mpi_communicator);
 *       DynamicSparsityPattern dsp(locally_relevant_dofs);
 *       DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, true);
 *       SparsityTools::distribute_sparsity_pattern(
 *           dsp, dof_handler.locally_owned_dofs(), mpi_communicator,
 *           locally_relevant_dofs);
 *       system_matrix.reinit(locally_owned_dofs, locally_owned_dofs, dsp,
 *                            mpi_communicator);
 *   
 *       assemble_system(time);
 *       solve_time_step();
 *   
 *       {
 *         TimerOutput::Scope t(computing_timer, "output");
 *         output_results();
 *       }
 *       computing_timer.print_summary();
 *       computing_timer.reset();
 *       pcout << std::endl;
 *   
 *       if ((timestep_number == 1) &&
 *           (pre_refinement_step < n_adaptive_pre_refinement_steps)) {
 *         refine_mesh(initial_global_refinement,
 *                     initial_global_refinement + n_adaptive_pre_refinement_steps);
 *         ++pre_refinement_step;
 *   
 *         pcout << std::endl;
 *   
 *         goto start_time_iteration;
 *       } else if ((timestep_number > 0) && (timestep_number % 5 == 0)) {
 *         refine_mesh(initial_global_refinement,
 *                     initial_global_refinement + n_adaptive_pre_refinement_steps);
 *       }
 *   
 *       old_locally_relevant_solution = locally_relevant_solution;
 *     }
 *   }
 *   } // namespace MPIHeatEquation
 *   
 *   int main(int argc, char *argv[]) {
 *     try {
 * @endcode
 * 
 * Added:
 * 
 * @code
 *       using namespace dealii;
 *   
 *       using namespace MPIHeatEquation;
 *   
 * @endcode
 * 
 * Added:
 * 
 * @code
 *       Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
 *   
 *       HeatEquation<2> heat_equation_solver;
 *       heat_equation_solver.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;
 *   }
 * @endcode
 
 
*/