diffusion_ldgh_pp.hxx

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#pragma once  // Ensure that file is included only once in a single compilation.
 
#include <HyperHDG/dense_la.hxx>
#include <HyperHDG/hypercube.hxx>
#include <tpp/quadrature/tensorial.hxx>
#include <tpp/shape_function/shape_function.hxx>
 
#include <algorithm>
#include <tuple>
 
namespace LocalSolver
{
/*!*************************************************************************************************
 * \brief   Default parameters for the diffusion equation, cf. below.
 *
 * \authors   Guido Kanschat, Heidelberg University, 2019--2020.
 * \authors   Andreas Rupp, Heidelberg University, 2019--2020.
 **************************************************************************************************/
template <unsigned int space_dimT, typename param_float_t = double>
struct DiffusionParametersDefault
{
  /*!***********************************************************************************************
   * \brief   Array containing hypernode types corresponding to Dirichlet boundary.
   ************************************************************************************************/
  static constexpr std::array<unsigned int, 0U> dirichlet_nodes{};
  /*!***********************************************************************************************
   * \brief   Array containing hypernode types corresponding to Neumann boundary.
   ************************************************************************************************/
  static constexpr std::array<unsigned int, 0U> neumann_nodes{};
  /*!***********************************************************************************************
   * \brief   Inverse diffusion coefficient in PDE as analytic function.
   ************************************************************************************************/
  static param_float_t inverse_diffusion_coeff(const Point<space_dimT, param_float_t>& point,
                                               const param_float_t time = 0.)
  {
    return 1.;
  }
  /*!***********************************************************************************************
   * \brief   Right-hand side in PDE as analytic function.
   ************************************************************************************************/
  static param_float_t right_hand_side(const Point<space_dimT, param_float_t>& point,
                                       const param_float_t time = 0.)
  {
    return 0.;
  }
  /*!***********************************************************************************************
   * \brief   Dirichlet values of solution as analytic function.
   ************************************************************************************************/
  static param_float_t dirichlet_value(const Point<space_dimT, param_float_t>& point,
                                       const param_float_t time = 0.)
  {
    return 0.;
  }
  /*!***********************************************************************************************
   * \brief   Neumann values of solution as analytic function.
   ************************************************************************************************/
  static param_float_t neumann_value(const Point<space_dimT, param_float_t>& point,
                                     const param_float_t time = 0.)
  {
    return 0.;
  }
  /*!***********************************************************************************************
   * \brief   Analytic result of PDE (for convergence tests).
   ************************************************************************************************/
  static param_float_t analytic_result(const Point<space_dimT, param_float_t>& point,
                                       const param_float_t time = 0.)
  {
    return 0.;
  }
};  // end of struct DiffusionParametersDefault
 
/*!*************************************************************************************************
 * \brief   Local solver diffusion equation on hypergraph.
 *
 * This class contains the local solver for an isotropic diffusion equation, i.e.,
 * \f[
 *  - \nabla \cdot ( d \nabla u ) = f \quad \text{ in } \Omega, \qquad
 *  u = u_\text D \quad \text{ on } \partial \Omega_\text D, \qquad
 *  - d \nabla u \cdot \nu = g_\text N \quad \text{ on } \partial \Omega_\text N
 * \f]
 * in a spatial domain \f$\Omega \subset \mathbb R^d\f$. Here, \f$d\f$ is the spatial dimension
 * \c space_dim, \f$\Omega\f$ is a regular graph (\c hyEdge_dimT = 1) or hypergraph whose
 * hyperedges are surfaces (\c hyEdge_dimT = 2) or volumes (\c hyEdge_dimT = 3) or hypervolumes (in
 * case of \c hyEdge_dimT > 3). \f$f\f$ and \f$d\f$ are scalars defined in the whole domain, the
 * Dirichlet and Neumann boundary data needs to be defined on their respective hypernodes.
 *
 * Moreover, this class provides the functions that approximate the condensed mass matrix. By this,
 * it is possible to approximate parabolic problems (in a very bad way) and to find good starting
 * values for the nonlinear eigenvalue problem.
 *
 * \tparam  hyEdge_dimT   Dimension of a hyperedge, i.e., 1 is for PDEs defined on graphs, 2 is for
 *                        PDEs defined on surfaces, and 3 is for PDEs defined on volumes.
 * \tparam  poly_deg      The polynomial degree of test and trial functions.
 * \tparam  quad_deg      The order of the quadrature rule.
 * \tparam  parametersT   Struct depending on templates \c space_dimTP and \c lSol_float_TP that
 *                        contains static parameter functions.
 *                        Defaults to above functions included in \c DiffusionParametersDefault.
 * \tparam  lSol_float_t  The floating point type calculations are executed in. Defaults to double.
 *
 * \authors   Guido Kanschat, Heidelberg University, 2019--2020.
 * \authors   Andreas Rupp, Heidelberg University, 2019--2020.
 **************************************************************************************************/
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename> typename parametersT = DiffusionParametersDefault,
          typename lSol_float_t = double>
class DiffusionPostprocess
{
 public:
  /*!***********************************************************************************************
   *  \brief  Define type of (hyperedge related) data that is stored in HyDataContainer.
   ************************************************************************************************/
  struct data_type
  {
  };
  /*!***********************************************************************************************
   *  \brief  Define type of node elements, especially with respect to nodal shape functions.
   ************************************************************************************************/
  struct node_element
  {
    typedef std::tuple<TPP::ShapeFunction<
      TPP::ShapeType::Tensorial<TPP::ShapeType::Legendre<poly_deg>, hyEdge_dimT - 1> > >
      functions;
  };
  /*!***********************************************************************************************
   *  \brief  Define how errors are evaluated.
   ************************************************************************************************/
  struct error_def
  {
    /*!*********************************************************************************************
     *  \brief  Define the typename returned by function errors.
     **********************************************************************************************/
    typedef std::array<lSol_float_t, 1U> error_t;
    /*!*********************************************************************************************
     *  \brief  Define how initial error is generated.
     **********************************************************************************************/
    static error_t initial_error()
    {
      std::array<lSol_float_t, 1U> summed_error;
      summed_error.fill(0.);
      return summed_error;
    }
    /*!*********************************************************************************************
     *  \brief  Define how local errors should be accumulated.
     **********************************************************************************************/
    static error_t sum_error(error_t& summed_error, const error_t& new_error)
    {
      for (unsigned int k = 0; k < summed_error.size(); ++k)
        summed_error[k] += new_error[k];
      return summed_error;
    }
    /*!*********************************************************************************************
     *  \brief  Define how global errors should be postprocessed.
     **********************************************************************************************/
    static error_t postprocess_error(error_t& summed_error)
    {
      for (unsigned int k = 0; k < summed_error.size(); ++k)
        summed_error[k] = std::sqrt(summed_error[k]);
      return summed_error;
    }
  };
 
  // -----------------------------------------------------------------------------------------------
  // Public, static constexpr functions
  // -----------------------------------------------------------------------------------------------
 
  /*!***********************************************************************************************
   * \brief   Dimension of hyper edge type that this object solves on.
   ************************************************************************************************/
  static constexpr unsigned int hyEdge_dim() { return hyEdge_dimT; }
  /*!***********************************************************************************************
   * \brief   Evaluate amount of global degrees of freedom per hypernode.
   *
   * This number must be equal to HyperNodeFactory::n_dofs_per_node() of the HyperNodeFactory
   * cooperating with this object.
   *
   * \retval  n_dofs        Number of global degrees of freedom per hypernode.
   ************************************************************************************************/
  static constexpr unsigned int n_glob_dofs_per_node()
  {
    return Hypercube<hyEdge_dimT - 1>::pow(poly_deg + 1);
  }
  /*!***********************************************************************************************
   * \brief   Dimension of of the solution evaluated with respect to a hyperedge.
   ************************************************************************************************/
  static constexpr unsigned int system_dimension() { return hyEdge_dimT + 1; }
  /*!***********************************************************************************************
   * \brief   Dimension of of the solution evaluated with respect to a hypernode.
   ************************************************************************************************/
  static constexpr unsigned int node_system_dimension() { return 1; }
 
 private:
  // -----------------------------------------------------------------------------------------------
  // Private, static constexpr functions
  // -----------------------------------------------------------------------------------------------
 
  /*!***********************************************************************************************
   * \brief   Number of local shape functions (with respect to all spatial dimensions).
   ************************************************************************************************/
  static constexpr unsigned int n_shape_fct_ = n_glob_dofs_per_node() * (poly_deg + 1);
  /*!***********************************************************************************************
   * \brief   Number oflocal  shape functions (with respect to a face / hypernode).
   ************************************************************************************************/
  static constexpr unsigned int n_shape_bdr_ = n_glob_dofs_per_node();
  /*!***********************************************************************************************
   * \brief   Number of (local) degrees of freedom per hyperedge.
   ************************************************************************************************/
  static constexpr unsigned int n_loc_dofs_ = (hyEdge_dimT + 1) * n_shape_fct_;
  /*!***********************************************************************************************
   * \brief   Dimension of of the solution evaluated with respect to a hypernode.
   *
   * This allows to the use of this quantity as template parameter in member functions.
   ************************************************************************************************/
  static constexpr unsigned int system_dim = system_dimension();
  /*!***********************************************************************************************
   * \brief   Dimension of of the solution evaluated with respect to a hypernode.
   *
   * This allows to the use of this quantity as template parameter in member functions.
   ************************************************************************************************/
  static constexpr unsigned int node_system_dim = node_system_dimension();
  /*!***********************************************************************************************
   * \brief   Find out whether a node is of Dirichlet type.
   ************************************************************************************************/
  template <typename parameters>
  static constexpr bool is_dirichlet(const unsigned int node_type)
  {
    return std::find(parameters::dirichlet_nodes.begin(), parameters::dirichlet_nodes.end(),
                     node_type) != parameters::dirichlet_nodes.end();
  }
 
  // -----------------------------------------------------------------------------------------------
  // Private, const members: Parameters and auxiliaries that help assembling matrices, etc.
  // -----------------------------------------------------------------------------------------------
 
  /*!***********************************************************************************************
   * \brief   (Globally constant) penalty parameter for HDG scheme.
   ************************************************************************************************/
  const lSol_float_t tau_;
  /*!***********************************************************************************************
   * \brief   An integrator helps to easily evaluate integrals (e.g. via quadrature).
   ************************************************************************************************/
  typedef TPP::Quadrature::Tensorial<
    TPP::Quadrature::GaussLegendre<quad_deg>,
    TPP::ShapeFunction<TPP::ShapeType::Tensorial<TPP::ShapeType::Legendre<poly_deg>, hyEdge_dimT> >,
    lSol_float_t>
    integrator;
  /*!***********************************************************************************************
   * \brief   An integrator helps to easily evaluate integrals of postprocessing.
   ************************************************************************************************/
  typedef TPP::Quadrature::Tensorial<
    TPP::Quadrature::GaussLegendre<quad_deg + 2>,
    TPP::ShapeFunction<
      TPP::ShapeType::Tensorial<TPP::ShapeType::Legendre<poly_deg + 1>, hyEdge_dimT> >,
    lSol_float_t>
    integrator_pp;
 
  // -----------------------------------------------------------------------------------------------
  // Private, internal functions for the local solver
  // -----------------------------------------------------------------------------------------------
 
  /*!***********************************************************************************************
   * \brief   Assemble local matrix for the local solver.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \param   time          Time, when the local matrix is evaluated.
   * \retval  loc_mat       Matrix of the local solver.
   ************************************************************************************************/
  template <typename hyEdgeT>
  inline SmallSquareMat<n_loc_dofs_, lSol_float_t> assemble_loc_matrix(
    hyEdgeT& hyper_edge,
    const lSol_float_t time) const;
  /*!***********************************************************************************************
   * \brief   Assemble local right-hand for the local solver (from skeletal).
   *
   * The right hand side needs the values of the global degrees of freedom. Note that we basically
   * follow the lines of
   *
   * B. Cockburn, J. Gopalakrishnan, and R. Lazarov.
   * Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for
   * second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009
   *
   * and discriminate between local solvers with respect to the skeletal variable and with respect
   * to the global right-hand side. This assembles the local right-hand side with respect to the
   * skeletal variable.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallMatT     The data type of the \c lambda values.
   * \param   lambda_values Global degrees of freedom associated to the hyperedge.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \retval  loc_rhs       Local right hand side of the locasl solver.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallMatT>
  inline SmallVec<n_loc_dofs_, lSol_float_t> assemble_rhs_from_lambda(
    const SmallMatT& lambda_values,
    hyEdgeT& hyper_edge) const;
  /*!***********************************************************************************************
   * \brief   Assemble local right-hand for the local solver (from global right-hand side).
   *
   * Note that we basically follow the lines of
   *
   * B. Cockburn, J. Gopalakrishnan, and R. Lazarov.
   * Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for
   * second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009
   *
   * and discriminate between local solvers with respect to the skeletal variable and with respect
   * to the global right-hand side. This assembles the local right-hand side with respect to the
   * global right-hand side. This function implicitly uses the parameters.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \param   time          Point of time, rhs is evaluated at
   * \retval  loc_rhs       Local right hand side of the locasl solver.
   ************************************************************************************************/
  template <typename hyEdgeT>
  inline SmallVec<n_loc_dofs_, lSol_float_t> assemble_rhs_from_global_rhs(
    hyEdgeT& hyper_edge,
    const lSol_float_t time) const;
  /*!***********************************************************************************************
   * \brief   Assemble local right-hand for the local solver (from volume function coefficients).
   *
   * Note that we basically follow the lines of
   *
   * B. Cockburn, J. Gopalakrishnan, and R. Lazarov.
   * Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for
   * second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009
   *
   * and discriminate between local solvers with respect to the skeletal variable and with respect
   * to the global right-hand side. This assembles the local right-hand side with respect to the
   * global right-hand side. This function implicitly uses the parameters.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallVecT     The data type of the coefficients.
   * \param   coeffs        Local coefficients of bulk variable u.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \retval  loc_rhs       Local right hand side of the locasl solver.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallVecT>
  inline SmallVec<n_loc_dofs_, lSol_float_t> assemble_rhs_from_coeffs(const SmallVecT& coeffs,
                                                                      hyEdgeT& hyper_edge) const;
  /*!***********************************************************************************************
   * \brief   Solve local problem (with right-hand side from skeletal).
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallMatT     The data type of \c lambda_values.
   * \param   lambda_values Global degrees of freedom associated to the hyperedge.
   * \param   solution_type Type of local problem that is to be solved.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \param   time          Point of time the problem is solved.
   * \retval  loc_sol       Solution of the local problem.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallMatT>
  inline SmallVec<n_loc_dofs_, lSol_float_t> solve_local_problem(const SmallMatT& lambda_values,
                                                                 const unsigned int solution_type,
                                                                 hyEdgeT& hyper_edge,
                                                                 const lSol_float_t time) const
  {
    try
    {
      SmallVec<n_loc_dofs_, lSol_float_t> rhs;
      if (solution_type == 0)
        rhs = assemble_rhs_from_lambda(lambda_values, hyper_edge);
      else if (solution_type == 1)
        rhs = assemble_rhs_from_lambda(lambda_values, hyper_edge) +
              assemble_rhs_from_global_rhs(hyper_edge, time);
      else
        hy_assert(0 == 1, "This has not been implemented!");
      return rhs / assemble_loc_matrix(hyper_edge, time);
    }
    catch (Wrapper::LAPACKexception& exc)
    {
      hy_assert(0 == 1, exc.what() << std::endl
                                   << "This can happen if quadrature is too inaccurate!");
      throw exc;
    }
  }
  /*!***********************************************************************************************
   * \brief   Evaluate primal variable at boundary.
   *
   * Function to evaluate primal variable of the solution. This function is needed to calculate
   * the local numerical fluxes.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \param   coeffs        Coefficients of the local solution.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \retval  bdr_coeffs    Coefficients of respective (dim-1) dimensional function at boundaries.
   ************************************************************************************************/
  template <typename hyEdgeT>
  inline SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> primal_at_boundary(
    const SmallVec<n_loc_dofs_, lSol_float_t>& coeffs,
    hyEdgeT& hyper_edge) const;
  /*!***********************************************************************************************
   * \brief   Evaluate dual variable at boundary.
   *
   * Function to evaluate dual variable of the solution. This function is needed to calculate the
   * local numerical fluxes.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \param   coeffs        Coefficients of the local solution.
   * \param   hyper_edge    The geometry of the considered hyperedge (of typename GeomT).
   * \retval  bdr_coeffs    Coefficients of respective (dim-1) dimensional function at boundaries.
   ************************************************************************************************/
  template <typename hyEdgeT>
  inline SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> dual_at_boundary(
    const SmallVec<n_loc_dofs_, lSol_float_t>& coeffs,
    hyEdgeT& hyper_edge) const;
 
 public:
  // -----------------------------------------------------------------------------------------------
  // Public functions (and one typedef) to be utilized by external functions.
  // -----------------------------------------------------------------------------------------------
 
  /*!***********************************************************************************************
   * \brief   Class is constructed using a single double indicating the penalty parameter.
   ************************************************************************************************/
  typedef lSol_float_t constructor_value_type;
  /*!***********************************************************************************************
   * \brief   Constructor for local solver.
   *
   * \param   tau           Penalty parameter of HDG scheme.
   ************************************************************************************************/
  DiffusionPostprocess(const constructor_value_type& tau = 1.) : tau_(tau) {}
  /*!***********************************************************************************************
   * \brief   Evaluate local contribution to matrix--vector multiplication.
   *
   * Execute matrix--vector multiplication y = A * x, where x represents the vector containing the
   * skeletal variable (adjacent to one hyperedge), and A is the condensed matrix arising from the
   * HDG discretization. This function does this multiplication (locally) for one hyperedge. The
   * hyperedge is no parameter, since all hyperedges are assumed to have the same properties.
   *
   * \tparam  hyEdgeT             The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallMatInT         Data type of \c lambda_values_in.
   * \tparam  SmallMatOutT        Data type of \c lambda_values_out.
   * \param   lambda_values_in    Local part of vector x.
   * \param   lambda_values_out   Local part that will be added to A * x.
   * \param   hyper_edge          The geometry of the considered hyperedge (of typename GeomT).
   * \param   time                Time.
   * \retval  vecAx               Local part of vector A * x.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallMatInT, typename SmallMatOutT>
  SmallMatOutT& trace_to_flux(const SmallMatInT& lambda_values_in,
                              SmallMatOutT& lambda_values_out,
                              hyEdgeT& hyper_edge,
                              const lSol_float_t time = 0.) const
  {
    hy_assert(lambda_values_in.size() == lambda_values_out.size() &&
                lambda_values_in.size() == 2 * hyEdge_dimT,
              "Both matrices must be of same size which corresponds to the number of faces!");
    for (unsigned int i = 0; i < lambda_values_in.size(); ++i)
      hy_assert(
        lambda_values_in[i].size() == lambda_values_out[i].size() &&
          lambda_values_in[i].size() == n_glob_dofs_per_node(),
        "Both matrices must be of same size which corresponds to the number of dofs per face!");
 
    using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
    SmallVec<n_loc_dofs_, lSol_float_t> coeffs =
      solve_local_problem(lambda_values_in, 0U, hyper_edge, time);
 
    SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> primals(
      primal_at_boundary(coeffs, hyper_edge)),
      duals(dual_at_boundary(coeffs, hyper_edge));
 
    for (unsigned int i = 0; i < lambda_values_out.size(); ++i)
      if (is_dirichlet<parameters>(hyper_edge.node_descriptor[i]))
        for (unsigned int j = 0; j < lambda_values_out[i].size(); ++j)
          lambda_values_out[i][j] = 0.;
      else
        for (unsigned int j = 0; j < lambda_values_out[i].size(); ++j)
          lambda_values_out[i][j] =
            duals(i, j) + tau_ * primals(i, j) -
            tau_ * lambda_values_in[i][j] * hyper_edge.geometry.face_area(i);
 
    return lambda_values_out;
  }
  /*!***********************************************************************************************
   * \brief   Fill an array with 1 if the node is Dirichlet and 0 otherwise.
   ************************************************************************************************/
  template <typename hyEdgeT>
  std::array<unsigned int, 2 * hyEdge_dimT> node_types(hyEdgeT& hyper_edge) const
  {
    using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
 
    std::array<unsigned int, 2 * hyEdge_dimT> result;
    result.fill(0);
 
    for (unsigned int i = 0; i < 2 * hyEdge_dimT; ++i)
      if (is_dirichlet<parameters>(hyper_edge.node_descriptor[i]))
        result[i] = 1;
 
    return result;
  }
  /*!***********************************************************************************************
   * \brief   Evaluate local contribution to residual \f$ A x - b \f$.
   *
   * \tparam  hyEdgeT             The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallMatInT         Data type of \c lambda_values_in.
   * \tparam  SmallMatOutT        Data type of \c lambda_values_out.
   * \param   lambda_values_in    Local part of vector x.
   * \param   lambda_values_out   Local part that will be added to A * x.
   * \param   hyper_edge          The geometry of the considered hyperedge (of typename hyEdgeT).
   * \param   time                Time at which analytic functions are evaluated.
   * \retval  vecAx               Local part of vector A * x - b.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallMatInT, typename SmallMatOutT>
  SmallMatOutT& residual_flux(const SmallMatInT& lambda_values_in,
                              SmallMatOutT& lambda_values_out,
                              hyEdgeT& hyper_edge,
                              const lSol_float_t time = 0.) const
  {
    hy_assert(lambda_values_in.size() == lambda_values_out.size() &&
                lambda_values_in.size() == 2 * hyEdge_dimT,
              "Both matrices must be of same size which corresponds to the number of faces!");
    for (unsigned int i = 0; i < lambda_values_in.size(); ++i)
      hy_assert(
        lambda_values_in[i].size() == lambda_values_out[i].size() &&
          lambda_values_in[i].size() == n_glob_dofs_per_node(),
        "Both matrices must be of same size which corresponds to the number of dofs per face!");
 
    using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
    SmallVec<n_loc_dofs_, lSol_float_t> coeffs =
      solve_local_problem(lambda_values_in, 1U, hyper_edge, time);
 
    SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> primals(
      primal_at_boundary(coeffs, hyper_edge)),
      duals(dual_at_boundary(coeffs, hyper_edge));
 
    for (unsigned int i = 0; i < lambda_values_out.size(); ++i)
    {
      if (is_dirichlet<parameters>(hyper_edge.node_descriptor[i]))
        for (unsigned int j = 0; j < lambda_values_out[i].size(); ++j)
          lambda_values_out[i][j] = 0.;
      else
        for (unsigned int j = 0; j < lambda_values_out[i].size(); ++j)
          lambda_values_out[i][j] =
            duals(i, j) + tau_ * primals(i, j) -
            tau_ * lambda_values_in[i][j] * hyper_edge.geometry.face_area(i);
    }
 
    return lambda_values_out;
  }
  /*!***********************************************************************************************
   * \brief   Calculate squared local contribution of post-processed L2 error.
   *
   * \tparam  hyEdgeT       The geometry type / typename of the considered hyEdge's geometry.
   * \tparam  SmallMatT     Data type of \c lambda_values.
   * \param   lambda_values The values of the skeletal variable's coefficients.
   * \param   hy_edge       The geometry of the considered hyperedge (of typename GeomT).
   * \param   time          Time at which anaytic functions are evaluated.
   * \retval  vec_b         Local part of vector b.
   ************************************************************************************************/
  template <typename hyEdgeT, typename SmallMatT>
  std::array<lSol_float_t, 1U> errors(const SmallMatT& lambda_values,
                                      hyEdgeT& hy_edge,
                                      const lSol_float_t time = 0.) const
  {
    hy_assert(lambda_values.size() == 2 * hyEdge_dimT, "Matrix must have appropriate size!");
    for (unsigned int i = 0; i < lambda_values.size(); ++i)
      hy_assert(lambda_values[i].size() == n_glob_dofs_per_node(),
                "Matrix must have appropriate size!");
 
    using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
    constexpr unsigned int n_shape_pp = Hypercube<hyEdge_dimT>::pow(poly_deg + 2);
 
    SmallVec<n_loc_dofs_, lSol_float_t> coefficients =
      solve_local_problem(lambda_values, 1U, hy_edge, time);
 
    SmallSquareMat<n_shape_pp, lSol_float_t> matrix;
    for (unsigned int i = 0; i < n_shape_pp; ++i)
      for (unsigned int j = 0; j < n_shape_pp; ++j)
        matrix(i, j) = integrator_pp::template integrate_vol_nablaphinablaphifunc<
          Point<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>,
          decltype(hyEdgeT::geometry), parameters::diffusion_coeff,
          Point<hyEdge_dimT, lSol_float_t> >(i, j, hy_edge.geometry, time);
    matrix(0, 0) = integrator_pp::integrate_vol_phiphi(0, 0, hy_edge.geometry);
 
    SmallVec<n_shape_pp, lSol_float_t> vector, sol;
    SmallVec<hyEdge_dimT, lSol_float_t> grad_int_vec;
 
    for (unsigned int i = 0; i < n_shape_pp; ++i)
      for (unsigned int j = 0; j < n_shape_fct_; ++j)
      {
        grad_int_vec = integrator_pp::template integrate_vol_nablaphiphi<
          Point<hyEdge_dimT, lSol_float_t>, decltype(hyEdgeT::geometry), poly_deg + 1, poly_deg>(
          i, j, hy_edge.geometry);
        for (unsigned int dim = 0; dim < hyEdge_dimT; ++dim)
          vector[i] -= coefficients[dim * n_shape_fct_ + j] * grad_int_vec[dim];
      }
    vector[0] = coefficients[hyEdge_dimT * n_shape_fct_] *
                integrator_pp::integrate_vol_phiphi(0, 0, hy_edge.geometry);
 
    sol = vector / matrix;
 
    return std::array<lSol_float_t, 1U>({integrator_pp::template integrate_vol_diffsquare_discana<
      Point<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>, decltype(hyEdgeT::geometry),
      parameters::analytic_result, Point<hyEdge_dimT, lSol_float_t> >(sol.data(), hy_edge.geometry,
                                                                      time)});
  }
  /*!***********************************************************************************************
   * \brief   Evaluate non-post-processed local reconstruction at tensorial products of abscissas.
   *
   * \tparam  absc_float_t      Floating type for the abscissa values.
   * \tparam  abscissas_sizeT   Size of the array of array of abscissas.
   * \tparam  input_array_t     Type of input array.
   * \tparam  hyEdgeT           The geometry type / typename of the considered hyEdge's geometry.
   * \param   abscissas         Abscissas of the supporting points.
   * \param   lambda_values     The values of the skeletal variable's coefficients.
   * \param   hyper_edge        The geometry of the considered hyperedge (of typename GeomT).
   * \param   time              Time at which analytic functions are evaluated.
   * \retval  function_values   Function values at quadrature points.
   ************************************************************************************************/
  template <typename abscissa_float_t,
            std::size_t abscissas_sizeT,
            class input_array_t,
            class hyEdgeT>
  std::array<std::array<lSol_float_t, Hypercube<hyEdge_dimT>::pow(abscissas_sizeT)>, system_dim>
  bulk_values(const std::array<abscissa_float_t, abscissas_sizeT>& abscissas,
              const input_array_t& lambda_values,
              hyEdgeT& hyper_edge,
              const lSol_float_t time = 0.) const;
};  // end of class DiffusionPostprocess
 
// -------------------------------------------------------------------------------------------------
// -------------------------------------------------------------------------------------------------
 
// -------------------------------------------------------------------------------------------------
// -------------------------------------------------------------------------------------------------
//
// IMPLEMENTATION OF MEMBER FUNCTIONS OF DiffusionPostprocess
//
// -------------------------------------------------------------------------------------------------
// -------------------------------------------------------------------------------------------------
 
// -------------------------------------------------------------------------------------------------
// assemble_loc_matrix
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT>
inline SmallSquareMat<
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_loc_dofs_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::
  assemble_loc_matrix(hyEdgeT& hyper_edge, const lSol_float_t time) const
{
  using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
  SmallSquareMat<n_loc_dofs_, lSol_float_t> local_mat;
  lSol_float_t vol_integral, face_integral, helper;
  SmallVec<hyEdge_dimT, lSol_float_t> grad_int_vec, normal_int_vec;
 
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
  {
    for (unsigned int j = 0; j < n_shape_fct_; ++j)
    {
      // Integral_element phi_i phi_j dx in diagonal blocks
      vol_integral = integrator::template integrate_vol_phiphifunc<
        Point<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>, decltype(hyEdgeT::geometry),
        parameters::inverse_diffusion_coeff, Point<hyEdge_dimT, lSol_float_t> >(
        i, j, hyper_edge.geometry, time);
      // Integral_element - nabla phi_i \vec phi_j dx
      // = Integral_element - div \vec phi_i phi_j dx in right upper and left lower blocks
      grad_int_vec =
        integrator::template integrate_vol_nablaphiphi<SmallVec<hyEdge_dimT, lSol_float_t>,
                                                       decltype(hyEdgeT::geometry)>(
          i, j, hyper_edge.geometry);
 
      face_integral = 0.;
      normal_int_vec = 0.;
      for (unsigned int face = 0; face < 2 * hyEdge_dimT; ++face)
      {
        helper = integrator::template integrate_bdr_phiphi<decltype(hyEdgeT::geometry)>(
          i, j, face, hyper_edge.geometry);
        face_integral += helper;
        normal_int_vec += helper * hyper_edge.geometry.local_normal(face);
      }
 
      local_mat(hyEdge_dimT * n_shape_fct_ + i, hyEdge_dimT * n_shape_fct_ + j) +=
        tau_ * face_integral;
      for (unsigned int dim = 0; dim < hyEdge_dimT; ++dim)
      {
        local_mat(dim * n_shape_fct_ + i, dim * n_shape_fct_ + j) += vol_integral;
        local_mat(hyEdge_dimT * n_shape_fct_ + i, dim * n_shape_fct_ + j) -= grad_int_vec[dim];
        local_mat(dim * n_shape_fct_ + i, hyEdge_dimT * n_shape_fct_ + j) -= grad_int_vec[dim];
        local_mat(hyEdge_dimT * n_shape_fct_ + i, dim * n_shape_fct_ + j) += normal_int_vec[dim];
      }
    }
  }
 
  return local_mat;
}  // end of DiffusionPostprocess::assemble_loc_matrix
 
// -------------------------------------------------------------------------------------------------
// assemble_rhs_from_lambda
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT, typename SmallMatT>
inline SmallVec<
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_loc_dofs_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::
  assemble_rhs_from_lambda(const SmallMatT& lambda_values, hyEdgeT& hyper_edge) const
{
  static_assert(std::is_same<typename SmallMatT::value_type::value_type, lSol_float_t>::value,
                "Lambda values should have same floating point arithmetics as local solver!");
  hy_assert(lambda_values.size() == 2 * hyEdge_dimT,
            "The size of the lambda values should be twice the dimension of a hyperedge.");
  for (unsigned int i = 0; i < 2 * hyEdge_dimT; ++i)
    hy_assert(lambda_values[i].size() == n_shape_bdr_,
              "The size of lambda should be the amount of ansatz functions at boundary.");
 
  SmallVec<n_loc_dofs_, lSol_float_t> right_hand_side;
  lSol_float_t integral;
 
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
    for (unsigned int j = 0; j < n_shape_bdr_; ++j)
      for (unsigned int face = 0; face < 2 * hyEdge_dimT; ++face)
      {
        integral = integrator::template integrate_bdr_phipsi<decltype(hyEdgeT::geometry)>(
          i, j, face, hyper_edge.geometry);
        right_hand_side[hyEdge_dimT * n_shape_fct_ + i] += tau_ * lambda_values[face][j] * integral;
        for (unsigned int dim = 0; dim < hyEdge_dimT; ++dim)
          right_hand_side[dim * n_shape_fct_ + i] -=
            hyper_edge.geometry.local_normal(face).operator[](dim) * lambda_values[face][j] *
            integral;
      }
 
  return right_hand_side;
}  // end of DiffusionPostprocess::assemble_rhs_from_lambda
 
// -------------------------------------------------------------------------------------------------
// assemble_rhs_from_global_rhs
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT>
inline SmallVec<
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_loc_dofs_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::
  assemble_rhs_from_global_rhs(hyEdgeT& hyper_edge, const lSol_float_t time) const
{
  using parameters = parametersT<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>;
  SmallVec<n_loc_dofs_, lSol_float_t> right_hand_side;
  lSol_float_t integral;
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
  {
    right_hand_side[hyEdge_dimT * n_shape_fct_ + i] = integrator::template integrate_vol_phifunc<
      Point<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>, decltype(hyEdgeT::geometry),
      parameters::right_hand_side, Point<hyEdge_dimT, lSol_float_t> >(i, hyper_edge.geometry, time);
    for (unsigned int face = 0; face < 2 * hyEdge_dimT; ++face)
    {
      if (!is_dirichlet<parameters>(hyper_edge.node_descriptor[face]))
        continue;
      integral = integrator::template integrate_bdr_phifunc<
        Point<decltype(hyEdgeT::geometry)::space_dim(), lSol_float_t>, decltype(hyEdgeT::geometry),
        parameters::dirichlet_value, Point<hyEdge_dimT, lSol_float_t> >(i, face,
                                                                        hyper_edge.geometry, time);
      right_hand_side[hyEdge_dimT * n_shape_fct_ + i] += tau_ * integral;
      for (unsigned int dim = 0; dim < hyEdge_dimT; ++dim)
        right_hand_side[dim * n_shape_fct_ + i] -=
          hyper_edge.geometry.local_normal(face).operator[](dim) * integral;
    }
  }
 
  return right_hand_side;
}  // end of DiffusionPostprocess::assemble_rhs_from_global_rhs
 
// -------------------------------------------------------------------------------------------------
// assemble_rhs_from_coeffs
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT, typename SmallVecT>
inline SmallVec<
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_loc_dofs_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::
  assemble_rhs_from_coeffs(const SmallVecT& coeffs, hyEdgeT& hyper_edge) const
{
  SmallVec<n_loc_dofs_, lSol_float_t> right_hand_side;
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
    for (unsigned int j = 0; j < n_shape_fct_; ++j)
      right_hand_side[hyEdge_dimT * n_shape_fct_ + i] +=
        coeffs[hyEdge_dimT * n_shape_fct_ + j] *
        integrator::template integrate_vol_phiphi(i, j, hyper_edge.geometry);
 
  return right_hand_side;
}  // end of DiffusionPostprocess::assemble_rhs_from_coeffs
 
// -------------------------------------------------------------------------------------------------
// primal_at_boundary
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT>
inline SmallMat<
  2 * hyEdge_dimT,
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_shape_bdr_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::
  primal_at_boundary(const SmallVec<n_loc_dofs_, lSol_float_t>& coeffs, hyEdgeT& hyper_edge) const
{
  SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> bdr_values;
 
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
    for (unsigned int j = 0; j < n_shape_bdr_; ++j)
      for (unsigned int face = 0; face < 2 * hyEdge_dimT; ++face)
        bdr_values(face, j) +=
          coeffs[hyEdge_dimT * n_shape_fct_ + i] *
          integrator::template integrate_bdr_phipsi<decltype(hyEdgeT::geometry)>(
            i, j, face, hyper_edge.geometry);
 
  return bdr_values;
}  // end of DiffusionPostprocess::primal_at_boundary
 
// -------------------------------------------------------------------------------------------------
// dual_at_boundary
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename hyEdgeT>
inline SmallMat<
  2 * hyEdge_dimT,
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::n_shape_bdr_,
  lSol_float_t>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::dual_at_boundary(
  const SmallVec<n_loc_dofs_, lSol_float_t>& coeffs,
  hyEdgeT& hyper_edge) const
{
  SmallMat<2 * hyEdge_dimT, n_shape_bdr_, lSol_float_t> bdr_values;
  lSol_float_t integral;
 
  for (unsigned int i = 0; i < n_shape_fct_; ++i)
    for (unsigned int j = 0; j < n_shape_bdr_; ++j)
      for (unsigned int face = 0; face < 2 * hyEdge_dimT; ++face)
      {
        integral = integrator::template integrate_bdr_phipsi<decltype(hyEdgeT::geometry)>(
          i, j, face, hyper_edge.geometry);
        for (unsigned int dim = 0; dim < hyEdge_dimT; ++dim)
          bdr_values(face, j) += hyper_edge.geometry.local_normal(face).operator[](dim) * integral *
                                 coeffs[dim * n_shape_fct_ + i];
      }
 
  return bdr_values;
}  // end of DiffusionPostprocess::dual_at_boundary
 
// -------------------------------------------------------------------------------------------------
// bulk_values
// -------------------------------------------------------------------------------------------------
 
template <unsigned int hyEdge_dimT,
          unsigned int poly_deg,
          unsigned int quad_deg,
          template <unsigned int, typename>
          typename parametersT,
          typename lSol_float_t>
template <typename abscissa_float_t,
          std::size_t abscissas_sizeT,
          class input_array_t,
          typename hyEdgeT>
std::array<
  std::array<lSol_float_t, Hypercube<hyEdge_dimT>::pow(abscissas_sizeT)>,
  DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::system_dim>
DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::bulk_values(
  const std::array<abscissa_float_t, abscissas_sizeT>& abscissas,
  const input_array_t& lambda_values,
  hyEdgeT& hyper_edge,
  const lSol_float_t time) const
{
  SmallVec<n_loc_dofs_, lSol_float_t> coefficients =
    solve_local_problem(lambda_values, 1U, hyper_edge, time);
  SmallVec<n_shape_fct_, lSol_float_t> coeffs;
  SmallVec<static_cast<unsigned int>(abscissas_sizeT), abscissa_float_t> helper(abscissas);
 
  std::array<
    std::array<lSol_float_t, Hypercube<hyEdge_dimT>::pow(abscissas_sizeT)>,
    DiffusionPostprocess<hyEdge_dimT, poly_deg, quad_deg, parametersT, lSol_float_t>::system_dim>
    point_vals;
 
  for (unsigned int d = 0; d < system_dim; ++d)
  {
    for (unsigned int i = 0; i < coeffs.size(); ++i)
      coeffs[i] = coefficients[d * n_shape_fct_ + i];
    for (unsigned int pt = 0; pt < Hypercube<hyEdge_dimT>::pow(abscissas_sizeT); ++pt)
      point_vals[d][pt] = integrator::shape_fun_t::template lin_comb_fct_val<float>(
        coeffs, Hypercube<hyEdge_dimT>::template tensorial_pt<Point<hyEdge_dimT> >(pt, helper));
  }
 
  return point_vals;
}
// end of DiffusionPostprocess::bulk_values
 
// -------------------------------------------------------------------------------------------------
// -------------------------------------------------------------------------------------------------
 
}  // namespace LocalSolver