step-30.h

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 = 0) const override
 *       {
 *         (void)points;
 *         AssertDimension(values.size(), points.size());
 *   
 *         std::fill(values.begin(), values.end(), 0.);
 *       }
 *     };
 *   
 *   
 *     template <int dim>
 *     class BoundaryValues : public Function<dim>
 *     {
 *     public:
 *       virtual void value_list(const std::vector<Point<dim>> &points,
 *                               std::vector<double>           &values,
 *                               const unsigned int /*component*/ = 0) const override
 *       {
 *         AssertDimension(values.size(), points.size());
 *   
 *         for (unsigned int i = 0; i < values.size(); ++i)
 *           {
 *             if (points[i][0] < 0.5)
 *               values[i] = 1.;
 *             else
 *               values[i] = 0.;
 *           }
 *       }
 *     };
 *   
 *   
 *     template <int dim>
 *     class Beta
 *     {
 *     public:
 * @endcode
 * 
 * The flow field is chosen to be a quarter circle with counterclockwise
 * flow direction and with the origin as midpoint for the right half of the
 * domain with positive @f$x@f$ values, whereas the flow simply goes to the left
 * in the left part of the domain at a velocity that matches the one coming
 * in from the right. In the circular part the magnitude of the flow
 * velocity is proportional to the distance from the origin. This is a
 * difference to @ref step_12 "step-12", where the magnitude was 1 everywhere. the new
 * definition leads to a linear variation of @f$\beta@f$ along each given face
 * of a cell. On the other hand, the solution @f$u(x,y)@f$ is exactly the same
 * as before.
 * 
 * @code
 *       void value_list(const std::vector<Point<dim>> &points,
 *                       std::vector<Point<dim>>       &values) const
 *       {
 *         AssertDimension(values.size(), points.size());
 *   
 *         for (unsigned int i = 0; i < points.size(); ++i)
 *           {
 *             if (points[i][0] > 0)
 *               {
 *                 values[i][0] = -points[i][1];
 *                 values[i][1] = points[i][0];
 *               }
 *             else
 *               {
 *                 values[i]    = Point<dim>();
 *                 values[i][0] = -points[i][1];
 *               }
 *           }
 *       }
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-ClassDGTransportEquation"></a> 
 * <h3>Class: DGTransportEquation</h3>
 *   
 
 * 
 * This declaration of this class is utterly unaffected by our current
 * changes.
 * 
 * @code
 *     template <int dim>
 *     class DGTransportEquation
 *     {
 *     public:
 *       DGTransportEquation();
 *   
 *       void assemble_cell_term(const FEValues<dim> &fe_v,
 *                               FullMatrix<double>  &ui_vi_matrix,
 *                               Vector<double>      &cell_vector) const;
 *   
 *       void assemble_boundary_term(const FEFaceValues<dim> &fe_v,
 *                                   FullMatrix<double>      &ui_vi_matrix,
 *                                   Vector<double>          &cell_vector) const;
 *   
 *       void assemble_face_term(const FEFaceValuesBase<dim> &fe_v,
 *                               const FEFaceValuesBase<dim> &fe_v_neighbor,
 *                               FullMatrix<double>          &ui_vi_matrix,
 *                               FullMatrix<double>          &ue_vi_matrix,
 *                               FullMatrix<double>          &ui_ve_matrix,
 *                               FullMatrix<double>          &ue_ve_matrix) const;
 *   
 *     private:
 *       const Beta<dim>           beta_function;
 *       const RHS<dim>            rhs_function;
 *       const BoundaryValues<dim> boundary_function;
 *     };
 *   
 *   
 *   
 * @endcode
 * 
 * Likewise, the constructor of the class as well as the functions
 * assembling the terms corresponding to cell interiors and boundary faces
 * are unchanged from before. The function that assembles face terms between
 * cells also did not change because all it does is operate on two objects
 * of type FEFaceValuesBase (which is the base class of both FEFaceValues
 * and FESubfaceValues). Where these objects come from, i.e. how they are
 * initialized, is of no concern to this function: it simply assumes that
 * the quadrature points on faces or subfaces represented by the two objects
 * correspond to the same points in physical space.
 * 
 * @code
 *     template <int dim>
 *     DGTransportEquation<dim>::DGTransportEquation()
 *       : beta_function()
 *       , rhs_function()
 *       , boundary_function()
 *     {}
 *   
 *   
 *   
 *     template <int dim>
 *     void DGTransportEquation<dim>::assemble_cell_term(
 *       const FEValues<dim> &fe_v,
 *       FullMatrix<double>  &ui_vi_matrix,
 *       Vector<double>      &cell_vector) const
 *     {
 *       const std::vector<double> &JxW = fe_v.get_JxW_values();
 *   
 *       std::vector<Point<dim>> beta(fe_v.n_quadrature_points);
 *       std::vector<double>     rhs(fe_v.n_quadrature_points);
 *   
 *       beta_function.value_list(fe_v.get_quadrature_points(), beta);
 *       rhs_function.value_list(fe_v.get_quadrature_points(), rhs);
 *   
 *       for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
 *         for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *           {
 *             for (unsigned int j = 0; j < fe_v.dofs_per_cell; ++j)
 *               ui_vi_matrix(i, j) -= beta[point] * fe_v.shape_grad(i, point) *
 *                                     fe_v.shape_value(j, point) * JxW[point];
 *   
 *             cell_vector(i) +=
 *               rhs[point] * fe_v.shape_value(i, point) * JxW[point];
 *           }
 *     }
 *   
 *   
 *     template <int dim>
 *     void DGTransportEquation<dim>::assemble_boundary_term(
 *       const FEFaceValues<dim> &fe_v,
 *       FullMatrix<double>      &ui_vi_matrix,
 *       Vector<double>          &cell_vector) const
 *     {
 *       const std::vector<double>         &JxW     = fe_v.get_JxW_values();
 *       const std::vector<Tensor<1, dim>> &normals = fe_v.get_normal_vectors();
 *   
 *       std::vector<Point<dim>> beta(fe_v.n_quadrature_points);
 *       std::vector<double>     g(fe_v.n_quadrature_points);
 *   
 *       beta_function.value_list(fe_v.get_quadrature_points(), beta);
 *       boundary_function.value_list(fe_v.get_quadrature_points(), g);
 *   
 *       for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
 *         {
 *           const double beta_n = beta[point] * normals[point];
 *           if (beta_n > 0)
 *             for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *               for (unsigned int j = 0; j < fe_v.dofs_per_cell; ++j)
 *                 ui_vi_matrix(i, j) += beta_n * fe_v.shape_value(j, point) *
 *                                       fe_v.shape_value(i, point) * JxW[point];
 *           else
 *             for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *               cell_vector(i) -=
 *                 beta_n * g[point] * fe_v.shape_value(i, point) * JxW[point];
 *         }
 *     }
 *   
 *   
 *     template <int dim>
 *     void DGTransportEquation<dim>::assemble_face_term(
 *       const FEFaceValuesBase<dim> &fe_v,
 *       const FEFaceValuesBase<dim> &fe_v_neighbor,
 *       FullMatrix<double>          &ui_vi_matrix,
 *       FullMatrix<double>          &ue_vi_matrix,
 *       FullMatrix<double>          &ui_ve_matrix,
 *       FullMatrix<double>          &ue_ve_matrix) const
 *     {
 *       const std::vector<double>         &JxW     = fe_v.get_JxW_values();
 *       const std::vector<Tensor<1, dim>> &normals = fe_v.get_normal_vectors();
 *   
 *       std::vector<Point<dim>> beta(fe_v.n_quadrature_points);
 *   
 *       beta_function.value_list(fe_v.get_quadrature_points(), beta);
 *   
 *       for (unsigned int point = 0; point < fe_v.n_quadrature_points; ++point)
 *         {
 *           const double beta_n = beta[point] * normals[point];
 *           if (beta_n > 0)
 *             {
 *               for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *                 for (unsigned int j = 0; j < fe_v.dofs_per_cell; ++j)
 *                   ui_vi_matrix(i, j) += beta_n * fe_v.shape_value(j, point) *
 *                                         fe_v.shape_value(i, point) * JxW[point];
 *   
 *               for (unsigned int k = 0; k < fe_v_neighbor.dofs_per_cell; ++k)
 *                 for (unsigned int j = 0; j < fe_v.dofs_per_cell; ++j)
 *                   ui_ve_matrix(k, j) -= beta_n * fe_v.shape_value(j, point) *
 *                                         fe_v_neighbor.shape_value(k, point) *
 *                                         JxW[point];
 *             }
 *           else
 *             {
 *               for (unsigned int i = 0; i < fe_v.dofs_per_cell; ++i)
 *                 for (unsigned int l = 0; l < fe_v_neighbor.dofs_per_cell; ++l)
 *                   ue_vi_matrix(i, l) += beta_n *
 *                                         fe_v_neighbor.shape_value(l, point) *
 *                                         fe_v.shape_value(i, point) * JxW[point];
 *   
 *               for (unsigned int k = 0; k < fe_v_neighbor.dofs_per_cell; ++k)
 *                 for (unsigned int l = 0; l < fe_v_neighbor.dofs_per_cell; ++l)
 *                   ue_ve_matrix(k, l) -=
 *                     beta_n * fe_v_neighbor.shape_value(l, point) *
 *                     fe_v_neighbor.shape_value(k, point) * JxW[point];
 *             }
 *         }
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-ClassDGMethod"></a> 
 * <h3>Class: DGMethod</h3>
 *   
 
 * 
 * This declaration is much like that of @ref step_12 "step-12". However, we introduce a
 * new routine (set_anisotropic_flags) and modify another one (refine_grid).
 * 
 * @code
 *     template <int dim>
 *     class DGMethod
 *     {
 *     public:
 *       DGMethod(const bool anisotropic);
 *   
 *       void run();
 *   
 *     private:
 *       void setup_system();
 *       void assemble_system();
 *       void solve(Vector<double> &solution);
 *       void refine_grid();
 *       void set_anisotropic_flags();
 *       void output_results(const unsigned int cycle) const;
 *   
 *       Triangulation<dim>   triangulation;
 *       const MappingQ1<dim> mapping;
 * @endcode
 * 
 * Again we want to use DG elements of degree 1 (but this is only
 * specified in the constructor). If you want to use a DG method of a
 * different degree replace 1 in the constructor by the new degree.
 * 
 * @code
 *       const unsigned int degree;
 *       const FE_DGQ<dim>  fe;
 *       DoFHandler<dim>    dof_handler;
 *   
 *       SparsityPattern      sparsity_pattern;
 *       SparseMatrix<double> system_matrix;
 * @endcode
 * 
 * This is new, the threshold value used in the evaluation of the
 * anisotropic jump indicator explained in the introduction. Its value is
 * set to 3.0 in the constructor, but it can easily be changed to a
 * different value greater than 1.
 * 
 * @code
 *       const double anisotropic_threshold_ratio;
 * @endcode
 * 
 * This is a bool flag indicating whether anisotropic refinement shall be
 * used or not. It is set by the constructor, which takes an argument of
 * the same name.
 * 
 * @code
 *       const bool anisotropic;
 *   
 *       const QGauss<dim>     quadrature;
 *       const QGauss<dim - 1> face_quadrature;
 *   
 *       Vector<double> solution2;
 *       Vector<double> right_hand_side;
 *   
 *       const DGTransportEquation<dim> dg;
 *     };
 *   
 *   
 *     template <int dim>
 *     DGMethod<dim>::DGMethod(const bool anisotropic)
 *       : mapping()
 * @endcode
 * 
 * Change here for DG methods of different degrees.
 * 
 * @code
 *       , degree(1)
 *       , fe(degree)
 *       , dof_handler(triangulation)
 *       , anisotropic_threshold_ratio(3.)
 *       , anisotropic(anisotropic)
 * @endcode
 * 
 * As beta is a linear function, we can choose the degree of the
 * quadrature for which the resulting integration is correct. Thus, we
 * choose to use <code>degree+1</code> Gauss points, which enables us to
 * integrate exactly polynomials of degree <code>2*degree+1</code>, enough
 * for all the integrals we will perform in this program.
 * 
 * @code
 *       , quadrature(degree + 1)
 *       , face_quadrature(degree + 1)
 *       , dg()
 *     {}
 *   
 *   
 *   
 *     template <int dim>
 *     void DGMethod<dim>::setup_system()
 *     {
 *       dof_handler.distribute_dofs(fe);
 *       sparsity_pattern.reinit(dof_handler.n_dofs(),
 *                               dof_handler.n_dofs(),
 *                               (GeometryInfo<dim>::faces_per_cell *
 *                                  GeometryInfo<dim>::max_children_per_face +
 *                                1) *
 *                                 fe.n_dofs_per_cell());
 *   
 *       DoFTools::make_flux_sparsity_pattern(dof_handler, sparsity_pattern);
 *   
 *       sparsity_pattern.compress();
 *   
 *       system_matrix.reinit(sparsity_pattern);
 *   
 *       solution2.reinit(dof_handler.n_dofs());
 *       right_hand_side.reinit(dof_handler.n_dofs());
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-Functionassemble_system"></a> 
 * <h4>Function: assemble_system</h4>
 *   
 
 * 
 * We proceed with the <code>assemble_system</code> function that implements
 * the DG discretization. This function does the same thing as the
 * <code>assemble_system</code> function from @ref step_12 "step-12" (but without
 * MeshWorker).  The four cases considered for the neighbor-relations of a
 * cell are the same as the isotropic case, namely a) cell is at the
 * boundary, b) there are finer neighboring cells, c) the neighbor is
 * neither coarser nor finer and d) the neighbor is coarser.  However, the
 * way in which we decide upon which case we have are modified in the way
 * described in the introduction.
 * 
 * @code
 *     template <int dim>
 *     void DGMethod<dim>::assemble_system()
 *     {
 *       const unsigned int dofs_per_cell = dof_handler.get_fe().n_dofs_per_cell();
 *       std::vector<types::global_dof_index> dofs(dofs_per_cell);
 *       std::vector<types::global_dof_index> dofs_neighbor(dofs_per_cell);
 *   
 *       const UpdateFlags update_flags = update_values | update_gradients |
 *                                        update_quadrature_points |
 *                                        update_JxW_values;
 *   
 *       const UpdateFlags face_update_flags =
 *         update_values | update_quadrature_points | update_JxW_values |
 *         update_normal_vectors;
 *   
 *       const UpdateFlags neighbor_face_update_flags = update_values;
 *   
 *       FEValues<dim>        fe_v(mapping, fe, quadrature, update_flags);
 *       FEFaceValues<dim>    fe_v_face(mapping,
 *                                   fe,
 *                                   face_quadrature,
 *                                   face_update_flags);
 *       FESubfaceValues<dim> fe_v_subface(mapping,
 *                                         fe,
 *                                         face_quadrature,
 *                                         face_update_flags);
 *       FEFaceValues<dim>    fe_v_face_neighbor(mapping,
 *                                            fe,
 *                                            face_quadrature,
 *                                            neighbor_face_update_flags);
 *   
 *   
 *       FullMatrix<double> ui_vi_matrix(dofs_per_cell, dofs_per_cell);
 *       FullMatrix<double> ue_vi_matrix(dofs_per_cell, dofs_per_cell);
 *   
 *       FullMatrix<double> ui_ve_matrix(dofs_per_cell, dofs_per_cell);
 *       FullMatrix<double> ue_ve_matrix(dofs_per_cell, dofs_per_cell);
 *   
 *       Vector<double> cell_vector(dofs_per_cell);
 *   
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 *         {
 *           ui_vi_matrix = 0;
 *           cell_vector  = 0;
 *   
 *           fe_v.reinit(cell);
 *   
 *           dg.assemble_cell_term(fe_v, ui_vi_matrix, cell_vector);
 *   
 *           cell->get_dof_indices(dofs);
 *   
 *           for (const auto face_no : cell->face_indices())
 *             {
 *               const auto face = cell->face(face_no);
 *   
 * @endcode
 * 
 * Case (a): The face is at the boundary.
 * 
 * @code
 *               if (face->at_boundary())
 *                 {
 *                   fe_v_face.reinit(cell, face_no);
 *   
 *                   dg.assemble_boundary_term(fe_v_face, ui_vi_matrix, cell_vector);
 *                 }
 *               else
 *                 {
 *                   Assert(cell->neighbor(face_no).state() == IteratorState::valid,
 *                          ExcInternalError());
 *                   const auto neighbor = cell->neighbor(face_no);
 *   
 * @endcode
 * 
 * Case (b): This is an internal face and the neighbor
 * is refined (which we can test by asking whether the
 * face of the current cell has children). In this
 * case, we will need to integrate over the
 * "sub-faces", i.e., the children of the face of the
 * current cell.
 *                 
 
 * 
 * (There is a slightly confusing corner case: If we
 * are in 1d -- where admittedly the current program
 * and its demonstration of anisotropic refinement is
 * not particularly relevant -- then the faces between
 * cells are always the same: they are just
 * vertices. In other words, in 1d, we do not want to
 * treat faces between cells of different level
 * differently. The condition `face->has_children()`
 * we check here ensures this: in 1d, this function
 * always returns `false`, and consequently in 1d we
 * will not ever go into this `if` branch. But we will
 * have to come back to this corner case below in case
 * (c).)
 * 
 * @code
 *                   if (face->has_children())
 *                     {
 * @endcode
 * 
 * We need to know, which of the neighbors faces points in
 * the direction of our cell. Using the @p
 * neighbor_face_no function we get this information for
 * both coarser and non-coarser neighbors.
 * 
 * @code
 *                       const unsigned int neighbor2 =
 *                         cell->neighbor_face_no(face_no);
 *   
 * @endcode
 * 
 * Now we loop over all subfaces, i.e. the children and
 * possibly grandchildren of the current face.
 * 
 * @code
 *                       for (unsigned int subface_no = 0;
 *                            subface_no < face->n_active_descendants();
 *                            ++subface_no)
 *                         {
 * @endcode
 * 
 * To get the cell behind the current subface we can
 * use the @p neighbor_child_on_subface function. it
 * takes care of all the complicated situations of
 * anisotropic refinement and non-standard faces.
 * 
 * @code
 *                           const auto neighbor_child =
 *                             cell->neighbor_child_on_subface(face_no, subface_no);
 *                           Assert(!neighbor_child->has_children(),
 *                                  ExcInternalError());
 *   
 * @endcode
 * 
 * The remaining part of this case is unchanged.
 * 
 * @code
 *                           ue_vi_matrix = 0;
 *                           ui_ve_matrix = 0;
 *                           ue_ve_matrix = 0;
 *   
 *                           fe_v_subface.reinit(cell, face_no, subface_no);
 *                           fe_v_face_neighbor.reinit(neighbor_child, neighbor2);
 *   
 *                           dg.assemble_face_term(fe_v_subface,
 *                                                 fe_v_face_neighbor,
 *                                                 ui_vi_matrix,
 *                                                 ue_vi_matrix,
 *                                                 ui_ve_matrix,
 *                                                 ue_ve_matrix);
 *   
 *                           neighbor_child->get_dof_indices(dofs_neighbor);
 *   
 *                           for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                             for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *                               {
 *                                 system_matrix.add(dofs[i],
 *                                                   dofs_neighbor[j],
 *                                                   ue_vi_matrix(i, j));
 *                                 system_matrix.add(dofs_neighbor[i],
 *                                                   dofs[j],
 *                                                   ui_ve_matrix(i, j));
 *                                 system_matrix.add(dofs_neighbor[i],
 *                                                   dofs_neighbor[j],
 *                                                   ue_ve_matrix(i, j));
 *                               }
 *                         }
 *                     }
 *                   else
 *                     {
 * @endcode
 * 
 * Case (c). We get here if this is an internal
 * face and if the neighbor is not further refined
 * (or, as mentioned above, we are in 1d in which
 * case we get here for every internal face). We
 * then need to decide whether we want to
 * integrate over the current face. If the
 * neighbor is in fact coarser, then we ignore the
 * face and instead handle it when we visit the
 * neighboring cell and look at the current face
 * (except in 1d, where as mentioned above this is
 * not happening):
 * 
 * @code
 *                       if (dim > 1 && cell->neighbor_is_coarser(face_no))
 *                         continue;
 *   
 * @endcode
 * 
 * On the other hand, if the neighbor is more
 * refined, then we have already handled the face
 * in case (b) above (except in 1d). So for 2d and
 * 3d, we just have to decide whether we want to
 * handle a face between cells at the same level
 * from the current side or from the neighboring
 * side.  We do this by introducing a tie-breaker:
 * We'll just take the cell with the smaller index
 * (within the current refinement level). In 1d,
 * we take either the coarser cell, or if they are
 * on the same level, the one with the smaller
 * index within that level. This leads to a
 * complicated condition that, hopefully, makes
 * sense given the description above:
 * 
 * @code
 *                       if (((dim > 1) && (cell->index() < neighbor->index())) ||
 *                           ((dim == 1) && ((cell->level() < neighbor->level()) ||
 *                                           ((cell->level() == neighbor->level()) &&
 *                                            (cell->index() < neighbor->index())))))
 *                         {
 * @endcode
 * 
 * Here we know, that the neighbor is not coarser so we
 * can use the usual @p neighbor_of_neighbor
 * function. However, we could also use the more
 * general @p neighbor_face_no function.
 * 
 * @code
 *                           const unsigned int neighbor2 =
 *                             cell->neighbor_of_neighbor(face_no);
 *   
 *                           ue_vi_matrix = 0;
 *                           ui_ve_matrix = 0;
 *                           ue_ve_matrix = 0;
 *   
 *                           fe_v_face.reinit(cell, face_no);
 *                           fe_v_face_neighbor.reinit(neighbor, neighbor2);
 *   
 *                           dg.assemble_face_term(fe_v_face,
 *                                                 fe_v_face_neighbor,
 *                                                 ui_vi_matrix,
 *                                                 ue_vi_matrix,
 *                                                 ui_ve_matrix,
 *                                                 ue_ve_matrix);
 *   
 *                           neighbor->get_dof_indices(dofs_neighbor);
 *   
 *                           for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                             for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *                               {
 *                                 system_matrix.add(dofs[i],
 *                                                   dofs_neighbor[j],
 *                                                   ue_vi_matrix(i, j));
 *                                 system_matrix.add(dofs_neighbor[i],
 *                                                   dofs[j],
 *                                                   ui_ve_matrix(i, j));
 *                                 system_matrix.add(dofs_neighbor[i],
 *                                                   dofs_neighbor[j],
 *                                                   ue_ve_matrix(i, j));
 *                               }
 *                         }
 *   
 * @endcode
 * 
 * We do not need to consider a case (d), as those
 * faces are treated 'from the other side within
 * case (b).
 * 
 * @code
 *                     }
 *                 }
 *             }
 *   
 *           for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *             for (unsigned int j = 0; j < dofs_per_cell; ++j)
 *               system_matrix.add(dofs[i], dofs[j], ui_vi_matrix(i, j));
 *   
 *           for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *             right_hand_side(dofs[i]) += cell_vector(i);
 *         }
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-Solver"></a> 
 * <h3>Solver</h3>
 *   
 
 * 
 * For this simple problem we use the simple Richardson iteration again. The
 * solver is completely unaffected by our anisotropic changes.
 * 
 * @code
 *     template <int dim>
 *     void DGMethod<dim>::solve(Vector<double> &solution)
 *     {
 *       SolverControl                    solver_control(1000, 1e-12, false, false);
 *       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);
 *     }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-Refinement"></a> 
 * <h3>Refinement</h3>
 *   
 
 * 
 * We refine the grid according to the same simple refinement criterion used
 * in @ref step_12 "step-12", namely an approximation to the gradient of the solution.
 * 
 * @code
 *     template <int dim>
 *     void DGMethod<dim>::refine_grid()
 *     {
 *       Vector<float> gradient_indicator(triangulation.n_active_cells());
 *   
 * @endcode
 * 
 * We approximate the gradient,
 * 
 * @code
 *       DerivativeApproximation::approximate_gradient(mapping,
 *                                                     dof_handler,
 *                                                     solution2,
 *                                                     gradient_indicator);
 *   
 * @endcode
 * 
 * and scale it to obtain an error indicator.
 * 
 * @code
 *       for (const auto &cell : triangulation.active_cell_iterators())
 *         gradient_indicator[cell->active_cell_index()] *=
 *           std::pow(cell->diameter(), 1 + 1.0 * dim / 2);
 * @endcode
 * 
 * Then we use this indicator to flag the 30 percent of the cells with
 * highest error indicator to be refined.
 * 
 * @code
 *       GridRefinement::refine_and_coarsen_fixed_number(triangulation,
 *                                                       gradient_indicator,
 *                                                       0.3,
 *                                                       0.1);
 * @endcode
 * 
 * Now the refinement flags are set for those cells with a large error
 * indicator. If nothing is done to change this, those cells will be
 * refined isotropically. If the @p anisotropic flag given to this
 * function is set, we now call the set_anisotropic_flags() function,
 * which uses the jump indicator to reset some of the refinement flags to
 * anisotropic refinement.
 * 
 * @code
 *       if (anisotropic)
 *         set_anisotropic_flags();
 * @endcode
 * 
 * Now execute the refinement considering anisotropic as well as isotropic
 * refinement flags.
 * 
 * @code
 *       triangulation.execute_coarsening_and_refinement();
 *     }
 *   
 * @endcode
 * 
 * Once an error indicator has been evaluated and the cells with largest
 * error are flagged for refinement we want to loop over the flagged cells
 * again to decide whether they need isotropic refinement or whether
 * anisotropic refinement is more appropriate. This is the anisotropic jump
 * indicator explained in the introduction.
 * 
 * @code
 *     template <int dim>
 *     void DGMethod<dim>::set_anisotropic_flags()
 *     {
 * @endcode
 * 
 * We want to evaluate the jump over faces of the flagged cells, so we
 * need some objects to evaluate values of the solution on faces.
 * 
 * @code
 *       UpdateFlags face_update_flags =
 *         UpdateFlags(update_values | update_JxW_values);
 *   
 *       FEFaceValues<dim>    fe_v_face(mapping,
 *                                   fe,
 *                                   face_quadrature,
 *                                   face_update_flags);
 *       FESubfaceValues<dim> fe_v_subface(mapping,
 *                                         fe,
 *                                         face_quadrature,
 *                                         face_update_flags);
 *       FEFaceValues<dim>    fe_v_face_neighbor(mapping,
 *                                            fe,
 *                                            face_quadrature,
 *                                            update_values);
 *   
 * @endcode
 * 
 * Now we need to loop over all active cells.
 * 
 * @code
 *       for (const auto &cell : dof_handler.active_cell_iterators())
 * @endcode
 * 
 * We only need to consider cells which are flagged for refinement.
 * 
 * @code
 *         if (cell->refine_flag_set())
 *           {
 *             std::array<double, dim> jump_in_coordinate_direction;
 *             std::array<double, dim> face_area_in_coordinate_direction;
 *   
 *             for (const auto face_no : cell->face_indices())
 *               {
 *                 const auto face = cell->face(face_no);
 *   
 *                 if (!face->at_boundary())
 *                   {
 *                     Assert(cell->neighbor(face_no).state() ==
 *                              IteratorState::valid,
 *                            ExcInternalError());
 *                     const auto neighbor = cell->neighbor(face_no);
 *   
 *                     std::vector<double> u(fe_v_face.n_quadrature_points);
 *                     std::vector<double> u_neighbor(fe_v_face.n_quadrature_points);
 *   
 * @endcode
 * 
 * The four cases of different neighbor relations seen in
 * the assembly routines are repeated much in the same way
 * here.
 * 
 * @code
 *                     if (face->has_children())
 *                       {
 * @endcode
 * 
 * The neighbor is refined.  First we store the
 * information, which of the neighbor's faces points in
 * the direction of our current cell. This property is
 * inherited to the children.
 * 
 * @code
 *                         unsigned int neighbor2 = cell->neighbor_face_no(face_no);
 * @endcode
 * 
 * Now we loop over all subfaces,
 * 
 * @code
 *                         for (unsigned int subface_no = 0;
 *                              subface_no < face->n_active_descendants();
 *                              ++subface_no)
 *                           {
 * @endcode
 * 
 * get an iterator pointing to the cell behind the
 * present subface...
 * 
 * @code
 *                             const auto neighbor_child =
 *                               cell->neighbor_child_on_subface(face_no,
 *                                                               subface_no);
 *                             Assert(!neighbor_child->has_children(),
 *                                    ExcInternalError());
 * @endcode
 * 
 * ... and reinit the respective FEFaceValues and
 * FESubFaceValues objects.
 * 
 * @code
 *                             fe_v_subface.reinit(cell, face_no, subface_no);
 *                             fe_v_face_neighbor.reinit(neighbor_child, neighbor2);
 * @endcode
 * 
 * We obtain the function values
 * 
 * @code
 *                             fe_v_subface.get_function_values(solution2, u);
 *                             fe_v_face_neighbor.get_function_values(solution2,
 *                                                                    u_neighbor);
 * @endcode
 * 
 * as well as the quadrature weights, multiplied by
 * the Jacobian determinant.
 * 
 * @code
 *                             const std::vector<double> &JxW =
 *                               fe_v_subface.get_JxW_values();
 * @endcode
 * 
 * Now we loop over all quadrature points
 * 
 * @code
 *                             for (unsigned int x = 0;
 *                                  x < fe_v_subface.n_quadrature_points;
 *                                  ++x)
 *                               {
 * @endcode
 * 
 * and integrate the absolute value of the jump
 * of the solution, i.e. the absolute value of
 * the difference between the function value
 * seen from the current cell and the
 * neighboring cell, respectively. We know, that
 * the first two faces are orthogonal to the
 * first coordinate direction on the unit cell,
 * the second two faces are orthogonal to the
 * second coordinate direction and so on, so we
 * accumulate these values into vectors with
 * <code>dim</code> components.
 * 
 * @code
 *                                 jump_in_coordinate_direction[face_no / 2] +=
 *                                   std::abs(u[x] - u_neighbor[x]) * JxW[x];
 * @endcode
 * 
 * We also sum up the scaled weights to obtain
 * the measure of the face.
 * 
 * @code
 *                                 face_area_in_coordinate_direction[face_no / 2] +=
 *                                   JxW[x];
 *                               }
 *                           }
 *                       }
 *                     else
 *                       {
 *                         if (!cell->neighbor_is_coarser(face_no))
 *                           {
 * @endcode
 * 
 * Our current cell and the neighbor have the same
 * refinement along the face under
 * consideration. Apart from that, we do much the
 * same as with one of the subcells in the above
 * case.
 * 
 * @code
 *                             unsigned int neighbor2 =
 *                               cell->neighbor_of_neighbor(face_no);
 *   
 *                             fe_v_face.reinit(cell, face_no);
 *                             fe_v_face_neighbor.reinit(neighbor, neighbor2);
 *   
 *                             fe_v_face.get_function_values(solution2, u);
 *                             fe_v_face_neighbor.get_function_values(solution2,
 *                                                                    u_neighbor);
 *   
 *                             const std::vector<double> &JxW =
 *                               fe_v_face.get_JxW_values();
 *   
 *                             for (unsigned int x = 0;
 *                                  x < fe_v_face.n_quadrature_points;
 *                                  ++x)
 *                               {
 *                                 jump_in_coordinate_direction[face_no / 2] +=
 *                                   std::abs(u[x] - u_neighbor[x]) * JxW[x];
 *                                 face_area_in_coordinate_direction[face_no / 2] +=
 *                                   JxW[x];
 *                               }
 *                           }
 *                         else // i.e. neighbor is coarser than cell
 *                           {
 * @endcode
 * 
 * Now the neighbor is actually coarser. This case
 * is new, in that it did not occur in the assembly
 * routine. Here, we have to consider it, but this
 * is not overly complicated. We simply use the @p
 * neighbor_of_coarser_neighbor function, which
 * again takes care of anisotropic refinement and
 * non-standard face orientation by itself.
 * 
 * @code
 *                             std::pair<unsigned int, unsigned int>
 *                               neighbor_face_subface =
 *                                 cell->neighbor_of_coarser_neighbor(face_no);
 *                             Assert(neighbor_face_subface.first < cell->n_faces(),
 *                                    ExcInternalError());
 *                             Assert(neighbor_face_subface.second <
 *                                      neighbor->face(neighbor_face_subface.first)
 *                                        ->n_active_descendants(),
 *                                    ExcInternalError());
 *                             Assert(neighbor->neighbor_child_on_subface(
 *                                      neighbor_face_subface.first,
 *                                      neighbor_face_subface.second) == cell,
 *                                    ExcInternalError());
 *   
 *                             fe_v_face.reinit(cell, face_no);
 *                             fe_v_subface.reinit(neighbor,
 *                                                 neighbor_face_subface.first,
 *                                                 neighbor_face_subface.second);
 *   
 *                             fe_v_face.get_function_values(solution2, u);
 *                             fe_v_subface.get_function_values(solution2,
 *                                                              u_neighbor);
 *   
 *                             const std::vector<double> &JxW =
 *                               fe_v_face.get_JxW_values();
 *   
 *                             for (unsigned int x = 0;
 *                                  x < fe_v_face.n_quadrature_points;
 *                                  ++x)
 *                               {
 *                                 jump_in_coordinate_direction[face_no / 2] +=
 *                                   std::abs(u[x] - u_neighbor[x]) * JxW[x];
 *                                 face_area_in_coordinate_direction[face_no / 2] +=
 *                                   JxW[x];
 *                               }
 *                           }
 *                       }
 *                   }
 *               }
 * @endcode
 * 
 * Now we analyze the size of the mean jumps, which we get dividing
 * the jumps by the measure of the respective faces.
 * 
 * @code
 *             std::array<double, dim> average_jumps;
 *             double                  sum_of_average_jumps = 0.;
 *             for (unsigned int i = 0; i < dim; ++i)
 *               {
 *                 average_jumps[i] = jump_in_coordinate_direction[i] /
 *                                    face_area_in_coordinate_direction[i];
 *                 sum_of_average_jumps += average_jumps[i];
 *               }
 *   
 * @endcode
 * 
 * Now we loop over the <code>dim</code> coordinate directions of
 * the unit cell and compare the average jump over the faces
 * orthogonal to that direction with the average jumps over faces
 * orthogonal to the remaining direction(s). If the first is larger
 * than the latter by a given factor, we refine only along hat
 * axis. Otherwise we leave the refinement flag unchanged, resulting
 * in isotropic refinement.
 * 
 * @code
 *             for (unsigned int i = 0; i < dim; ++i)
 *               if (average_jumps[i] > anisotropic_threshold_ratio *
 *                                        (sum_of_average_jumps - average_jumps[i]))
 *                 cell->set_refine_flag(RefinementCase<dim>::cut_axis(i));
 *           }
 *     }
 *   
 * @endcode
 * 
 * 
 * <a name="step_30-TheRest"></a> 
 * <h3>The Rest</h3>
 *   
 
 * 
 * The remaining part of the program very much follows the scheme of
 * previous tutorial programs. We output the mesh in VTU format (just
 * as we did in @ref step_1 "step-1", for example), and the visualization output
 * in VTU format as we almost always do.
 * 
 * @code
 *     template <int dim>
 *     void DGMethod<dim>::output_results(const unsigned int cycle) const
 *     {
 *       std::string refine_type;
 *       if (anisotropic)
 *         refine_type = ".aniso";
 *       else
 *         refine_type = ".iso";
 *   
 *       {
 *         const std::string filename =
 *           "grid-" + std::to_string(cycle) + refine_type + ".svg";
 *         std::cout << "   Writing grid to <" << filename << ">..." << std::endl;
 *         std::ofstream svg_output(filename);
 *   
 *         GridOut grid_out;
 *         grid_out.write_svg(triangulation, svg_output);
 *       }
 *   
 *       {
 *         const std::string filename =
 *           "sol-" + std::to_string(cycle) + refine_type + ".vtu";
 *         std::cout << "   Writing solution to <" << filename << ">..."
 *                   << std::endl;
 *         std::ofstream gnuplot_output(filename);
 *   
 *         DataOut<dim> data_out;
 *         data_out.attach_dof_handler(dof_handler);
 *         data_out.add_data_vector(solution2, "u");
 *   
 *         data_out.build_patches(degree);
 *   
 *         data_out.write_vtu(gnuplot_output);
 *       }
 *     }
 *   
 *   
 *   
 *     template <int dim>
 *     void DGMethod<dim>::run()
 *     {
 *       for (unsigned int cycle = 0; cycle < 6; ++cycle)
 *         {
 *           std::cout << "Cycle " << cycle << ':' << std::endl;
 *   
 *           if (cycle == 0)
 *             {
 * @endcode
 * 
 * Create the rectangular domain.
 * 
 * @code
 *               Point<dim> p1, p2;
 *               p1[0] = -1.;
 *               for (unsigned int i = 0; i < dim; ++i)
 *                 p2[i] = 1.;
 * @endcode
 * 
 * Adjust the number of cells in different directions to obtain
 * completely isotropic cells for the original mesh.
 * 
 * @code
 *               std::vector<unsigned int> repetitions(dim, 1);
 *               repetitions[0] = 2;
 *               GridGenerator::subdivided_hyper_rectangle(triangulation,
 *                                                         repetitions,
 *                                                         p1,
 *                                                         p2);
 *   
 *               triangulation.refine_global(5 - dim);
 *             }
 *           else
 *             refine_grid();
 *   
 *   
 *           std::cout << "   Number of active cells:       "
 *                     << triangulation.n_active_cells() << std::endl;
 *   
 *           setup_system();
 *   
 *           std::cout << "   Number of degrees of freedom: " << dof_handler.n_dofs()
 *                     << std::endl;
 *   
 *           Timer assemble_timer;
 *           assemble_system();
 *           std::cout << "   Time of assemble_system: " << assemble_timer.cpu_time()
 *                     << std::endl;
 *           solve(solution2);
 *   
 *           output_results(cycle);
 *   
 *           std::cout << std::endl;
 *         }
 *     }
 *   } // namespace Step30
 *   
 *   
 *   
 *   int main()
 *   {
 *     try
 *       {
 *         using namespace Step30;
 *   
 * @endcode
 * 
 * If you want to run the program in 3d, simply change the following
 * line to <code>const unsigned int dim = 3;</code>.
 * 
 * @code
 *         const unsigned int dim = 2;
 *   
 *         {
 * @endcode
 * 
 * First, we perform a run with isotropic refinement.
 * 
 * @code
 *           std::cout << "Performing a " << dim
 *                     << "D run with isotropic refinement..." << std::endl
 *                     << "------------------------------------------------"
 *                     << std::endl;
 *           DGMethod<dim> dgmethod_iso(false);
 *           dgmethod_iso.run();
 *         }
 *   
 *         {
 * @endcode
 * 
 * Now we do a second run, this time with anisotropic refinement.
 * 
 * @code
 *           std::cout << std::endl
 *                     << "Performing a " << dim
 *                     << "D run with anisotropic refinement..." << std::endl
 *                     << "--------------------------------------------------"
 *                     << std::endl;
 *           DGMethod<dim> dgmethod_aniso(true);
 *           dgmethod_aniso.run();
 *         }
 *       }
 *     catch (std::exception &exc)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Exception on processing: " << std::endl
 *                   << exc.what() << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       }
 *     catch (...)
 *       {
 *         std::cerr << std::endl
 *                   << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         std::cerr << "Unknown exception!" << std::endl
 *                   << "Aborting!" << std::endl
 *                   << "----------------------------------------------------"
 *                   << std::endl;
 *         return 1;
 *       };
 *   
 *     return 0;
 *   }
 * @endcode
<a name="step_30-Results"></a><h1>Results</h1>
 
 
 
The output of this program consist of the console output, the SVG
files containing the grids, and the solutions given in VTU format.
@code
Performing a 2D run with isotropic refinement...
------------------------------------------------
Cycle 0:
   Number of active cells:       128
   Number of degrees of freedom: 512
   Time of assemble_system: 0.092049
   Writing grid to <grid-0.iso.svg>...
   Writing solution to <sol-0.iso.vtu>...
 
Cycle 1:
   Number of active cells:       239
   Number of degrees of freedom: 956
   Time of assemble_system: 0.109519
   Writing grid to <grid-1.iso.svg>...
   Writing solution to <sol-1.iso.vtu>...
 
Cycle 2:
   Number of active cells:       491
   Number of degrees of freedom: 1964
   Time of assemble_system: 0.08303
   Writing grid to <grid-2.iso.svg>...
   Writing solution to <sol-2.iso.vtu>...
 
Cycle 3:
   Number of active cells:       1031
   Number of degrees of freedom: 4124
   Time of assemble_system: 0.278987
   Writing grid to <grid-3.iso.svg>...
   Writing solution to <sol-3.iso.vtu>...
 
Cycle 4:
   Number of active cells:       2027
   Number of degrees of freedom: 8108
   Time of assemble_system: 0.305869
   Writing grid to <grid-4.iso.svg>...
   Writing solution to <sol-4.iso.vtu>...
 
Cycle 5:
   Number of active cells:       4019
   Number of degrees of freedom: 16076
   Time of assemble_system: 0.47616
   Writing grid to <grid-5.iso.svg>...
   Writing solution to <sol-5.iso.vtu>...
 
 
Performing a 2D run with anisotropic refinement...
--------------------------------------------------
Cycle 0:
   Number of active cells:       128
   Number of degrees of freedom: 512
   Time of assemble_system: 0.052866
   Writing grid to <grid-0.aniso.svg>...
   Writing solution to <sol-0.aniso.vtu>...
 
Cycle 1:
   Number of active cells:       171
   Number of degrees of freedom: 684
   Time of assemble_system: 0.050917
   Writing grid to <grid-1.aniso.svg>...
   Writing solution to <sol-1.aniso.vtu>...
 
Cycle 2:
   Number of active cells:       255
   Number of degrees of freedom: 1020
   Time of assemble_system: 0.064132
   Writing grid to <grid-2.aniso.svg>...
   Writing solution to <sol-2.aniso.vtu>...
 
Cycle 3:
   Number of active cells:       394
   Number of degrees of freedom: 1576
   Time of assemble_system: 0.119849
   Writing grid to <grid-3.aniso.svg>...
   Writing solution to <sol-3.aniso.vtu>...
 
Cycle 4:
   Number of active cells:       648
   Number of degrees of freedom: 2592
   Time of assemble_system: 0.218244
   Writing grid to <grid-4.aniso.svg>...
   Writing solution to <sol-4.aniso.vtu>...
 
Cycle 5:
   Number of active cells:       1030
   Number of degrees of freedom: 4120
   Time of assemble_system: 0.128121
   Writing grid to <grid-5.aniso.svg>...
   Writing solution to <sol-5.aniso.vtu>...
@endcode
 
This text output shows the reduction in the number of cells which results from
the successive application of anisotropic refinement. After the last refinement
step the savings have accumulated so much that almost four times as many cells
and thus degrees of freedom are needed in the isotropic case. The time needed for assembly
scales with a similar factor.
 
The first interesting part is of course to see how the meshes look like.
On the left are the isotropically refined ones, on the right the
anisotropic ones (colors indicate the refinement level of cells):
 
<table width="80%" align="center">
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-0.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-0.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-1.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-1.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-2.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-2.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-3.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-3.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-4.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-4.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-5.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.grid-5.aniso.9.2.png" alt="">
    </td>
  </tr>
</table>
 
 
The other interesting thing is, of course, to see the solution on these
two sequences of meshes. Here they are, on the refinement cycles 1 and 4,
clearly showing that the solution is indeed composed of <i>discontinuous</i> piecewise
polynomials:
 
<table width="60%" align="center">
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.sol-1.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.sol-1.aniso.9.2.png" alt="">
    </td>
  </tr>
  <tr>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.sol-4.iso.9.2.png" alt="">
    </td>
    <td align="center">
      <img src="https://www.dealii.org/images/steps/developer/step-30.sol-4.aniso.9.2.png" alt="">
    </td>
  </tr>
</table>
 
We see, that the solution on the anisotropically refined mesh is very similar to
the solution obtained on the isotropically refined mesh. Thus the anisotropic
indicator seems to effectively select the appropriate cells for anisotropic
refinement.
 
The pictures also explain why the mesh is refined as it is.
In the whole left part of the domain refinement is only performed along the
@f$y@f$-axis of cells. In the right part of the domain the refinement is dominated by
isotropic refinement, as the anisotropic feature of the solution - the jump from
one to zero - is not well aligned with the mesh where the advection direction
takes a turn. However, at the bottom and closest (to the observer) parts of the
quarter circle this jumps again becomes more and more aligned
with the mesh and the refinement algorithm reacts by creating anisotropic cells
of increasing aspect ratio.
 
It might seem that the necessary alignment of anisotropic features and the
coarse mesh can decrease performance significantly for real world
problems. That is not wrong in general: If one were, for example, to apply
anisotropic refinement to problems in which shocks appear (e.g., the
equations solved in @ref step_69 "step-69"), then it many cases the shock is not aligned
with the mesh and anisotropic refinement will help little unless one also
introduces techniques to move the mesh in alignment with the shocks.
On the other hand, many steep features of solutions are due to boundary layers.
In those cases, the mesh is already aligned with the anisotropic features
because it is of course aligned with the boundary itself, and anisotropic
refinement will almost always increase the efficiency of computations on
adapted grids for these cases.
 *
 *
<a name="step_30-PlainProg"></a>
<h1> The plain program</h1>
@include "step-30.cc"
*/