step-51.h

Go to the documentation of this file.

 
 = 0) const override
 *       {
 *         double sum = 0;
 *         for (unsigned int i = 0; i < this->n_source_centers; ++i)
 *           {
 *             const Tensor<1, dim> x_minus_xi = p - this->source_centers[i];
 *             sum +=
 *               std::exp(-x_minus_xi.norm_square() / (this->width * this->width));
 *           }
 *   
 *         return sum /
 *                std::pow(2. * numbers::PI * this->width * this->width, dim / 2.);
 *       }
 *   
 *       virtual Tensor<1, dim>
 *       gradient(const Point<dim> &p,
 *                const unsigned int /*component*/ = 0) const override
 *       {
 *         Tensor<1, dim> sum;
 *         for (unsigned int i = 0; i < this->n_source_centers; ++i)
 *           {
 *             const Tensor<1, dim> x_minus_xi = p - this->source_centers[i];
 *   
 *             sum +=
 *               (-2 / (this->width * this->width) *
 *                std::exp(-x_minus_xi.norm_square() / (this->width * this->width)) *
 *                x_minus_xi);
 *           }
 *   
 *         return sum /
 *                std::pow(2. * numbers::PI * this->width * this->width, dim / 2.);
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * This class implements a function where the scalar solution and its negative
 * gradient are collected together. This function is used when computing the
 * error of the HDG approximation and its implementation is to simply call
 * value and gradient function of the Solution class.
 * 
 * @code
 *     template <int dim>
 *     class SolutionAndGradient : public Function<dim>, protected SolutionBase<dim>
 *     {
 *     public:
 *       SolutionAndGradient()
 *         : Function<dim>(dim + 1)
 *       {}
 *   
 *       virtual void vector_value(const Point<dim> &p,
 *                                 Vector<double>   &v) const override
 *       {
 *         AssertDimension(v.size(), dim + 1);
 *         Solution<dim>  solution;
 *         Tensor<1, dim> grad = solution.gradient(p);
 *         for (unsigned int d = 0; d < dim; ++d)
 *           v[d] = -grad[d];
 *         v[dim] = solution.value(p);
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * Next comes the implementation of the convection velocity. As described in
 * the introduction, we choose a velocity field that is @f$(y, -x)@f$ in 2d and
 * @f$(y, -x, 1)@f$ in 3d. This gives a divergence-free velocity field.
 * 
 * @code
 *     template <int dim>
 *     class ConvectionVelocity : public TensorFunction<1, dim>
 *     {
 *     public:
 *       ConvectionVelocity()
 *         : TensorFunction<1, dim>()
 *       {}
 *   
 *       virtual Tensor<1, dim> value(const Point<dim> &p) const override
 *       {
 *         Tensor<1, dim> convection;
 *         switch (dim)
 *           {
 *             case 1:
 *               convection[0] = 1;
 *               break;
 *             case 2:
 *               convection[0] = p[1];
 *               convection[1] = -p[0];
 *               break;
 *             case 3:
 *               convection[0] = p[1];
 *               convection[1] = -p[0];
 *               convection[2] = 1;
 *               break;
 *             default:
 *               DEAL_II_NOT_IMPLEMENTED();
 *           }
 *         return convection;
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * The last function we implement is the right hand side for the
 * manufactured solution. It is very similar to @ref step_7 "step-7", with the exception
 * that we now have a convection term instead of the reaction term. Since
 * the velocity field is incompressible, i.e., @f$\nabla \cdot \mathbf{c} =
 * 0@f$, the advection term simply reads @f$\mathbf{c} \nabla u@f$.
 * 
 * @code
 *     template <int dim>
 *     class RightHandSide : public Function<dim>, protected SolutionBase<dim>
 *     {
 *     public:
 *       virtual double value(const Point<dim> &p,
 *                            const unsigned int /*component*/ = 0) const override
 *       {
 *         ConvectionVelocity<dim> convection_velocity;
 *         Tensor<1, dim>          convection = convection_velocity.value(p);
 *         double                  sum        = 0;
 *         for (unsigned int i = 0; i < this->n_source_centers; ++i)
 *           {
 *             const Tensor<1, dim> x_minus_xi = p - this->source_centers[i];
 *   
 *             sum +=
 *               ((2 * dim - 2 * convection * x_minus_xi -
 *                 4 * x_minus_xi.norm_square() / (this->width * this->width)) /
 *                (this->width * this->width) *
 *                std::exp(-x_minus_xi.norm_square() / (this->width * this->width)));
 *           }
 *   
 *         return sum /
 *                std::pow(2. * numbers::PI * this->width * this->width, dim / 2.);
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-TheHDGsolverclass"></a> 
 * <h3>The HDG solver class</h3>
 * 
 
 * 
 * The HDG solution procedure follows closely that of @ref step_7 "step-7". The major
 * difference is the use of three different sets of DoFHandler and FE
 * objects, along with the ChunkSparseMatrix and the corresponding solutions
 * vectors. We also use WorkStream to enable a multithreaded local solution
 * process which exploits the embarrassingly parallel nature of the local
 * solver. For WorkStream, we define the local operations on a cell and a
 * copy function into the global matrix and vector. We do this both for the
 * assembly (which is run twice, once when we generate the system matrix and
 * once when we compute the element-interior solutions from the skeleton
 * values) and for the postprocessing where we extract a solution that
 * converges at higher order.
 * 
 * @code
 *     template <int dim>
 *     class HDG
 *     {
 *     public:
 *       enum RefinementMode
 *       {
 *         global_refinement,
 *         adaptive_refinement
 *       };
 *   
 *       HDG(const unsigned int degree, const RefinementMode refinement_mode);
 *       void run();
 *   
 *     private:
 *       void setup_system();
 *       void assemble_system(const bool reconstruct_trace = false);
 *       void solve();
 *       void postprocess();
 *       void refine_grid(const unsigned int cycle);
 *       void output_results(const unsigned int cycle);
 *   
 * @endcode
 * 
 * Data for the assembly and solution of the primal variables.
 * 
 * @code
 *       struct PerTaskData;
 *       struct ScratchData;
 *   
 * @endcode
 * 
 * Post-processing the solution to obtain @f$u^*@f$ is an element-by-element
 * procedure; as such, we do not need to assemble any global data and do
 * not declare any 'task data' for WorkStream to use.
 * 
 * @code
 *       struct PostProcessScratchData;
 *   
 * @endcode
 * 
 * The following three functions are used by WorkStream to do the actual
 * work of the program.
 * 
 * @code
 *       void assemble_system_one_cell(
 *         const typename DoFHandler<dim>::active_cell_iterator &cell,
 *         ScratchData                                          &scratch,
 *         PerTaskData                                          &task_data);
 *   
 *       void copy_local_to_global(const PerTaskData &data);
 *   
 *       void postprocess_one_cell(
 *         const typename DoFHandler<dim>::active_cell_iterator &cell,
 *         PostProcessScratchData                               &scratch,
 *         unsigned int                                         &empty_data);
 *   
 *   
 *       Triangulation<dim> triangulation;
 *   
 * @endcode
 * 
 * The 'local' solutions are interior to each element.  These
 * represent the primal solution field @f$u@f$ as well as the auxiliary
 * field @f$\mathbf{q}@f$.
 * 
 * @code
 *       const FESystem<dim> fe_local;
 *       DoFHandler<dim>     dof_handler_local;
 *       Vector<double>      solution_local;
 *   
 * @endcode
 * 
 * The new finite element type and corresponding <code>DoFHandler</code> are
 * used for the global skeleton solution that couples the element-level
 * local solutions.
 * 
 * @code
 *       const FE_FaceQ<dim> fe;
 *       DoFHandler<dim>     dof_handler;
 *       Vector<double>      solution;
 *       Vector<double>      system_rhs;
 *   
 * @endcode
 * 
 * As stated in the introduction, HDG solutions can be post-processed to
 * attain superconvergence rates of @f$\mathcal{O}(h^{p+2})@f$.  The
 * post-processed solution is a discontinuous finite element solution
 * representing the primal variable on the interior of each cell.  We define
 * a FE type of degree @f$p+1@f$ to represent this post-processed solution,
 * which we only use for output after constructing it.
 * 
 * @code
 *       const FE_DGQ<dim> fe_u_post;
 *       DoFHandler<dim>   dof_handler_u_post;
 *       Vector<double>    solution_u_post;
 *   
 * @endcode
 * 
 * The degrees of freedom corresponding to the skeleton strongly enforce
 * Dirichlet boundary conditions, just as in a continuous Galerkin finite
 * element method. We can enforce the boundary conditions in an analogous
 * manner via an AffineConstraints object. In addition, hanging nodes are
 * handled in the same way as for continuous finite elements: For the face
 * elements which only define degrees of freedom on the face, this process
 * sets the solution on the refined side to coincide with the
 * representation on the coarse side.
 *     
 
 * 
 * Note that for HDG, the elimination of hanging nodes is not the only
 * possibility &mdash; in terms of the HDG theory, one could also use the
 * unknowns from the refined side and express the local solution on the
 * coarse side through the trace values on the refined side. However, such
 * a setup is not as easily implemented in terms of deal.II loops and not
 * further analyzed.
 * 
 * @code
 *       AffineConstraints<double> constraints;
 *   
 * @endcode
 * 
 * The usage of the ChunkSparseMatrix class is similar to the usual sparse
 * matrices: You need a sparsity pattern of type ChunkSparsityPattern and
 * the actual matrix object. When creating the sparsity pattern, we just
 * have to additionally pass the size of local blocks.
 * 
 * @code
 *       ChunkSparsityPattern      sparsity_pattern;
 *       ChunkSparseMatrix<double> system_matrix;
 *   
 * @endcode
 * 
 * Same as @ref step_7 "step-7":
 * 
 * @code
 *       const RefinementMode refinement_mode;
 *       ConvergenceTable     convergence_table;
 *     };
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-TheHDGclassimplementation"></a> 
 * <h3>The HDG class implementation</h3>
 * 
 
 * 
 * 
 * <a name="step_51-Constructor"></a> 
 * <h4>Constructor</h4>
 * The constructor is similar to those in other examples, with the exception
 * of handling multiple DoFHandler and FiniteElement objects. Note that we
 * create a system of finite elements for the local DG part, including the
 * gradient/flux part and the scalar part.
 * 
 * @code
 *     template <int dim>
 *     HDG<dim>::HDG(const unsigned int degree, const RefinementMode refinement_mode)
 *       : fe_local(FE_DGQ<dim>(degree) ^ dim, FE_DGQ<dim>(degree))
 *       , dof_handler_local(triangulation)
 *       , fe(degree)
 *       , dof_handler(triangulation)
 *       , fe_u_post(degree + 1)
 *       , dof_handler_u_post(triangulation)
 *       , refinement_mode(refinement_mode)
 *     {}
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGsetup_system"></a> 
 * <h4>HDG::setup_system</h4>
 * The system for an HDG solution is setup in an analogous manner to most
 * of the other tutorial programs.  We are careful to distribute dofs with
 * all of our DoFHandler objects.  The @p solution and @p system_matrix
 * objects go with the global skeleton solution.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::setup_system()
 *     {
 *       dof_handler_local.distribute_dofs(fe_local);
 *       dof_handler.distribute_dofs(fe);
 *       dof_handler_u_post.distribute_dofs(fe_u_post);
 *   
 *       std::cout << "   Number of degrees of freedom: " << dof_handler.n_dofs()
 *                 << std::endl;
 *   
 *       solution.reinit(dof_handler.n_dofs());
 *       system_rhs.reinit(dof_handler.n_dofs());
 *   
 *       solution_local.reinit(dof_handler_local.n_dofs());
 *       solution_u_post.reinit(dof_handler_u_post.n_dofs());
 *   
 *       constraints.clear();
 *       DoFTools::make_hanging_node_constraints(dof_handler, constraints);
 *       std::map<types::boundary_id, const Function<dim> *> boundary_functions;
 *       Solution<dim>                                       solution_function;
 *       boundary_functions[0] = &solution_function;
 *       VectorTools::project_boundary_values(dof_handler,
 *                                            boundary_functions,
 *                                            QGauss<dim - 1>(fe.degree + 1),
 *                                            constraints);
 *       constraints.close();
 *   
 * @endcode
 * 
 * When creating the chunk sparsity pattern, we first create the usual
 * dynamic sparsity pattern and then set the chunk size, which is equal
 * to the number of dofs on a face, when copying this into the final
 * sparsity pattern.
 * 
 * @code
 *       {
 *         DynamicSparsityPattern dsp(dof_handler.n_dofs());
 *         DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, false);
 *         sparsity_pattern.copy_from(dsp, fe.n_dofs_per_face());
 *       }
 *       system_matrix.reinit(sparsity_pattern);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGPerTaskData"></a> 
 * <h4>HDG::PerTaskData</h4>
 * Next comes the definition of the local data structures for the parallel
 * assembly. The first structure @p PerTaskData contains the local vector
 * and matrix that are written into the global matrix, whereas the
 * ScratchData contains all data that we need for the local assembly. There
 * is one variable worth noting here, namely the boolean variable @p
 * trace_reconstruct. As mentioned in the introduction, we solve the HDG
 * system in two steps. First, we create a linear system for the skeleton
 * system where we condense the local part into it via the Schur complement
 * @f$D-CA^{-1}B@f$. Then, we solve for the local part using the skeleton
 * solution. For these two steps, we need the same matrices on the elements
 * twice, which we want to compute by two assembly steps. Since most of the
 * code is similar, we do this with the same function but only switch
 * between the two based on a flag that we set when starting the
 * assembly. Since we need to pass this information on to the local worker
 * routines, we store it once in the task data.
 * 
 * @code
 *     template <int dim>
 *     struct HDG<dim>::PerTaskData
 *     {
 *       FullMatrix<double>                   cell_matrix;
 *       Vector<double>                       cell_vector;
 *       std::vector<types::global_dof_index> dof_indices;
 *   
 *       bool trace_reconstruct;
 *   
 *       PerTaskData(const unsigned int n_dofs, const bool trace_reconstruct)
 *         : cell_matrix(n_dofs, n_dofs)
 *         , cell_vector(n_dofs)
 *         , dof_indices(n_dofs)
 *         , trace_reconstruct(trace_reconstruct)
 *       {}
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGScratchData"></a> 
 * <h4>HDG::ScratchData</h4>
 * @p ScratchData contains persistent data for each
 * thread within WorkStream.  The FEValues, matrix,
 * and vector objects should be familiar by now.  There are two objects that
 * need to be discussed: `std::vector<std::vector<unsigned int> >
 * fe_local_support_on_face` and `std::vector<std::vector<unsigned int> >
 * fe_support_on_face`.  These are used to indicate whether or not the finite
 * elements chosen have support (non-zero values) on a given face of the
 * reference cell for the local part associated to @p fe_local and the
 * skeleton part @p fe. We extract this information in the
 * constructor and store it once for all cells that we work on.  Had we not
 * stored this information, we would be forced to assemble a large number of
 * zero terms on each cell, which would significantly slow the program.
 * 
 * @code
 *     template <int dim>
 *     struct HDG<dim>::ScratchData
 *     {
 *       FEValues<dim>     fe_values_local;
 *       FEFaceValues<dim> fe_face_values_local;
 *       FEFaceValues<dim> fe_face_values;
 *   
 *       FullMatrix<double> ll_matrix;
 *       FullMatrix<double> lf_matrix;
 *       FullMatrix<double> fl_matrix;
 *       FullMatrix<double> tmp_matrix;
 *       Vector<double>     l_rhs;
 *       Vector<double>     tmp_rhs;
 *   
 *       std::vector<Tensor<1, dim>> q_phi;
 *       std::vector<double>         q_phi_div;
 *       std::vector<double>         u_phi;
 *       std::vector<Tensor<1, dim>> u_phi_grad;
 *       std::vector<double>         tr_phi;
 *       std::vector<double>         trace_values;
 *   
 *       std::vector<std::vector<unsigned int>> fe_local_support_on_face;
 *       std::vector<std::vector<unsigned int>> fe_support_on_face;
 *   
 *       ConvectionVelocity<dim> convection_velocity;
 *       RightHandSide<dim>      right_hand_side;
 *       const Solution<dim>     exact_solution;
 *   
 *       ScratchData(const FiniteElement<dim> &fe,
 *                   const FiniteElement<dim> &fe_local,
 *                   const QGauss<dim>        &quadrature_formula,
 *                   const QGauss<dim - 1>    &face_quadrature_formula,
 *                   const UpdateFlags         local_flags,
 *                   const UpdateFlags         local_face_flags,
 *                   const UpdateFlags         flags)
 *         : fe_values_local(fe_local, quadrature_formula, local_flags)
 *         , fe_face_values_local(fe_local,
 *                                face_quadrature_formula,
 *                                local_face_flags)
 *         , fe_face_values(fe, face_quadrature_formula, flags)
 *         , ll_matrix(fe_local.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
 *         , lf_matrix(fe_local.n_dofs_per_cell(), fe.n_dofs_per_cell())
 *         , fl_matrix(fe.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
 *         , tmp_matrix(fe.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
 *         , l_rhs(fe_local.n_dofs_per_cell())
 *         , tmp_rhs(fe_local.n_dofs_per_cell())
 *         , q_phi(fe_local.n_dofs_per_cell())
 *         , q_phi_div(fe_local.n_dofs_per_cell())
 *         , u_phi(fe_local.n_dofs_per_cell())
 *         , u_phi_grad(fe_local.n_dofs_per_cell())
 *         , tr_phi(fe.n_dofs_per_cell())
 *         , trace_values(face_quadrature_formula.size())
 *         , fe_local_support_on_face(GeometryInfo<dim>::faces_per_cell)
 *         , fe_support_on_face(GeometryInfo<dim>::faces_per_cell)
 *         , exact_solution()
 *       {
 *         for (const unsigned int face_no : GeometryInfo<dim>::face_indices())
 *           for (unsigned int i = 0; i < fe_local.n_dofs_per_cell(); ++i)
 *             {
 *               if (fe_local.has_support_on_face(i, face_no))
 *                 fe_local_support_on_face[face_no].push_back(i);
 *             }
 *   
 *         for (const unsigned int face_no : GeometryInfo<dim>::face_indices())
 *           for (unsigned int i = 0; i < fe.n_dofs_per_cell(); ++i)
 *             {
 *               if (fe.has_support_on_face(i, face_no))
 *                 fe_support_on_face[face_no].push_back(i);
 *             }
 *       }
 *   
 *       ScratchData(const ScratchData &sd)
 *         : fe_values_local(sd.fe_values_local.get_fe(),
 *                           sd.fe_values_local.get_quadrature(),
 *                           sd.fe_values_local.get_update_flags())
 *         , fe_face_values_local(sd.fe_face_values_local.get_fe(),
 *                                sd.fe_face_values_local.get_quadrature(),
 *                                sd.fe_face_values_local.get_update_flags())
 *         , fe_face_values(sd.fe_face_values.get_fe(),
 *                          sd.fe_face_values.get_quadrature(),
 *                          sd.fe_face_values.get_update_flags())
 *         , ll_matrix(sd.ll_matrix)
 *         , lf_matrix(sd.lf_matrix)
 *         , fl_matrix(sd.fl_matrix)
 *         , tmp_matrix(sd.tmp_matrix)
 *         , l_rhs(sd.l_rhs)
 *         , tmp_rhs(sd.tmp_rhs)
 *         , q_phi(sd.q_phi)
 *         , q_phi_div(sd.q_phi_div)
 *         , u_phi(sd.u_phi)
 *         , u_phi_grad(sd.u_phi_grad)
 *         , tr_phi(sd.tr_phi)
 *         , trace_values(sd.trace_values)
 *         , fe_local_support_on_face(sd.fe_local_support_on_face)
 *         , fe_support_on_face(sd.fe_support_on_face)
 *         , exact_solution()
 *       {}
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGPostProcessScratchData"></a> 
 * <h4>HDG::PostProcessScratchData</h4>
 * @p PostProcessScratchData contains the data used by WorkStream
 * when post-processing the local solution @f$u^*@f$.  It is similar, but much
 * simpler, than @p ScratchData.
 * 
 * @code
 *     template <int dim>
 *     struct HDG<dim>::PostProcessScratchData
 *     {
 *       FEValues<dim> fe_values_local;
 *       FEValues<dim> fe_values;
 *   
 *       std::vector<double>         u_values;
 *       std::vector<Tensor<1, dim>> u_gradients;
 *       FullMatrix<double>          cell_matrix;
 *   
 *       Vector<double> cell_rhs;
 *       Vector<double> cell_sol;
 *   
 *       PostProcessScratchData(const FiniteElement<dim> &fe,
 *                              const FiniteElement<dim> &fe_local,
 *                              const QGauss<dim>        &quadrature_formula,
 *                              const UpdateFlags         local_flags,
 *                              const UpdateFlags         flags)
 *         : fe_values_local(fe_local, quadrature_formula, local_flags)
 *         , fe_values(fe, quadrature_formula, flags)
 *         , u_values(quadrature_formula.size())
 *         , u_gradients(quadrature_formula.size())
 *         , cell_matrix(fe.n_dofs_per_cell(), fe.n_dofs_per_cell())
 *         , cell_rhs(fe.n_dofs_per_cell())
 *         , cell_sol(fe.n_dofs_per_cell())
 *       {}
 *   
 *       PostProcessScratchData(const PostProcessScratchData &sd)
 *         : fe_values_local(sd.fe_values_local.get_fe(),
 *                           sd.fe_values_local.get_quadrature(),
 *                           sd.fe_values_local.get_update_flags())
 *         , fe_values(sd.fe_values.get_fe(),
 *                     sd.fe_values.get_quadrature(),
 *                     sd.fe_values.get_update_flags())
 *         , u_values(sd.u_values)
 *         , u_gradients(sd.u_gradients)
 *         , cell_matrix(sd.cell_matrix)
 *         , cell_rhs(sd.cell_rhs)
 *         , cell_sol(sd.cell_sol)
 *       {}
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGassemble_system"></a> 
 * <h4>HDG::assemble_system</h4>
 * The @p assemble_system function is similar to the one on @ref step_32 "step-32", where
 * the quadrature formula and the update flags are set up, and then
 * <code>WorkStream</code> is used to do the work in a multi-threaded
 * manner.  The @p trace_reconstruct input parameter is used to decide
 * whether we are solving for the global skeleton solution (false) or the
 * local solution (true).
 *   
 
 * 
 * One thing worth noting for the multi-threaded execution of assembly is
 * the fact that the local computations in `assemble_system_one_cell()` call
 * into BLAS and LAPACK functions if those are available in deal.II. Thus,
 * the underlying BLAS/LAPACK library must support calls from multiple
 * threads at the same time. Most implementations do support this, but some
 * libraries need to be built in a specific way to avoid problems. For
 * example, OpenBLAS compiled without multithreading inside the BLAS/LAPACK
 * calls needs to built with a flag called `USE_LOCKING` set to true.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::assemble_system(const bool trace_reconstruct)
 *     {
 *       const QGauss<dim>     quadrature_formula(fe.degree + 1);
 *       const QGauss<dim - 1> face_quadrature_formula(fe.degree + 1);
 *   
 *       const UpdateFlags local_flags(update_values | update_gradients |
 *                                     update_JxW_values | update_quadrature_points);
 *   
 *       const UpdateFlags local_face_flags(update_values);
 *   
 *       const UpdateFlags flags(update_values | update_normal_vectors |
 *                               update_quadrature_points | update_JxW_values);
 *   
 *       PerTaskData task_data(fe.n_dofs_per_cell(), trace_reconstruct);
 *       ScratchData scratch(fe,
 *                           fe_local,
 *                           quadrature_formula,
 *                           face_quadrature_formula,
 *                           local_flags,
 *                           local_face_flags,
 *                           flags);
 *   
 *       WorkStream::run(dof_handler.begin_active(),
 *                       dof_handler.end(),
 *                       *this,
 *                       &HDG<dim>::assemble_system_one_cell,
 *                       &HDG<dim>::copy_local_to_global,
 *                       scratch,
 *                       task_data);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGassemble_system_one_cell"></a> 
 * <h4>HDG::assemble_system_one_cell</h4>
 * The real work of the HDG program is done by @p assemble_system_one_cell.
 * Assembling the local matrices @f$A, B, C@f$ is done here, along with the
 * local contributions of the global matrix @f$D@f$.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::assemble_system_one_cell(
 *       const typename DoFHandler<dim>::active_cell_iterator &cell,
 *       ScratchData                                          &scratch,
 *       PerTaskData                                          &task_data)
 *     {
 * @endcode
 * 
 * Construct iterator for dof_handler_local for FEValues reinit function.
 * 
 * @code
 *       const typename DoFHandler<dim>::active_cell_iterator loc_cell =
 *         cell->as_dof_handler_iterator(dof_handler_local);
 *   
 *       const unsigned int n_q_points =
 *         scratch.fe_values_local.get_quadrature().size();
 *       const unsigned int n_face_q_points =
 *         scratch.fe_face_values_local.get_quadrature().size();
 *   
 *       const unsigned int loc_dofs_per_cell =
 *         scratch.fe_values_local.get_fe().n_dofs_per_cell();
 *   
 *       const FEValuesExtractors::Vector fluxes(0);
 *       const FEValuesExtractors::Scalar scalar(dim);
 *   
 *       scratch.ll_matrix = 0;
 *       scratch.l_rhs     = 0;
 *       if (!task_data.trace_reconstruct)
 *         {
 *           scratch.lf_matrix     = 0;
 *           scratch.fl_matrix     = 0;
 *           task_data.cell_matrix = 0;
 *           task_data.cell_vector = 0;
 *         }
 *       scratch.fe_values_local.reinit(loc_cell);
 *   
 * @endcode
 * 
 * We first compute the cell-interior contribution to @p ll_matrix matrix
 * (referred to as matrix @f$A@f$ in the introduction) corresponding to
 * local-local coupling, as well as the local right-hand-side vector.  We
 * store the values at each quadrature point for the basis functions, the
 * right-hand-side value, and the convection velocity, in order to have
 * quick access to these fields.
 * 
 * @code
 *       for (unsigned int q = 0; q < n_q_points; ++q)
 *         {
 *           const double rhs_value = scratch.right_hand_side.value(
 *             scratch.fe_values_local.quadrature_point(q));
 *           const Tensor<1, dim> convection = scratch.convection_velocity.value(
 *             scratch.fe_values_local.quadrature_point(q));
 *           const double JxW = scratch.fe_values_local.JxW(q);
 *           for (unsigned int k = 0; k < loc_dofs_per_cell; ++k)
 *             {
 *               scratch.q_phi[k] = scratch.fe_values_local[fluxes].value(k, q);
 *               scratch.q_phi_div[k] =
 *                 scratch.fe_values_local[fluxes].divergence(k, q);
 *               scratch.u_phi[k] = scratch.fe_values_local[scalar].value(k, q);
 *               scratch.u_phi_grad[k] =
 *                 scratch.fe_values_local[scalar].gradient(k, q);
 *             }
 *           for (unsigned int i = 0; i < loc_dofs_per_cell; ++i)
 *             {
 *               for (unsigned int j = 0; j < loc_dofs_per_cell; ++j)
 *                 scratch.ll_matrix(i, j) +=
 *                   (scratch.q_phi[i] * scratch.q_phi[j] -
 *                    scratch.q_phi_div[i] * scratch.u_phi[j] +
 *                    scratch.u_phi[i] * scratch.q_phi_div[j] -
 *                    (scratch.u_phi_grad[i] * convection) * scratch.u_phi[j]) *
 *                   JxW;
 *               scratch.l_rhs(i) += scratch.u_phi[i] * rhs_value * JxW;
 *             }
 *         }
 *   
 * @endcode
 * 
 * Face terms are assembled on all faces of all elements. This is in
 * contrast to more traditional DG methods, where each face is only visited
 * once in the assembly procedure.
 * 
 * @code
 *       for (const auto face_no : cell->face_indices())
 *         {
 *           scratch.fe_face_values_local.reinit(loc_cell, face_no);
 *           scratch.fe_face_values.reinit(cell, face_no);
 *   
 * @endcode
 * 
 * The already obtained @f$\hat{u}@f$ values are needed when solving for the
 * local variables.
 * 
 * @code
 *           if (task_data.trace_reconstruct)
 *             scratch.fe_face_values.get_function_values(solution,
 *                                                        scratch.trace_values);
 *   
 *           for (unsigned int q = 0; q < n_face_q_points; ++q)
 *             {
 *               const double     JxW = scratch.fe_face_values.JxW(q);
 *               const Point<dim> quadrature_point =
 *                 scratch.fe_face_values.quadrature_point(q);
 *               const Tensor<1, dim> normal =
 *                 scratch.fe_face_values.normal_vector(q);
 *               const Tensor<1, dim> convection =
 *                 scratch.convection_velocity.value(quadrature_point);
 *   
 * @endcode
 * 
 * Here we compute the stabilization parameter discussed in the
 * introduction: since the diffusion is one and the diffusion
 * length scale is set to 1/5, it simply results in a contribution
 * of 5 for the diffusion part and the magnitude of convection
 * through the element boundary in a centered scheme for the
 * convection part.
 * 
 * @code
 *               const double tau_stab = (5. + std::abs(convection * normal));
 *   
 * @endcode
 * 
 * We store the non-zero flux and scalar values, making use of the
 * support_on_face information we created in @p ScratchData.
 * 
 * @code
 *               for (unsigned int k = 0;
 *                    k < scratch.fe_local_support_on_face[face_no].size();
 *                    ++k)
 *                 {
 *                   const unsigned int kk =
 *                     scratch.fe_local_support_on_face[face_no][k];
 *                   scratch.q_phi[k] =
 *                     scratch.fe_face_values_local[fluxes].value(kk, q);
 *                   scratch.u_phi[k] =
 *                     scratch.fe_face_values_local[scalar].value(kk, q);
 *                 }
 *   
 * @endcode
 * 
 * When @p trace_reconstruct=false, we are preparing to assemble the
 * system for the skeleton variable @f$\hat{u}@f$. If this is the case,
 * we must assemble all local matrices associated with the problem:
 * local-local, local-face, face-local, and face-face.  The
 * face-face matrix is stored as @p TaskData::cell_matrix, so that
 * it can be assembled into the global system by @p
 * copy_local_to_global.
 * 
 * @code
 *               if (!task_data.trace_reconstruct)
 *                 {
 *                   for (unsigned int k = 0;
 *                        k < scratch.fe_support_on_face[face_no].size();
 *                        ++k)
 *                     scratch.tr_phi[k] = scratch.fe_face_values.shape_value(
 *                       scratch.fe_support_on_face[face_no][k], q);
 *                   for (unsigned int i = 0;
 *                        i < scratch.fe_local_support_on_face[face_no].size();
 *                        ++i)
 *                     for (unsigned int j = 0;
 *                          j < scratch.fe_support_on_face[face_no].size();
 *                          ++j)
 *                       {
 *                         const unsigned int ii =
 *                           scratch.fe_local_support_on_face[face_no][i];
 *                         const unsigned int jj =
 *                           scratch.fe_support_on_face[face_no][j];
 *                         scratch.lf_matrix(ii, jj) +=
 *                           ((scratch.q_phi[i] * normal +
 *                             (convection * normal - tau_stab) * scratch.u_phi[i]) *
 *                            scratch.tr_phi[j]) *
 *                           JxW;
 *   
 * @endcode
 * 
 * Note the sign of the face_no-local matrix.  We negate
 * the sign during assembly here so that we can use the
 * FullMatrix::mmult with addition when computing the
 * Schur complement.
 * 
 * @code
 *                         scratch.fl_matrix(jj, ii) -=
 *                           ((scratch.q_phi[i] * normal +
 *                             tau_stab * scratch.u_phi[i]) *
 *                            scratch.tr_phi[j]) *
 *                           JxW;
 *                       }
 *   
 *                   for (unsigned int i = 0;
 *                        i < scratch.fe_support_on_face[face_no].size();
 *                        ++i)
 *                     for (unsigned int j = 0;
 *                          j < scratch.fe_support_on_face[face_no].size();
 *                          ++j)
 *                       {
 *                         const unsigned int ii =
 *                           scratch.fe_support_on_face[face_no][i];
 *                         const unsigned int jj =
 *                           scratch.fe_support_on_face[face_no][j];
 *                         task_data.cell_matrix(ii, jj) +=
 *                           ((convection * normal - tau_stab) * scratch.tr_phi[i] *
 *                            scratch.tr_phi[j]) *
 *                           JxW;
 *                       }
 *   
 *                   if (cell->face(face_no)->at_boundary() &&
 *                       (cell->face(face_no)->boundary_id() == 1))
 *                     {
 *                       const double neumann_value =
 *                         -scratch.exact_solution.gradient(quadrature_point) *
 *                           normal +
 *                         convection * normal *
 *                           scratch.exact_solution.value(quadrature_point);
 *                       for (unsigned int i = 0;
 *                            i < scratch.fe_support_on_face[face_no].size();
 *                            ++i)
 *                         {
 *                           const unsigned int ii =
 *                             scratch.fe_support_on_face[face_no][i];
 *                           task_data.cell_vector(ii) +=
 *                             scratch.tr_phi[i] * neumann_value * JxW;
 *                         }
 *                     }
 *                 }
 *   
 * @endcode
 * 
 * This last term adds the contribution of the term @f$\left<w,\tau
 * u_h\right>_{\partial \mathcal T}@f$ to the local matrix. As opposed
 * to the face matrices above, we need it in both assembly stages.
 * 
 * @code
 *               for (unsigned int i = 0;
 *                    i < scratch.fe_local_support_on_face[face_no].size();
 *                    ++i)
 *                 for (unsigned int j = 0;
 *                      j < scratch.fe_local_support_on_face[face_no].size();
 *                      ++j)
 *                   {
 *                     const unsigned int ii =
 *                       scratch.fe_local_support_on_face[face_no][i];
 *                     const unsigned int jj =
 *                       scratch.fe_local_support_on_face[face_no][j];
 *                     scratch.ll_matrix(ii, jj) +=
 *                       tau_stab * scratch.u_phi[i] * scratch.u_phi[j] * JxW;
 *                   }
 *   
 * @endcode
 * 
 * When @p trace_reconstruct=true, we are solving for the local
 * solutions on an element by element basis.  The local
 * right-hand-side is calculated by replacing the basis functions @p
 * tr_phi in the @p lf_matrix computation by the computed values @p
 * trace_values.  Of course, the sign of the matrix is now minus
 * since we have moved everything to the other side of the equation.
 * 
 * @code
 *               if (task_data.trace_reconstruct)
 *                 for (unsigned int i = 0;
 *                      i < scratch.fe_local_support_on_face[face_no].size();
 *                      ++i)
 *                   {
 *                     const unsigned int ii =
 *                       scratch.fe_local_support_on_face[face_no][i];
 *                     scratch.l_rhs(ii) -=
 *                       (scratch.q_phi[i] * normal +
 *                        scratch.u_phi[i] * (convection * normal - tau_stab)) *
 *                       scratch.trace_values[q] * JxW;
 *                   }
 *             }
 *         }
 *   
 * @endcode
 * 
 * Once assembly of all of the local contributions is complete, we must
 * either: (1) assemble the global system, or (2) compute the local solution
 * values and save them. In either case, the first step is to invert the
 * local-local matrix.
 * 
 * @code
 *       scratch.ll_matrix.gauss_jordan();
 *   
 * @endcode
 * 
 * For (1), we compute the Schur complement and add it to the @p
 * cell_matrix, matrix @f$D@f$ in the introduction.
 * 
 * @code
 *       if (task_data.trace_reconstruct == false)
 *         {
 *           scratch.fl_matrix.mmult(scratch.tmp_matrix, scratch.ll_matrix);
 *           scratch.tmp_matrix.vmult_add(task_data.cell_vector, scratch.l_rhs);
 *           scratch.tmp_matrix.mmult(task_data.cell_matrix,
 *                                    scratch.lf_matrix,
 *                                    true);
 *           cell->get_dof_indices(task_data.dof_indices);
 *         }
 * @endcode
 * 
 * For (2), we are simply solving (ll_matrix).(solution_local) = (l_rhs).
 * Hence, we multiply @p l_rhs by our already inverted local-local matrix
 * and store the result using the <code>set_dof_values</code> function.
 * 
 * @code
 *       else
 *         {
 *           scratch.ll_matrix.vmult(scratch.tmp_rhs, scratch.l_rhs);
 *           loc_cell->set_dof_values(scratch.tmp_rhs, solution_local);
 *         }
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGcopy_local_to_global"></a> 
 * <h4>HDG::copy_local_to_global</h4>
 * If we are in the first step of the solution, i.e. @p trace_reconstruct=false,
 * then we assemble the local matrices into the global system.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::copy_local_to_global(const PerTaskData &data)
 *     {
 *       if (data.trace_reconstruct == false)
 *         constraints.distribute_local_to_global(data.cell_matrix,
 *                                                data.cell_vector,
 *                                                data.dof_indices,
 *                                                system_matrix,
 *                                                system_rhs);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGsolve"></a> 
 * <h4>HDG::solve</h4>
 * The skeleton solution is solved for by using a BiCGStab solver with
 * identity preconditioner.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::solve()
 *     {
 *       SolverControl                  solver_control(system_matrix.m() * 10,
 *                                    1e-11 * system_rhs.l2_norm());
 *       SolverBicgstab<Vector<double>> solver(solver_control);
 *       solver.solve(system_matrix, solution, system_rhs, PreconditionIdentity());
 *   
 *       std::cout << "   Number of BiCGStab iterations: "
 *                 << solver_control.last_step() << std::endl;
 *   
 *       system_matrix.clear();
 *       sparsity_pattern.reinit(0, 0, 0, 1);
 *   
 *       constraints.distribute(solution);
 *   
 * @endcode
 * 
 * Once we have solved for the skeleton solution,
 * we can solve for the local solutions in an element-by-element
 * fashion.  We do this by re-using the same @p assemble_system function
 * but switching @p trace_reconstruct to true.
 * 
 * @code
 *       assemble_system(true);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGpostprocess"></a> 
 * <h4>HDG::postprocess</h4>
 * 
 
 * 
 * The postprocess method serves two purposes. First, we want to construct a
 * post-processed scalar variables in the element space of degree @f$p+1@f$ that
 * we hope will converge at order @f$p+2@f$. This is again an element-by-element
 * process and only involves the scalar solution as well as the gradient on
 * the local cell. To do this, we introduce the already defined scratch data
 * together with some update flags and run the work stream to do this in
 * parallel.
 *   
 
 * 
 * Secondly, we want to compute discretization errors just as we did in
 * @ref step_7 "step-7". The overall procedure is similar with calls to
 * VectorTools::integrate_difference. The difference is in how we compute
 * the errors for the scalar variable and the gradient variable. In @ref step_7 "step-7",
 * we did this by computing @p L2_norm or @p H1_seminorm
 * contributions. Here, we have a DoFHandler with these two contributions
 * computed and sorted by their vector component, <code>[0, dim)</code> for
 * the
 * gradient and @p dim for the scalar. To compute their value, we hence use
 * a ComponentSelectFunction with either of them, together with the @p
 * SolutionAndGradient class introduced above that contains the analytic
 * parts of either of them. Eventually, we also compute the L2-error of the
 * post-processed solution and add the results into the convergence table.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::postprocess()
 *     {
 *       {
 *         const QGauss<dim> quadrature_formula(fe_u_post.degree + 1);
 *         const UpdateFlags local_flags(update_values);
 *         const UpdateFlags flags(update_values | update_gradients |
 *                                 update_JxW_values);
 *   
 *         PostProcessScratchData scratch(
 *           fe_u_post, fe_local, quadrature_formula, local_flags, flags);
 *   
 *         WorkStream::run(
 *           dof_handler_u_post.begin_active(),
 *           dof_handler_u_post.end(),
 *           [this](const typename DoFHandler<dim>::active_cell_iterator &cell,
 *                  PostProcessScratchData                               &scratch,
 *                  unsigned int                                         &data) {
 *             this->postprocess_one_cell(cell, scratch, data);
 *           },
 *           std::function<void(const unsigned int &)>(),
 *           scratch,
 *           0U);
 *       }
 *   
 *       Vector<float> difference_per_cell(triangulation.n_active_cells());
 *   
 *       ComponentSelectFunction<dim> value_select(dim, dim + 1);
 *       VectorTools::integrate_difference(dof_handler_local,
 *                                         solution_local,
 *                                         SolutionAndGradient<dim>(),
 *                                         difference_per_cell,
 *                                         QGauss<dim>(fe.degree + 2),
 *                                         VectorTools::L2_norm,
 *                                         &value_select);
 *       const double L2_error =
 *         VectorTools::compute_global_error(triangulation,
 *                                           difference_per_cell,
 *                                           VectorTools::L2_norm);
 *   
 *       ComponentSelectFunction<dim> gradient_select(
 *         std::pair<unsigned int, unsigned int>(0, dim), dim + 1);
 *       VectorTools::integrate_difference(dof_handler_local,
 *                                         solution_local,
 *                                         SolutionAndGradient<dim>(),
 *                                         difference_per_cell,
 *                                         QGauss<dim>(fe.degree + 2),
 *                                         VectorTools::L2_norm,
 *                                         &gradient_select);
 *       const double grad_error =
 *         VectorTools::compute_global_error(triangulation,
 *                                           difference_per_cell,
 *                                           VectorTools::L2_norm);
 *   
 *       VectorTools::integrate_difference(dof_handler_u_post,
 *                                         solution_u_post,
 *                                         Solution<dim>(),
 *                                         difference_per_cell,
 *                                         QGauss<dim>(fe.degree + 3),
 *                                         VectorTools::L2_norm);
 *       const double post_error =
 *         VectorTools::compute_global_error(triangulation,
 *                                           difference_per_cell,
 *                                           VectorTools::L2_norm);
 *   
 *       convergence_table.add_value("cells", triangulation.n_active_cells());
 *       convergence_table.add_value("dofs", dof_handler.n_dofs());
 *   
 *       convergence_table.add_value("val L2", L2_error);
 *       convergence_table.set_scientific("val L2", true);
 *       convergence_table.set_precision("val L2", 3);
 *   
 *       convergence_table.add_value("grad L2", grad_error);
 *       convergence_table.set_scientific("grad L2", true);
 *       convergence_table.set_precision("grad L2", 3);
 *   
 *       convergence_table.add_value("val L2-post", post_error);
 *       convergence_table.set_scientific("val L2-post", true);
 *       convergence_table.set_precision("val L2-post", 3);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGpostprocess_one_cell"></a> 
 * <h4>HDG::postprocess_one_cell</h4>
 *   
 
 * 
 * This is the actual work done for the postprocessing. According to the
 * discussion in the introduction, we need to set up a system that projects
 * the gradient part of the DG solution onto the gradient of the
 * post-processed variable. Moreover, we need to set the average of the new
 * post-processed variable to equal the average of the scalar DG solution
 * on the cell.
 *   
 
 * 
 * More technically speaking, the projection of the gradient is a system
 * that would potentially fills our @p dofs_per_cell times @p dofs_per_cell
 * matrix but is singular (the sum of all rows would be zero because the
 * constant function has zero gradient). Therefore, we take one row away and
 * use it for imposing the average of the scalar value. We pick the first
 * row for the scalar part, even though we could pick any row for @f$\mathcal
 * Q_{-p}@f$ elements. However, had we used FE_DGP elements instead, the first
 * row would correspond to the constant part already and deleting e.g. the
 * last row would give us a singular system. This way, our program can also
 * be used for those elements.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::postprocess_one_cell(
 *       const typename DoFHandler<dim>::active_cell_iterator &cell,
 *       PostProcessScratchData                               &scratch,
 *       unsigned int &)
 *     {
 *       const typename DoFHandler<dim>::active_cell_iterator loc_cell =
 *         cell->as_dof_handler_iterator(dof_handler_local);
 *   
 *       scratch.fe_values_local.reinit(loc_cell);
 *       scratch.fe_values.reinit(cell);
 *   
 *       const FEValuesExtractors::Vector fluxes(0);
 *       const FEValuesExtractors::Scalar scalar(dim);
 *   
 *       const unsigned int n_q_points = scratch.fe_values.get_quadrature().size();
 *       const unsigned int dofs_per_cell = scratch.fe_values.dofs_per_cell;
 *   
 *       scratch.fe_values_local[scalar].get_function_values(solution_local,
 *                                                           scratch.u_values);
 *       scratch.fe_values_local[fluxes].get_function_values(solution_local,
 *                                                           scratch.u_gradients);
 *   
 *       double sum = 0;
 *       for (unsigned int i = 1; i < dofs_per_cell; ++i)
 *         {
 *           for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *             {
 *               sum = 0;
 *               for (unsigned int q = 0; q < n_q_points; ++q)
 *                 sum += (scratch.fe_values.shape_grad(i, q) *
 *                         scratch.fe_values.shape_grad(j, q)) *
 *                        scratch.fe_values.JxW(q);
 *               scratch.cell_matrix(i, j) = sum;
 *             }
 *   
 *           sum = 0;
 *           for (unsigned int q = 0; q < n_q_points; ++q)
 *             sum -= (scratch.fe_values.shape_grad(i, q) * scratch.u_gradients[q]) *
 *                    scratch.fe_values.JxW(q);
 *           scratch.cell_rhs(i) = sum;
 *         }
 *       for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *         {
 *           sum = 0;
 *           for (unsigned int q = 0; q < n_q_points; ++q)
 *             sum += scratch.fe_values.shape_value(j, q) * scratch.fe_values.JxW(q);
 *           scratch.cell_matrix(0, j) = sum;
 *         }
 *       {
 *         sum = 0;
 *         for (unsigned int q = 0; q < n_q_points; ++q)
 *           sum += scratch.u_values[q] * scratch.fe_values.JxW(q);
 *         scratch.cell_rhs(0) = sum;
 *       }
 *   
 * @endcode
 * 
 * Having assembled all terms, we can again go on and solve the linear
 * system. We invert the matrix and then multiply the inverse by the
 * right hand side. An alternative (and more numerically stable) method
 * would have been to only factorize the matrix and apply the factorization.
 * 
 * @code
 *       scratch.cell_matrix.gauss_jordan();
 *       scratch.cell_matrix.vmult(scratch.cell_sol, scratch.cell_rhs);
 *       cell->distribute_local_to_global(scratch.cell_sol, solution_u_post);
 *     }
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGoutput_results"></a> 
 * <h4>HDG::output_results</h4>
 * We have 3 sets of results that we would like to output:  the local
 * solution, the post-processed local solution, and the skeleton solution. The
 * former 2 both 'live' on element volumes, whereas the latter lives on
 * codimension-1 surfaces
 * of the triangulation.  Our @p output_results function writes all local solutions
 * to the same vtk file, even though they correspond to different
 * DoFHandler objects.  The graphical output for the skeleton
 * variable is done through use of the DataOutFaces class.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::output_results(const unsigned int cycle)
 *     {
 *       std::string filename;
 *       switch (refinement_mode)
 *         {
 *           case global_refinement:
 *             filename = "solution-global";
 *             break;
 *           case adaptive_refinement:
 *             filename = "solution-adaptive";
 *             break;
 *           default:
 *             DEAL_II_NOT_IMPLEMENTED();
 *         }
 *   
 *       std::string face_out(filename);
 *       face_out += "-face";
 *   
 *       filename += "-q" + Utilities::int_to_string(fe.degree, 1);
 *       filename += "-" + Utilities::int_to_string(cycle, 2);
 *       filename += ".vtk";
 *       std::ofstream output(filename);
 *   
 *       DataOut<dim> data_out;
 *   
 * @endcode
 * 
 * We first define the names and types of the local solution,
 * and add the data to @p data_out.
 * 
 * @code
 *       std::vector<std::string> names(dim, "gradient");
 *       names.emplace_back("solution");
 *       std::vector<DataComponentInterpretation::DataComponentInterpretation>
 *         component_interpretation(
 *           dim + 1, DataComponentInterpretation::component_is_part_of_vector);
 *       component_interpretation[dim] =
 *         DataComponentInterpretation::component_is_scalar;
 *       data_out.add_data_vector(dof_handler_local,
 *                                solution_local,
 *                                names,
 *                                component_interpretation);
 *   
 * @endcode
 * 
 * The second data item we add is the post-processed solution.
 * In this case, it is a single scalar variable belonging to
 * a different DoFHandler.
 * 
 * @code
 *       std::vector<std::string> post_name(1, "u_post");
 *       std::vector<DataComponentInterpretation::DataComponentInterpretation>
 *         post_comp_type(1, DataComponentInterpretation::component_is_scalar);
 *       data_out.add_data_vector(dof_handler_u_post,
 *                                solution_u_post,
 *                                post_name,
 *                                post_comp_type);
 *   
 *       data_out.build_patches(fe.degree);
 *       data_out.write_vtk(output);
 *   
 *       face_out += "-q" + Utilities::int_to_string(fe.degree, 1);
 *       face_out += "-" + Utilities::int_to_string(cycle, 2);
 *       face_out += ".vtk";
 *       std::ofstream face_output(face_out);
 *   
 * @endcode
 * 
 * The <code>DataOutFaces</code> class works analogously to the
 * <code>DataOut</code> class when we have a <code>DoFHandler</code> that
 * defines the solution on the skeleton of the triangulation.  We treat it
 * as such here, and the code is similar to that above.
 * 
 * @code
 *       DataOutFaces<dim>        data_out_face(false);
 *       std::vector<std::string> face_name(1, "u_hat");
 *       std::vector<DataComponentInterpretation::DataComponentInterpretation>
 *         face_component_type(1, DataComponentInterpretation::component_is_scalar);
 *   
 *       data_out_face.add_data_vector(dof_handler,
 *                                     solution,
 *                                     face_name,
 *                                     face_component_type);
 *   
 *       data_out_face.build_patches(fe.degree);
 *       data_out_face.write_vtk(face_output);
 *     }
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGrefine_grid"></a> 
 * <h4>HDG::refine_grid</h4>
 * 
 
 * 
 * We implement two different refinement cases for HDG, just as in
 * <code>@ref step_7 "step-7"</code>: adaptive_refinement and global_refinement.  The
 * global_refinement option recreates the entire triangulation every
 * time. This is because we want to use a finer sequence of meshes than what
 * we would get with one refinement step, namely 2, 3, 4, 6, 8, 12, 16, ...
 * elements per direction.
 * 
 
 * 
 * The adaptive_refinement mode uses the <code>KellyErrorEstimator</code> to
 * give a decent indication of the non-regular regions in the scalar local
 * solutions.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::refine_grid(const unsigned int cycle)
 *     {
 *       if (cycle == 0)
 *         {
 *           GridGenerator::subdivided_hyper_cube(triangulation, 2, -1, 1);
 *           triangulation.refine_global(3 - dim);
 *         }
 *       else
 *         switch (refinement_mode)
 *           {
 *             case global_refinement:
 *               {
 *                 triangulation.clear();
 *                 GridGenerator::subdivided_hyper_cube(triangulation,
 *                                                      2 + (cycle % 2),
 *                                                      -1,
 *                                                      1);
 *                 triangulation.refine_global(3 - dim + cycle / 2);
 *                 break;
 *               }
 *   
 *             case adaptive_refinement:
 *               {
 *                 Vector<float> estimated_error_per_cell(
 *                   triangulation.n_active_cells());
 *   
 *                 const FEValuesExtractors::Scalar scalar(dim);
 *                 std::map<types::boundary_id, const Function<dim> *>
 *                   neumann_boundary;
 *                 KellyErrorEstimator<dim>::estimate(dof_handler_local,
 *                                                    QGauss<dim - 1>(fe.degree + 1),
 *                                                    neumann_boundary,
 *                                                    solution_local,
 *                                                    estimated_error_per_cell,
 *                                                    fe_local.component_mask(
 *                                                      scalar));
 *   
 *                 GridRefinement::refine_and_coarsen_fixed_number(
 *                   triangulation, estimated_error_per_cell, 0.3, 0.);
 *   
 *                 triangulation.execute_coarsening_and_refinement();
 *   
 *                 break;
 *               }
 *   
 *             default:
 *               {
 *                 DEAL_II_NOT_IMPLEMENTED();
 *               }
 *           }
 *   
 * @endcode
 * 
 * Just as in @ref step_7 "step-7", we set the boundary indicator of two of the faces to 1
 * where we want to specify Neumann boundary conditions instead of Dirichlet
 * conditions. Since we re-create the triangulation every time for global
 * refinement, the flags are set in every refinement step, not just at the
 * beginning.
 * 
 * @code
 *       for (const auto &cell : triangulation.cell_iterators())
 *         for (const auto &face : cell->face_iterators())
 *           if (face->at_boundary())
 *             if ((std::fabs(face->center()[0] - (-1)) < 1e-12) ||
 *                 (std::fabs(face->center()[1] - (-1)) < 1e-12))
 *               face->set_boundary_id(1);
 *     }
 *   
 * @endcode
 * 
 * 
 * <a name="step_51-HDGrun"></a> 
 * <h4>HDG::run</h4>
 * The functionality here is basically the same as <code>@ref step_7 "step-7"</code>.
 * We loop over 10 cycles, refining the grid on each one.  At the end,
 * convergence tables are created.
 * 
 * @code
 *     template <int dim>
 *     void HDG<dim>::run()
 *     {
 *       for (unsigned int cycle = 0; cycle < 10; ++cycle)
 *         {
 *           std::cout << "Cycle " << cycle << ':' << std::endl;
 *   
 *           refine_grid(cycle);
 *           setup_system();
 *           assemble_system(false);
 *           solve();
 *           postprocess();
 *           output_results(cycle);
 *         }
 *   
 * @endcode
 * 
 * There is one minor change for the convergence table compared to @ref step_7 "step-7":
 * Since we did not refine our mesh by a factor two in each cycle (but
 * rather used the sequence 2, 3, 4, 6, 8, 12, ...), we need to tell the
 * convergence rate evaluation about this. We do this by setting the
 * number of cells as a reference column and additionally specifying the
 * dimension of the problem, which gives the necessary information for the
 * relation between number of cells and mesh size.
 * 
 * @code
 *       if (refinement_mode == global_refinement)
 *         {
 *           convergence_table.evaluate_convergence_rates(
 *             "val L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
 *           convergence_table.evaluate_convergence_rates(
 *             "grad L2", "cells", ConvergenceTable::reduction_rate_log2, dim);
 *           convergence_table.evaluate_convergence_rates(
 *             "val L2-post", "cells", ConvergenceTable::reduction_rate_log2, dim);
 *         }
 *       convergence_table.write_text(std::cout);
 *     }
 *   
 *   } // end of namespace Step51
 *   
 *   
 *   
 *   int main()
 *   {
 *     const unsigned int dim = 2;
 *   
 *     try
 *       {
 * @endcode
 * 
 * Now for the three calls to the main class in complete analogy to
 * @ref step_7 "step-7".
 * 
 * @code
 *         {
 *           std::cout << "Solving with Q1 elements, adaptive refinement"
 *                     << std::endl
 *                     << "============================================="
 *                     << std::endl
 *                     << std::endl;
 *   
 *           Step51::HDG<dim> hdg_problem(1, Step51::HDG<dim>::adaptive_refinement);
 *           hdg_problem.run();
 *   
 *           std::cout << std::endl;
 *         }
 *   
 *         {
 *           std::cout << "Solving with Q1 elements, global refinement" << std::endl
 *                     << "===========================================" << std::endl
 *                     << std::endl;
 *   
 *           Step51::HDG<dim> hdg_problem(1, Step51::HDG<dim>::global_refinement);
 *           hdg_problem.run();
 *   
 *           std::cout << std::endl;
 *         }
 *   
 *         {
 *           std::cout << "Solving with Q3 elements, global refinement" << std::endl
 *                     << "===========================================" << std::endl
 *                     << std::endl;
 *   
 *           Step51::HDG<dim> hdg_problem(3, Step51::HDG<dim>::global_refinement);
 *           hdg_problem.run();
 *   
 *           std::cout << std::endl;
 *         }
 *       }
 *     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_51-Results"></a><h1>Results</h1>
 
 
<a name="step_51-Programoutput"></a><h3>Program output</h3>
 
 
We first have a look at the output generated by the program when run in 2D. In
the four images below, we show the solution for polynomial degree @f$p=1@f$
and cycles 2, 3, 4, and 8 of the program. In the plots, we overlay the data
generated from the internal data (DG part) with the skeleton part (@f$\hat{u}@f$)
into the same plot. We had to generate two different data sets because cells
and faces represent different geometric entities, the combination of which (in
the same file) is not supported in the VTK output of deal.II.
 
The images show the distinctive features of HDG: The cell solution (colored
surfaces) is discontinuous between the cells. The solution on the skeleton
variable sits on the faces and ties together the local parts. The skeleton
solution is not continuous on the vertices where the faces meet, even though
its values are quite close along lines in the same coordinate direction. The
skeleton solution can be interpreted as a rubber spring between the two sides
that balances the jumps in the solution (or rather, the flux @f$\kappa \nabla u
+ \mathbf{c} u@f$). From the picture at the top left, it is clear that
the bulk solution frequently over- and undershoots and that the
skeleton variable in indeed a better approximation to the exact
solution; this explains why we can get a better solution using a
postprocessing step.
 
As the mesh is refined, the jumps between the cells get
small (we represent a smooth solution), and the skeleton solution approaches
the interior parts. For cycle 8, there is no visible difference in the two
variables. We also see how boundary conditions are implemented weakly and that
the interior variables do not exactly satisfy boundary conditions. On the
lower and left boundaries, we set Neumann boundary conditions, whereas we set
Dirichlet conditions on the right and top boundaries.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_2.png" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_3.png" alt=""></td>
  </tr>
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_4.png" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_8.png" alt=""></td>
  </tr>
</table>
 
Next, we have a look at the post-processed solution, again at cycles 2, 3, 4,
and 8. This is a discontinuous solution that is locally described by second
order polynomials. While the solution does not look very good on the mesh of
cycle two, it looks much better for cycles three and four. As shown by the
convergence table below, we find that is also converges more quickly to the
analytical solution.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_2.png" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_3.png" alt=""></td>
  </tr>
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_4.png" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_8.png" alt=""></td>
  </tr>
</table>
 
Finally, we look at the solution for @f$p=3@f$ at cycle 2. Despite the coarse
mesh with only 64 cells, the post-processed solution is similar in quality
to the linear solution (not post-processed) at cycle 8 with 4,096
cells. This clearly shows the superiority of high order methods for smooth
solutions.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_q3_2.png" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_q3_2.png" alt=""></td>
  </tr>
</table>
 
<a name="step_51-Convergencetables"></a><h4>Convergence tables</h4>
 
 
When the program is run, it also outputs information about the respective
steps and convergence tables with errors in the various components in the
end. In 2D, the convergence tables look the following:
 
@code
Q1 elements, adaptive refinement:
cells dofs   val L2    grad L2  val L2-post
   16    80 1.804e+01 2.207e+01   1.798e+01
   31   170 9.874e+00 1.322e+01   9.798e+00
   61   314 7.452e-01 3.793e+00   4.891e-01
  121   634 3.240e-01 1.511e+00   2.616e-01
  238  1198 8.585e-02 8.212e-01   1.808e-02
  454  2290 4.802e-02 5.178e-01   2.195e-02
  898  4378 2.561e-02 2.947e-01   4.318e-03
 1720  7864 1.306e-02 1.664e-01   2.978e-03
 3271 14638 7.025e-03 9.815e-02   1.075e-03
 6217 27214 4.119e-03 6.407e-02   9.975e-04
 
Q1 elements, global refinement:
cells dofs      val L2        grad L2      val L2-post
   16    80 1.804e+01    - 2.207e+01    - 1.798e+01    -
   36   168 6.125e+00 2.66 9.472e+00 2.09 6.084e+00 2.67
   64   288 9.785e-01 6.38 4.260e+00 2.78 7.102e-01 7.47
  144   624 2.730e-01 3.15 1.866e+00 2.04 6.115e-02 6.05
  256  1088 1.493e-01 2.10 1.046e+00 2.01 2.880e-02 2.62
  576  2400 6.965e-02 1.88 4.846e-01 1.90 9.204e-03 2.81
 1024  4224 4.018e-02 1.91 2.784e-01 1.93 4.027e-03 2.87
 2304  9408 1.831e-02 1.94 1.264e-01 1.95 1.236e-03 2.91
 4096 16640 1.043e-02 1.96 7.185e-02 1.96 5.306e-04 2.94
 9216 37248 4.690e-03 1.97 3.228e-02 1.97 1.599e-04 2.96
 
Q3 elements, global refinement:
cells dofs      val L2        grad L2      val L2-post
   16   160 3.613e-01    - 1.891e+00    - 3.020e-01    -
   36   336 6.411e-02 4.26 5.081e-01 3.24 3.238e-02 5.51
   64   576 3.480e-02 2.12 2.533e-01 2.42 5.277e-03 6.31
  144  1248 8.297e-03 3.54 5.924e-02 3.58 6.330e-04 5.23
  256  2176 2.254e-03 4.53 1.636e-02 4.47 1.403e-04 5.24
  576  4800 4.558e-04 3.94 3.277e-03 3.96 1.844e-05 5.01
 1024  8448 1.471e-04 3.93 1.052e-03 3.95 4.378e-06 5.00
 2304 18816 2.956e-05 3.96 2.104e-04 3.97 5.750e-07 5.01
 4096 33280 9.428e-06 3.97 6.697e-05 3.98 1.362e-07 5.01
 9216 74496 1.876e-06 3.98 1.330e-05 3.99 1.788e-08 5.01
@endcode
 
 
One can see the error reduction upon grid refinement, and for the cases where
global refinement was performed, also the convergence rates. The quadratic
convergence rates of Q1 elements in the @f$L_2@f$ norm for both the scalar
variable and the gradient variable is apparent, as is the cubic rate for the
postprocessed scalar variable in the @f$L_2@f$ norm. Note this distinctive
feature of an HDG solution. In typical continuous finite elements, the
gradient of the solution of order @f$p@f$ converges at rate @f$p@f$ only, as
opposed to @f$p+1@f$ for the actual solution. Even though superconvergence
results for finite elements are also available (e.g. superconvergent patch
recovery first introduced by Zienkiewicz and Zhu), these are typically limited
to structured meshes and other special cases. For Q3 HDG variables, the scalar
variable and gradient converge at fourth order and the postprocessed scalar
variable at fifth order.
 
The same convergence rates are observed in 3d.
@code
Q1 elements, adaptive refinement:
cells   dofs    val L2    grad L2  val L2-post
     8     144 7.122e+00 1.941e+01   6.102e+00
    29     500 3.309e+00 1.023e+01   2.145e+00
   113    1792 2.204e+00 1.023e+01   1.912e+00
   379    5732 6.085e-01 5.008e+00   2.233e-01
  1317   19412 1.543e-01 1.464e+00   4.196e-02
  4579   64768 5.058e-02 5.611e-01   9.521e-03
 14596  199552 2.129e-02 3.122e-01   4.569e-03
 46180  611400 1.033e-02 1.622e-01   1.684e-03
144859 1864212 5.007e-03 8.371e-02   7.364e-04
451060 5684508 2.518e-03 4.562e-02   3.070e-04
 
Q1 elements, global refinement:
cells   dofs       val L2          grad L2       val L2-post
     8     144 7.122e+00    - 1.941e+01     - 6.102e+00    -
    27     432 5.491e+00 0.64 2.184e+01 -0.29 4.448e+00 0.78
    64     960 3.646e+00 1.42 1.299e+01  1.81 3.306e+00 1.03
   216    3024 1.595e+00 2.04 8.550e+00  1.03 1.441e+00 2.05
   512    6912 6.922e-01 2.90 5.306e+00  1.66 2.511e-01 6.07
  1728   22464 2.915e-01 2.13 2.490e+00  1.87 8.588e-02 2.65
  4096   52224 1.684e-01 1.91 1.453e+00  1.87 4.055e-02 2.61
 13824  172800 7.972e-02 1.84 6.861e-01  1.85 1.335e-02 2.74
 32768  405504 4.637e-02 1.88 3.984e-01  1.89 5.932e-03 2.82
110592 1354752 2.133e-02 1.92 1.830e-01  1.92 1.851e-03 2.87
 
Q3 elements, global refinement:
cells   dofs       val L2        grad L2      val L2-post
     8     576 5.670e+00    - 1.868e+01    - 5.462e+00    -
    27    1728 1.048e+00 4.16 6.988e+00 2.42 8.011e-01 4.73
    64    3840 2.831e-01 4.55 2.710e+00 3.29 1.363e-01 6.16
   216   12096 7.883e-02 3.15 7.721e-01 3.10 2.158e-02 4.55
   512   27648 3.642e-02 2.68 3.305e-01 2.95 5.231e-03 4.93
  1728   89856 8.546e-03 3.58 7.581e-02 3.63 7.640e-04 4.74
  4096  208896 2.598e-03 4.14 2.313e-02 4.13 1.783e-04 5.06
 13824  691200 5.314e-04 3.91 4.697e-03 3.93 2.355e-05 4.99
 32768 1622016 1.723e-04 3.91 1.517e-03 3.93 5.602e-06 4.99
110592 5419008 3.482e-05 3.94 3.055e-04 3.95 7.374e-07 5.00
@endcode
 
<a name="step_51-Comparisonwithcontinuousfiniteelements"></a><h3>Comparison with continuous finite elements</h3>
 
 
<a name="step_51-Resultsfor2D"></a><h4>Results for 2D</h4>
 
 
The convergence tables verify the expected convergence rates stated in the
introduction. Now, we want to show a quick comparison of the computational
efficiency of the HDG method compared to a usual finite element (continuous
Galkerin) method on the problem of this tutorial. Of course, stability aspects
of the HDG method compared to continuous finite elements for
transport-dominated problems are also important in practice, which is an
aspect not seen on a problem with smooth analytic solution. In the picture
below, we compare the @f$L_2@f$ error as a function of the number of degrees of
freedom (left) and of the computing time spent in the linear solver (right)
for two space dimensions of continuous finite elements (CG) and the hybridized
discontinuous Galerkin method presented in this tutorial. As opposed to the
tutorial where we only use unpreconditioned BiCGStab, the times shown in the
figures below use the Trilinos algebraic multigrid preconditioner in
TrilinosWrappers::PreconditionAMG. For the HDG part, a wrapper around
ChunkSparseMatrix for the trace variable has been used in order to utilize the
block structure in the matrix on the finest level.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_plain.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_plain.png" width="400" alt=""></td>
  </tr>
</table>
 
The results in the graphs show that the HDG method is slower than continuous
finite elements at @f$p=1@f$, about equally fast for cubic elements and
faster for sixth order elements. However, we have seen above that the HDG
method actually produces solutions which are more accurate than what is
represented in the original variables. Therefore, in the next two plots below
we instead display the error of the post-processed solution for HDG (denoted
by @f$p=1^*@f$ for example). We now see a clear advantage of HDG for the same
amount of work for both @f$p=3@f$ and @f$p=6@f$, and about the same quality
for @f$p=1@f$.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_post.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_post.png" width="400" alt=""></td>
  </tr>
</table>
 
Since the HDG method actually produces results converging as
@f$h^{p+2}@f$, we should compare it to a continuous Galerkin
solution with the same asymptotic convergence behavior, i.e., FE_Q with degree
@f$p+1@f$. If we do this, we get the convergence curves below. We see that
CG with second order polynomials is again clearly better than HDG with
linears. However, the advantage of HDG for higher orders remains.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_postb.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_postb.png" width="400" alt=""></td>
  </tr>
</table>
 
The results are in line with properties of DG methods in general: Best
performance is typically not achieved for linear elements, but rather at
somewhat higher order, usually around @f$p=3@f$. This is because of a
volume-to-surface effect for discontinuous solutions with too much of the
solution living on the surfaces and hence duplicating work when the elements
are linear. Put in other words, DG methods are often most efficient when used
at relatively high order, despite their focus on a discontinuous (and hence,
seemingly low accurate) representation of solutions.
 
<a name="step_51-Resultsfor3D"></a><h4>Results for 3D</h4>
 
 
We now show the same figures in 3D: The first row shows the number of degrees
of freedom and computing time versus the @f$L_2@f$ error in the scalar variable
@f$u@f$ for CG and HDG at order @f$p@f$, the second row shows the
post-processed HDG solution instead of the original one, and the third row
compares the post-processed HDG solution with CG at order @f$p+1@f$. In 3D,
the volume-to-surface effect makes the cost of HDG somewhat higher and the CG
solution is clearly better than HDG for linears by any metric. For cubics, HDG
and CG are of similar quality, whereas HDG is again more efficient for sixth
order polynomials. One can alternatively also use the combination of FE_DGP
and FE_FaceP instead of (FE_DGQ, FE_FaceQ), which do not use tensor product
polynomials of degree @f$p@f$ but Legendre polynomials of <i>complete</i>
degree @f$p@f$. There are fewer degrees of freedom on the skeleton variable
for FE_FaceP for a given mesh size, but the solution quality (error vs. number
of DoFs) is very similar to the results for FE_FaceQ.
 
<table align="center">
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_plain.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_plain.png" width="400" alt=""></td>
  </tr>
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_post.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_post.png" width="400" alt=""></td>
  </tr>
  <tr>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_postb.png" width="400" alt=""></td>
    <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_postb.png" width="400" alt=""></td>
  </tr>
</table>
 
One final note on the efficiency comparison: We tried to use general-purpose
sparse matrix structures and similar solvers (optimal AMG preconditioners for
both without particular tuning of the AMG parameters on any of them) to give a
fair picture of the cost versus accuracy of two methods, on a toy example. It
should be noted however that geometric multigrid (GMG) for continuous finite
elements is about a factor four to five faster for @f$p=3@f$ and @f$p=6@f$. As of
2019, optimal-complexity iterative solvers for HDG are still under development
in the research community. Also, there are other implementation aspects for CG
available such as fast matrix-free approaches as shown in @ref step_37 "step-37" that make
higher order continuous elements more competitive. Again, it is not clear to
the authors of the tutorial whether similar improvements could be made for
HDG. We refer to <a href="https://dx.doi.org/10.1137/16M110455X">Kronbichler
and Wall (2018)</a> for a recent efficiency evaluation.
 
 
<a name="step_51-Possibilitiesforimprovements"></a><h3>Possibilities for improvements</h3>
 
 
As already mentioned in the introduction, one possibility is to implement
another post-processing technique as discussed in the literature.
 
A second item that is not done optimally relates to the performance of this
program, which is of course an issue in practical applications (weighing in
also the better solution quality of (H)DG methods for transport-dominated
problems). Let us look at
the computing time of the tutorial program and the share of the individual
components:
 
<table align="center" class="doxtable">
  <tr>
    <th>&nbsp;</th>
    <th>&nbsp;</th>
    <th>Setup</th>
    <th>Assemble</th>
    <th>Solve</th>
    <th>Trace reconstruct</th>
    <th>Post-processing</th>
    <th>Output</th>
  </tr>
  <tr>
    <th>&nbsp;</th>
    <th>Total time</th>
    <th colspan="6">Relative share</th>
  </tr>
  <tr>
    <td align="left">2D, Q1, cycle 9, 37,248 dofs</td>
    <td align="center">5.34s</td>
    <td align="center">0.7%</td>
    <td align="center">1.2%</td>
    <td align="center">89.5%</td>
    <td align="center">0.9%</td>
    <td align="center">2.3%</td>
    <td align="center">5.4%</td>
  </tr>
  <tr>
    <td align="left">2D, Q3, cycle 9, 74,496 dofs</td>
    <td align="center">22.2s</td>
    <td align="center">0.4%</td>
    <td align="center">4.3%</td>
    <td align="center">84.1%</td>
    <td align="center">4.1%</td>
    <td align="center">3.5%</td>
    <td align="center">3.6%</td>
  </tr>
  <tr>
    <td align="left">3D, Q1, cycle 7, 172,800 dofs</td>
    <td align="center">9.06s</td>
    <td align="center">3.1%</td>
    <td align="center">8.9%</td>
    <td align="center">42.7%</td>
    <td align="center">7.0%</td>
    <td align="center">20.6%</td>
    <td align="center">17.7%</td>
  </tr>
  <tr>
    <td align="left">3D, Q3, cycle 7, 691,200 dofs</td>
    <td align="center">516s</td>
    <td align="center">0.6%</td>
    <td align="center">34.5%</td>
    <td align="center">13.4%</td>
    <td align="center">32.8%</td>
    <td align="center">17.1%</td>
    <td align="center">1.5%</td>
  </tr>
</table>
 
As can be seen from the table, the solver and assembly calls dominate the
runtime of the program. This also gives a clear indication of where
improvements would make the most sense:
 
<ol>
  <li> Better linear solvers: We use a BiCGStab iterative solver without
  preconditioner, where the number of iteration increases with increasing
  problem size (the number of iterations for Q1 elements and global
  refinements starts at 35 for the small sizes but increase up to 701 for the
  largest size). To do better, one could for example use an algebraic
  multigrid preconditioner from Trilinos, or some more advanced variants as
  the one discussed in <a
  href="https://dx.doi.org/10.1137/16M110455X">Kronbichler and Wall
  (2018)</a>. For diffusion-dominated problems such as the problem at hand
  with finer meshes, such a solver can be designed that uses the matrix-vector
  products from the more efficient ChunkSparseMatrix on the finest level, as
  long as we are not working in parallel with MPI. For MPI-parallelized
  computations, a standard TrilinosWrappers::SparseMatrix can be used.
 
  <li> Speed up assembly by pre-assembling parts that do not change from one
  cell to another (those that do neither contain variable coefficients nor
  mapping-dependent terms).
</ol>
 *
 *
<a name="step_51-PlainProg"></a>
<h1> The plain program</h1>
@include "step-51.cc"
*/