advection_reaction_estimator.h

Go to the documentation of this file.

 
 * @endcode
 
 
<a name="ann-main.cc"></a>
<h1>Annotated version of main.cc</h1>
 * 
 * 
 * 
 * 
 * @code
 *   #include "include/DG_advection_reaction.h"
 *   
 *   int
 *   main(int argc, char **argv)
 *   {
 *     try
 *       {
 *         std::string par_name = "";
 *         if (argc > 1)
 *           {
 *             par_name = argv[1];
 *           }
 *         else
 *           {
 *             par_name = "parameters.prm";
 *           }
 *         deallog.depth_console(2);
 *         AdvectionReaction<2> problem;
 *         problem.initialize_params(par_name);
 *         problem.run();
 *       }
 *     catch (std::exception &exc)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << exc.what() << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       }
 *     catch (const theta_exc &theta_range)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << theta_range.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
 
 
<a name="ann-source/DG_advection_reaction.cc"></a>
<h1>Annotated version of source/DG_advection_reaction.cc</h1>
 * 
 * 
 * 
 * 
 * @code
 *   /* ---------------------------------------------------------------------
 *    *
 *    * Copyright (C) 2009 - 2021 by the deal.II authors
 *    *
 *    * This file is part of the deal.II library.
 *    *
 *    * The deal.II library is free software; you can use it, redistribute
 *    * it, and/or modify it under the terms of the GNU Lesser General
 *    * Public License as published by the Free Software Foundation; either
 *    * version 2.1 of the License, or (at your option) any later version.
 *    * The full text of the license can be found in the file LICENSE.md at
 *    * the top level directory of deal.II.
 *    *
 *    * ---------------------------------------------------------------------
 *    *
 *    * Author: Marco Feder, SISSA, 2021
 *    *
 *    */
 *   
 *   #include "../include/DG_advection_reaction.h"
 *   
 * @endcode
 * 
 * Compute and returns the wind field b
 * 
 * @code
 *   template <int dim>
 *   Tensor<1, dim>
 *   beta(const Point<dim> &p)
 *   {
 *     Assert(dim > 1, ExcNotImplemented());
 *     (void)p; // suppress warnings from p
 *     Tensor<1, dim> wind_field;
 *     wind_field[0] = 1.0;
 *     wind_field[1] = 1.0;
 *   
 *     return wind_field;
 *   }
 *   
 * @endcode
 * 
 * 
 * <a name="TheScratchDataandCopyDataclasses"></a> 
 * <h3>The ScratchData and CopyData classes</h3>
 * 
 
 * 
 * The following objects are the scratch and copy objects we use in the call
 * to MeshWorker::mesh_loop(). The new object is the FEInterfaceValues object,
 * that works similar to FEValues or FEFacesValues, except that it acts on
 * an interface between two cells and allows us to assemble the interface
 * terms in our weak form.
 * 
 * @code
 *   template <int dim>
 *   struct ScratchData
 *   {
 *     ScratchData(const Mapping<dim>        &mapping,
 *                 const FiniteElement<dim>  &fe,
 *                 const Quadrature<dim>     &quadrature,
 *                 const Quadrature<dim - 1> &quadrature_face,
 *                 const UpdateFlags          update_flags = update_values |
 *                                                  update_gradients |
 *                                                  update_quadrature_points |
 *                                                  update_JxW_values,
 *                 const UpdateFlags interface_update_flags =
 *                   update_values | update_gradients | update_quadrature_points |
 *                   update_JxW_values | update_normal_vectors)
 *       : fe_values(mapping, fe, quadrature, update_flags)
 *       , fe_interface_values(mapping, fe, quadrature_face, 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_interface_values.get_mapping(),
 *                             scratch_data.fe_interface_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 CopyDataFace
 *   {
 *     FullMatrix<double>                   cell_matrix;
 *     std::vector<types::global_dof_index> joint_dof_indices;
 *     std::array<double, 2>                values;
 *     std::array<unsigned int, 2>          cell_indices;
 *   };
 *   
 *   
 *   
 *   struct CopyData
 *   {
 *     FullMatrix<double>                   cell_matrix;
 *     Vector<double>                       cell_rhs;
 *     std::vector<types::global_dof_index> local_dof_indices;
 *     std::vector<CopyDataFace>            face_data;
 *   
 *     double       value;
 *     double       value_estimator;
 *     unsigned int cell_index;
 *   
 *     FullMatrix<double> cell_mass_matrix;
 *     Vector<double>     cell_mass_rhs;
 *   
 *     template <class Iterator>
 *     void
 *     reinit(const Iterator &cell, unsigned int dofs_per_cell)
 *     {
 *       cell_matrix.reinit(dofs_per_cell, dofs_per_cell);
 *       cell_mass_matrix.reinit(dofs_per_cell, dofs_per_cell);
 *   
 *       cell_rhs.reinit(dofs_per_cell);
 *       cell_mass_rhs.reinit(dofs_per_cell);
 *   
 *       local_dof_indices.resize(dofs_per_cell);
 *       cell->get_dof_indices(local_dof_indices);
 *     }
 *   };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Auxiliaryfunction"></a> 
 * <h3>Auxiliary function</h3>
 * This auxiliary function is taken from @ref step_74 "step-74" and it's used to
 * compute the jump of the finite element function @f$u_h@f$ on a face.
 * 
 * @code
 *   template <int dim>
 *   void
 *   get_function_jump(const FEInterfaceValues<dim> &fe_iv,
 *                     const Vector<double>         &solution,
 *                     std::vector<double>          &jump)
 *   {
 *     const unsigned int                 n_q = fe_iv.n_quadrature_points;
 *     std::array<std::vector<double>, 2> face_values;
 *     jump.resize(n_q);
 *     for (unsigned int i = 0; i < 2; ++i)
 *       {
 *         face_values[i].resize(n_q);
 *         fe_iv.get_fe_face_values(i).get_function_values(solution, face_values[i]);
 *       }
 *     for (unsigned int q = 0; q < n_q; ++q)
 *       jump[q] = face_values[0][q] - face_values[1][q];
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   AdvectionReaction<dim>::AdvectionReaction()
 *     : mapping()
 *     , dof_handler(triangulation)
 *   {
 *     Assert(dim > 1, ExcMessage("Not implemented in 1D."));
 *     add_parameter("Finite element degree", fe_degree);
 *     add_parameter("Problem constants", constants);
 *     add_parameter("Output filename", output_filename);
 *     add_parameter("Use direct solver", use_direct_solver);
 *     add_parameter("Number of refinement cycles", n_refinement_cycles);
 *     add_parameter("Number of global refinement", n_global_refinements);
 *     add_parameter("Refinement", refinement);
 *     add_parameter("Exact solution expression", exact_solution_expression);
 *     add_parameter("Boundary conditions expression",
 *                   boundary_conditions_expression);
 *     add_parameter("Theta", theta);
 *     add_parameter("Advection coefficient expression",
 *                   advection_coefficient_expression);
 *     add_parameter("Right hand side expression", rhs_expression);
 *   
 *     this->prm.enter_subsection("Error table");
 *     error_table.add_parameters(this->prm);
 *     this->prm.leave_subsection();
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::initialize_params(const std::string &filename)
 *   {
 *     ParameterAcceptor::initialize(filename,
 *                                   "last_used_parameters.prm",
 *                                   ParameterHandler::Short);
 *     if (theta < 0.0 || theta > 10.0 || std::abs(theta) < 1e-12)
 *       {
 *         throw(
 *           theta_exc("Theta parameter is not in a suitable range: see paper by "
 *                     "Brezzi, Marini, Suli for an extended discussion"));
 *       }
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::parse_string(const std::string &parameters)
 *   {
 *     ParameterAcceptor::prm.parse_input_from_string(parameters);
 *     ParameterAcceptor::parse_all_parameters();
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::setup_system()
 *   {
 * @endcode
 * 
 * first need to distribute the DoFs.
 * 
 * @code
 *     if (!fe)
 *       {
 *         fe              = std::make_unique<FE_DGQ<dim>>(fe_degree);
 *         const auto vars = dim == 2 ? "x,y" : "x,y,z";
 *         exact_solution.initialize(vars, exact_solution_expression, constants);
 *         rhs.initialize(vars, rhs_expression, constants);
 *         advection_coeff.initialize(vars,
 *                                    advection_coefficient_expression,
 *                                    constants);
 *         boundary_conditions.initialize(vars,
 *                                        boundary_conditions_expression,
 *                                        constants);
 *       }
 *     dof_handler.distribute_dofs(*fe);
 *   
 * @endcode
 * 
 * To build the sparsity pattern for DG discretizations, we can call the
 * function analogue to DoFTools::make_sparsity_pattern, which is called
 * DoFTools::make_flux_sparsity_pattern:
 * 
 * @code
 *     DynamicSparsityPattern dsp(dof_handler.n_dofs());
 *     DoFTools::make_flux_sparsity_pattern(dof_handler,
 *                                          dsp); // DG sparsity pattern generator
 *     sparsity_pattern.copy_from(dsp);
 *   
 * @endcode
 * 
 * Finally, we set up the structure of all components of the linear system.
 * 
 * @code
 *     system_matrix.reinit(sparsity_pattern);
 *     solution.reinit(dof_handler.n_dofs());
 *     right_hand_side.reinit(dof_handler.n_dofs());
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * in the call to  MeshWorker::mesh_loop() we only need to specify what should
 * happen on
 * each cell, each boundary face, and each interior face. These three tasks
 * are handled by the lambda functions inside the function below.
 * 
 
 * 
 * 
 * @code
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::assemble_system()
 *   {
 *     using Iterator = typename DoFHandler<dim>::active_cell_iterator;
 *   
 *     const QGauss<dim>     quadrature      = fe->tensor_degree() + 1;
 *     const QGauss<dim - 1> quadrature_face = fe->tensor_degree() + 1;
 *   
 * @endcode
 * 
 * This is the function that will be executed for each cell.
 * 
 * @code
 *     const auto cell_worker = [&](const Iterator   &cell,
 *                                  ScratchData<dim> &scratch_data,
 *                                  CopyData         &copy_data) {
 *       FEValues<dim> fe_values_continuous(*fe,
 *                                          quadrature,
 *                                          update_values | update_gradients |
 *                                            update_quadrature_points |
 *                                            update_JxW_values);
 *   
 *       const unsigned int n_dofs =
 *         scratch_data.fe_values.get_fe().n_dofs_per_cell();
 *       copy_data.reinit(cell, n_dofs);
 *       scratch_data.fe_values.reinit(cell);
 *   
 *       const auto &q_points = scratch_data.fe_values.get_quadrature_points();
 *   
 *       const FEValues<dim>       &fe_v = scratch_data.fe_values;
 *       const std::vector<double> &JxW  = fe_v.get_JxW_values();
 *   
 *       for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
 *         {
 *           auto beta_q = beta(q_points[point]);
 *           for (unsigned int i = 0; i < n_dofs; ++i)
 *             {
 *               for (unsigned int j = 0; j < n_dofs; ++j)
 *                 {
 *                   copy_data.cell_matrix(i, j) +=
 *                     (-beta_q                                    // -\beta
 *                        * fe_v.shape_grad(i, point)              // \nabla \phi_i
 *                        * fe_v.shape_value(j, point)             // \phi_j
 *                      + advection_coeff.value(q_points[point]) * // gamma
 *                          fe_v.shape_value(i, point)             // phi_i
 *                          * fe_v.shape_value(j, point)           // phi_j
 *                      ) *
 *                     JxW[point]; // dx
 *                 }
 *               copy_data.cell_rhs(i) += rhs.value(q_points[point])   // f(x_q)
 *                                        * fe_v.shape_value(i, point) // phi_i(x_q)
 *                                        * JxW[point];                // dx
 *             }
 *         }
 *     };
 *   
 * @endcode
 * 
 * This is the function called for boundary faces and consists of a normal
 * integration using FEFaceValues. New is the logic to decide if the term
 * goes into the system matrix (outflow) or the right-hand side (inflow).
 * 
 * @code
 *     const auto boundary_worker = [&](const Iterator     &cell,
 *                                      const unsigned int &face_no,
 *                                      ScratchData<dim>   &scratch_data,
 *                                      CopyData           &copy_data) {
 *       scratch_data.fe_interface_values.reinit(cell, face_no);
 *       const FEFaceValuesBase<dim> &fe_face =
 *         scratch_data.fe_interface_values.get_fe_face_values(0);
 *   
 *       const auto &q_points = fe_face.get_quadrature_points();
 *   
 *       const unsigned int n_facet_dofs        = fe_face.get_fe().n_dofs_per_cell();
 *       const std::vector<double>         &JxW = fe_face.get_JxW_values();
 *       const std::vector<Tensor<1, dim>> &normals = fe_face.get_normal_vectors();
 *   
 *       std::vector<double> g(q_points.size());
 *       exact_solution.value_list(q_points, g);
 *   
 *       for (unsigned int point = 0; point < q_points.size(); ++point)
 *         {
 *           const double beta_dot_n = beta(q_points[point]) * normals[point];
 *   
 *           if (beta_dot_n > 0)
 *             {
 *               for (unsigned int i = 0; i < n_facet_dofs; ++i)
 *                 for (unsigned int j = 0; j < n_facet_dofs; ++j)
 *                   copy_data.cell_matrix(i, j) +=
 *                     fe_face.shape_value(i,
 *                                         point)      // \phi_i
 *                     * fe_face.shape_value(j, point) // \phi_j
 *                     * beta_dot_n                    // \beta . n
 *                     * JxW[point];                   // dx
 *             }
 *           else
 *             for (unsigned int i = 0; i < n_facet_dofs; ++i)
 *               copy_data.cell_rhs(i) += -fe_face.shape_value(i, point) // \phi_i
 *                                        * g[point]                     // g*/
 *                                        * beta_dot_n                   // \beta . n
 *                                        * JxW[point];                  // dx
 *         }
 *     };
 *   
 * @endcode
 * 
 * This is the function called on interior faces. The arguments specify
 * cells, face and subface indices (for adaptive refinement). We just pass
 * them along to the reinit() function of FEInterfaceValues.
 * 
 * @code
 *     const 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_iv = scratch_data.fe_interface_values;
 *       fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
 *       const auto &q_points = fe_iv.get_quadrature_points();
 *   
 *       copy_data.face_data.emplace_back();
 *       CopyDataFace &copy_data_face = copy_data.face_data.back();
 *   
 *       const unsigned int n_dofs        = fe_iv.n_current_interface_dofs();
 *       copy_data_face.joint_dof_indices = fe_iv.get_interface_dof_indices();
 *   
 *       copy_data_face.cell_matrix.reinit(n_dofs, n_dofs);
 *   
 *       const std::vector<double>         &JxW     = fe_iv.get_JxW_values();
 *       const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
 *   
 *       for (unsigned int qpoint = 0; qpoint < q_points.size(); ++qpoint)
 *         {
 *           const double beta_dot_n = beta(q_points[qpoint]) * normals[qpoint];
 *           for (unsigned int i = 0; i < n_dofs; ++i)
 *             {
 *               for (unsigned int j = 0; j < n_dofs; ++j)
 *                 {
 *                   copy_data_face.cell_matrix(i, j) +=
 *                     (beta(q_points[qpoint]) * normals[qpoint] *
 *                        fe_iv.average_of_shape_values(j, qpoint) *
 *                        fe_iv.jump_in_shape_values(i, qpoint) +
 *                      theta * std::abs(beta_dot_n) *
 *                        fe_iv.jump_in_shape_values(j, qpoint) *
 *                        fe_iv.jump_in_shape_values(i, qpoint)) *
 *                     JxW[qpoint];
 *                 }
 *             }
 *         }
 *     };
 *   
 * @endcode
 * 
 * The following lambda function will handle copying the data from the
 * cell and face assembly into the global matrix and right-hand side.
 *   
 
 * 
 * While we would not need an AffineConstraints object, because there are
 * no hanging node constraints in DG discretizations, we use an empty
 * object here as this allows us to use its `copy_local_to_global`
 * functionality.
 * 
 * @code
 *     const AffineConstraints<double> constraints;
 *   
 *     const auto copier = [&](const CopyData &c) {
 *       constraints.distribute_local_to_global(c.cell_matrix,
 *                                              c.cell_rhs,
 *                                              c.local_dof_indices,
 *                                              system_matrix,
 *                                              right_hand_side);
 *   
 *       for (auto &cdf : c.face_data)
 *         {
 *           constraints.distribute_local_to_global(cdf.cell_matrix,
 *                                                  cdf.joint_dof_indices,
 *                                                  system_matrix);
 *         }
 *     };
 *   
 *     ScratchData<dim> scratch_data(mapping, *fe, quadrature, quadrature_face);
 *     CopyData         copy_data;
 *   
 * @endcode
 * 
 * Here, we finally handle the assembly. We pass in ScratchData and
 * CopyData objects, the lambda functions from above, an specify that we
 * want to assemble interior faces once.
 * 
 * @code
 *     MeshWorker::mesh_loop(dof_handler.begin_active(),
 *                           dof_handler.end(),
 *                           cell_worker,
 *                           copier,
 *                           scratch_data,
 *                           copy_data,
 *                           MeshWorker::assemble_own_cells |
 *                             MeshWorker::assemble_boundary_faces |
 *                             MeshWorker::assemble_own_interior_faces_once,
 *                           boundary_worker,
 *                           face_worker);
 *   }
 *   
 *   
 *   
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::solve()
 *   {
 *     if (use_direct_solver)
 *       {
 *         SparseDirectUMFPACK system_matrix_inverse;
 *         system_matrix_inverse.initialize(system_matrix);
 *         system_matrix_inverse.vmult(solution, right_hand_side);
 *       }
 *     else
 *       {
 * @endcode
 * 
 * Here we have a classic iterative solver, as done in many tutorials:
 * 
 * @code
 *         SolverControl                               solver_control(1000, 1e-15);
 *         SolverRichardson<Vector<double>>            solver(solver_control);
 *         PreconditionBlockSSOR<SparseMatrix<double>> preconditioner;
 *         preconditioner.initialize(system_matrix, fe->n_dofs_per_cell());
 *         solver.solve(system_matrix, solution, right_hand_side, preconditioner);
 *         std::cout << "  Solver converged in " << solver_control.last_step()
 *                   << " iterations." << std::endl;
 *       }
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Meshrefinement"></a> 
 * <h3>Mesh refinement</h3>
 * We refine the grid according the proposed estimator or with an approximation
 * to the gradient of the solution. The first option is the default one (you can
 * see it in the header file)
 * 
 * @code
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::refine_grid()
 *   {
 *     if (refinement == "residual")
 *       {
 * @endcode
 * 
 * If the `refinement` string is `"residual"`, then we first compute the
 * local projection
 * 
 * @code
 *         compute_local_projection_and_estimate();
 * @endcode
 * 
 * We then set the refinement fraction and as usual we execute the
 * refinement.
 * 
 * @code
 *         const double refinement_fraction = 0.6;
 *         GridRefinement::refine_and_coarsen_fixed_fraction(
 *           triangulation, error_indicator_per_cell, refinement_fraction, 0.0);
 *         triangulation.execute_coarsening_and_refinement();
 *       }
 *     else if (refinement == "gradient")
 *       {
 *         Vector<float> gradient_indicator(triangulation.n_active_cells());
 *   
 * @endcode
 * 
 * Now the approximate gradients are computed
 * 
 * @code
 *         DerivativeApproximation::approximate_gradient(mapping,
 *                                                       dof_handler,
 *                                                       solution,
 *                                                       gradient_indicator);
 *   
 * @endcode
 * 
 * and they are cell-wise scaled by the factor @f$h^{1+d/2}@f$
 * 
 * @code
 *         unsigned int cell_no = 0;
 *         for (const auto &cell : dof_handler.active_cell_iterators())
 *           gradient_indicator(cell_no++) *=
 *             std::pow(cell->diameter(), 1 + 1.0 * dim / 2);
 *   
 * @endcode
 * 
 * Finally they serve as refinement indicator.
 * 
 * @code
 *         GridRefinement::refine_and_coarsen_fixed_fraction(triangulation,
 *                                                           gradient_indicator,
 *                                                           0.25,
 *                                                           0.0);
 *   
 *         triangulation.execute_coarsening_and_refinement();
 *         std::cout << gradient_indicator.l2_norm() << '\n';
 *       }
 *     else if (refinement == "global")
 *       {
 *         triangulation.refine_global(
 *           1); // just for testing on uniformly refined meshes
 *       }
 *     else
 *       {
 *         Assert(false, ExcInternalError());
 *       }
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * The output of this program consists of a vtk file of the adaptively
 * refined grids and the numerical solutions.
 * 
 * @code
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::output_results(const unsigned int cycle) const
 *   {
 *     const std::string filename = "solution-" + std::to_string(cycle) + ".vtk";
 *     std::cout << "  Writing solution to <" << filename << ">" << std::endl;
 *     std::ofstream output(filename);
 *   
 *     DataOut<dim> data_out;
 *     data_out.attach_dof_handler(dof_handler);
 *     data_out.add_data_vector(solution, "u", DataOut<dim>::type_dof_data);
 *     data_out.build_patches(mapping);
 *     data_out.write_vtk(output);
 *   }
 *   
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::compute_error()
 *   {
 *     error_table.error_from_exact(
 *       mapping,
 *       dof_handler,
 *       solution,
 *       exact_solution); // be careful: a FD approximation of the gradient is used
 * @endcode
 * 
 * to compute the H^1 norm if Solution<dim> doesn't
 * implements the Gradient function
 * 
 * @code
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Computetheenergynorm"></a> 
 * <h3>Compute the energy norm</h3>
 * The energy norm is defined as @f$ |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 +
 * \sum_{F \in \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2
 * \Bigr)^{\frac{1}{2}}@f$ Notice that in the current case we have @f$c_f = \frac{|b
 * \cdot n|}{2}@f$ Like in the assembly, all the contributions are handled
 * separately by using ScratchData and CopyData objects.
 * 
 * @code
 *   template <int dim>
 *   double
 *   AdvectionReaction<dim>::compute_energy_norm()
 *   {
 *     energy_norm_square_per_cell.reinit(triangulation.n_active_cells());
 *   
 *     using Iterator = typename DoFHandler<dim>::active_cell_iterator;
 *   
 * @endcode
 * 
 * We start off by adding cell contributions
 * 
 * @code
 *     const auto cell_worker = [&](const Iterator   &cell,
 *                                  ScratchData<dim> &scratch_data,
 *                                  CopyData         &copy_data) {
 *       const unsigned int n_dofs =
 *         scratch_data.fe_values.get_fe().n_dofs_per_cell();
 *       copy_data.reinit(cell, n_dofs);
 *       scratch_data.fe_values.reinit(cell);
 *   
 *       copy_data.cell_index = cell->active_cell_index();
 *   
 *       const auto &q_points = scratch_data.fe_values.get_quadrature_points();
 *       const FEValues<dim>       &fe_v = scratch_data.fe_values;
 *       const std::vector<double> &JxW  = fe_v.get_JxW_values();
 *   
 *       double              error_square_norm{0.0};
 *       std::vector<double> sol_u(fe_v.n_quadrature_points);
 *       fe_v.get_function_values(solution, sol_u);
 *   
 *       for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
 *         {
 *           const double diff =
 *             (sol_u[point] - exact_solution.value(q_points[point]));
 *           error_square_norm += diff * diff * JxW[point];
 *         }
 *       copy_data.value = error_square_norm;
 *     };
 *   
 * @endcode
 * 
 * Here we add contributions coming from the internal faces
 * 
 * @code
 *     const 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_iv = scratch_data.fe_interface_values;
 *       fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
 *   
 *       copy_data.face_data.emplace_back();
 *       CopyDataFace &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 auto                &q_points   = fe_iv.get_quadrature_points();
 *       const unsigned             n_q_points = q_points.size();
 *       const std::vector<double> &JxW        = fe_iv.get_JxW_values();
 *       std::vector<double>        g(n_q_points);
 *   
 *       std::vector<double> jump(n_q_points);
 *       get_function_jump(fe_iv, solution, jump);
 *   
 *       const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
 *   
 *       double error_jump_square{0.0};
 *       for (unsigned int point = 0; point < n_q_points; ++point)
 *         {
 *           const double beta_dot_n =
 *             theta * std::abs(beta(q_points[point]) * normals[point]);
 *           error_jump_square +=
 *             beta_dot_n * jump[point] * jump[point] * JxW[point];
 *         }
 *   
 *       copy_data.value = error_jump_square;
 *     };
 *   
 * @endcode
 * 
 * Finally, we add the boundary contributions
 * 
 * @code
 *     const auto boundary_worker = [&](const Iterator     &cell,
 *                                      const unsigned int &face_no,
 *                                      ScratchData<dim>   &scratch_data,
 *                                      CopyData           &copy_data) {
 *       scratch_data.fe_interface_values.reinit(cell, face_no);
 *       const FEFaceValuesBase<dim> &fe_fv =
 *         scratch_data.fe_interface_values.get_fe_face_values(0);
 *       const auto                &q_points   = fe_fv.get_quadrature_points();
 *       const unsigned             n_q_points = q_points.size();
 *       const std::vector<double> &JxW        = fe_fv.get_JxW_values();
 *   
 *       std::vector<double> g(n_q_points);
 *   
 *       std::vector<double> sol_u(n_q_points);
 *       fe_fv.get_function_values(solution, sol_u);
 *   
 *       const std::vector<Tensor<1, dim>> &normals = fe_fv.get_normal_vectors();
 *   
 *       double difference_norm_square = 0.;
 *       for (unsigned int point = 0; point < q_points.size(); ++point)
 *         {
 *           const double beta_dot_n =
 *             theta * std::abs(beta(q_points[point]) * normals[point]);
 *           const double diff =
 *             (boundary_conditions.value(q_points[point]) - sol_u[point]);
 *           difference_norm_square += beta_dot_n * diff * diff * JxW[point];
 *         }
 *       copy_data.value = difference_norm_square;
 *     };
 *   
 *     const auto copier = [&](const auto &copy_data) {
 *       if (copy_data.cell_index != numbers::invalid_unsigned_int)
 *         {
 *           energy_norm_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)
 *           energy_norm_square_per_cell[cdf.cell_indices[j]] += cdf.values[j];
 *     };
 *   
 *     ScratchData<dim> scratch_data(mapping,
 *                                   *fe,
 *                                   QGauss<dim>{fe->tensor_degree() + 1},
 *                                   QGauss<dim - 1>{fe->tensor_degree() + 1});
 *   
 *     CopyData copy_data;
 *   
 *     MeshWorker::mesh_loop(dof_handler.begin_active(),
 *                           dof_handler.end(),
 *                           cell_worker,
 *                           copier,
 *                           scratch_data,
 *                           copy_data,
 *                           MeshWorker::assemble_own_cells |
 *                             MeshWorker::assemble_own_interior_faces_once |
 *                             MeshWorker::assemble_boundary_faces,
 *                           boundary_worker,
 *                           face_worker);
 *   
 *     const double energy_error = std::sqrt(energy_norm_square_per_cell.l1_norm());
 *     return energy_error;
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="Computingtheestimator"></a> 
 * <h3>Computing the estimator</h3>
 * In the estimator, we have to compute the term @f$||f- c u_h - \Pi(f- c
 * u_h)||_{T}^{2}@f$ over a generic cell @f$T@f$. To achieve this, we first need to
 * compute the projection involving the finite element function @f$u_h@f$. Using the
 * definition of orthogonal projection, we're required to solve cellwise
 * @f$(v_h,f-c u_h)_T = (v_h,\Pi)_T \qquad \forall v_h \in V_h@f$ for @f$\Pi@f$, which
 * means that we have to build a mass-matrix on each cell. Once we have the
 * projection, which is a finite element function, we can add its contribution
 * in the <code>cell_worker</code> lambda. As done in @ref step_74 "step-74", the square of the
 * error indicator is computed.
 * 
 
 * 
 * 
 * @code
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::compute_local_projection_and_estimate()
 *   {
 * @endcode
 * 
 * Compute the term @f$||f-c u_h - \Pi(f- cu_h)||_T^2@f$
 * 
 * @code
 *     using Iterator = typename DoFHandler<dim>::active_cell_iterator;
 *     error_indicator_per_cell.reinit(triangulation.n_active_cells());
 *   
 *     const auto cell_worker = [&](const Iterator   &cell,
 *                                  ScratchData<dim> &scratch_data,
 *                                  CopyData         &copy_data) {
 *       const unsigned int n_dofs =
 *         scratch_data.fe_values.get_fe().n_dofs_per_cell();
 *   
 *       copy_data.reinit(cell, n_dofs);
 *       scratch_data.fe_values.reinit(cell);
 *       copy_data.cell_index = cell->active_cell_index();
 *   
 *       const auto    &q_points   = scratch_data.fe_values.get_quadrature_points();
 *       const unsigned n_q_points = q_points.size();
 *   
 *       const FEValues<dim>       &fe_v = scratch_data.fe_values;
 *       const std::vector<double> &JxW  = fe_v.get_JxW_values();
 *   
 *       std::vector<double> sol_u_at_quadrature_points(fe_v.n_quadrature_points);
 *       fe_v.get_function_values(solution, sol_u_at_quadrature_points);
 *   
 * @endcode
 * 
 * Compute local L^2 projection of  @f$f- c u_h@f$ over the local finite element
 * space
 * 
 * @code
 *       for (unsigned int point = 0; point < n_q_points; ++point)
 *         {
 *           for (unsigned int i = 0; i < n_dofs; ++i)
 *             {
 *               for (unsigned int j = 0; j < n_dofs; ++j)
 *                 {
 *                   copy_data.cell_mass_matrix(i, j) +=
 *                     fe_v.shape_value(i, point) * // phi_i(x_q)
 *                     fe_v.shape_value(j, point) * // phi_j(x_q)
 *                     JxW[point];                  // dx(x_q)
 *                 }
 *               copy_data.cell_mass_rhs(i) +=
 *                 (rhs.value(q_points[point]) * // f(x_q)
 *                    fe_v.shape_value(i, point) // phi_i(x_q)
 *                  - advection_coeff.value(q_points[point]) *
 *                      fe_v.shape_value(i, point) *         // c*phi_i(x_q)
 *                      sol_u_at_quadrature_points[point]) * // u_h(x_q)
 *                 JxW[point];                               // dx
 *             }
 *         }
 *   
 *       FullMatrix<double> inverse(fe_v.n_quadrature_points,
 *                                  fe_v.n_quadrature_points);
 *       inverse.invert(copy_data.cell_mass_matrix);
 *       Vector<double> proj(fe_v.n_quadrature_points); // projection of (f-c*U_h) on
 * @endcode
 * 
 * the local fe_space
 * 
 * @code
 *       inverse.vmult(proj, copy_data.cell_mass_rhs);  // M^{-1}*rhs = proj
 *   
 *       double square_norm_over_cell = 0.0;
 *       for (unsigned int point = 0; point < n_q_points; ++point)
 *         {
 *           const double diff = rhs.value(q_points[point]) -
 *                               sol_u_at_quadrature_points[point] - proj[point];
 *           square_norm_over_cell += diff * diff * JxW[point];
 *         }
 *       copy_data.value_estimator = square_norm_over_cell;
 *     };
 *   
 * @endcode
 * 
 * Finally we have the boundary term with @f$||\beta (g-u_h^+)||^2@f$
 * 
 * @code
 *     const auto boundary_worker = [&](const Iterator     &cell,
 *                                      const unsigned int &face_no,
 *                                      ScratchData<dim>   &scratch_data,
 *                                      CopyData           &copy_data) {
 *       scratch_data.fe_interface_values.reinit(cell, face_no);
 *       const FEFaceValuesBase<dim> &fe_fv =
 *         scratch_data.fe_interface_values.get_fe_face_values(0);
 *       const auto                &q_points   = fe_fv.get_quadrature_points();
 *       const unsigned             n_q_points = q_points.size();
 *       const std::vector<double> &JxW        = fe_fv.get_JxW_values();
 *   
 *       std::vector<double> g(n_q_points);
 *       exact_solution.value_list(q_points, g);
 *   
 *       std::vector<double> sol_u(n_q_points);
 *       fe_fv.get_function_values(solution, sol_u);
 *   
 *       const std::vector<Tensor<1, dim>> &normals = fe_fv.get_normal_vectors();
 *   
 *       double square_norm_over_bdary_face = 0.;
 *       for (unsigned int point = 0; point < q_points.size(); ++point)
 *         {
 *           const double beta_dot_n = beta(q_points[point]) * normals[point];
 *   
 *           if (beta_dot_n < 0) //\partial_{-T} \cap \partial_{- \Omega}
 *             {
 *               const double diff =
 *                 std::abs(beta_dot_n) * (g[point] - sol_u[point]);
 *               square_norm_over_bdary_face += diff * diff * JxW[point];
 *             }
 *         }
 *       copy_data.value_estimator += square_norm_over_bdary_face;
 *     };
 *   
 * @endcode
 * 
 * Then compute the interior face terms with @f$|| \sqrt{b \cdot n}[u_h]||^2@f$
 * 
 * @code
 *     const 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_iv = scratch_data.fe_interface_values;
 *       fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
 *   
 *       copy_data.face_data.emplace_back();
 *       CopyDataFace &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 auto    &q_points   = fe_iv.get_quadrature_points();
 *       const unsigned n_q_points = q_points.size();
 *   
 *       const std::vector<double> &JxW = fe_iv.get_JxW_values();
 *       std::vector<double>        g(n_q_points);
 *   
 *       std::vector<double> jump(n_q_points);
 *       get_function_jump(fe_iv, solution, jump);
 *   
 *       const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
 *   
 *       double error_jump_square{0.0};
 *       for (unsigned int point = 0; point < n_q_points; ++point)
 *         {
 *           const double beta_dot_n = beta(q_points[point]) * normals[point];
 *           if (beta_dot_n < 0)
 *             {
 *               error_jump_square +=
 *                 std::abs(beta_dot_n) * jump[point] * jump[point] * JxW[point];
 *             }
 *         }
 *   
 *       copy_data_face.values[0] = error_jump_square;
 *       copy_data_face.values[1] = copy_data_face.values[0];
 *     };
 *   
 *     ScratchData<dim> scratch_data(mapping,
 *                                   *fe,
 *                                   QGauss<dim>{fe->tensor_degree() + 1},
 *                                   QGauss<dim - 1>{fe->tensor_degree() + 1});
 *   
 *     const auto copier = [&](const auto &copy_data) {
 *       if (copy_data.cell_index != numbers::invalid_unsigned_int)
 *         {
 *           error_indicator_per_cell[copy_data.cell_index] +=
 *             copy_data.value_estimator;
 *         }
 *       for (auto &cdf : copy_data.face_data)
 *         {
 *           for (unsigned int j = 0; j < 2; ++j)
 *             {
 *               error_indicator_per_cell[cdf.cell_indices[j]] += cdf.values[j];
 *             }
 *         }
 *     };
 *   
 * @endcode
 * 
 * Here, we finally handle the assembly of the Mass matrix (M)_{ij} = (\phi_j,
 * \phi_i)_T. We pass in ScratchData and CopyData objects
 * 
 * @code
 *     CopyData copy_data;
 *     MeshWorker::mesh_loop(dof_handler.begin_active(),
 *                           dof_handler.end(),
 *                           cell_worker,
 *                           copier,
 *                           scratch_data,
 *                           copy_data,
 *                           MeshWorker::assemble_own_cells |
 *                             MeshWorker::assemble_boundary_faces |
 *                             MeshWorker::assemble_own_interior_faces_once,
 *                           boundary_worker,
 *                           face_worker);
 *   }
 *   
 *   
 *   
 * @endcode
 * 
 * Usual <code>run</code> function, which runs over several refinement cycles
 * 
 * @code
 *   template <int dim>
 *   void
 *   AdvectionReaction<dim>::run()
 *   {
 *     std::vector<double> energy_errors;
 *     std::vector<int>    dofs_hist;
 *     for (unsigned int cycle = 0; cycle < n_refinement_cycles; ++cycle)
 *       {
 *         std::cout << "Cycle " << cycle << std::endl;
 *   
 *         if (cycle == 0)
 *           {
 *             GridGenerator::hyper_cube(triangulation);
 *             triangulation.refine_global(n_global_refinements);
 *           }
 *         else
 *           {
 *             refine_grid();
 *           }
 *         std::cout << "  Number of active cells:       "
 *                   << triangulation.n_active_cells() << std::endl;
 *         std::cout << "  Number of degrees of freedom: " << dof_handler.n_dofs()
 *                   << std::endl;
 *   
 *         setup_system();
 *         assemble_system();
 *         solve();
 *         compute_error();
 *         output_results(cycle);
 *   
 *         energy_errors.emplace_back(compute_energy_norm());
 *         dofs_hist.emplace_back(triangulation.n_active_cells());
 *       }
 *     error_table.output_table(std::cout);
 *   
 *     for (unsigned int i = 0; i < n_refinement_cycles; ++i)
 *       std::cout << "Cycle " << i << "\t" << energy_errors[i] << std::endl;
 *   }
 * @endcode
 * 
 * Explicit instantiation
 * 
 * @code
 *   template class AdvectionReaction<1>;
 *   template class AdvectionReaction<2>;
 *   template class AdvectionReaction<3>;
 * @endcode
 
 
*/