step-74.h

Go to the documentation of this file.

) const
 *     {
 *       using numbers::PI;
 *       for (unsigned int i = 0; i < values.size(); ++i)
 *         values[i] =
 *           std::sin(2. * PI * points[i][0]) * std::sin(2. * PI * points[i][1]);
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     Tensor<1, dim>
 *     SmoothSolution<dim>::gradient(const Point<dim> &point,
 *                                   const unsigned int /*component*/) const
 *     {
 *       Tensor<1, dim> return_value;
 *       using numbers::PI;
 *       return_value[0] =
 *         2. * PI * std::cos(2. * PI * point[0]) * std::sin(2. * PI * point[1]);
 *       return_value[1] =
 *         2. * PI * std::sin(2. * PI * point[0]) * std::cos(2. * PI * point[1]);
 *       return return_value;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * The corresponding right-hand side of the smooth function:
 * 
 * @code
 *     template <int dim>
 *     class SmoothRightHandSide : public Function<dim>
 *     {
 *     public:
 *       SmoothRightHandSide()
 *         : Function<dim>()
 *       {}
 *   
 *       virtual void value_list(const std::vector<Point<dim>> &points,
 *                               std::vector<double>           &values,
 *                               const unsigned int /*component*/) const override;
 *     };
 *   
 *   
 *   
 *     template <int dim>
 *     void
 *     SmoothRightHandSide<dim>::value_list(const std::vector<Point<dim>> &points,
 *                                          std::vector<double>           &values,
 *                                          const unsigned int /*component*/) const
 *     {
 *       using numbers::PI;
 *       for (unsigned int i = 0; i < values.size(); ++i)
 *         values[i] = 8. * PI * PI * std::sin(2. * PI * points[i][0]) *
 *                     std::sin(2. * PI * points[i][1]);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * The right-hand side that corresponds to the function
 * Functions::LSingularityFunction, where we
 * assume that the diffusion coefficient @f$\nu = 1@f$:
 * 
 * @code
 *     template <int dim>
 *     class SingularRightHandSide : public Function<dim>
 *     {
 *     public:
 *       SingularRightHandSide()
 *         : Function<dim>()
 *       {}
 *   
 *       virtual void value_list(const std::vector<Point<dim>> &points,
 *                               std::vector<double>           &values,
 *                               const unsigned int /*component*/) const override;
 *   
 *     private:
 *       const Functions::LSingularityFunction ref;
 *     };
 *   
 *   
 *   
 *     template <int dim>
 *     void
 *     SingularRightHandSide<dim>::value_list(const std::vector<Point<dim>> &points,
 *                                            std::vector<double>           &values,
 *                                            const unsigned int /*component*/) const
 *     {
 *       for (unsigned int i = 0; i < values.size(); ++i)
 *         values[i] = -ref.laplacian(points[i]);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Auxiliaryfunctions"></a> 
 * <h3>Auxiliary functions</h3>
 * This function computes the penalty @f$\sigma@f$.
 * 
 * @code
 *     double get_penalty_factor(const unsigned int fe_degree,
 *                               const double       cell_extent_left,
 *                               const double       cell_extent_right)
 *     {
 *       const unsigned int degree = std::max(1U, fe_degree);
 *       return degree * (degree + 1.) * 0.5 *
 *              (1. / cell_extent_left + 1. / cell_extent_right);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-TheCopyData"></a> 
 * <h3>The CopyData</h3>
 * In the following, we define "Copy" objects for the MeshWorker::mesh_loop(),
 * which is essentially the same as @ref step_12 "step-12". Note that the
 * "Scratch" object is not defined here because we use
 * MeshWorker::ScratchData<dim> instead. (The use of "Copy" and "Scratch"
 * objects is extensively explained in the WorkStream namespace documentation.
 * 
 * @code
 *     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;
 *       unsigned int                         cell_index;
 *   
 *   
 *       template <class Iterator>
 *       void reinit(const Iterator &cell, const unsigned int dofs_per_cell)
 *       {
 *         cell_matrix.reinit(dofs_per_cell, dofs_per_cell);
 *         cell_rhs.reinit(dofs_per_cell);
 *         local_dof_indices.resize(dofs_per_cell);
 *         cell->get_dof_indices(local_dof_indices);
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-TheSIPGLaplaceclass"></a> 
 * <h3>The SIPGLaplace class</h3>
 * After these preparations, we proceed with the main class of this program,
 * called `SIPGLaplace`. The overall structure of the class is as in many
 * of the other tutorial programs. Major differences will only come up in the
 * implementation of the assemble functions, since we use FEInterfaceValues to
 * assemble face terms.
 * 
 * @code
 *     template <int dim>
 *     class SIPGLaplace
 *     {
 *     public:
 *       SIPGLaplace(const TestCase &test_case);
 *       void run();
 *   
 *     private:
 *       void setup_system();
 *       void assemble_system();
 *       void solve();
 *       void refine_grid();
 *       void output_results(const unsigned int cycle) const;
 *   
 *       void   compute_errors();
 *       void   compute_error_estimate();
 *       double compute_energy_norm_error();
 *   
 *       Triangulation<dim>    triangulation;
 *       const unsigned int    degree;
 *       const QGauss<dim>     quadrature;
 *       const QGauss<dim - 1> face_quadrature;
 *       const QGauss<dim>     quadrature_overintegration;
 *       const QGauss<dim - 1> face_quadrature_overintegration;
 *       const MappingQ1<dim>  mapping;
 *   
 *       using ScratchData = MeshWorker::ScratchData<dim>;
 *   
 *       const FE_DGQ<dim> fe;
 *       DoFHandler<dim>   dof_handler;
 *   
 *       SparsityPattern      sparsity_pattern;
 *       SparseMatrix<double> system_matrix;
 *       Vector<double>       solution;
 *       Vector<double>       system_rhs;
 *   
 * @endcode
 * 
 * The remainder of the class's members are used for the following:
 * - Vectors to store error estimator square and energy norm square per
 * cell.
 * - Print convergence rate and errors on the screen.
 * - The fiffusion coefficient @f$\nu@f$ is set to 1.
 * - Members that store information about the test case to be computed.
 * 
 * @code
 *       Vector<double> estimated_error_square_per_cell;
 *       Vector<double> energy_norm_square_per_cell;
 *   
 *       ConvergenceTable convergence_table;
 *   
 *       const double diffusion_coefficient = 1.;
 *   
 *       const TestCase                       test_case;
 *       std::unique_ptr<const Function<dim>> exact_solution;
 *       std::unique_ptr<const Function<dim>> rhs_function;
 *     };
 *   
 * @endcode
 * 
 * The constructor here takes the test case as input and then
 * determines the correct solution and right-hand side classes. The
 * remaining member variables are initialized in the obvious way.
 * 
 * @code
 *     template <int dim>
 *     SIPGLaplace<dim>::SIPGLaplace(const TestCase &test_case)
 *       : degree(3)
 *       , quadrature(degree + 1)
 *       , face_quadrature(degree + 1)
 *       , quadrature_overintegration(degree + 2)
 *       , face_quadrature_overintegration(degree + 2)
 *       , mapping()
 *       , fe(degree)
 *       , dof_handler(triangulation)
 *       , test_case(test_case)
 *     {
 *       if (test_case == TestCase::convergence_rate)
 *         {
 *           exact_solution = std::make_unique<const SmoothSolution<dim>>();
 *           rhs_function   = std::make_unique<const SmoothRightHandSide<dim>>();
 *         }
 *   
 *       else if (test_case == TestCase::l_singularity)
 *         {
 *           exact_solution =
 *             std::make_unique<const Functions::LSingularityFunction>();
 *           rhs_function = std::make_unique<const SingularRightHandSide<dim>>();
 *         }
 *       else
 *         AssertThrow(false, ExcNotImplemented());
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void SIPGLaplace<dim>::setup_system()
 *     {
 *       dof_handler.distribute_dofs(fe);
 *       DynamicSparsityPattern dsp(dof_handler.n_dofs());
 *       DoFTools::make_flux_sparsity_pattern(dof_handler, dsp);
 *       sparsity_pattern.copy_from(dsp);
 *   
 *       system_matrix.reinit(sparsity_pattern);
 *       solution.reinit(dof_handler.n_dofs());
 *       system_rhs.reinit(dof_handler.n_dofs());
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Theassemble_systemfunction"></a> 
 * <h3>The assemble_system function</h3>
 * The assemble function here is similar to that in @ref step_12 "step-12" and @ref step_47 "step-47".
 * Different from assembling by hand, we just need to focus
 * on assembling on each cell, each boundary face, and each
 * interior face. The loops over cells and faces are handled
 * automatically by MeshWorker::mesh_loop().
 *   
 
 * 
 * The function starts by defining a local (lambda) function that is
 * used to integrate the cell terms:
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::assemble_system()
 *     {
 *       const auto cell_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEValues<dim> &fe_v          = scratch_data.reinit(cell);
 *           const unsigned int   dofs_per_cell = fe_v.dofs_per_cell;
 *           copy_data.reinit(cell, dofs_per_cell);
 *   
 *           const std::vector<Point<dim>> &q_points =
 *             scratch_data.get_quadrature_points();
 *           const unsigned int         n_q_points = q_points.size();
 *           const std::vector<double> &JxW        = scratch_data.get_JxW_values();
 *   
 *           std::vector<double> rhs(n_q_points);
 *           rhs_function->value_list(q_points, rhs);
 *   
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *               {
 *                 for (unsigned int j = 0; j < fe_v.dofs_per_cell; ++j)
 *                   copy_data.cell_matrix(i, j) +=
 *                     diffusion_coefficient *     // nu
 *                     fe_v.shape_grad(i, point) * // grad v_h
 *                     fe_v.shape_grad(j, point) * // grad u_h
 *                     JxW[point];                 // dx
 *   
 *                 copy_data.cell_rhs(i) += fe_v.shape_value(i, point) * // v_h
 *                                          rhs[point] *                 // f
 *                                          JxW[point];                  // dx
 *               }
 *         };
 *   
 * @endcode
 * 
 * Next, we need a function that assembles face integrals on the boundary:
 * 
 * @code
 *       const auto boundary_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             const unsigned int                                   &face_no,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEFaceValuesBase<dim> &fe_fv = scratch_data.reinit(cell, face_no);
 *   
 *           const std::vector<Point<dim>> &q_points =
 *             scratch_data.get_quadrature_points();
 *           const unsigned int n_q_points    = q_points.size();
 *           const unsigned int dofs_per_cell = fe_fv.dofs_per_cell;
 *   
 *           const std::vector<double>         &JxW = scratch_data.get_JxW_values();
 *           const std::vector<Tensor<1, dim>> &normals =
 *             scratch_data.get_normal_vectors();
 *   
 *           std::vector<double> g(n_q_points);
 *           exact_solution->value_list(q_points, g);
 *   
 *           const double extent1 = cell->measure() / cell->face(face_no)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent1);
 *   
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             {
 *               for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                 for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *                   copy_data.cell_matrix(i, j) +=
 *                     (-diffusion_coefficient *        // - nu
 *                        fe_fv.shape_value(i, point) * // v_h
 *                        (fe_fv.shape_grad(j, point) * // (grad u_h .
 *                         normals[point])              //  n)
 *   
 *                      - diffusion_coefficient *         // - nu
 *                          (fe_fv.shape_grad(i, point) * // (grad v_h .
 *                           normals[point]) *            //  n)
 *                          fe_fv.shape_value(j, point)   // u_h
 *   
 *                      + diffusion_coefficient * penalty * // + nu sigma
 *                          fe_fv.shape_value(i, point) *   // v_h
 *                          fe_fv.shape_value(j, point)     // u_h
 *   
 *                      ) *
 *                     JxW[point]; // dx
 *   
 *               for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                 copy_data.cell_rhs(i) +=
 *                   (-diffusion_coefficient *        // - nu
 *                      (fe_fv.shape_grad(i, point) * // (grad v_h .
 *                       normals[point]) *            //  n)
 *                      g[point]                      // g
 *   
 *   
 *                    + diffusion_coefficient * penalty *        // + nu sigma
 *                        fe_fv.shape_value(i, point) * g[point] // v_h g
 *   
 *                    ) *
 *                   JxW[point]; // dx
 *             }
 *         };
 *   
 * @endcode
 * 
 * Finally, a function that assembles face integrals on interior
 * faces. To reinitialize FEInterfaceValues, we need to pass
 * cells, face and subface indices (for adaptive refinement) to
 * the reinit() function of FEInterfaceValues:
 * 
 * @code
 *       const auto face_worker =
 *         [&](const typename DoFHandler<dim>::cell_iterator &cell,
 *             const unsigned int                            &f,
 *             const unsigned int                            &sf,
 *             const typename DoFHandler<dim>::cell_iterator &ncell,
 *             const unsigned int                            &nf,
 *             const unsigned int                            &nsf,
 *             ScratchData                                   &scratch_data,
 *             CopyData                                      &copy_data) {
 *           const FEInterfaceValues<dim> &fe_iv =
 *             scratch_data.reinit(cell, f, sf, ncell, nf, nsf);
 *   
 *           copy_data.face_data.emplace_back();
 *           CopyDataFace      &copy_data_face = copy_data.face_data.back();
 *           const unsigned int n_dofs_face    = 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_face, n_dofs_face);
 *   
 *           const std::vector<double>         &JxW     = fe_iv.get_JxW_values();
 *           const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
 *   
 *           const double extent1 = cell->measure() / cell->face(f)->measure();
 *           const double extent2 = ncell->measure() / ncell->face(nf)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent2);
 *   
 *           for (const unsigned int point : fe_iv.quadrature_point_indices())
 *             {
 *               for (const unsigned int i : fe_iv.dof_indices())
 *                 for (const unsigned int j : fe_iv.dof_indices())
 *                   copy_data_face.cell_matrix(i, j) +=
 *                     (-diffusion_coefficient *                     // - nu
 *                        fe_iv.jump_in_shape_values(i, point) *     // [v_h]
 *                        (fe_iv.average_of_shape_gradients(j,       
 *                                                          point) * // ({grad u_h} .
 *                         normals[point])                           //  n)
 *   
 *                      - diffusion_coefficient * // - nu
 *                          (fe_iv.average_of_shape_gradients(i,
 *                                                            point) * // (grad v_h .
 *                           normals[point]) *                         //  n)
 *                          fe_iv.jump_in_shape_values(j, point)       // [u_h]
 *   
 *                      + diffusion_coefficient * penalty *        // + nu sigma
 *                          fe_iv.jump_in_shape_values(i, point) * // [v_h]
 *                          fe_iv.jump_in_shape_values(j, point)   // [u_h]
 *   
 *                      ) *
 *                     JxW[point]; // dx
 *             }
 *         };
 *   
 * @endcode
 * 
 * The following lambda function will then copy data into the
 * global matrix and right-hand side.  Though there are no hanging
 * node constraints in DG discretization, we define an empty
 * AffineConstraints object that allows us to use the
 * AffineConstraints::distribute_local_to_global() functionality.
 * 
 * @code
 *       AffineConstraints<double> constraints;
 *       constraints.close();
 *       const auto copier = [&](const CopyData &c) {
 *         constraints.distribute_local_to_global(c.cell_matrix,
 *                                                c.cell_rhs,
 *                                                c.local_dof_indices,
 *                                                system_matrix,
 *                                                system_rhs);
 *   
 * @endcode
 * 
 * Copy data from interior face assembly to the global matrix.
 * 
 * @code
 *         for (const CopyDataFace &cdf : c.face_data)
 *           {
 *             constraints.distribute_local_to_global(cdf.cell_matrix,
 *                                                    cdf.joint_dof_indices,
 *                                                    system_matrix);
 *           }
 *       };
 *   
 *   
 * @endcode
 * 
 * With the assembly functions defined, we can now create
 * ScratchData and CopyData objects, and pass them together with
 * the lambda functions above to MeshWorker::mesh_loop(). In
 * addition, we need to specify that we want to assemble on
 * interior faces exactly once.
 * 
 * @code
 *       const UpdateFlags cell_flags = update_values | update_gradients |
 *                                      update_quadrature_points | update_JxW_values;
 *       const UpdateFlags face_flags = update_values | update_gradients |
 *                                      update_quadrature_points |
 *                                      update_normal_vectors | update_JxW_values;
 *   
 *       ScratchData scratch_data(
 *         mapping, fe, quadrature, cell_flags, face_quadrature, face_flags);
 *       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
 * 
 * 
 * <a name="step_74-Thesolveandoutput_resultsfunction"></a> 
 * <h3>The solve() and output_results() function</h3>
 * The following two functions are entirely standard and without difficulty.
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::solve()
 *     {
 *       SparseDirectUMFPACK A_direct;
 *       A_direct.initialize(system_matrix);
 *       A_direct.vmult(solution, system_rhs);
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void SIPGLaplace<dim>::output_results(const unsigned int cycle) const
 *     {
 *       const std::string filename = "sol_Q" + Utilities::int_to_string(degree, 1) +
 *                                    "-" + Utilities::int_to_string(cycle, 2) +
 *                                    ".vtu";
 *       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_vtu(output);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Thecompute_error_estimatefunction"></a> 
 * <h3>The compute_error_estimate() function</h3>
 * The assembly of the error estimator here is quite similar to
 * that of the global matrix and right-had side and can be handled
 * by the MeshWorker::mesh_loop() framework. To understand what
 * each of the local (lambda) functions is doing, recall first that
 * the local cell residual is defined as
 * @f$h_K^2 \left\| f + \nu \Delta u_h \right\|_K^2@f$:
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::compute_error_estimate()
 *     {
 *       const auto cell_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEValues<dim> &fe_v = scratch_data.reinit(cell);
 *   
 *           copy_data.cell_index = cell->active_cell_index();
 *   
 *           const std::vector<Point<dim>> &q_points = fe_v.get_quadrature_points();
 *           const unsigned int             n_q_points = q_points.size();
 *           const std::vector<double>     &JxW        = fe_v.get_JxW_values();
 *   
 *           std::vector<Tensor<2, dim>> hessians(n_q_points);
 *           fe_v.get_function_hessians(solution, hessians);
 *   
 *           std::vector<double> rhs(n_q_points);
 *           rhs_function->value_list(q_points, rhs);
 *   
 *           const double hk                   = cell->diameter();
 *           double       residual_norm_square = 0;
 *   
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             {
 *               const double residual =
 *                 rhs[point] + diffusion_coefficient * trace(hessians[point]);
 *               residual_norm_square += residual * residual * JxW[point];
 *             }
 *           copy_data.value = hk * hk * residual_norm_square;
 *         };
 *   
 * @endcode
 * 
 * Next compute boundary terms @f$\sum_{f\in \partial K \cap \partial \Omega}
 * \sigma \left\| [  u_h-g_D ]  \right\|_f^2  @f$:
 * 
 * @code
 *       const auto boundary_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             const unsigned int                                   &face_no,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEFaceValuesBase<dim> &fe_fv = scratch_data.reinit(cell, face_no);
 *   
 *           const std::vector<Point<dim>> &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 double extent1 = cell->measure() / cell->face(face_no)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent1);
 *   
 *           double difference_norm_square = 0.;
 *           for (unsigned int point = 0; point < q_points.size(); ++point)
 *             {
 *               const double diff = (g[point] - sol_u[point]);
 *               difference_norm_square += diff * diff * JxW[point];
 *             }
 *           copy_data.value += penalty * difference_norm_square;
 *         };
 *   
 * @endcode
 * 
 * And finally interior face terms @f$\sum_{f\in \partial K}\lbrace \sigma
 * \left\| [u_h]  \right\|_f^2   +  h_f \left\|  [\nu \nabla u_h \cdot
 * \mathbf n ] \right\|_f^2 \rbrace@f$:
 * 
 * @code
 *       const auto face_worker =
 *         [&](const typename DoFHandler<dim>::cell_iterator &cell,
 *             const unsigned int                            &f,
 *             const unsigned int                            &sf,
 *             const typename DoFHandler<dim>::cell_iterator &ncell,
 *             const unsigned int                            &nf,
 *             const unsigned int                            &nsf,
 *             ScratchData                                   &scratch_data,
 *             CopyData                                      &copy_data) {
 *           const FEInterfaceValues<dim> &fe_iv =
 *             scratch_data.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 std::vector<double>         &JxW     = fe_iv.get_JxW_values();
 *           const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
 *   
 *           const std::vector<Point<dim>> &q_points = fe_iv.get_quadrature_points();
 *           const unsigned int             n_q_points = q_points.size();
 *   
 *           std::vector<double> jump(n_q_points);
 *           fe_iv.get_jump_in_function_values(solution, jump);
 *   
 *           std::vector<Tensor<1, dim>> grad_jump(n_q_points);
 *           fe_iv.get_jump_in_function_gradients(solution, grad_jump);
 *   
 *           const double h = cell->face(f)->diameter();
 *   
 *           const double extent1 = cell->measure() / cell->face(f)->measure();
 *           const double extent2 = ncell->measure() / ncell->face(nf)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent2);
 *   
 *           double flux_jump_square = 0;
 *           double u_jump_square    = 0;
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             {
 *               u_jump_square += jump[point] * jump[point] * JxW[point];
 *               const double flux_jump = grad_jump[point] * normals[point];
 *               flux_jump_square +=
 *                 diffusion_coefficient * flux_jump * flux_jump * JxW[point];
 *             }
 *           copy_data_face.values[0] =
 *             0.5 * h * (flux_jump_square + penalty * u_jump_square);
 *           copy_data_face.values[1] = copy_data_face.values[0];
 *         };
 *   
 * @endcode
 * 
 * Having computed local contributions for each cell, we still
 * need a way to copy these into the global vector that will hold
 * the error estimators for all cells:
 * 
 * @code
 *       const 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 (const CopyDataFace &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];
 *       };
 *   
 * @endcode
 * 
 * After all of this set-up, let's do the actual work: We resize
 * the vector into which the results will be written, and then
 * drive the whole process using the MeshWorker::mesh_loop()
 * function.
 * 
 * @code
 *       estimated_error_square_per_cell.reinit(triangulation.n_active_cells());
 *   
 *       const UpdateFlags cell_flags =
 *         update_hessians | update_quadrature_points | update_JxW_values;
 *       const UpdateFlags face_flags = update_values | update_gradients |
 *                                      update_quadrature_points |
 *                                      update_JxW_values | update_normal_vectors;
 *   
 *       ScratchData scratch_data(
 *         mapping, fe, quadrature, cell_flags, face_quadrature, face_flags);
 *   
 *       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);
 *     }
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Thecompute_energy_norm_errorfunction"></a> 
 * <h3>The compute_energy_norm_error() function</h3>
 * Next, we evaluate the accuracy in terms of the energy norm.
 * This function is similar to the assembling of the error estimator above.
 * Here we compute the square of the energy norm defined by
 * @f[
 * \|u \|_{1,h}^2 = \sum_{K \in \Gamma_h} \nu\|\nabla u \|_K^2 +
 * \sum_{f \in F_i} \sigma \| [ u ] \|_f^2 +
 * \sum_{f \in F_b} \sigma  \|u\|_f^2.
 * @f]
 * Therefore the corresponding error is
 * @f[
 * \|u -u_h \|_{1,h}^2 = \sum_{K \in \Gamma_h} \nu\|\nabla (u_h - u)  \|_K^2
 * + \sum_{f \in F_i} \sigma  \|[ u_h ] \|_f^2 + \sum_{f \in F_b}\sigma
 * \|u_h-g_D\|_f^2.
 * @f]
 * 
 * @code
 *     template <int dim>
 *     double SIPGLaplace<dim>::compute_energy_norm_error()
 *     {
 *       energy_norm_square_per_cell.reinit(triangulation.n_active_cells());
 *   
 * @endcode
 * 
 * Assemble @f$\sum_{K \in \Gamma_h} \nu\|\nabla (u_h - u)  \|_K^2 @f$.
 * 
 * @code
 *       const auto cell_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEValues<dim> &fe_v = scratch_data.reinit(cell);
 *   
 *           copy_data.cell_index = cell->active_cell_index();
 *   
 *           const std::vector<Point<dim>> &q_points = fe_v.get_quadrature_points();
 *           const unsigned int             n_q_points = q_points.size();
 *           const std::vector<double>     &JxW        = fe_v.get_JxW_values();
 *   
 *           std::vector<Tensor<1, dim>> grad_u(n_q_points);
 *           fe_v.get_function_gradients(solution, grad_u);
 *   
 *           std::vector<Tensor<1, dim>> grad_exact(n_q_points);
 *           exact_solution->gradient_list(q_points, grad_exact);
 *   
 *           double norm_square = 0;
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             {
 *               norm_square +=
 *                 (grad_u[point] - grad_exact[point]).norm_square() * JxW[point];
 *             }
 *           copy_data.value = diffusion_coefficient * norm_square;
 *         };
 *   
 * @endcode
 * 
 * Assemble @f$\sum_{f \in F_b}\sigma  \|u_h-g_D\|_f^2@f$.
 * 
 * @code
 *       const auto boundary_worker =
 *         [&](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *             const unsigned int                                   &face_no,
 *             ScratchData                                          &scratch_data,
 *             CopyData                                             &copy_data) {
 *           const FEFaceValuesBase<dim> &fe_fv = scratch_data.reinit(cell, face_no);
 *   
 *           const std::vector<Point<dim>> &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 double extent1 = cell->measure() / cell->face(face_no)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent1);
 *   
 *           double difference_norm_square = 0.;
 *           for (unsigned int point = 0; point < q_points.size(); ++point)
 *             {
 *               const double diff = (g[point] - sol_u[point]);
 *               difference_norm_square += diff * diff * JxW[point];
 *             }
 *           copy_data.value += penalty * difference_norm_square;
 *         };
 *   
 * @endcode
 * 
 * Assemble @f$\sum_{f \in F_i} \sigma  \| [ u_h ] \|_f^2@f$.
 * 
 * @code
 *       const auto face_worker =
 *         [&](const typename DoFHandler<dim>::cell_iterator &cell,
 *             const unsigned int                            &f,
 *             const unsigned int                            &sf,
 *             const typename DoFHandler<dim>::cell_iterator &ncell,
 *             const unsigned int                            &nf,
 *             const unsigned int                            &nsf,
 *             ScratchData                                   &scratch_data,
 *             CopyData                                      &copy_data) {
 *           const FEInterfaceValues<dim> &fe_iv =
 *             scratch_data.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 std::vector<double> &JxW = fe_iv.get_JxW_values();
 *   
 *           const std::vector<Point<dim>> &q_points = fe_iv.get_quadrature_points();
 *           const unsigned int             n_q_points = q_points.size();
 *   
 *           std::vector<double> jump(n_q_points);
 *           fe_iv.get_jump_in_function_values(solution, jump);
 *   
 *           const double extent1 = cell->measure() / cell->face(f)->measure();
 *           const double extent2 = ncell->measure() / ncell->face(nf)->measure();
 *           const double penalty = get_penalty_factor(degree, extent1, extent2);
 *   
 *           double u_jump_square = 0;
 *           for (unsigned int point = 0; point < n_q_points; ++point)
 *             {
 *               u_jump_square += jump[point] * jump[point] * JxW[point];
 *             }
 *           copy_data_face.values[0] = 0.5 * penalty * u_jump_square;
 *           copy_data_face.values[1] = copy_data_face.values[0];
 *         };
 *   
 *       const auto copier = [&](const CopyData &copy_data) {
 *         if (copy_data.cell_index != numbers::invalid_unsigned_int)
 *           energy_norm_square_per_cell[copy_data.cell_index] += copy_data.value;
 *         for (const CopyDataFace &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];
 *       };
 *   
 *       const UpdateFlags cell_flags =
 *         update_gradients | update_quadrature_points | update_JxW_values;
 *       UpdateFlags face_flags =
 *         update_values | update_quadrature_points | update_JxW_values;
 *   
 *       const ScratchData scratch_data(mapping,
 *                                      fe,
 *                                      quadrature_overintegration,
 *                                      cell_flags,
 *                                      face_quadrature_overintegration,
 *                                      face_flags);
 *   
 *       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="step_74-Therefine_gridfunction"></a> 
 * <h3>The refine_grid() function</h3>
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::refine_grid()
 *     {
 *       const double refinement_fraction = 0.1;
 *   
 *       GridRefinement::refine_and_coarsen_fixed_number(
 *         triangulation, estimated_error_square_per_cell, refinement_fraction, 0.);
 *   
 *       triangulation.execute_coarsening_and_refinement();
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Thecompute_errorsfunction"></a> 
 * <h3>The compute_errors() function</h3>
 * We compute three errors in the @f$L_2@f$ norm, @f$H_1@f$ seminorm, and
 * the energy norm, respectively. These are then printed to screen,
 * but also stored in a table that records how these errors decay
 * with mesh refinement and which can be output in one step at the
 * end of the program.
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::compute_errors()
 *     {
 *       double L2_error, H1_error, energy_error;
 *   
 *       {
 *         Vector<float> difference_per_cell(triangulation.n_active_cells());
 *         VectorTools::integrate_difference(mapping,
 *                                           dof_handler,
 *                                           solution,
 *                                           *(exact_solution.get()),
 *                                           difference_per_cell,
 *                                           quadrature_overintegration,
 *                                           VectorTools::L2_norm);
 *   
 *         L2_error = VectorTools::compute_global_error(triangulation,
 *                                                      difference_per_cell,
 *                                                      VectorTools::L2_norm);
 *         convergence_table.add_value("L2", L2_error);
 *       }
 *   
 *       {
 *         Vector<float> difference_per_cell(triangulation.n_active_cells());
 *         VectorTools::integrate_difference(mapping,
 *                                           dof_handler,
 *                                           solution,
 *                                           *(exact_solution.get()),
 *                                           difference_per_cell,
 *                                           quadrature_overintegration,
 *                                           VectorTools::H1_seminorm);
 *   
 *         H1_error = VectorTools::compute_global_error(triangulation,
 *                                                      difference_per_cell,
 *                                                      VectorTools::H1_seminorm);
 *         convergence_table.add_value("H1", H1_error);
 *       }
 *   
 *       {
 *         energy_error = compute_energy_norm_error();
 *         convergence_table.add_value("Energy", energy_error);
 *       }
 *   
 *       std::cout << "  Error in the L2 norm         : " << L2_error << std::endl
 *                 << "  Error in the H1 seminorm     : " << H1_error << std::endl
 *                 << "  Error in the energy norm     : " << energy_error
 *                 << std::endl;
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Therunfunction"></a> 
 * <h3>The run() function</h3>
 * 
 * @code
 *     template <int dim>
 *     void SIPGLaplace<dim>::run()
 *     {
 *       const unsigned int max_cycle =
 *         (test_case == TestCase::convergence_rate ? 6 : 20);
 *       for (unsigned int cycle = 0; cycle < max_cycle; ++cycle)
 *         {
 *           std::cout << "Cycle " << cycle << std::endl;
 *   
 *           switch (test_case)
 *             {
 *               case TestCase::convergence_rate:
 *                 {
 *                   if (cycle == 0)
 *                     {
 *                       GridGenerator::hyper_cube(triangulation);
 *   
 *                       triangulation.refine_global(2);
 *                     }
 *                   else
 *                     {
 *                       triangulation.refine_global(1);
 *                     }
 *                   break;
 *                 }
 *   
 *               case TestCase::l_singularity:
 *                 {
 *                   if (cycle == 0)
 *                     {
 *                       GridGenerator::hyper_L(triangulation);
 *                       triangulation.refine_global(3);
 *                     }
 *                   else
 *                     {
 *                       refine_grid();
 *                     }
 *                   break;
 *                 }
 *   
 *               default:
 *                 {
 *                   DEAL_II_NOT_IMPLEMENTED();
 *                 }
 *             }
 *   
 *           std::cout << "  Number of active cells       : "
 *                     << triangulation.n_active_cells() << std::endl;
 *           setup_system();
 *   
 *           std::cout << "  Number of degrees of freedom : " << dof_handler.n_dofs()
 *                     << std::endl;
 *   
 *           assemble_system();
 *           solve();
 *           output_results(cycle);
 *           {
 *             convergence_table.add_value("cycle", cycle);
 *             convergence_table.add_value("cells", triangulation.n_active_cells());
 *             convergence_table.add_value("dofs", dof_handler.n_dofs());
 *           }
 *           compute_errors();
 *   
 *           if (test_case == TestCase::l_singularity)
 *             {
 *               compute_error_estimate();
 *               std::cout << "  Estimated error              : "
 *                         << std::sqrt(estimated_error_square_per_cell.l1_norm())
 *                         << std::endl;
 *   
 *               convergence_table.add_value(
 *                 "Estimator",
 *                 std::sqrt(estimated_error_square_per_cell.l1_norm()));
 *             }
 *           std::cout << std::endl;
 *         }
 *   
 * @endcode
 * 
 * Having run all of our computations, let us tell the convergence
 * table how to format its data and output it to screen:
 * 
 * @code
 *       convergence_table.set_precision("L2", 3);
 *       convergence_table.set_precision("H1", 3);
 *       convergence_table.set_precision("Energy", 3);
 *   
 *       convergence_table.set_scientific("L2", true);
 *       convergence_table.set_scientific("H1", true);
 *       convergence_table.set_scientific("Energy", true);
 *   
 *       if (test_case == TestCase::convergence_rate)
 *         {
 *           convergence_table.evaluate_convergence_rates(
 *             "L2", ConvergenceTable::reduction_rate_log2);
 *           convergence_table.evaluate_convergence_rates(
 *             "H1", ConvergenceTable::reduction_rate_log2);
 *         }
 *       if (test_case == TestCase::l_singularity)
 *         {
 *           convergence_table.set_precision("Estimator", 3);
 *           convergence_table.set_scientific("Estimator", true);
 *         }
 *   
 *       std::cout << "degree = " << degree << std::endl;
 *       convergence_table.write_text(
 *         std::cout, TableHandler::TextOutputFormat::org_mode_table);
 *     }
 *   } // namespace Step74
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_74-Themainfunction"></a> 
 * <h3>The main() function</h3>
 * The following <code>main</code> function is similar to previous examples as
 * well, and need not be commented on.
 * 
 * @code
 *   int main()
 *   {
 *     try
 *       {
 *         using namespace dealii;
 *         using namespace Step74;
 *   
 *         const TestCase test_case = TestCase::l_singularity;
 *   
 *         SIPGLaplace<2> problem(test_case);
 *         problem.run();
 *       }
 *     catch (std::exception &exc)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << exc.what() << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       }
 *     catch (...)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Unknown exception!" << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       };
 *   
 *     return 0;
 *   }
 * @endcode
<a name="step_74-Results"></a><h1>Results</h1>
 
 
The output of this program consist of the console output and
solutions in vtu format.
 
In the first test case, when you run the program, the screen output should look like the following:
@code
Cycle 0
  Number of active cells       : 16
  Number of degrees of freedom : 256
  Error in the L2 norm         : 0.00193285
  Error in the H1 seminorm     : 0.106087
  Error in the energy norm     : 0.150625
 
Cycle 1
  Number of active cells       : 64
  Number of degrees of freedom : 1024
  Error in the L2 norm         : 9.60497e-05
  Error in the H1 seminorm     : 0.0089954
  Error in the energy norm     : 0.0113265
 
Cycle 2
.
.
.
@endcode
 
When using the smooth case with polynomial degree 3, the convergence
table will look like this:
<table align="center" class="doxtable">
  <tr>
    <th>cycle</th>
    <th>n_cells</th>
    <th>n_dofs</th>
    <th>L2 </th>
    <th>rate</th>
    <th>H1</th>
    <th>rate</th>
    <th>Energy</th>
  </tr>
  <tr>
    <td align="center">0</td>
    <td align="right">16</td>
    <td align="right">256</td>
    <td align="center">1.933e-03</td>
    <td>&nbsp;</td>
    <td align="center">1.061e-01</td>
    <td>&nbsp;</td>
    <td align="center">1.506e-01</td>
  </tr>
  <tr>
    <td align="center">1</td>
    <td align="right">64</td>
    <td align="right">1024</td>
    <td align="center">9.605e-05</td>
    <td align="center">4.33</td>
    <td align="center">8.995e-03</td>
    <td align="center">3.56</td>
    <td align="center">1.133e-02</td>
  </tr>
  <tr>
    <td align="center">2</td>
    <td align="right">256</td>
    <td align="right">4096</td>
    <td align="center">5.606e-06</td>
    <td align="center">4.10</td>
    <td align="center">9.018e-04</td>
    <td align="center">3.32</td>
    <td align="center">9.736e-04</td>
  </tr>
  <tr>
    <td align="center">3</td>
    <td align="right">1024</td>
    <td align="right">16384</td>
    <td align="center">3.484e-07</td>
    <td align="center">4.01</td>
    <td align="center">1.071e-04</td>
    <td align="center">3.07</td>
    <td align="center">1.088e-04</td>
  </tr>
  <tr>
    <td align="center">4</td>
    <td align="right">4096</td>
    <td align="right">65536</td>
    <td align="center">2.179e-08</td>
    <td align="center">4.00</td>
    <td align="center">1.327e-05</td>
    <td align="center">3.01</td>
    <td align="center">1.331e-05</td>
  </tr>
  <tr>
    <td align="center">5</td>
    <td align="right">16384</td>
    <td align="right">262144</td>
    <td align="center">1.363e-09</td>
    <td align="center">4.00</td>
    <td align="center">1.656e-06</td>
    <td align="center">3.00</td>
    <td align="center">1.657e-06</td>
  </tr>
</table>
 
Theoretically, for polynomial degree @f$p@f$, the order of convergence in @f$L_2@f$
norm and @f$H^1@f$ seminorm should be @f$p+1@f$ and @f$p@f$, respectively. Our numerical
results are in good agreement with theory.
 
In the second test case, when you run the program, the screen output should look like the following:
@code
Cycle 0
  Number of active cells       : 192
  Number of degrees of freedom : 3072
  Error in the L2 norm         : 0.000323585
  Error in the H1 seminorm     : 0.0296202
  Error in the energy norm     : 0.0420478
  Estimated error              : 0.136067
 
Cycle 1
  Number of active cells       : 249
  Number of degrees of freedom : 3984
  Error in the L2 norm         : 0.000114739
  Error in the H1 seminorm     : 0.0186571
  Error in the energy norm     : 0.0264879
  Estimated error              : 0.0857186
 
Cycle 2
.
.
.
@endcode
 
The following figure provides a log-log plot of the errors versus
the number of degrees of freedom for this test case on the L-shaped
domain. In order to interpret it, let @f$n@f$ be the number of degrees of
freedom, then on uniformly refined meshes, @f$h@f$ is of order
@f$1/\sqrt{n}@f$ in 2D. Combining the theoretical results in the previous case,
we see that if the solution is sufficiently smooth,
we can expect the error in the @f$L_2@f$ norm to be of order @f$O(n^{-\frac{p+1}{2}})@f$
and in @f$H^1@f$ seminorm to be @f$O(n^{-\frac{p}{2}})@f$. It is not a priori
clear that one would get the same kind of behavior as a function of
@f$n@f$ on adaptively refined meshes like the ones we use for this second
test case, but one can certainly hope. Indeed, from the figure, we see
that the SIPG with adaptive mesh refinement produces asymptotically
the kinds of hoped-for results:
 
<img width="600px" src="https://www.dealii.org/images/steps/developer/step-74.log-log-plot.png" alt="">
 
In addition, we observe that the error estimator decreases
at almost the same rate as the errors in the energy norm and @f$H^1@f$ seminorm,
and one order lower than the @f$L_2@f$ error. This suggests
its ability to predict regions with large errors.
 
While this tutorial is focused on the implementation, the @ref step_59 "step-59" tutorial program achieves an efficient
large-scale solver in terms of computing time with matrix-free solution techniques.
Note that the @ref step_59 "step-59" tutorial does not work with meshes containing hanging nodes at this moment,
because the multigrid interface matrices are not as easily determined,
but that is merely the lack of some interfaces in deal.II, nothing fundamental.
 *
 *
<a name="step_74-PlainProg"></a>
<h1> The plain program</h1>
@include "step-74.cc"
*/