This program was contributed by Marco Feder <marco.feder@sissa.it>.
It comes without any warranty or support by its authors or the authors of deal.II.

This program is part of the deal.II code gallery and consists of the following files (click to inspect):

Pictures from this code gallery program

Annotated version of README.md

A posteriori error estimator for first order hyperbolic problems

Running the code:

As in the tutorial programs, type

cmake -DDEAL_II_DIR=/path/to/deal.II .

on the command line to configure the program. After that you can compile with make and run with either make run or using

./DG_advection_reaction

on the command line.

Parameter file :

If you run ./DG_advection_reaction parameters.prm, an error message will tell you that a parameter file has been created for you. You can open it and change some useful parameters like the number of refinement cycles, the advection coefficient, and others. If you don't specify anything, then the default values used for the test case (see paragraph below) will be used.

The problem:

This program solves the problem, for \(\Omega \in \mathbb{R^2}\)

\[ \begin{cases} b \cdot \nabla u + c u = f \qquad \text{in } \Omega \\ \qquad \qquad u=g \qquad \text{on } \partial_{-}\Omega \end{cases} \]

where \(g \in L^2(\partial_{-}\Omega)\) and \(\partial_{-}\Omega=\{ x \in \partial \Omega: b(x)\cdot n(x) <0\}\) is the inflow part of the boundary, with \(b=(b_1,b_2) \in \mathbb{R^2}\). As we know from classical DG theory, we need to ensure that

\[ c(x) - \frac{1}{2}\nabla \cdot b \geq \gamma_0 >0 \]

for some positive \(\gamma_0\) so that we have coercivity in \(L^2\) at the continuous level. Discrete coercivity is achieved by using a stronger norm which takes care of jumps, see Di Pietro and Ern [2] for details.

The weak formulation:

As trial space we choose \(V_h = \{ v_h \in L^2(\Omega): v_h \in P^1(\mathbb{T_h})\} \notin H^1(\Omega)\). If we integrate by parts and sum over all cells

\[ \sum_{T \in \mathbb{T}_h} \Bigl( (-u,\beta \cdot \nabla v_h) _T + (c u,v_h)_T + \bigl<(b \cdot n) u ,v_h \bigr>_{\partial T} \Bigr) = (f,v_h)_{\Omega} \]

and use the so-called DG magic formula and exploit the property \([bu]_{\mathbb{F}^i} = 0\) where \(\mathbb{F}^i\) are set of internal faces we obtain the (unstable!) formulation:

Find \(u_h \in V_h\):

\[ a_h(u_h,v_h) + b_h(u_h,v_h)=l(v_h) \qquad \forall v_h \in V_h \]

where

\[ a_h(u,v_h)=\sum_{T \in \mathbb{T}_h} \Bigl( (-u,b \cdot \nabla v_h) _T + (c u,v_h)_T \Bigr) \]

\[ b_h(u,v_h)= \sum_{F \not \in \partial_{-}\Omega} \bigl< \{ bu\}, [v_h]\bigr>_F \]

\[ l(v_h)= (f,v_h)_{\Omega} - \sum_{F \in \partial_{-}\Omega} \bigl< (b \cdot n) g,v_h \bigr>_F \]

It's well known this formulation is coercive only in \(L^2\), hence the formulation is unstable as we don't "see" the derivatives. To stabilize this, we can use a jump-penalty term, i.e. our \(b_h\) is replaced by:

\[ b_h^s(u_h,v_h)=b_h(u_h,v_h)+ \sum_{F \in \mathbb{F}^i} \bigl< c_F [u_h],[v_h] \bigr> \]

where \(c_F>0\) is a function on each edge such that \(c_F \geq \theta |b \cdot n|\) for some positive \(\theta\). In this program, \(\theta=\frac{1}{2}\) and \(c_F = \frac{1}{2} |b \cdot n|\), which corresponds to an upwind formulation. Notice that consistency is trivially achieved, as \([u]_{\mathbb{F}^i} =0\). This formulation is stable in the energy norm

\[ |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2 \Bigr)^{\frac{1}{2}} \]

(well defined on \(H^1(\Omega) + V_h\)) and moreover we have the a-priori bound:

\[ |||u-u_h||| \leq C h^{k+\frac{1}{2}}||u||_{k+1,\Omega} \]

valid for \(u \in H^{k+1}(\Omega)\).

See Brezzi-Marini-Süli [3] for more details.

A-posteriori error estimator:

The estimator is the one proposed by Georgoulis, Edward Hall and Charalambos Makridakis in [3]. This approach is quite different with respect to other works in the field, as the authors are trying to develop an estimator for the original hyperbolic problem, rather than taking the hyperbolic regime as the vanishing diffusivity limit.

The reliability is:

\[ |||u-u_h|||^2 \leq C || \sqrt{b \cdot n}[u_h]||_{\Gamma^{-}}^2 + C \sum_{T \in \mathbb{T}_h}\Bigl( ||\beta (g-u_h^+)||_{\partial_{-}T \cap \partial_{-} \Omega}^2 +||f-c u_h - \Pi(f- cu_h)||_T^2 \Bigr) \]

where:

\(\Pi\) is the (local) \(L^2\) orthogonal projection onto \(V_h\)
\(\Gamma\) is the skeleton of the mesh
\(c\) is constant
\(\beta = |b \cdot n|\)
\(u_h^+\) is the interior trace from the current cell \(T\) of a the finite element function \(u_h\).

Test case:

The following test case has been taken from [3]. Consider:

\(c=1\)
\(b=(1,1)\)
\(f\) to be such that the exact solution is \(u(x,y)=\tanh(100(x+y-\frac{1}{2}))\) This solution has an internal layer along the line \(y=\frac{1}{2} -x\), hence we would like to see that part of the domain to be much more refined than the rest.

The next image is the 3D view of the numerical solution:

Screenshot

More interestingly, we see that the estimator has been able to capture the layer. Here a bulk-chasing criterion is used, with bottom fraction ´0.5´ and no coarsening. This mesh is obtained after 12 refinement cycles. Screenshot

If we look at the decrease of the energy norm of the error in the globally refined case and in the adaptively case, with respect to the DoFs, we obtain:

Screenshot

References

[1] Emmanuil H. Georgoulis, Edward Hall and Charalambos Makridakis (2013), Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems. DOI: 10.1007/978-3-319-01818-8_8 [2] Di Pietro, Daniele Antonio and Ern, Alexandre (2012), Mathematical Aspects of Discontinuous Galerkin Methods. ISBN: 978-3-642-22980-0 [3] Franco Brezzi, Luisa Donatella Marini and Endre Süli (2004) Discontinuous Galerkin Methods for First-Order Hyperbolic Problems. DOI: 10.1142/S0218202504003866

Annotated version of include/DG_advection_reaction.h

    /* -----------------------------------------------------------------------------
*    *
*    * SPDX-License-Identifier: LGPL-2.1-or-later
*    * Copyright (C) 2015 by Marco Feder
*    *
*    * This file is part of the deal.II code gallery.
*    *
*    * -----------------------------------------------------------------------------
*    */
    
    #ifndef INCLUDE_DG_UPWIND_H_
    #define INCLUDE_DG_UPWIND_H_
    
    
    #include <deal.II/base/function.h>
    #include <deal.II/base/quadrature_lib.h>
    
    #include <deal.II/dofs/dof_handler.h>
    #include <deal.II/dofs/dof_tools.h>
    
    #include <deal.II/fe/fe_dgq.h>
    #include <deal.II/fe/fe_q.h>
    #include <deal.II/fe/fe_values.h>
    #include <deal.II/fe/mapping_q1.h>
    
    #include <deal.II/grid/grid_generator.h>
    #include <deal.II/grid/grid_out.h>
    #include <deal.II/grid/grid_refinement.h>
    #include <deal.II/grid/tria.h>
    
    #include <deal.II/lac/dynamic_sparsity_pattern.h>
    #include <deal.II/lac/full_matrix.h>
    #include <deal.II/lac/sparse_matrix.h>
    #include <deal.II/lac/vector.h>
    
    #include <deal.II/numerics/data_out.h>
    #include <deal.II/numerics/vector_tools.h>

This header is needed for FEInterfaceValues to compute integrals on interfaces:

#include <deal.II/fe/fe_interface_values.h>

Solver

  #include <deal.II/lac/precondition_block.h>
  #include <deal.II/lac/solver_richardson.h>
  #include <deal.II/lac/sparse_direct.h>

We are going to use gradients as refinement indicator.

#include <deal.II/numerics/derivative_approximation.h>

Using using the mesh_loop from the MeshWorker framework

  #include <deal.II/base/convergence_table.h>
  
  #include <deal.II/meshworker/mesh_loop.h>

To enable parameter handling

  #include <deal.II/base/function_parser.h>
  #include <deal.II/base/parameter_acceptor.h>
  #include <deal.II/base/parameter_handler.h>
  #include <deal.II/base/parsed_convergence_table.h>
  #include <deal.II/base/symbolic_function.h>
  
  #include <deal.II/meshworker/copy_data.h>
  #include <deal.II/meshworker/mesh_loop.h>
  #include <deal.II/meshworker/scratch_data.h>
  
  #include <fstream>
  #include <iostream>
  using namespace dealii;
  

This is a struct used only for throwing an exception when theta parameter is not okay.

  struct theta_exc
  {
    std::string message;
    theta_exc(std::string &&s)
      : message{std::move(s)} {};
    const char *
    what() const
    {
      return message.c_str();
    }
  };
  
  
  template <int dim>
  class AdvectionReaction : ParameterAcceptor
  {
  public:
    AdvectionReaction();
    void
    initialize_params(const std::string &filename);
    void
    run();
  
  private:
    using Iterator = typename DoFHandler<dim>::active_cell_iterator;
    void
    parse_string(const std::string &parameters);
    void
    setup_system();
    void
    assemble_system();
    void
    solve();
    void
    refine_grid();
    void
    output_results(const unsigned int cycle) const;
    void
    compute_error();
    double
    compute_energy_norm();
    void
    compute_local_projection_and_estimate();
  
    Triangulation<dim>   triangulation;
    const MappingQ1<dim> mapping;
  

Furthermore we want to use DG elements.

    std::unique_ptr<FE_DGQ<dim>> fe;
    DoFHandler<dim>              dof_handler;
  
    SparsityPattern      sparsity_pattern;
    SparseMatrix<double> system_matrix;
  
    Vector<double> solution;
    Vector<double> right_hand_side;
    Vector<double> energy_norm_square_per_cell;
    Vector<double> error_indicator_per_cell;
  

So far we declared the usual objects. Hereafter we declare FunctionParser<dim> objects

    FunctionParser<dim> exact_solution;
    FunctionParser<dim> boundary_conditions;
    FunctionParser<dim> rhs;
    FunctionParser<dim> advection_coeff;
  
    unsigned int fe_degree = 1;
  

and then we define default values that will be parsed from the following strings

    std::string exact_solution_expression =
      "tanh(100*(x+y-0.5))"; // internal layer solution
    std::string rhs_expression =
      "-200*tanh(100*x + 100*y - 50.0)^2 + tanh(100*x + 100*y - 50.0) + 200";
    std::string advection_coefficient_expression = "1.0";
    std::string boundary_conditions_expression   = "tanh(100*x + 100*y - 50.0)";
    std::string refinement                       = "residual";
    std::string output_filename = "DG_advection_reaction_estimator";
    std::map<std::string, double> constants;
    ParsedConvergenceTable        error_table;
  
    bool         use_direct_solver    = true;
    unsigned int n_refinement_cycles  = 8;
    unsigned int n_global_refinements = 3;
    double theta = 0.5; // default is 0.5 so that I have classical upwind flux
  };
  
  #endif /* INCLUDE_DG_UPWIND_H_ */

Annotated version of main.cc

    /* -----------------------------------------------------------------------------
*    *
*    * SPDX-License-Identifier: LGPL-2.1-or-later
*    * Copyright (C) 2015 by Marco Feder
*    *
*    * This file is part of the deal.II code gallery.
*    *
*    * -----------------------------------------------------------------------------
*    */
    
    #include "include/DG_advection_reaction.h"
    
    int
    main(int argc, char **argv)
    {
      try
        {
          std::string par_name = "";
          if (argc > 1)
            {
              par_name = argv[1];
            }
          else
            {
              par_name = "parameters.prm";
            }
          deallog.depth_console(2);
          AdvectionReaction<2> problem;
          problem.initialize_params(par_name);
          problem.run();
        }
      catch (std::exception &exc)
        {
          std::cerr << std::endl
                    << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          std::cerr << "Exception on processing: " << std::endl
                    << exc.what() << std::endl
                    << "Aborting!" << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          return 1;
        }
      catch (const theta_exc &theta_range)
        {
          std::cerr << std::endl
                    << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          std::cerr << "Exception on processing: " << std::endl
                    << theta_range.what() << std::endl
                    << "Aborting!" << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          return 1;
        }
      catch (...)
        {
          std::cerr << std::endl
                    << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          std::cerr << "Unknown exception!" << std::endl
                    << "Aborting!" << std::endl
                    << "----------------------------------------------------"
                    << std::endl;
          return 1;
        }
    
      return 0;
    }

Annotated version of source/DG_advection_reaction.cc

    /* -----------------------------------------------------------------------------
*    *
*    * SPDX-License-Identifier: LGPL-2.1-or-later
*    * Copyright (C) 2015 by Marco Feder
*    *
*    * This file is part of the deal.II code gallery.
*    *
*    * -----------------------------------------------------------------------------
*    */
    
    #include "../include/DG_advection_reaction.h"
    

Compute and returns the wind field b

  template <int dim>
  Tensor<1, dim>
  beta(const Point<dim> &p)
  {
    Assert(dim > 1, ExcNotImplemented());
    (void)p; // suppress warnings from p
    Tensor<1, dim> wind_field;
    wind_field[0] = 1.0;
    wind_field[1] = 1.0;
  
    return wind_field;
  }
  

The ScratchData and CopyData classes

The following objects are the scratch and copy objects we use in the call to MeshWorker::mesh_loop(). The new object is the FEInterfaceValues object, that works similar to FEValues or FEFacesValues, except that it acts on an interface between two cells and allows us to assemble the interface terms in our weak form.

  template <int dim>
  struct ScratchData
  {
    ScratchData(const Mapping<dim>        &mapping,
                const FiniteElement<dim>  &fe,
                const Quadrature<dim>     &quadrature,
                const Quadrature<dim - 1> &quadrature_face,
                const UpdateFlags          update_flags = update_values |
                                                 update_gradients |
                                                 update_quadrature_points |
                                                 update_JxW_values,
                const UpdateFlags interface_update_flags =
                  update_values | update_gradients | update_quadrature_points |
                  update_JxW_values | update_normal_vectors)
      : fe_values(mapping, fe, quadrature, update_flags)
      , fe_interface_values(mapping, fe, quadrature_face, interface_update_flags)
    {}
  
    ScratchData(const ScratchData<dim> &scratch_data)
      : fe_values(scratch_data.fe_values.get_mapping(),
                  scratch_data.fe_values.get_fe(),
                  scratch_data.fe_values.get_quadrature(),
                  scratch_data.fe_values.get_update_flags())
      , fe_interface_values(scratch_data.fe_interface_values.get_mapping(),
                            scratch_data.fe_interface_values.get_fe(),
                            scratch_data.fe_interface_values.get_quadrature(),
                            scratch_data.fe_interface_values.get_update_flags())
    {}
  
    FEValues<dim>          fe_values;
    FEInterfaceValues<dim> fe_interface_values;
  };
  
  
  
  struct CopyDataFace
  {
    FullMatrix<double>                   cell_matrix;
    std::vector<types::global_dof_index> joint_dof_indices;
    std::array<double, 2>                values;
    std::array<unsigned int, 2>          cell_indices;
  };
  
  
  
  struct CopyData
  {
    FullMatrix<double>                   cell_matrix;
    Vector<double>                       cell_rhs;
    std::vector<types::global_dof_index> local_dof_indices;
    std::vector<CopyDataFace>            face_data;
  
    double       value;
    double       value_estimator;
    unsigned int cell_index;
  
    FullMatrix<double> cell_mass_matrix;
    Vector<double>     cell_mass_rhs;
  
    template <class Iterator>
    void
    reinit(const Iterator &cell, unsigned int dofs_per_cell)
    {
      cell_matrix.reinit(dofs_per_cell, dofs_per_cell);
      cell_mass_matrix.reinit(dofs_per_cell, dofs_per_cell);
  
      cell_rhs.reinit(dofs_per_cell);
      cell_mass_rhs.reinit(dofs_per_cell);
  
      local_dof_indices.resize(dofs_per_cell);
      cell->get_dof_indices(local_dof_indices);
    }
  };
  
  
  

Auxiliary function

This auxiliary function is taken from step-74 and it's used to compute the jump of the finite element function \(u_h\) on a face.

  template <int dim>
  void
  get_function_jump(const FEInterfaceValues<dim> &fe_iv,
                    const Vector<double>         &solution,
                    std::vector<double>          &jump)
  {
    const unsigned int                 n_q = fe_iv.n_quadrature_points;
    std::array<std::vector<double>, 2> face_values;
    jump.resize(n_q);
    for (unsigned int i = 0; i < 2; ++i)
      {
        face_values[i].resize(n_q);
        fe_iv.get_fe_face_values(i).get_function_values(solution, face_values[i]);
      }
    for (unsigned int q = 0; q < n_q; ++q)
      jump[q] = face_values[0][q] - face_values[1][q];
  }
  
  
  
  template <int dim>
  AdvectionReaction<dim>::AdvectionReaction()
    : mapping()
    , dof_handler(triangulation)
  {
    Assert(dim > 1, ExcMessage("Not implemented in 1D."));
    add_parameter("Finite element degree", fe_degree);
    add_parameter("Problem constants", constants);
    add_parameter("Output filename", output_filename);
    add_parameter("Use direct solver", use_direct_solver);
    add_parameter("Number of refinement cycles", n_refinement_cycles);
    add_parameter("Number of global refinement", n_global_refinements);
    add_parameter("Refinement", refinement);
    add_parameter("Exact solution expression", exact_solution_expression);
    add_parameter("Boundary conditions expression",
                  boundary_conditions_expression);
    add_parameter("Theta", theta);
    add_parameter("Advection coefficient expression",
                  advection_coefficient_expression);
    add_parameter("Right hand side expression", rhs_expression);
  
    this->prm.enter_subsection("Error table");
    error_table.add_parameters(this->prm);
    this->prm.leave_subsection();
  }
  
  
  
  template <int dim>
  void
  AdvectionReaction<dim>::initialize_params(const std::string &filename)
  {
    ParameterAcceptor::initialize(filename,
                                  "last_used_parameters.prm",
                                  ParameterHandler::Short);
    if (theta < 0.0 || theta > 10.0 || std::abs(theta) < 1e-12)
      {
        throw(
          theta_exc("Theta parameter is not in a suitable range: see paper by "
                    "Brezzi, Marini, Suli for an extended discussion"));
      }
  }
  
  
  
  template <int dim>
  void
  AdvectionReaction<dim>::parse_string(const std::string &parameters)
  {
    ParameterAcceptor::prm.parse_input_from_string(parameters);
    ParameterAcceptor::parse_all_parameters();
  }
  
  
  
  template <int dim>
  void
  AdvectionReaction<dim>::setup_system()
  {

first need to distribute the DoFs.

    if (!fe)
      {
        fe              = std::make_unique<FE_DGQ<dim>>(fe_degree);
        const auto vars = dim == 2 ? "x,y" : "x,y,z";
        exact_solution.initialize(vars, exact_solution_expression, constants);
        rhs.initialize(vars, rhs_expression, constants);
        advection_coeff.initialize(vars,
                                   advection_coefficient_expression,
                                   constants);
        boundary_conditions.initialize(vars,
                                       boundary_conditions_expression,
                                       constants);
      }
    dof_handler.distribute_dofs(*fe);
  

To build the sparsity pattern for DG discretizations, we can call the function analogue to DoFTools::make_sparsity_pattern, which is called DoFTools::make_flux_sparsity_pattern:

    DynamicSparsityPattern dsp(dof_handler.n_dofs());
    DoFTools::make_flux_sparsity_pattern(dof_handler,
                                         dsp); // DG sparsity pattern generator
    sparsity_pattern.copy_from(dsp);
  

Finally, we set up the structure of all components of the linear system.

    system_matrix.reinit(sparsity_pattern);
    solution.reinit(dof_handler.n_dofs());
    right_hand_side.reinit(dof_handler.n_dofs());
  }

in the call to MeshWorker::mesh_loop() we only need to specify what should happen on each cell, each boundary face, and each interior face. These three tasks are handled by the lambda functions inside the function below.

  template <int dim>
  void
  AdvectionReaction<dim>::assemble_system()
  {
    using Iterator = typename DoFHandler<dim>::active_cell_iterator;
  
    const QGauss<dim>     quadrature      = fe->tensor_degree() + 1;
    const QGauss<dim - 1> quadrature_face = fe->tensor_degree() + 1;
  

This is the function that will be executed for each cell.

    const auto cell_worker = [&](const Iterator   &cell,
                                 ScratchData<dim> &scratch_data,
                                 CopyData         &copy_data) {
      FEValues<dim> fe_values_continuous(*fe,
                                         quadrature,
                                         update_values | update_gradients |
                                           update_quadrature_points |
                                           update_JxW_values);
  
      const unsigned int n_dofs =
        scratch_data.fe_values.get_fe().n_dofs_per_cell();
      copy_data.reinit(cell, n_dofs);
      scratch_data.fe_values.reinit(cell);
  
      const auto &q_points = scratch_data.fe_values.get_quadrature_points();
  
      const FEValues<dim>       &fe_v = scratch_data.fe_values;
      const std::vector<double> &JxW  = fe_v.get_JxW_values();
  
      for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
        {
          auto beta_q = beta(q_points[point]);
          for (unsigned int i = 0; i < n_dofs; ++i)
            {
              for (unsigned int j = 0; j < n_dofs; ++j)
                {
                  copy_data.cell_matrix(i, j) +=
                    (-beta_q                                    // -\beta
                       * fe_v.shape_grad(i, point)              // \nabla \phi_i
                       * fe_v.shape_value(j, point)             // \phi_j
                     + advection_coeff.value(q_points[point]) * // gamma
                         fe_v.shape_value(i, point)             // phi_i
                         * fe_v.shape_value(j, point)           // phi_j
                     ) *
                    JxW[point]; // dx
                }
              copy_data.cell_rhs(i) += rhs.value(q_points[point])   // f(x_q)
                                       * fe_v.shape_value(i, point) // phi_i(x_q)
                                       * JxW[point];                // dx
            }
        }
    };
  

This is the function called for boundary faces and consists of a normal integration using FEFaceValues. New is the logic to decide if the term goes into the system matrix (outflow) or the right-hand side (inflow).

    const auto boundary_worker = [&](const Iterator     &cell,
                                     const unsigned int &face_no,
                                     ScratchData<dim>   &scratch_data,
                                     CopyData           &copy_data) {
      scratch_data.fe_interface_values.reinit(cell, face_no);
      const FEFaceValuesBase<dim> &fe_face =
        scratch_data.fe_interface_values.get_fe_face_values(0);
  
      const auto &q_points = fe_face.get_quadrature_points();
  
      const unsigned int n_facet_dofs        = fe_face.get_fe().n_dofs_per_cell();
      const std::vector<double>         &JxW = fe_face.get_JxW_values();
      const std::vector<Tensor<1, dim>> &normals = fe_face.get_normal_vectors();
  
      std::vector<double> g(q_points.size());
      exact_solution.value_list(q_points, g);
  
      for (unsigned int point = 0; point < q_points.size(); ++point)
        {
          const double beta_dot_n = beta(q_points[point]) * normals[point];
  
          if (beta_dot_n > 0)
            {
              for (unsigned int i = 0; i < n_facet_dofs; ++i)
                for (unsigned int j = 0; j < n_facet_dofs; ++j)
                  copy_data.cell_matrix(i, j) +=
                    fe_face.shape_value(i,
                                        point)      // \phi_i
                    * fe_face.shape_value(j, point) // \phi_j
                    * beta_dot_n                    // \beta . n
                    * JxW[point];                   // dx
            }
          else
            for (unsigned int i = 0; i < n_facet_dofs; ++i)
              copy_data.cell_rhs(i) += -fe_face.shape_value(i, point) // \phi_i
                                       * g[point]                     // g*/
                                       * beta_dot_n                   // \beta . n
                                       * JxW[point];                  // dx
        }
    };
  

This is the function called on interior faces. The arguments specify cells, face and subface indices (for adaptive refinement). We just pass them along to the reinit() function of FEInterfaceValues.

    const auto face_worker = [&](const Iterator     &cell,
                                 const unsigned int &f,
                                 const unsigned int &sf,
                                 const Iterator     &ncell,
                                 const unsigned int &nf,
                                 const unsigned int &nsf,
                                 ScratchData<dim>   &scratch_data,
                                 CopyData           &copy_data) {
      FEInterfaceValues<dim> &fe_iv = scratch_data.fe_interface_values;
      fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
      const auto &q_points = fe_iv.get_quadrature_points();
  
      copy_data.face_data.emplace_back();
      CopyDataFace &copy_data_face = copy_data.face_data.back();
  
      const unsigned int n_dofs        = fe_iv.n_current_interface_dofs();
      copy_data_face.joint_dof_indices = fe_iv.get_interface_dof_indices();
  
      copy_data_face.cell_matrix.reinit(n_dofs, n_dofs);
  
      const std::vector<double>         &JxW     = fe_iv.get_JxW_values();
      const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
  
      for (unsigned int qpoint = 0; qpoint < q_points.size(); ++qpoint)
        {
          const double beta_dot_n = beta(q_points[qpoint]) * normals[qpoint];
          for (unsigned int i = 0; i < n_dofs; ++i)
            {
              for (unsigned int j = 0; j < n_dofs; ++j)
                {
                  copy_data_face.cell_matrix(i, j) +=
                    (beta(q_points[qpoint]) * normals[qpoint] *
                       fe_iv.average_of_shape_values(j, qpoint) *
                       fe_iv.jump_in_shape_values(i, qpoint) +
                     theta * std::abs(beta_dot_n) *
                       fe_iv.jump_in_shape_values(j, qpoint) *
                       fe_iv.jump_in_shape_values(i, qpoint)) *
                    JxW[qpoint];
                }
            }
        }
    };
  

The following lambda function will handle copying the data from the cell and face assembly into the global matrix and right-hand side.

While we would not need an AffineConstraints object, because there are no hanging node constraints in DG discretizations, we use an empty object here as this allows us to use its copy_local_to_global functionality.

    const AffineConstraints<double> constraints;
  
    const auto copier = [&](const CopyData &c) {
      constraints.distribute_local_to_global(c.cell_matrix,
                                             c.cell_rhs,
                                             c.local_dof_indices,
                                             system_matrix,
                                             right_hand_side);
  
      for (auto &cdf : c.face_data)
        {
          constraints.distribute_local_to_global(cdf.cell_matrix,
                                                 cdf.joint_dof_indices,
                                                 system_matrix);
        }
    };
  
    ScratchData<dim> scratch_data(mapping, *fe, quadrature, quadrature_face);
    CopyData         copy_data;
  

Here, we finally handle the assembly. We pass in ScratchData and CopyData objects, the lambda functions from above, an specify that we want to assemble interior faces once.

    MeshWorker::mesh_loop(dof_handler.begin_active(),
                          dof_handler.end(),
                          cell_worker,
                          copier,
                          scratch_data,
                          copy_data,
                          MeshWorker::assemble_own_cells |
                            MeshWorker::assemble_boundary_faces |
                            MeshWorker::assemble_own_interior_faces_once,
                          boundary_worker,
                          face_worker);
  }
  
  
  
  template <int dim>
  void
  AdvectionReaction<dim>::solve()
  {
    if (use_direct_solver)
      {
        SparseDirectUMFPACK system_matrix_inverse;
        system_matrix_inverse.initialize(system_matrix);
        system_matrix_inverse.vmult(solution, right_hand_side);
      }
    else
      {

Here we have a classic iterative solver, as done in many tutorials:

        SolverControl                               solver_control(1000, 1e-15);
        SolverRichardson<Vector<double>>            solver(solver_control);
        PreconditionBlockSSOR<SparseMatrix<double>> preconditioner;
        preconditioner.initialize(system_matrix, fe->n_dofs_per_cell());
        solver.solve(system_matrix, solution, right_hand_side, preconditioner);
        std::cout << "  Solver converged in " << solver_control.last_step()
                  << " iterations." << std::endl;
      }
  }
  
  
  

Mesh refinement

We refine the grid according the proposed estimator or with an approximation to the gradient of the solution. The first option is the default one (you can see it in the header file)

  template <int dim>
  void
  AdvectionReaction<dim>::refine_grid()
  {
    if (refinement == "residual")
      {

If the refinement string is "residual", then we first compute the local projection

compute_local_projection_and_estimate();

We then set the refinement fraction and as usual we execute the refinement.

        const double refinement_fraction = 0.6;
        GridRefinement::refine_and_coarsen_fixed_fraction(
          triangulation, error_indicator_per_cell, refinement_fraction, 0.0);
        triangulation.execute_coarsening_and_refinement();
      }
    else if (refinement == "gradient")
      {
        Vector<float> gradient_indicator(triangulation.n_active_cells());
  

Now the approximate gradients are computed

        DerivativeApproximation::approximate_gradient(mapping,
                                                      dof_handler,
                                                      solution,
                                                      gradient_indicator);
  

and they are cell-wise scaled by the factor \(h^{1+d/2}\)

        unsigned int cell_no = 0;
        for (const auto &cell : dof_handler.active_cell_iterators())
          gradient_indicator(cell_no++) *=
            std::pow(cell->diameter(), 1 + 1.0 * dim / 2);
  

Finally they serve as refinement indicator.

        GridRefinement::refine_and_coarsen_fixed_fraction(triangulation,
                                                          gradient_indicator,
                                                          0.25,
                                                          0.0);
  
        triangulation.execute_coarsening_and_refinement();
        std::cout << gradient_indicator.l2_norm() << '\n';
      }
    else if (refinement == "global")
      {
        triangulation.refine_global(
          1); // just for testing on uniformly refined meshes
      }
    else
      {
        Assert(false, ExcInternalError());
      }
  }
  
  
  

The output of this program consists of a vtk file of the adaptively refined grids and the numerical solutions.

  template <int dim>
  void
  AdvectionReaction<dim>::output_results(const unsigned int cycle) const
  {
    const std::string filename = "solution-" + std::to_string(cycle) + ".vtk";
    std::cout << "  Writing solution to <" << filename << ">" << std::endl;
    std::ofstream output(filename);
  
    DataOut<dim> data_out;
    data_out.attach_dof_handler(dof_handler);
    data_out.add_data_vector(solution, "u", DataOut<dim>::type_dof_data);
    data_out.build_patches(mapping);
    data_out.write_vtk(output);
  }
  
  template <int dim>
  void
  AdvectionReaction<dim>::compute_error()
  {
    error_table.error_from_exact(
      mapping,
      dof_handler,
      solution,
      exact_solution); // be careful: a FD approximation of the gradient is used

to compute the H^1 norm if Solution<dim> doesn't implements the Gradient function

}

Compute the energy norm

The energy norm is defined as \( |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2 \Bigr)^{\frac{1}{2}}\) Notice that in the current case we have \(c_f = \frac{|b \cdot n|}{2}\) Like in the assembly, all the contributions are handled separately by using ScratchData and CopyData objects.

  template <int dim>
  double
  AdvectionReaction<dim>::compute_energy_norm()
  {
    energy_norm_square_per_cell.reinit(triangulation.n_active_cells());
  
    using Iterator = typename DoFHandler<dim>::active_cell_iterator;
  

We start off by adding cell contributions

    const auto cell_worker = [&](const Iterator   &cell,
                                 ScratchData<dim> &scratch_data,
                                 CopyData         &copy_data) {
      const unsigned int n_dofs =
        scratch_data.fe_values.get_fe().n_dofs_per_cell();
      copy_data.reinit(cell, n_dofs);
      scratch_data.fe_values.reinit(cell);
  
      copy_data.cell_index = cell->active_cell_index();
  
      const auto &q_points = scratch_data.fe_values.get_quadrature_points();
      const FEValues<dim>       &fe_v = scratch_data.fe_values;
      const std::vector<double> &JxW  = fe_v.get_JxW_values();
  
      double              error_square_norm{0.0};
      std::vector<double> sol_u(fe_v.n_quadrature_points);
      fe_v.get_function_values(solution, sol_u);
  
      for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
        {
          const double diff =
            (sol_u[point] - exact_solution.value(q_points[point]));
          error_square_norm += diff * diff * JxW[point];
        }
      copy_data.value = error_square_norm;
    };
  

Here we add contributions coming from the internal faces

    const auto face_worker = [&](const Iterator     &cell,
                                 const unsigned int &f,
                                 const unsigned int &sf,
                                 const Iterator     &ncell,
                                 const unsigned int &nf,
                                 const unsigned int &nsf,
                                 ScratchData<dim>   &scratch_data,
                                 CopyData           &copy_data) {
      FEInterfaceValues<dim> &fe_iv = scratch_data.fe_interface_values;
      fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
  
      copy_data.face_data.emplace_back();
      CopyDataFace &copy_data_face   = copy_data.face_data.back();
      copy_data_face.cell_indices[0] = cell->active_cell_index();
      copy_data_face.cell_indices[1] = ncell->active_cell_index();
  
      const auto                &q_points   = fe_iv.get_quadrature_points();
      const unsigned             n_q_points = q_points.size();
      const std::vector<double> &JxW        = fe_iv.get_JxW_values();
      std::vector<double>        g(n_q_points);
  
      std::vector<double> jump(n_q_points);
      get_function_jump(fe_iv, solution, jump);
  
      const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
  
      double error_jump_square{0.0};
      for (unsigned int point = 0; point < n_q_points; ++point)
        {
          const double beta_dot_n =
            theta * std::abs(beta(q_points[point]) * normals[point]);
          error_jump_square +=
            beta_dot_n * jump[point] * jump[point] * JxW[point];
        }
  
      copy_data.value = error_jump_square;
    };
  

Finally, we add the boundary contributions

    const auto boundary_worker = [&](const Iterator     &cell,
                                     const unsigned int &face_no,
                                     ScratchData<dim>   &scratch_data,
                                     CopyData           &copy_data) {
      scratch_data.fe_interface_values.reinit(cell, face_no);
      const FEFaceValuesBase<dim> &fe_fv =
        scratch_data.fe_interface_values.get_fe_face_values(0);
      const auto                &q_points   = fe_fv.get_quadrature_points();
      const unsigned             n_q_points = q_points.size();
      const std::vector<double> &JxW        = fe_fv.get_JxW_values();
  
      std::vector<double> g(n_q_points);
  
      std::vector<double> sol_u(n_q_points);
      fe_fv.get_function_values(solution, sol_u);
  
      const std::vector<Tensor<1, dim>> &normals = fe_fv.get_normal_vectors();
  
      double difference_norm_square = 0.;
      for (unsigned int point = 0; point < q_points.size(); ++point)
        {
          const double beta_dot_n =
            theta * std::abs(beta(q_points[point]) * normals[point]);
          const double diff =
            (boundary_conditions.value(q_points[point]) - sol_u[point]);
          difference_norm_square += beta_dot_n * diff * diff * JxW[point];
        }
      copy_data.value = difference_norm_square;
    };
  
    const auto copier = [&](const auto &copy_data) {
      if (copy_data.cell_index != numbers::invalid_unsigned_int)
        {
          energy_norm_square_per_cell[copy_data.cell_index] += copy_data.value;
        }
      for (auto &cdf : copy_data.face_data)
        for (unsigned int j = 0; j < 2; ++j)
          energy_norm_square_per_cell[cdf.cell_indices[j]] += cdf.values[j];
    };
  
    ScratchData<dim> scratch_data(mapping,
                                  *fe,
                                  QGauss<dim>{fe->tensor_degree() + 1},
                                  QGauss<dim - 1>{fe->tensor_degree() + 1});
  
    CopyData copy_data;
  
    MeshWorker::mesh_loop(dof_handler.begin_active(),
                          dof_handler.end(),
                          cell_worker,
                          copier,
                          scratch_data,
                          copy_data,
                          MeshWorker::assemble_own_cells |
                            MeshWorker::assemble_own_interior_faces_once |
                            MeshWorker::assemble_boundary_faces,
                          boundary_worker,
                          face_worker);
  
    const double energy_error = std::sqrt(energy_norm_square_per_cell.l1_norm());
    return energy_error;
  }
  
  
  

Computing the estimator

In the estimator, we have to compute the term \(||f- c u_h - \Pi(f- c u_h)||_{T}^{2}\) over a generic cell \(T\). To achieve this, we first need to compute the projection involving the finite element function \(u_h\). Using the definition of orthogonal projection, we're required to solve cellwise \((v_h,f-c u_h)_T = (v_h,\Pi)_T \qquad \forall v_h \in V_h\) for \(\Pi\), which means that we have to build a mass-matrix on each cell. Once we have the projection, which is a finite element function, we can add its contribution in the cell_worker lambda. As done in step-74, the square of the error indicator is computed.

  template <int dim>
  void
  AdvectionReaction<dim>::compute_local_projection_and_estimate()
  {

Compute the term \(||f-c u_h - \Pi(f- cu_h)||_T^2\)

    using Iterator = typename DoFHandler<dim>::active_cell_iterator;
    error_indicator_per_cell.reinit(triangulation.n_active_cells());
  
    const auto cell_worker = [&](const Iterator   &cell,
                                 ScratchData<dim> &scratch_data,
                                 CopyData         &copy_data) {
      const unsigned int n_dofs =
        scratch_data.fe_values.get_fe().n_dofs_per_cell();
  
      copy_data.reinit(cell, n_dofs);
      scratch_data.fe_values.reinit(cell);
      copy_data.cell_index = cell->active_cell_index();
  
      const auto    &q_points   = scratch_data.fe_values.get_quadrature_points();
      const unsigned n_q_points = q_points.size();
  
      const FEValues<dim>       &fe_v = scratch_data.fe_values;
      const std::vector<double> &JxW  = fe_v.get_JxW_values();
  
      std::vector<double> sol_u_at_quadrature_points(fe_v.n_quadrature_points);
      fe_v.get_function_values(solution, sol_u_at_quadrature_points);
  

Compute local L^2 projection of \(f- c u_h\) over the local finite element space

      for (unsigned int point = 0; point < n_q_points; ++point)
        {
          for (unsigned int i = 0; i < n_dofs; ++i)
            {
              for (unsigned int j = 0; j < n_dofs; ++j)
                {
                  copy_data.cell_mass_matrix(i, j) +=
                    fe_v.shape_value(i, point) * // phi_i(x_q)
                    fe_v.shape_value(j, point) * // phi_j(x_q)
                    JxW[point];                  // dx(x_q)
                }
              copy_data.cell_mass_rhs(i) +=
                (rhs.value(q_points[point]) * // f(x_q)
                   fe_v.shape_value(i, point) // phi_i(x_q)
                 - advection_coeff.value(q_points[point]) *
                     fe_v.shape_value(i, point) *         // c*phi_i(x_q)
                     sol_u_at_quadrature_points[point]) * // u_h(x_q)
                JxW[point];                               // dx
            }
        }
  
      FullMatrix<double> inverse(fe_v.n_quadrature_points,
                                 fe_v.n_quadrature_points);
      inverse.invert(copy_data.cell_mass_matrix);
      Vector<double> proj(fe_v.n_quadrature_points); // projection of (f-c*U_h) on

the local fe_space

      inverse.vmult(proj, copy_data.cell_mass_rhs);  // M^{-1}*rhs = proj
  
      double square_norm_over_cell = 0.0;
      for (unsigned int point = 0; point < n_q_points; ++point)
        {
          const double diff = rhs.value(q_points[point]) -
                              sol_u_at_quadrature_points[point] - proj[point];
          square_norm_over_cell += diff * diff * JxW[point];
        }
      copy_data.value_estimator = square_norm_over_cell;
    };
  

Finally we have the boundary term with \(||\beta (g-u_h^+)||^2\)

    const auto boundary_worker = [&](const Iterator     &cell,
                                     const unsigned int &face_no,
                                     ScratchData<dim>   &scratch_data,
                                     CopyData           &copy_data) {
      scratch_data.fe_interface_values.reinit(cell, face_no);
      const FEFaceValuesBase<dim> &fe_fv =
        scratch_data.fe_interface_values.get_fe_face_values(0);
      const auto                &q_points   = fe_fv.get_quadrature_points();
      const unsigned             n_q_points = q_points.size();
      const std::vector<double> &JxW        = fe_fv.get_JxW_values();
  
      std::vector<double> g(n_q_points);
      exact_solution.value_list(q_points, g);
  
      std::vector<double> sol_u(n_q_points);
      fe_fv.get_function_values(solution, sol_u);
  
      const std::vector<Tensor<1, dim>> &normals = fe_fv.get_normal_vectors();
  
      double square_norm_over_bdary_face = 0.;
      for (unsigned int point = 0; point < q_points.size(); ++point)
        {
          const double beta_dot_n = beta(q_points[point]) * normals[point];
  
          if (beta_dot_n < 0) //\partial_{-T} \cap \partial_{- \Omega}
            {
              const double diff =
                std::abs(beta_dot_n) * (g[point] - sol_u[point]);
              square_norm_over_bdary_face += diff * diff * JxW[point];
            }
        }
      copy_data.value_estimator += square_norm_over_bdary_face;
    };
  

Then compute the interior face terms with \(|| \sqrt{b \cdot n}[u_h]||^2\)

    const auto face_worker = [&](const Iterator     &cell,
                                 const unsigned int &f,
                                 const unsigned int &sf,
                                 const Iterator     &ncell,
                                 const unsigned int &nf,
                                 const unsigned int &nsf,
                                 ScratchData<dim>   &scratch_data,
                                 CopyData           &copy_data) {
      FEInterfaceValues<dim> &fe_iv = scratch_data.fe_interface_values;
      fe_iv.reinit(cell, f, sf, ncell, nf, nsf);
  
      copy_data.face_data.emplace_back();
      CopyDataFace &copy_data_face   = copy_data.face_data.back();
      copy_data_face.cell_indices[0] = cell->active_cell_index();
      copy_data_face.cell_indices[1] = ncell->active_cell_index();
  
      const auto    &q_points   = fe_iv.get_quadrature_points();
      const unsigned n_q_points = q_points.size();
  
      const std::vector<double> &JxW = fe_iv.get_JxW_values();
      std::vector<double>        g(n_q_points);
  
      std::vector<double> jump(n_q_points);
      get_function_jump(fe_iv, solution, jump);
  
      const std::vector<Tensor<1, dim>> &normals = fe_iv.get_normal_vectors();
  
      double error_jump_square{0.0};
      for (unsigned int point = 0; point < n_q_points; ++point)
        {
          const double beta_dot_n = beta(q_points[point]) * normals[point];
          if (beta_dot_n < 0)
            {
              error_jump_square +=
                std::abs(beta_dot_n) * jump[point] * jump[point] * JxW[point];
            }
        }
  
      copy_data_face.values[0] = error_jump_square;
      copy_data_face.values[1] = copy_data_face.values[0];
    };
  
    ScratchData<dim> scratch_data(mapping,
                                  *fe,
                                  QGauss<dim>{fe->tensor_degree() + 1},
                                  QGauss<dim - 1>{fe->tensor_degree() + 1});
  
    const auto copier = [&](const auto &copy_data) {
      if (copy_data.cell_index != numbers::invalid_unsigned_int)
        {
          error_indicator_per_cell[copy_data.cell_index] +=
            copy_data.value_estimator;
        }
      for (auto &cdf : copy_data.face_data)
        {
          for (unsigned int j = 0; j < 2; ++j)
            {
              error_indicator_per_cell[cdf.cell_indices[j]] += cdf.values[j];
            }
        }
    };
  

Here, we finally handle the assembly of the Mass matrix (M)_{ij} = (\phi_j, \phi_i)_T. We pass in ScratchData and CopyData objects

    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);
  }
  
  
  

Usual run function, which runs over several refinement cycles

  template <int dim>
  void
  AdvectionReaction<dim>::run()
  {
    std::vector<double> energy_errors;
    std::vector<int>    dofs_hist;
    for (unsigned int cycle = 0; cycle < n_refinement_cycles; ++cycle)
      {
        std::cout << "Cycle " << cycle << std::endl;
  
        if (cycle == 0)
          {
            GridGenerator::hyper_cube(triangulation);
            triangulation.refine_global(n_global_refinements);
          }
        else
          {
            refine_grid();
          }
        std::cout << "  Number of active cells:       "
                  << triangulation.n_active_cells() << std::endl;
        std::cout << "  Number of degrees of freedom: " << dof_handler.n_dofs()
                  << std::endl;
  
        setup_system();
        assemble_system();
        solve();
        compute_error();
        output_results(cycle);
  
        energy_errors.emplace_back(compute_energy_norm());
        dofs_hist.emplace_back(triangulation.n_active_cells());
      }
    error_table.output_table(std::cout);
  
    for (unsigned int i = 0; i < n_refinement_cycles; ++i)
      std::cout << "Cycle " << i << "\t" << energy_errors[i] << std::endl;
  }

Explicit instantiation

  template class AdvectionReaction<1>;
  template class AdvectionReaction<2>;
  template class AdvectionReaction<3>;