Reference documentation for deal.II version 9.6.0
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step-61.h
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1
732,
733 *   const unsigned int /*component*/) const
734 *   {
735 *   return 0;
736 *   }
737 *  
738 *  
739 *  
740 *   template <int dim>
741 *   class RightHandSide : public Function<dim>
742 *   {
743 *   public:
744 *   virtual double value(const Point<dim> &p,
745 *   const unsigned int component = 0) const override;
746 *   };
747 *  
748 *  
749 *  
750 *   template <int dim>
751 *   double RightHandSide<dim>::value(const Point<dim> &p,
752 *   const unsigned int /*component*/) const
753 *   {
754 *   return (2 * numbers::PI * numbers::PI * std::sin(numbers::PI * p[0]) *
755 *   std::sin(numbers::PI * p[1]));
756 *   }
757 *  
758 *  
759 *  
760 * @endcode
761 *
762 * The class that implements the exact pressure solution has an
763 * oddity in that we implement it as a vector-valued one with two
764 * components. (We say that it has two components in the constructor
765 * where we call the constructor of the base Function class.) In the
766 * `value()` function, we do not test for the value of the
767 * `component` argument, which implies that we return the same value
768 * for both components of the vector-valued function. We do this
769 * because we describe the finite element in use in this program as
770 * a vector-valued system that contains the interior and the
771 * interface pressures, and when we compute errors, we will want to
772 * use the same pressure solution to test both of these components.
773 *
774 * @code
775 *   template <int dim>
776 *   class ExactPressure : public Function<dim>
777 *   {
778 *   public:
779 *   ExactPressure()
780 *   : Function<dim>(2)
781 *   {}
782 *  
783 *   virtual double value(const Point<dim> &p,
784 *   const unsigned int component) const override;
785 *   };
786 *  
787 *  
788 *  
789 *   template <int dim>
790 *   double ExactPressure<dim>::value(const Point<dim> &p,
791 *   const unsigned int /*component*/) const
792 *   {
793 *   return std::sin(numbers::PI * p[0]) * std::sin(numbers::PI * p[1]);
794 *   }
795 *  
796 *  
797 *  
798 *   template <int dim>
799 *   class ExactVelocity : public TensorFunction<1, dim>
800 *   {
801 *   public:
802 *   ExactVelocity()
804 *   {}
805 *  
806 *   virtual Tensor<1, dim> value(const Point<dim> &p) const override;
807 *   };
808 *  
809 *  
810 *  
811 *   template <int dim>
812 *   Tensor<1, dim> ExactVelocity<dim>::value(const Point<dim> &p) const
813 *   {
814 *   Tensor<1, dim> return_value;
815 *   return_value[0] = -numbers::PI * std::cos(numbers::PI * p[0]) *
816 *   std::sin(numbers::PI * p[1]);
817 *   return_value[1] = -numbers::PI * std::sin(numbers::PI * p[0]) *
818 *   std::cos(numbers::PI * p[1]);
819 *   return return_value;
820 *   }
821 *  
822 *  
823 *  
824 * @endcode
825 *
826 *
827 * <a name="step_61-WGDarcyEquationclassimplementation"></a>
828 * <h3>WGDarcyEquation class implementation</h3>
829 *
830
831 *
832 *
833 * <a name="step_61-WGDarcyEquationWGDarcyEquation"></a>
834 * <h4>WGDarcyEquation::WGDarcyEquation</h4>
835 *
836
837 *
838 * In this constructor, we create a finite element space for vector valued
839 * functions, which will here include the ones used for the interior and
840 * interface pressures, @f$p^\circ@f$ and @f$p^\partial@f$.
841 *
842 * @code
843 *   template <int dim>
844 *   WGDarcyEquation<dim>::WGDarcyEquation(const unsigned int degree)
845 *   : fe(FE_DGQ<dim>(degree), FE_FaceQ<dim>(degree))
846 *   , dof_handler(triangulation)
847 *   , fe_dgrt(degree)
848 *   , dof_handler_dgrt(triangulation)
849 *   {}
850 *  
851 *  
852 *  
853 * @endcode
854 *
855 *
856 * <a name="step_61-WGDarcyEquationmake_grid"></a>
857 * <h4>WGDarcyEquation::make_grid</h4>
858 *
859
860 *
861 * We generate a mesh on the unit square domain and refine it.
862 *
863 * @code
864 *   template <int dim>
865 *   void WGDarcyEquation<dim>::make_grid()
866 *   {
869 *  
870 *   std::cout << " Number of active cells: " << triangulation.n_active_cells()
871 *   << std::endl
872 *   << " Total number of cells: " << triangulation.n_cells()
873 *   << std::endl;
874 *   }
875 *  
876 *  
877 *  
878 * @endcode
879 *
880 *
881 * <a name="step_61-WGDarcyEquationsetup_system"></a>
882 * <h4>WGDarcyEquation::setup_system</h4>
883 *
884
885 *
886 * After we have created the mesh above, we distribute degrees of
887 * freedom and resize matrices and vectors. The only piece of
888 * interest in this function is how we interpolate the boundary
889 * values for the pressure. Since the pressure consists of interior
890 * and interface components, we need to make sure that we only
891 * interpolate onto that component of the vector-valued solution
892 * space that corresponds to the interface pressures (as these are
893 * the only ones that are defined on the boundary of the domain). We
894 * do this via a component mask object for only the interface
895 * pressures.
896 *
897 * @code
898 *   template <int dim>
899 *   void WGDarcyEquation<dim>::setup_system()
900 *   {
901 *   dof_handler.distribute_dofs(fe);
902 *   dof_handler_dgrt.distribute_dofs(fe_dgrt);
903 *  
904 *   std::cout << " Number of pressure degrees of freedom: "
905 *   << dof_handler.n_dofs() << std::endl;
906 *  
907 *   solution.reinit(dof_handler.n_dofs());
908 *   system_rhs.reinit(dof_handler.n_dofs());
909 *  
910 *  
911 *   {
912 *   constraints.clear();
913 *   const FEValuesExtractors::Scalar interface_pressure(1);
914 *   const ComponentMask interface_pressure_mask =
915 *   fe.component_mask(interface_pressure);
917 *   0,
918 *   BoundaryValues<dim>(),
919 *   constraints,
920 *   interface_pressure_mask);
921 *   constraints.close();
922 *   }
923 *  
924 *  
925 * @endcode
926 *
927 * In the bilinear form, there is no integration term over faces
928 * between two neighboring cells, so we can just use
929 * <code>DoFTools::make_sparsity_pattern</code> to calculate the sparse
930 * matrix.
931 *
932 * @code
933 *   DynamicSparsityPattern dsp(dof_handler.n_dofs());
934 *   DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints);
935 *   sparsity_pattern.copy_from(dsp);
936 *  
937 *   system_matrix.reinit(sparsity_pattern);
938 *   }
939 *  
940 *  
941 *  
942 * @endcode
943 *
944 *
945 * <a name="step_61-WGDarcyEquationassemble_system"></a>
946 * <h4>WGDarcyEquation::assemble_system</h4>
947 *
948
949 *
950 * This function is more interesting. As detailed in the
951 * introduction, the assembly of the linear system requires us to
952 * evaluate the weak gradient of the shape functions, which is an
953 * element in the Raviart-Thomas space. As a consequence, we need to
954 * define a Raviart-Thomas finite element object, and have FEValues
955 * objects that evaluate it at quadrature points. We then need to
956 * compute the matrix @f$C^K@f$ on every cell @f$K@f$, for which we need the
957 * matrices @f$M^K@f$ and @f$G^K@f$ mentioned in the introduction.
958 *
959
960 *
961 * A point that may not be obvious is that in all previous tutorial
962 * programs, we have always called FEValues::reinit() with a cell
963 * iterator from a DoFHandler. This is so that one can call
964 * functions such as FEValuesBase::get_function_values() that
965 * extract the values of a finite element function (represented by a
966 * vector of DoF values) on the quadrature points of a cell. For
967 * this operation to work, one needs to know which vector elements
968 * correspond to the degrees of freedom on a given cell -- i.e.,
969 * exactly the kind of information and operation provided by the
970 * DoFHandler class.
971 *
972
973 *
974 * We could create a DoFHandler object for the "broken" Raviart-Thomas space
975 * (using the FE_DGRaviartThomas class), but we really don't want to here: At
976 * least in the current function, we have no need for any globally defined
977 * degrees of freedom associated with this broken space, but really only
978 * need to reference the shape functions of such a space on the current
979 * cell. As a consequence, we use the fact that one can call
980 * FEValues::reinit() also with cell iterators into Triangulation
981 * objects (rather than DoFHandler objects). In this case, FEValues
982 * can of course only provide us with information that only
983 * references information about cells, rather than degrees of freedom
984 * enumerated on these cells. So we can't use
985 * FEValuesBase::get_function_values(), but we can use
986 * FEValues::shape_value() to obtain the values of shape functions
987 * at quadrature points on the current cell. It is this kind of
988 * functionality we will make use of below. The variable that will
989 * give us this information about the Raviart-Thomas functions below
990 * is then the `fe_values_rt` (and corresponding `fe_face_values_rt`)
991 * object.
992 *
993
994 *
995 * Given this introduction, the following declarations should be
996 * pretty obvious:
997 *
998 * @code
999 *   template <int dim>
1000 *   void WGDarcyEquation<dim>::assemble_system()
1001 *   {
1002 *   const QGauss<dim> quadrature_formula(fe_dgrt.degree + 1);
1003 *   const QGauss<dim - 1> face_quadrature_formula(fe_dgrt.degree + 1);
1004 *  
1005 *   FEValues<dim> fe_values(fe,
1006 *   quadrature_formula,
1008 *   update_JxW_values);
1009 *   FEFaceValues<dim> fe_face_values(fe,
1010 *   face_quadrature_formula,
1013 *   update_JxW_values);
1014 *  
1015 *   FEValues<dim> fe_values_dgrt(fe_dgrt,
1016 *   quadrature_formula,
1019 *   update_JxW_values);
1020 *   FEFaceValues<dim> fe_face_values_dgrt(fe_dgrt,
1021 *   face_quadrature_formula,
1022 *   update_values |
1025 *   update_JxW_values);
1026 *  
1027 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1028 *   const unsigned int dofs_per_cell_dgrt = fe_dgrt.n_dofs_per_cell();
1029 *  
1030 *   const unsigned int n_q_points = fe_values.get_quadrature().size();
1031 *   const unsigned int n_q_points_dgrt = fe_values_dgrt.get_quadrature().size();
1032 *  
1033 *   const unsigned int n_face_q_points = fe_face_values.get_quadrature().size();
1034 *  
1035 *   RightHandSide<dim> right_hand_side;
1036 *   std::vector<double> right_hand_side_values(n_q_points);
1037 *  
1038 *   const Coefficient<dim> coefficient;
1039 *   std::vector<Tensor<2, dim>> coefficient_values(n_q_points);
1040 *  
1041 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1042 *  
1043 *  
1044 * @endcode
1045 *
1046 * Next, let us declare the various cell matrices discussed in the
1047 * introduction:
1048 *
1049 * @code
1050 *   FullMatrix<double> cell_matrix_M(dofs_per_cell_dgrt, dofs_per_cell_dgrt);
1051 *   FullMatrix<double> cell_matrix_G(dofs_per_cell_dgrt, dofs_per_cell);
1052 *   FullMatrix<double> cell_matrix_C(dofs_per_cell, dofs_per_cell_dgrt);
1054 *   Vector<double> cell_rhs(dofs_per_cell);
1055 *   Vector<double> cell_solution(dofs_per_cell);
1056 *  
1057 * @endcode
1058 *
1059 * We need <code>FEValuesExtractors</code> to access the @p interior and
1060 * @p face component of the shape functions.
1061 *
1062 * @code
1063 *   const FEValuesExtractors::Vector velocities(0);
1064 *   const FEValuesExtractors::Scalar pressure_interior(0);
1065 *   const FEValuesExtractors::Scalar pressure_face(1);
1066 *  
1067 * @endcode
1068 *
1069 * This finally gets us in position to loop over all cells. On
1070 * each cell, we will first calculate the various cell matrices
1071 * used to construct the local matrix -- as they depend on the
1072 * cell in question, they need to be re-computed on each cell. We
1073 * need shape functions for the Raviart-Thomas space as well, for
1074 * which we need to create first an iterator to the cell of the
1075 * triangulation, which we can obtain by assignment from the cell
1076 * pointing into the DoFHandler.
1077 *
1078 * @code
1079 *   for (const auto &cell : dof_handler.active_cell_iterators())
1080 *   {
1081 *   fe_values.reinit(cell);
1082 *  
1083 *   const typename Triangulation<dim>::active_cell_iterator cell_dgrt =
1084 *   cell;
1085 *   fe_values_dgrt.reinit(cell_dgrt);
1086 *  
1087 *   right_hand_side.value_list(fe_values.get_quadrature_points(),
1088 *   right_hand_side_values);
1089 *   coefficient.value_list(fe_values.get_quadrature_points(),
1090 *   coefficient_values);
1091 *  
1092 * @endcode
1093 *
1094 * The first cell matrix we will compute is the @ref GlossMassMatrix "mass matrix"
1095 * for the Raviart-Thomas space. Hence, we need to loop over
1096 * all the quadrature points for the velocity FEValues object.
1097 *
1098 * @code
1099 *   cell_matrix_M = 0;
1100 *   for (unsigned int q = 0; q < n_q_points_dgrt; ++q)
1101 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1102 *   {
1103 *   const Tensor<1, dim> v_i = fe_values_dgrt[velocities].value(i, q);
1104 *   for (unsigned int k = 0; k < dofs_per_cell_dgrt; ++k)
1105 *   {
1106 *   const Tensor<1, dim> v_k =
1107 *   fe_values_dgrt[velocities].value(k, q);
1108 *   cell_matrix_M(i, k) += (v_i * v_k * fe_values_dgrt.JxW(q));
1109 *   }
1110 *   }
1111 * @endcode
1112 *
1113 * Next we take the inverse of this matrix by using
1114 * FullMatrix::gauss_jordan(). It will be used to calculate
1115 * the coefficient matrix @f$C^K@f$ later. It is worth recalling
1116 * later that `cell_matrix_M` actually contains the *inverse*
1117 * of @f$M^K@f$ after this call.
1118 *
1119 * @code
1120 *   cell_matrix_M.gauss_jordan();
1121 *  
1122 * @endcode
1123 *
1124 * From the introduction, we know that the right hand side
1125 * @f$G^K@f$ of the equation that defines @f$C^K@f$ is the difference
1126 * between a face integral and a cell integral. Here, we
1127 * approximate the negative of the contribution in the
1128 * interior. Each component of this matrix is the integral of
1129 * a product between a basis function of the polynomial space
1130 * and the divergence of a basis function of the
1131 * Raviart-Thomas space. These basis functions are defined in
1132 * the interior.
1133 *
1134 * @code
1135 *   cell_matrix_G = 0;
1136 *   for (unsigned int q = 0; q < n_q_points; ++q)
1137 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1138 *   {
1139 *   const double div_v_i =
1140 *   fe_values_dgrt[velocities].divergence(i, q);
1141 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
1142 *   {
1143 *   const double phi_j_interior =
1144 *   fe_values[pressure_interior].value(j, q);
1145 *  
1146 *   cell_matrix_G(i, j) -=
1147 *   (div_v_i * phi_j_interior * fe_values.JxW(q));
1148 *   }
1149 *   }
1150 *  
1151 *  
1152 * @endcode
1153 *
1154 * Next, we approximate the integral on faces by quadrature.
1155 * Each component is the integral of a product between a basis function
1156 * of the polynomial space and the dot product of a basis function of
1157 * the Raviart-Thomas space and the normal vector. So we loop over all
1158 * the faces of the element and obtain the normal vector.
1159 *
1160 * @code
1161 *   for (const auto &face : cell->face_iterators())
1162 *   {
1163 *   fe_face_values.reinit(cell, face);
1164 *   fe_face_values_dgrt.reinit(cell_dgrt, face);
1165 *  
1166 *   for (unsigned int q = 0; q < n_face_q_points; ++q)
1167 *   {
1168 *   const Tensor<1, dim> &normal = fe_face_values.normal_vector(q);
1169 *  
1170 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1171 *   {
1172 *   const Tensor<1, dim> v_i =
1173 *   fe_face_values_dgrt[velocities].value(i, q);
1174 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
1175 *   {
1176 *   const double phi_j_face =
1177 *   fe_face_values[pressure_face].value(j, q);
1178 *  
1179 *   cell_matrix_G(i, j) +=
1180 *   ((v_i * normal) * phi_j_face * fe_face_values.JxW(q));
1181 *   }
1182 *   }
1183 *   }
1184 *   }
1185 *  
1186 * @endcode
1187 *
1188 * @p cell_matrix_C is then the matrix product between the
1189 * transpose of @f$G^K@f$ and the inverse of the mass matrix
1190 * (where this inverse is stored in @p cell_matrix_M):
1191 *
1192 * @code
1193 *   cell_matrix_G.Tmmult(cell_matrix_C, cell_matrix_M);
1194 *  
1195 * @endcode
1196 *
1197 * Finally we can compute the local matrix @f$A^K@f$. Element
1198 * @f$A^K_{ij}@f$ is given by @f$\int_{E} \sum_{k,l} C_{ik} C_{jl}
1199 * (\mathbf{K} \mathbf{v}_k) \cdot \mathbf{v}_l
1200 * \mathrm{d}x@f$. We have calculated the coefficients @f$C@f$ in
1201 * the previous step, and so obtain the following after
1202 * suitably re-arranging the loops:
1203 *
1204 * @code
1205 *   local_matrix = 0;
1206 *   for (unsigned int q = 0; q < n_q_points_dgrt; ++q)
1207 *   {
1208 *   for (unsigned int k = 0; k < dofs_per_cell_dgrt; ++k)
1209 *   {
1210 *   const Tensor<1, dim> v_k =
1211 *   fe_values_dgrt[velocities].value(k, q);
1212 *   for (unsigned int l = 0; l < dofs_per_cell_dgrt; ++l)
1213 *   {
1214 *   const Tensor<1, dim> v_l =
1215 *   fe_values_dgrt[velocities].value(l, q);
1216 *  
1217 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1218 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
1219 *   local_matrix(i, j) +=
1220 *   (coefficient_values[q] * cell_matrix_C[i][k] * v_k) *
1221 *   cell_matrix_C[j][l] * v_l * fe_values_dgrt.JxW(q);
1222 *   }
1223 *   }
1224 *   }
1225 *  
1226 * @endcode
1227 *
1228 * Next, we calculate the right hand side, @f$\int_{K} f q \mathrm{d}x@f$:
1229 *
1230 * @code
1231 *   cell_rhs = 0;
1232 *   for (unsigned int q = 0; q < n_q_points; ++q)
1233 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1234 *   {
1235 *   cell_rhs(i) += (fe_values[pressure_interior].value(i, q) *
1236 *   right_hand_side_values[q] * fe_values.JxW(q));
1237 *   }
1238 *  
1239 * @endcode
1240 *
1241 * The last step is to distribute components of the local
1242 * matrix into the system matrix and transfer components of
1243 * the cell right hand side into the system right hand side:
1244 *
1245 * @code
1246 *   cell->get_dof_indices(local_dof_indices);
1247 *   constraints.distribute_local_to_global(
1248 *   local_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
1249 *   }
1250 *   }
1251 *  
1252 *  
1253 *  
1254 * @endcode
1255 *
1256 *
1257 * <a name="step_61-WGDarcyEquationdimsolve"></a>
1258 * <h4>WGDarcyEquation<dim>::solve</h4>
1259 *
1260
1261 *
1262 * This step is rather trivial and the same as in many previous
1263 * tutorial programs:
1264 *
1265 * @code
1266 *   template <int dim>
1267 *   void WGDarcyEquation<dim>::solve()
1268 *   {
1269 *   SolverControl solver_control(1000, 1e-8 * system_rhs.l2_norm());
1270 *   SolverCG<Vector<double>> solver(solver_control);
1271 *   solver.solve(system_matrix, solution, system_rhs, PreconditionIdentity());
1272 *   constraints.distribute(solution);
1273 *   }
1274 *  
1275 *  
1276 * @endcode
1277 *
1278 *
1279 * <a name="step_61-WGDarcyEquationdimcompute_postprocessed_velocity"></a>
1280 * <h4>WGDarcyEquation<dim>::compute_postprocessed_velocity</h4>
1281 *
1282
1283 *
1284 * In this function, compute the velocity field from the pressure
1285 * solution previously computed. The
1286 * velocity is defined as @f$\mathbf{u}_h = \mathbf{Q}_h \left(
1287 * -\mathbf{K}\nabla_{w,d}p_h \right)@f$, which requires us to compute
1288 * many of the same terms as in the assembly of the system matrix.
1289 * There are also the matrices @f$E^K,D^K@f$ we need to assemble (see
1290 * the introduction) but they really just follow the same kind of
1291 * pattern.
1292 *
1293
1294 *
1295 * Computing the same matrices here as we have already done in the
1296 * `assemble_system()` function is of course wasteful in terms of
1297 * CPU time. Likewise, we copy some of the code from there to this
1298 * function, and this is also generally a poor idea. A better
1299 * implementation might provide for a function that encapsulates
1300 * this duplicated code. One could also think of using the classic
1301 * trade-off between computing efficiency and memory efficiency to
1302 * only compute the @f$C^K@f$ matrices once per cell during the
1303 * assembly, storing them somewhere on the side, and re-using them
1304 * here. (This is what @ref step_51 "step-51" does, for example, where the
1305 * `assemble_system()` function takes an argument that determines
1306 * whether the local matrices are recomputed, and a similar approach
1307 * -- maybe with storing local matrices elsewhere -- could be
1308 * adapted for the current program.)
1309 *
1310 * @code
1311 *   template <int dim>
1312 *   void WGDarcyEquation<dim>::compute_postprocessed_velocity()
1313 *   {
1314 *   darcy_velocity.reinit(dof_handler_dgrt.n_dofs());
1315 *  
1316 *   const QGauss<dim> quadrature_formula(fe_dgrt.degree + 1);
1317 *   const QGauss<dim - 1> face_quadrature_formula(fe_dgrt.degree + 1);
1318 *  
1319 *   FEValues<dim> fe_values(fe,
1320 *   quadrature_formula,
1322 *   update_JxW_values);
1323 *  
1324 *   FEFaceValues<dim> fe_face_values(fe,
1325 *   face_quadrature_formula,
1328 *   update_JxW_values);
1329 *  
1330 *   FEValues<dim> fe_values_dgrt(fe_dgrt,
1331 *   quadrature_formula,
1334 *   update_JxW_values);
1335 *  
1336 *   FEFaceValues<dim> fe_face_values_dgrt(fe_dgrt,
1337 *   face_quadrature_formula,
1338 *   update_values |
1341 *   update_JxW_values);
1342 *  
1343 *   const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1344 *   const unsigned int dofs_per_cell_dgrt = fe_dgrt.n_dofs_per_cell();
1345 *  
1346 *   const unsigned int n_q_points = fe_values.get_quadrature().size();
1347 *   const unsigned int n_q_points_dgrt = fe_values_dgrt.get_quadrature().size();
1348 *  
1349 *   const unsigned int n_face_q_points = fe_face_values.get_quadrature().size();
1350 *  
1351 *  
1352 *   std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1353 *   std::vector<types::global_dof_index> local_dof_indices_dgrt(
1354 *   dofs_per_cell_dgrt);
1355 *  
1356 *   FullMatrix<double> cell_matrix_M(dofs_per_cell_dgrt, dofs_per_cell_dgrt);
1357 *   FullMatrix<double> cell_matrix_G(dofs_per_cell_dgrt, dofs_per_cell);
1358 *   FullMatrix<double> cell_matrix_C(dofs_per_cell, dofs_per_cell_dgrt);
1359 *   FullMatrix<double> cell_matrix_D(dofs_per_cell_dgrt, dofs_per_cell_dgrt);
1360 *   FullMatrix<double> cell_matrix_E(dofs_per_cell_dgrt, dofs_per_cell_dgrt);
1361 *  
1362 *   Vector<double> cell_solution(dofs_per_cell);
1363 *   Vector<double> cell_velocity(dofs_per_cell_dgrt);
1364 *  
1365 *   const Coefficient<dim> coefficient;
1366 *   std::vector<Tensor<2, dim>> coefficient_values(n_q_points_dgrt);
1367 *  
1368 *   const FEValuesExtractors::Vector velocities(0);
1369 *   const FEValuesExtractors::Scalar pressure_interior(0);
1370 *   const FEValuesExtractors::Scalar pressure_face(1);
1371 *  
1372 * @endcode
1373 *
1374 * In the introduction, we explained how to calculate the numerical velocity
1375 * on the cell. We need the pressure solution values on each cell,
1376 * coefficients of the Gram matrix and coefficients of the @f$L_2@f$ projection.
1377 * We have already calculated the global solution, so we will extract the
1378 * cell solution from the global solution. The coefficients of the Gram
1379 * matrix have been calculated when we assembled the system matrix for the
1380 * pressures. We will do the same way here. For the coefficients of the
1381 * projection, we do matrix multiplication, i.e., the inverse of the Gram
1382 * matrix times the matrix with @f$(\mathbf{K} \mathbf{w}, \mathbf{w})@f$ as
1383 * components. Then, we multiply all these coefficients and call them beta.
1384 * The numerical velocity is the product of beta and the basis functions of
1385 * the Raviart-Thomas space.
1386 *
1387 * @code
1389 *   cell = dof_handler.begin_active(),
1390 *   endc = dof_handler.end(), cell_dgrt = dof_handler_dgrt.begin_active();
1391 *   for (; cell != endc; ++cell, ++cell_dgrt)
1392 *   {
1393 *   fe_values.reinit(cell);
1394 *   fe_values_dgrt.reinit(cell_dgrt);
1395 *  
1396 *   coefficient.value_list(fe_values_dgrt.get_quadrature_points(),
1397 *   coefficient_values);
1398 *  
1399 * @endcode
1400 *
1401 * The component of this <code>cell_matrix_E</code> is the integral of
1402 * @f$(\mathbf{K} \mathbf{w}, \mathbf{w})@f$. <code>cell_matrix_M</code> is
1403 * the Gram matrix.
1404 *
1405 * @code
1406 *   cell_matrix_M = 0;
1407 *   cell_matrix_E = 0;
1408 *   for (unsigned int q = 0; q < n_q_points_dgrt; ++q)
1409 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1410 *   {
1411 *   const Tensor<1, dim> v_i = fe_values_dgrt[velocities].value(i, q);
1412 *   for (unsigned int k = 0; k < dofs_per_cell_dgrt; ++k)
1413 *   {
1414 *   const Tensor<1, dim> v_k =
1415 *   fe_values_dgrt[velocities].value(k, q);
1416 *  
1417 *   cell_matrix_E(i, k) +=
1418 *   (coefficient_values[q] * v_i * v_k * fe_values_dgrt.JxW(q));
1419 *  
1420 *   cell_matrix_M(i, k) += (v_i * v_k * fe_values_dgrt.JxW(q));
1421 *   }
1422 *   }
1423 *  
1424 * @endcode
1425 *
1426 * To compute the matrix @f$D@f$ mentioned in the introduction, we
1427 * then need to evaluate @f$D=M^{-1}E@f$ as explained in the
1428 * introduction:
1429 *
1430 * @code
1431 *   cell_matrix_M.gauss_jordan();
1432 *   cell_matrix_M.mmult(cell_matrix_D, cell_matrix_E);
1433 *  
1434 * @endcode
1435 *
1436 * Then we also need, again, to compute the matrix @f$C@f$ that is
1437 * used to evaluate the weak discrete gradient. This is the
1438 * exact same code as used in the assembly of the system
1439 * matrix, so we just copy it from there:
1440 *
1441 * @code
1442 *   cell_matrix_G = 0;
1443 *   for (unsigned int q = 0; q < n_q_points; ++q)
1444 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1445 *   {
1446 *   const double div_v_i =
1447 *   fe_values_dgrt[velocities].divergence(i, q);
1448 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
1449 *   {
1450 *   const double phi_j_interior =
1451 *   fe_values[pressure_interior].value(j, q);
1452 *  
1453 *   cell_matrix_G(i, j) -=
1454 *   (div_v_i * phi_j_interior * fe_values.JxW(q));
1455 *   }
1456 *   }
1457 *  
1458 *   for (const auto &face : cell->face_iterators())
1459 *   {
1460 *   fe_face_values.reinit(cell, face);
1461 *   fe_face_values_dgrt.reinit(cell_dgrt, face);
1462 *  
1463 *   for (unsigned int q = 0; q < n_face_q_points; ++q)
1464 *   {
1465 *   const Tensor<1, dim> &normal = fe_face_values.normal_vector(q);
1466 *  
1467 *   for (unsigned int i = 0; i < dofs_per_cell_dgrt; ++i)
1468 *   {
1469 *   const Tensor<1, dim> v_i =
1470 *   fe_face_values_dgrt[velocities].value(i, q);
1471 *   for (unsigned int j = 0; j < dofs_per_cell; ++j)
1472 *   {
1473 *   const double phi_j_face =
1474 *   fe_face_values[pressure_face].value(j, q);
1475 *  
1476 *   cell_matrix_G(i, j) +=
1477 *   ((v_i * normal) * phi_j_face * fe_face_values.JxW(q));
1478 *   }
1479 *   }
1480 *   }
1481 *   }
1482 *   cell_matrix_G.Tmmult(cell_matrix_C, cell_matrix_M);
1483 *  
1484 * @endcode
1485 *
1486 * Finally, we need to extract the pressure unknowns that
1487 * correspond to the current cell:
1488 *
1489 * @code
1490 *   cell->get_dof_values(solution, cell_solution);
1491 *  
1492 * @endcode
1493 *
1494 * We are now in a position to compute the local velocity
1495 * unknowns (with respect to the Raviart-Thomas space we are
1496 * projecting the term @f$-\mathbf K \nabla_{w,d} p_h@f$ into):
1497 *
1498 * @code
1499 *   cell_velocity = 0;
1500 *   for (unsigned int k = 0; k < dofs_per_cell_dgrt; ++k)
1501 *   for (unsigned int j = 0; j < dofs_per_cell_dgrt; ++j)
1502 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1503 *   cell_velocity(k) +=
1504 *   -(cell_solution(i) * cell_matrix_C(i, j) * cell_matrix_D(k, j));
1505 *  
1506 * @endcode
1507 *
1508 * We compute Darcy velocity.
1509 * This is same as cell_velocity but used to graph Darcy velocity.
1510 *
1511 * @code
1512 *   cell_dgrt->get_dof_indices(local_dof_indices_dgrt);
1513 *   for (unsigned int k = 0; k < dofs_per_cell_dgrt; ++k)
1514 *   for (unsigned int j = 0; j < dofs_per_cell_dgrt; ++j)
1515 *   for (unsigned int i = 0; i < dofs_per_cell; ++i)
1516 *   darcy_velocity(local_dof_indices_dgrt[k]) +=
1517 *   -(cell_solution(i) * cell_matrix_C(i, j) * cell_matrix_D(k, j));
1518 *   }
1519 *   }
1520 *  
1521 *  
1522 *  
1523 * @endcode
1524 *
1525 *
1526 * <a name="step_61-WGDarcyEquationdimcompute_pressure_error"></a>
1527 * <h4>WGDarcyEquation<dim>::compute_pressure_error</h4>
1528 *
1529
1530 *
1531 * This part is to calculate the @f$L_2@f$ error of the pressure. We
1532 * define a vector that holds the norm of the error on each cell.
1533 * Next, we use VectorTool::integrate_difference() to compute the
1534 * error in the @f$L_2@f$ norm on each cell. However, we really only
1535 * care about the error in the interior component of the solution
1536 * vector (we can't even evaluate the interface pressures at the
1537 * quadrature points because these are all located in the interior
1538 * of cells) and consequently have to use a weight function that
1539 * ensures that the interface component of the solution variable is
1540 * ignored. This is done by using the ComponentSelectFunction whose
1541 * arguments indicate which component we want to select (component
1542 * zero, i.e., the interior pressures) and how many components there
1543 * are in total (two).
1544 *
1545 * @code
1546 *   template <int dim>
1547 *   void WGDarcyEquation<dim>::compute_pressure_error()
1548 *   {
1549 *   Vector<float> difference_per_cell(triangulation.n_active_cells());
1550 *   const ComponentSelectFunction<dim> select_interior_pressure(0, 2);
1551 *   VectorTools::integrate_difference(dof_handler,
1552 *   solution,
1553 *   ExactPressure<dim>(),
1554 *   difference_per_cell,
1555 *   QGauss<dim>(fe.degree + 2),
1557 *   &select_interior_pressure);
1558 *  
1559 *   const double L2_error = difference_per_cell.l2_norm();
1560 *   std::cout << "L2_error_pressure " << L2_error << std::endl;
1561 *   }
1562 *  
1563 *  
1564 *  
1565 * @endcode
1566 *
1567 *
1568 * <a name="step_61-WGDarcyEquationdimcompute_velocity_error"></a>
1569 * <h4>WGDarcyEquation<dim>::compute_velocity_error</h4>
1570 *
1571
1572 *
1573 * In this function, we evaluate @f$L_2@f$ errors for the velocity on
1574 * each cell, and @f$L_2@f$ errors for the flux on faces. The function
1575 * relies on the `compute_postprocessed_velocity()` function having
1576 * previous computed, which computes the velocity field based on the
1577 * pressure solution that has previously been computed.
1578 *
1579
1580 *
1581 * We are going to evaluate velocities on each cell and calculate
1582 * the difference between numerical and exact velocities.
1583 *
1584 * @code
1585 *   template <int dim>
1586 *   void WGDarcyEquation<dim>::compute_velocity_errors()
1587 *   {
1588 *   const QGauss<dim> quadrature_formula(fe_dgrt.degree + 1);
1589 *   const QGauss<dim - 1> face_quadrature_formula(fe_dgrt.degree + 1);
1590 *  
1591 *   FEValues<dim> fe_values_dgrt(fe_dgrt,
1592 *   quadrature_formula,
1595 *   update_JxW_values);
1596 *  
1597 *   FEFaceValues<dim> fe_face_values_dgrt(fe_dgrt,
1598 *   face_quadrature_formula,
1599 *   update_values |
1602 *   update_JxW_values);
1603 *  
1604 *   const unsigned int n_q_points_dgrt = fe_values_dgrt.get_quadrature().size();
1605 *   const unsigned int n_face_q_points_dgrt =
1606 *   fe_face_values_dgrt.get_quadrature().size();
1607 *  
1608 *   std::vector<Tensor<1, dim>> velocity_values(n_q_points_dgrt);
1609 *   std::vector<Tensor<1, dim>> velocity_face_values(n_face_q_points_dgrt);
1610 *  
1611 *   const FEValuesExtractors::Vector velocities(0);
1612 *  
1613 *   const ExactVelocity<dim> exact_velocity;
1614 *  
1615 *   double L2_err_velocity_cell_sqr_global = 0;
1616 *   double L2_err_flux_sqr = 0;
1617 *  
1618 * @endcode
1619 *
1620 * Having previously computed the postprocessed velocity, we here
1621 * only have to extract the corresponding values on each cell and
1622 * face and compare it to the exact values.
1623 *
1624 * @code
1625 *   for (const auto &cell_dgrt : dof_handler_dgrt.active_cell_iterators())
1626 *   {
1627 *   fe_values_dgrt.reinit(cell_dgrt);
1628 *  
1629 * @endcode
1630 *
1631 * First compute the @f$L_2@f$ error between the postprocessed velocity
1632 * field and the exact one:
1633 *
1634 * @code
1635 *   fe_values_dgrt[velocities].get_function_values(darcy_velocity,
1636 *   velocity_values);
1637 *   double L2_err_velocity_cell_sqr_local = 0;
1638 *   for (unsigned int q = 0; q < n_q_points_dgrt; ++q)
1639 *   {
1640 *   const Tensor<1, dim> velocity = velocity_values[q];
1641 *   const Tensor<1, dim> true_velocity =
1642 *   exact_velocity.value(fe_values_dgrt.quadrature_point(q));
1643 *  
1644 *   L2_err_velocity_cell_sqr_local +=
1645 *   ((velocity - true_velocity) * (velocity - true_velocity) *
1646 *   fe_values_dgrt.JxW(q));
1647 *   }
1648 *   L2_err_velocity_cell_sqr_global += L2_err_velocity_cell_sqr_local;
1649 *  
1650 * @endcode
1651 *
1652 * For reconstructing the flux we need the size of cells and
1653 * faces. Since fluxes are calculated on faces, we have the
1654 * loop over all four faces of each cell. To calculate the
1655 * face velocity, we extract values at the quadrature points from the
1656 * `darcy_velocity` which we have computed previously. Then, we
1657 * calculate the squared velocity error in normal direction. Finally, we
1658 * calculate the @f$L_2@f$ flux error on the cell by appropriately scaling
1659 * with face and cell areas and add it to the global error.
1660 *
1661 * @code
1662 *   const double cell_area = cell_dgrt->measure();
1663 *   for (const auto &face_dgrt : cell_dgrt->face_iterators())
1664 *   {
1665 *   const double face_length = face_dgrt->measure();
1666 *   fe_face_values_dgrt.reinit(cell_dgrt, face_dgrt);
1667 *   fe_face_values_dgrt[velocities].get_function_values(
1668 *   darcy_velocity, velocity_face_values);
1669 *  
1670 *   double L2_err_flux_face_sqr_local = 0;
1671 *   for (unsigned int q = 0; q < n_face_q_points_dgrt; ++q)
1672 *   {
1673 *   const Tensor<1, dim> velocity = velocity_face_values[q];
1674 *   const Tensor<1, dim> true_velocity =
1675 *   exact_velocity.value(fe_face_values_dgrt.quadrature_point(q));
1676 *  
1677 *   const Tensor<1, dim> &normal =
1678 *   fe_face_values_dgrt.normal_vector(q);
1679 *  
1680 *   L2_err_flux_face_sqr_local +=
1681 *   ((velocity * normal - true_velocity * normal) *
1682 *   (velocity * normal - true_velocity * normal) *
1683 *   fe_face_values_dgrt.JxW(q));
1684 *   }
1685 *   const double err_flux_each_face =
1686 *   L2_err_flux_face_sqr_local / face_length * cell_area;
1687 *   L2_err_flux_sqr += err_flux_each_face;
1688 *   }
1689 *   }
1690 *  
1691 * @endcode
1692 *
1693 * After adding up errors over all cells and faces, we take the
1694 * square root and get the @f$L_2@f$ errors of velocity and
1695 * flux. These we output to screen.
1696 *
1697 * @code
1698 *   const double L2_err_velocity_cell =
1699 *   std::sqrt(L2_err_velocity_cell_sqr_global);
1700 *   const double L2_err_flux_face = std::sqrt(L2_err_flux_sqr);
1701 *  
1702 *   std::cout << "L2_error_vel: " << L2_err_velocity_cell << std::endl
1703 *   << "L2_error_flux: " << L2_err_flux_face << std::endl;
1704 *   }
1705 *  
1706 *  
1707 * @endcode
1708 *
1709 *
1710 * <a name="step_61-WGDarcyEquationoutput_results"></a>
1711 * <h4>WGDarcyEquation::output_results</h4>
1712 *
1713
1714 *
1715 * We have two sets of results to output: the interior solution and
1716 * the skeleton solution. We use <code>DataOut</code> to visualize
1717 * interior results. The graphical output for the skeleton results
1718 * is done by using the DataOutFaces class.
1719 *
1720
1721 *
1722 * In both of the output files, both the interior and the face
1723 * variables are stored. For the interface output, the output file
1724 * simply contains the interpolation of the interior pressures onto
1725 * the faces, but because it is undefined which of the two interior
1726 * pressure variables you get from the two adjacent cells, it is
1727 * best to ignore the interior pressure in the interface output
1728 * file. Conversely, for the cell interior output file, it is of
1729 * course impossible to show any interface pressures @f$p^\partial@f$,
1730 * because these are only available on interfaces and not cell
1731 * interiors. Consequently, you will see them shown as an invalid
1732 * value (such as an infinity).
1733 *
1734
1735 *
1736 * For the cell interior output, we also want to output the velocity
1737 * variables. This is a bit tricky since it lives on the same mesh
1738 * but uses a different DoFHandler object (the pressure variables live
1739 * on the `dof_handler` object, the Darcy velocity on the `dof_handler_dgrt`
1740 * object). Fortunately, there are variations of the
1741 * DataOut::add_data_vector() function that allow specifying which
1742 * DoFHandler a vector corresponds to, and consequently we can visualize
1743 * the data from both DoFHandler objects within the same file.
1744 *
1745 * @code
1746 *   template <int dim>
1747 *   void WGDarcyEquation<dim>::output_results() const
1748 *   {
1749 *   {
1750 *   DataOut<dim> data_out;
1751 *  
1752 * @endcode
1753 *
1754 * First attach the pressure solution to the DataOut object:
1755 *
1756 * @code
1757 *   const std::vector<std::string> solution_names = {"interior_pressure",
1758 *   "interface_pressure"};
1759 *   data_out.add_data_vector(dof_handler, solution, solution_names);
1760 *  
1761 * @endcode
1762 *
1763 * Then do the same with the Darcy velocity field, and continue
1764 * with writing everything out into a file.
1765 *
1766 * @code
1767 *   const std::vector<std::string> velocity_names(dim, "velocity");
1768 *   const std::vector<
1770 *   velocity_component_interpretation(
1772 *   data_out.add_data_vector(dof_handler_dgrt,
1773 *   darcy_velocity,
1774 *   velocity_names,
1775 *   velocity_component_interpretation);
1776 *  
1777 *   data_out.build_patches(fe.degree);
1778 *   std::ofstream output("solution_interior.vtu");
1779 *   data_out.write_vtu(output);
1780 *   }
1781 *  
1782 *   {
1783 *   DataOutFaces<dim> data_out_faces(false);
1784 *   data_out_faces.attach_dof_handler(dof_handler);
1785 *   data_out_faces.add_data_vector(solution, "Pressure_Face");
1786 *   data_out_faces.build_patches(fe.degree);
1787 *   std::ofstream face_output("solution_interface.vtu");
1788 *   data_out_faces.write_vtu(face_output);
1789 *   }
1790 *   }
1791 *  
1792 *  
1793 * @endcode
1794 *
1795 *
1796 * <a name="step_61-WGDarcyEquationrun"></a>
1797 * <h4>WGDarcyEquation::run</h4>
1798 *
1799
1800 *
1801 * This is the final function of the main class. It calls the other functions
1802 * of our class.
1803 *
1804 * @code
1805 *   template <int dim>
1806 *   void WGDarcyEquation<dim>::run()
1807 *   {
1808 *   std::cout << "Solving problem in " << dim << " space dimensions."
1809 *   << std::endl;
1810 *   make_grid();
1811 *   setup_system();
1812 *   assemble_system();
1813 *   solve();
1814 *   compute_postprocessed_velocity();
1815 *   compute_pressure_error();
1816 *   compute_velocity_errors();
1817 *   output_results();
1818 *   }
1819 *  
1820 *   } // namespace Step61
1821 *  
1822 *  
1823 * @endcode
1824 *
1825 *
1826 * <a name="step_61-Thecodemaincodefunction"></a>
1827 * <h3>The <code>main</code> function</h3>
1828 *
1829
1830 *
1831 * This is the main function. We can change the dimension here to run in 3d.
1832 *
1833 * @code
1834 *   int main()
1835 *   {
1836 *   try
1837 *   {
1838 *   Step61::WGDarcyEquation<2> wg_darcy(0);
1839 *   wg_darcy.run();
1840 *   }
1841 *   catch (std::exception &exc)
1842 *   {
1843 *   std::cerr << std::endl
1844 *   << std::endl
1845 *   << "----------------------------------------------------"
1846 *   << std::endl;
1847 *   std::cerr << "Exception on processing: " << std::endl
1848 *   << exc.what() << std::endl
1849 *   << "Aborting!" << std::endl
1850 *   << "----------------------------------------------------"
1851 *   << std::endl;
1852 *   return 1;
1853 *   }
1854 *   catch (...)
1855 *   {
1856 *   std::cerr << std::endl
1857 *   << std::endl
1858 *   << "----------------------------------------------------"
1859 *   << std::endl;
1860 *   std::cerr << "Unknown exception!" << std::endl
1861 *   << "Aborting!" << std::endl
1862 *   << "----------------------------------------------------"
1863 *   << std::endl;
1864 *   return 1;
1865 *   }
1866 *  
1867 *   return 0;
1868 *   }
1869 * @endcode
1870<a name="step_61-Results"></a><h1>Results</h1>
1871
1872
1873We run the program with a right hand side that will produce the
1874solution @f$p = \sin(\pi x) \sin(\pi y)@f$ and with homogeneous Dirichlet
1875boundary conditions in the domain @f$\Omega = (0,1)^2@f$. In addition, we
1876choose the coefficient matrix in the differential operator
1877@f$\mathbf{K}@f$ as the identity matrix. We test this setup using
1878@f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$, @f$\mbox{WG}(Q_1,Q_1;RT_{[1]})@f$ and
1879@f$\mbox{WG}(Q_2,Q_2;RT_{[2]})@f$ element combinations, which one can
1880select by using the appropriate constructor argument for the
1881`WGDarcyEquation` object in `main()`. We will then visualize pressure
1882values in interiors of cells and on faces. We want to see that the
1883pressure maximum is around 1 and the minimum is around 0. With mesh
1884refinement, the convergence rates of pressure, velocity and flux
1885should then be around 1 for @f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$ , 2 for
1886@f$\mbox{WG}(Q_1,Q_1;RT_{[1]})@f$, and 3 for
1887@f$\mbox{WG}(Q_2,Q_2;RT_{[2]})@f$.
1888
1889
1890<a name="step_61-TestresultsoniWGQsub0subQsub0subRTsub0subi"></a><h3>Test results on <i>WG(Q<sub>0</sub>,Q<sub>0</sub>;RT<sub>[0]</sub>)</i></h3>
1891
1892
1893The following figures show interior pressures and face pressures using the
1894@f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$ element. The mesh is refined 2 times (top)
1895and 4 times (bottom), respectively. (This number can be adjusted in the
1896`make_grid()` function.) When the mesh is coarse, one can see
1897the face pressures @f$p^\partial@f$ neatly between the values of the interior
1898pressures @f$p^\circ@f$ on the two adjacent cells.
1899
1900<table align="center">
1901 <tr>
1902 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg000_2d_2.png" alt=""></td>
1903 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg000_3d_2.png" alt=""></td>
1904 </tr>
1905 <tr>
1906 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg000_2d_4.png" alt=""></td>
1907 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg000_3d_4.png" alt=""></td>
1908 </tr>
1909</table>
1910
1911From the figures, we can see that with the mesh refinement, the maximum and
1912minimum pressure values are approaching the values we expect.
1913Since the mesh is a rectangular mesh and numbers of cells in each direction is even, we
1914have symmetric solutions. From the 3d figures on the right,
1915we can see that on @f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$, the pressure is a constant
1916in the interior of the cell, as expected.
1917
1918<a name="step_61-Convergencetableforik0i"></a><h4>Convergence table for <i>k=0</i></h4>
1919
1920
1921We run the code with differently refined meshes (chosen in the `make_grid()` function)
1922and get the following convergence rates of pressure,
1923velocity, and flux (as defined in the introduction).
1924
1925<table align="center" class="doxtable">
1926 <tr>
1927 <th>number of refinements </th><th> @f$\|p-p_h^\circ\|@f$ </th><th> @f$\|\mathbf{u}-\mathbf{u}_h\|@f$ </th><th> @f$\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|@f$ </th>
1928 </tr>
1929 <tr>
1930 <td> 2 </td><td> 1.587e-01 </td><td> 5.113e-01 </td><td> 7.062e-01 </td>
1931 </tr>
1932 <tr>
1933 <td> 3 </td><td> 8.000e-02 </td><td> 2.529e-01 </td><td> 3.554e-01 </td>
1934 </tr>
1935 <tr>
1936 <td> 4 </td><td> 4.006e-02 </td><td> 1.260e-01 </td><td> 1.780e-01 </td>
1937 </tr>
1938 <tr>
1939 <td> 5 </td><td> 2.004e-02 </td><td> 6.297e-02 </td><td> 8.902e-02 </td>
1940 </tr>
1941 <tr>
1942 <th>Conv.rate </th><th> 1.00 </th><th> 1.00 </th><th> 1.00 </th>
1943 </tr>
1944</table>
1945
1946We can see that the convergence rates of @f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$ are around 1.
1947This, of course, matches our theoretical expectations.
1948
1949
1950<a name="step_61-TestresultsoniWGQsub1subQsub1subRTsub1subi"></a><h3>Test results on <i>WG(Q<sub>1</sub>,Q<sub>1</sub>;RT<sub>[1]</sub>)</i></h3>
1951
1952
1953We can repeat the experiment from above using the next higher polynomial
1954degree:
1955The following figures are interior pressures and face pressures implemented using
1956@f$\mbox{WG}(Q_1,Q_1;RT_{[1]})@f$. The mesh is refined 4 times. Compared to the
1957previous figures using
1958@f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$, on each cell, the solution is no longer constant
1959on each cell, as we now use bilinear polynomials to do the approximation.
1960Consequently, there are 4 pressure values in one interior, 2 pressure values on
1961each face.
1962
1963<table align="center">
1964 <tr>
1965 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg111_2d_4.png" alt=""></td>
1966 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg111_3d_4.png" alt=""></td>
1967 </tr>
1968</table>
1969
1970Compared to the corresponding image for the @f$\mbox{WG}(Q_0,Q_0;RT_{[0]})@f$
1971combination, the solution is now substantially more accurate and, in
1972particular so close to being continuous at the interfaces that we can
1973no longer distinguish the interface pressures @f$p^\partial@f$ from the
1974interior pressures @f$p^\circ@f$ on the adjacent cells.
1975
1976<a name="step_61-Convergencetableforik1i"></a><h4>Convergence table for <i>k=1</i></h4>
1977
1978
1979The following are the convergence rates of pressure, velocity, and flux
1980we obtain from using the @f$\mbox{WG}(Q_1,Q_1;RT_{[1]})@f$ element combination:
1981
1982<table align="center" class="doxtable">
1983 <tr>
1984 <th>number of refinements </th><th> @f$\|p-p_h^\circ\|@f$ </th><th> @f$\|\mathbf{u}-\mathbf{u}_h\|@f$ </th><th> @f$\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|@f$ </th>
1985 </tr>
1986 <tr>
1987 <td> 2 </td><td> 1.613e-02 </td><td> 5.093e-02 </td><td> 7.167e-02 </td>
1988 </tr>
1989 <tr>
1990 <td> 3 </td><td> 4.056e-03 </td><td> 1.276e-02 </td><td> 1.802e-02 </td>
1991 </tr>
1992 <tr>
1993 <td> 4 </td><td> 1.015e-03 </td><td> 3.191e-03 </td><td> 4.512e-03 </td>
1994 </tr>
1995 <tr>
1996 <td> 5 </td><td> 2.540e-04 </td><td> 7.979e-04 </td><td> 1.128e-03 </td>
1997 </tr>
1998 <tr>
1999 <th>Conv.rate </th><th> 2.00 </th><th> 2.00 </th><th> 2.00 </th>
2000 </tr>
2001</table>
2002
2003The convergence rates of @f$WG(Q_1,Q_1;RT_{[1]})@f$ are around 2, as expected.
2004
2005
2006
2007<a name="step_61-TestresultsoniWGQsub2subQsub2subRTsub2subi"></a><h3>Test results on <i>WG(Q<sub>2</sub>,Q<sub>2</sub>;RT<sub>[2]</sub>)</i></h3>
2008
2009
2010Let us go one polynomial degree higher.
2011The following are interior pressures and face pressures implemented using
2012@f$WG(Q_2,Q_2;RT_{[2]})@f$, with mesh size @f$h = 1/32@f$ (i.e., 5 global mesh
2013refinement steps). In the program, we use
2014`data_out_face.build_patches(fe.degree)` when generating graphical output
2015(see the documentation of DataOut::build_patches()), which here implies that
2016we divide each 2d cell interior into 4 subcells in order to provide a better
2017visualization of the quadratic polynomials.
2018<table align="center">
2019 <tr>
2020 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg222_2d_5.png" alt=""></td>
2021 <td><img src="https://www.dealii.org/images/steps/developer/step-61.wg222_3d_5.png" alt=""></td>
2022 </tr>
2023</table>
2024
2025
2026<a name="step_61-Convergencetableforik2i"></a><h4>Convergence table for <i>k=2</i></h4>
2027
2028
2029As before, we can generate convergence data for the
2030@f$L_2@f$ errors of pressure, velocity, and flux
2031using the @f$\mbox{WG}(Q_2,Q_2;RT_{[2]})@f$ combination:
2032
2033<table align="center" class="doxtable">
2034 <tr>
2035 <th>number of refinements </th><th> @f$\|p-p_h^\circ\|@f$ </th><th> @f$\|\mathbf{u}-\mathbf{u}_h\|@f$ </th><th> @f$\|(\mathbf{u}-\mathbf{u}_h) \cdot \mathbf{n}\|@f$ </th>
2036 </tr>
2037 <tr>
2038 <td> 2 </td><td> 1.072e-03 </td><td> 3.375e-03 </td><td> 4.762e-03 </td>
2039 </tr>
2040 <tr>
2041 <td> 3 </td><td> 1.347e-04 </td><td> 4.233e-04 </td><td> 5.982e-04 </td>
2042 </tr>
2043 <tr>
2044 <td> 4 </td><td> 1.685e-05 </td><td> 5.295e-05 </td><td> 7.487e-05 </td>
2045 </tr>
2046 <tr>
2047 <td> 5 </td><td> 2.107e-06 </td><td> 6.620e-06 </td><td> 9.362e-06 </td>
2048 </tr>
2049 <tr>
2050 <th>Conv.rate </th><th> 3.00 </th><th> 3.00 </th><th> 3.00 </th>
2051 </tr>
2052</table>
2053
2054Once more, the convergence rates of @f$\mbox{WG}(Q_2,Q_2;RT_{[2]})@f$ is
2055as expected, with values around 3.
2056 *
2057 *
2058<a name="step_61-PlainProg"></a>
2059<h1> The plain program</h1>
2060@include "step-61.cc"
2061*/
std::vector< bool > component_mask
void add_data_vector(const VectorType &data, const std::vector< std::string > &names, const DataVectorType type=type_automatic, const std::vector< DataComponentInterpretation::DataComponentInterpretation > &data_component_interpretation={})
virtual void build_patches(const unsigned int n_subdivisions=0)
Definition data_out.cc:1062
const unsigned int dofs_per_cell
const SmartPointer< const FiniteElement< dim, spacedim >, FEValuesBase< dim, spacedim > > fe
const Quadrature< dim > quadrature
Definition fe_values.h:172
void reinit(const TriaIterator< DoFCellAccessor< dim, spacedim, level_dof_access > > &cell)
void gauss_jordan()
virtual RangeNumberType value(const Point< dim > &p, const unsigned int component=0) const
Definition point.h:111
virtual value_type value(const Point< dim > &p) const
TensorFunction(const time_type initial_time=time_type(0.0))
unsigned int n_active_cells() const
void refine_global(const unsigned int times=1)
unsigned int n_cells() const
DerivativeForm< 1, spacedim, dim, Number > transpose(const DerivativeForm< 1, dim, spacedim, Number > &DF)
Point< 2 > first
Definition grid_out.cc:4623
typename ActiveSelector::active_cell_iterator active_cell_iterator
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:564
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints={}, const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
@ update_values
Shape function values.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt K
void approximate(const SynchronousIterators< std::tuple< typename DoFHandler< dim, spacedim >::active_cell_iterator, Vector< float >::iterator > > &cell, const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof_handler, const InputVector &solution, const unsigned int component)
void interpolate(const DoFHandler< dim, spacedim > &dof1, const InVector &u1, const DoFHandler< dim, spacedim > &dof2, OutVector &u2)
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
void refine(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold, const unsigned int max_to_mark=numbers::invalid_unsigned_int)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
Definition divergence.h:471
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:191
SymmetricTensor< 2, dim, Number > E(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > w(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
constexpr ReturnType< rank, T >::value_type & extract(T &t, const ArrayType &indices)
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask={})
void integrate_difference(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const ReadVector< Number > &fe_function, const Function< spacedim, Number > &exact_solution, OutVector &difference, const Quadrature< dim > &q, const NormType &norm, const Function< spacedim, double > *weight=nullptr, const double exponent=2.)
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
void copy(const T *begin, const T *end, U *dest)
int(& functions)(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
Definition loop.h:70
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
static constexpr double PI
Definition numbers.h:259
::VectorizedArray< Number, width > cos(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation