595 *
for (
unsigned int i = 0; i < this->n_source_centers; ++i)
599 *
std::exp(-x_minus_xi.norm_square() / (this->width * this->width));
608 *
const unsigned int = 0) const override
611 *
for (
unsigned int i = 0; i < this->n_source_centers; ++i)
616 * (-2 / (this->width * this->width) *
617 *
std::exp(-x_minus_xi.norm_square() / (this->width * this->width)) *
630 * This
class implements a function where the
scalar solution and its
negative
631 *
gradient are collected together. This function is used when computing the
632 * error of the HDG approximation and its implementation is to simply call
637 *
class SolutionAndGradient :
public Function<dim>,
protected SolutionBase<dim>
640 * SolutionAndGradient()
648 * Solution<dim> solution;
650 *
for (
unsigned int d = 0;
d < dim; ++
d)
652 * v[dim] = solution.value(p);
660 * Next comes the implementation of the convection velocity. As described in
661 * the introduction, we choose a velocity field that is @f$(y, -x)@f$ in 2
d and
662 * @f$(y, -x, 1)@f$ in 3
d. This gives a divergence-free velocity field.
669 * ConvectionVelocity()
682 * convection[0] = p[1];
683 * convection[1] = -p[0];
686 * convection[0] = p[1];
687 * convection[1] = -p[0];
701 * The last function we implement is the right hand side
for the
702 * manufactured solution. It is very similar to @ref step_7
"step-7", with the exception
703 * that we now have a convection term instead of the reaction term. Since
704 * the velocity field is incompressible, i.e., @f$\nabla \cdot \mathbf{c} =
705 * 0@f$, the advection term simply reads @f$\mathbf{c} \nabla u@f$.
709 *
class RightHandSide :
public Function<dim>,
protected SolutionBase<dim>
713 *
const unsigned int = 0) const override
715 * ConvectionVelocity<dim> convection_velocity;
718 *
for (
unsigned int i = 0; i < this->n_source_centers; ++i)
723 * ((2 * dim - 2 * convection * x_minus_xi -
724 * 4 * x_minus_xi.norm_square() / (this->width * this->width)) /
725 * (this->width * this->width) *
726 *
std::exp(-x_minus_xi.norm_square() / (this->width * this->width)));
739 * <a name=
"step_51-TheHDGsolverclass"></a>
740 * <h3>The HDG solver
class</h3>
744 * The HDG solution procedure follows closely that of @ref step_7
"step-7". The major
745 * difference is the use of three different sets of
DoFHandler and FE
747 * vectors. We also use
WorkStream to enable a multithreaded local solution
748 * process which exploits the embarrassingly
parallel nature of the local
749 * solver. For
WorkStream, we define the local operations on a cell and a
750 *
copy function into the global
matrix and vector. We
do this both
for the
751 * assembly (which is run twice, once when we generate the system matrix and
752 * once when we compute the element-interior solutions from the skeleton
753 * values) and
for the postprocessing where we
extract a solution that
754 * converges at higher order.
761 *
enum RefinementMode
764 * adaptive_refinement
767 * HDG(
const unsigned int degree,
const RefinementMode refinement_mode);
771 *
void setup_system();
772 *
void assemble_system(
const bool reconstruct_trace =
false);
774 *
void postprocess();
775 *
void refine_grid(
const unsigned int cycle);
776 *
void output_results(
const unsigned int cycle);
780 * Data
for the assembly and solution of the primal variables.
783 *
struct PerTaskData;
784 *
struct ScratchData;
788 * Post-processing the solution to obtain @f$u^*@f$ is an element-by-element
789 * procedure; as such, we
do not need to
assemble any global data and
do
790 * not declare any
'task data' for WorkStream to use.
793 *
struct PostProcessScratchData;
798 * work of the program.
801 *
void assemble_system_one_cell(
803 * ScratchData &scratch,
804 * PerTaskData &task_data);
806 *
void copy_local_to_global(
const PerTaskData &data);
808 *
void postprocess_one_cell(
810 * PostProcessScratchData &scratch,
811 *
unsigned int &empty_data);
818 * The
'local' solutions are interior to each element. These
819 * represent the primal solution field @f$u@f$ as well as the auxiliary
820 * field @f$\mathbf{q}@f$.
829 * The
new finite element type and corresponding <code>
DoFHandler</code> are
830 * used
for the global skeleton solution that couples the element-
level
841 * As stated in the introduction, HDG solutions can be post-processed to
842 * attain superconvergence rates of @f$\mathcal{O}(h^{p+2})@f$. The
843 * post-processed solution is a discontinuous finite element solution
844 * representing the primal variable on the interior of each cell. We define
845 * a FE type of degree @f$p+1@f$ to represent
this post-processed solution,
846 * which we only use
for output after constructing it.
855 * The degrees of freedom corresponding to the skeleton strongly enforce
856 * Dirichlet boundary conditions, just as in a continuous Galerkin finite
857 * element method. We can enforce the boundary conditions in an analogous
859 * handled in the same way as
for continuous finite elements: For the face
860 * elements which only define degrees of freedom on the face,
this process
861 * sets the solution on the refined side to coincide with the
862 * representation on the coarse side.
866 * Note that
for HDG, the elimination of hanging nodes is not the only
867 * possibility — in terms of the HDG theory, one could also use the
868 * unknowns from the refined side and express the local solution on the
869 * coarse side through the
trace values on the refined side. However, such
870 * a setup is not as easily implemented in terms of deal.II loops and not
880 * the actual
matrix object. When creating the sparsity pattern, we just
881 * have to additionally pass the size of local blocks.
889 * Same as @ref step_7
"step-7":
892 *
const RefinementMode refinement_mode;
899 * <a name=
"step_51-TheHDGclassimplementation"></a>
900 * <h3>The HDG
class implementation</h3>
905 * <a name=
"step_51-Constructor"></a>
906 * <h4>Constructor</h4>
907 * The constructor is similar to those in other examples, with the exception
909 * create a system of finite elements
for the local DG part, including the
914 * HDG<dim>::HDG(
const unsigned int degree,
const RefinementMode refinement_mode)
919 * , fe_u_post(degree + 1)
921 * , refinement_mode(refinement_mode)
929 * <a name=
"step_51-HDGsetup_system"></a>
930 * <h4>HDG::setup_system</h4>
931 * The system
for an HDG solution is setup in an analogous manner to most
932 * of the other tutorial programs. We are careful to distribute dofs with
933 * all of our
DoFHandler objects. The @p solution and @p system_matrix
934 * objects go with the global skeleton solution.
938 *
void HDG<dim>::setup_system()
940 * dof_handler_local.distribute_dofs(fe_local);
941 * dof_handler.distribute_dofs(fe);
942 * dof_handler_u_post.distribute_dofs(fe_u_post);
944 * std::cout <<
" Number of degrees of freedom: " << dof_handler.n_dofs()
947 * solution.reinit(dof_handler.n_dofs());
948 * system_rhs.reinit(dof_handler.n_dofs());
950 * solution_local.reinit(dof_handler_local.n_dofs());
951 * solution_u_post.reinit(dof_handler_u_post.n_dofs());
953 * constraints.
clear();
955 * std::map<types::boundary_id, const Function<dim> *> boundary_functions;
956 * Solution<dim> solution_function;
957 * boundary_functions[0] = &solution_function;
959 * boundary_functions,
962 * constraints.close();
966 * When creating the chunk sparsity pattern, we
first create the usual
967 *
dynamic sparsity pattern and then set the chunk size, which is
equal
968 * to the number of dofs on a face, when copying
this into the
final
975 * sparsity_pattern.copy_from(dsp, fe.n_dofs_per_face());
977 * system_matrix.reinit(sparsity_pattern);
985 * <a name=
"step_51-HDGPerTaskData"></a>
986 * <h4>HDG::PerTaskData</h4>
987 * Next comes the definition of the local data structures
for the
parallel
988 * assembly. The
first structure @p PerTaskData contains the local vector
989 * and
matrix that are written into the global
matrix, whereas the
990 * ScratchData contains all data that we need
for the local assembly. There
991 * is one variable worth noting here, namely the
boolean variable @p
992 * trace_reconstruct. As mentioned in the introduction, we solve the HDG
993 * system in two steps. First, we create a linear system
for the skeleton
994 * system where we condense the local part into it via the Schur complement
995 * @f$D-CA^{-1}B@f$. Then, we solve
for the local part
using the skeleton
996 * solution. For these two steps, we need the same matrices on the elements
997 * twice, which we want to compute by two assembly steps. Since most of the
998 * code is similar, we
do this with the same function but only
switch
999 * between the two based on a flag that we set when starting the
1000 * assembly. Since we need to pass
this information on to the local worker
1001 * routines, we store it once in the task data.
1004 *
template <
int dim>
1005 *
struct HDG<dim>::PerTaskData
1009 * std::vector<types::global_dof_index> dof_indices;
1011 *
bool trace_reconstruct;
1013 * PerTaskData(
const unsigned int n_dofs,
const bool trace_reconstruct)
1015 * , cell_vector(n_dofs)
1016 * , dof_indices(n_dofs)
1017 * , trace_reconstruct(trace_reconstruct)
1026 * <a name=
"step_51-HDGScratchData"></a>
1027 * <h4>HDG::ScratchData</h4>
1028 * @p ScratchData contains persistent data
for each
1030 * and vector objects should be familiar by now. There are two objects that
1031 * need to be discussed: `std::vector<std::vector<unsigned int> >
1032 * fe_local_support_on_face` and `std::vector<std::vector<unsigned int> >
1033 * fe_support_on_face`. These are used to indicate whether or not the finite
1034 * elements chosen have support (non-zero values) on a given face of the
1035 * reference cell
for the local part associated to @p fe_local and the
1036 * skeleton part @p fe. We
extract this information in the
1037 * constructor and store it once
for all cells that we work on. Had we not
1038 * stored
this information, we would be forced to
assemble a large number of
1039 * zero terms on each cell, which would significantly slow the program.
1042 *
template <
int dim>
1043 *
struct HDG<dim>::ScratchData
1056 * std::vector<Tensor<1, dim>> q_phi;
1057 * std::vector<double> q_phi_div;
1058 * std::vector<double> u_phi;
1059 * std::vector<Tensor<1, dim>> u_phi_grad;
1060 * std::vector<double> tr_phi;
1061 * std::vector<double> trace_values;
1063 * std::vector<std::vector<unsigned int>> fe_local_support_on_face;
1064 * std::vector<std::vector<unsigned int>> fe_support_on_face;
1066 * ConvectionVelocity<dim> convection_velocity;
1067 * RightHandSide<dim> right_hand_side;
1068 *
const Solution<dim> exact_solution;
1077 * : fe_values_local(fe_local, quadrature_formula, local_flags)
1078 * , fe_face_values_local(fe_local,
1079 * face_quadrature_formula,
1081 * , fe_face_values(fe, face_quadrature_formula, flags)
1082 * , ll_matrix(fe_local.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
1083 * , lf_matrix(fe_local.n_dofs_per_cell(), fe.n_dofs_per_cell())
1084 * , fl_matrix(fe.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
1085 * , tmp_matrix(fe.n_dofs_per_cell(), fe_local.n_dofs_per_cell())
1086 * , l_rhs(fe_local.n_dofs_per_cell())
1087 * , tmp_rhs(fe_local.n_dofs_per_cell())
1088 * , q_phi(fe_local.n_dofs_per_cell())
1089 * , q_phi_div(fe_local.n_dofs_per_cell())
1090 * , u_phi(fe_local.n_dofs_per_cell())
1091 * , u_phi_grad(fe_local.n_dofs_per_cell())
1092 * , tr_phi(fe.n_dofs_per_cell())
1093 * , trace_values(face_quadrature_formula.size())
1096 * , exact_solution()
1098 *
for (
const unsigned int face_no :
GeometryInfo<dim>::face_indices())
1099 * for (unsigned
int i = 0; i < fe_local.n_dofs_per_cell(); ++i)
1101 *
if (fe_local.has_support_on_face(i, face_no))
1102 * fe_local_support_on_face[face_no].push_back(i);
1105 *
for (
const unsigned int face_no :
GeometryInfo<dim>::face_indices())
1106 * for (unsigned
int i = 0; i < fe.n_dofs_per_cell(); ++i)
1108 *
if (fe.has_support_on_face(i, face_no))
1109 * fe_support_on_face[face_no].push_back(i);
1113 * ScratchData(
const ScratchData &sd)
1114 * : fe_values_local(sd.fe_values_local.get_fe(),
1115 * sd.fe_values_local.get_quadrature(),
1116 * sd.fe_values_local.get_update_flags())
1117 * , fe_face_values_local(sd.fe_face_values_local.get_fe(),
1118 * sd.fe_face_values_local.get_quadrature(),
1119 * sd.fe_face_values_local.get_update_flags())
1120 * , fe_face_values(sd.fe_face_values.get_fe(),
1121 * sd.fe_face_values.get_quadrature(),
1122 * sd.fe_face_values.get_update_flags())
1123 * , ll_matrix(sd.ll_matrix)
1124 * , lf_matrix(sd.lf_matrix)
1125 * , fl_matrix(sd.fl_matrix)
1126 * , tmp_matrix(sd.tmp_matrix)
1128 * , tmp_rhs(sd.tmp_rhs)
1130 * , q_phi_div(sd.q_phi_div)
1132 * , u_phi_grad(sd.u_phi_grad)
1133 * , tr_phi(sd.tr_phi)
1134 * , trace_values(sd.trace_values)
1135 * , fe_local_support_on_face(sd.fe_local_support_on_face)
1136 * , fe_support_on_face(sd.fe_support_on_face)
1137 * , exact_solution()
1146 * <a name=
"step_51-HDGPostProcessScratchData"></a>
1147 * <h4>HDG::PostProcessScratchData</h4>
1148 * @p PostProcessScratchData contains the data used by
WorkStream
1149 * when post-processing the local solution @f$u^*@f$. It is similar, but much
1150 * simpler, than @p ScratchData.
1153 *
template <
int dim>
1154 *
struct HDG<dim>::PostProcessScratchData
1159 * std::vector<double> u_values;
1160 * std::vector<Tensor<1, dim>> u_gradients;
1171 * : fe_values_local(fe_local, quadrature_formula, local_flags)
1172 * , fe_values(fe, quadrature_formula, flags)
1173 * , u_values(quadrature_formula.size())
1174 * , u_gradients(quadrature_formula.size())
1175 * ,
cell_matrix(fe.n_dofs_per_cell(), fe.n_dofs_per_cell())
1176 * , cell_rhs(fe.n_dofs_per_cell())
1177 * , cell_sol(fe.n_dofs_per_cell())
1180 * PostProcessScratchData(
const PostProcessScratchData &sd)
1181 * : fe_values_local(sd.fe_values_local.get_fe(),
1182 * sd.fe_values_local.get_quadrature(),
1183 * sd.fe_values_local.get_update_flags())
1184 * , fe_values(sd.fe_values.get_fe(),
1185 * sd.fe_values.get_quadrature(),
1186 * sd.fe_values.get_update_flags())
1187 * , u_values(sd.u_values)
1188 * , u_gradients(sd.u_gradients)
1190 * , cell_rhs(sd.cell_rhs)
1191 * , cell_sol(sd.cell_sol)
1200 * <a name=
"step_51-HDGassemble_system"></a>
1201 * <h4>HDG::assemble_system</h4>
1202 * The @p assemble_system function is similar to the one on @ref step_32
"step-32", where
1203 * the quadrature formula and the update flags are set up, and then
1204 * <code>
WorkStream</code> is used to
do the work in a multi-threaded
1205 * manner. The @p trace_reconstruct input parameter is used to decide
1206 * whether we are solving
for the global skeleton solution (
false) or the
1207 * local solution (true).
1211 * One thing worth noting for the multi-threaded execution of assembly is
1212 * the fact that the local computations in `assemble_system_one_cell()` call
1213 * into BLAS and LAPACK functions if those are available in deal.II. Thus,
1214 * the underlying BLAS/LAPACK library must support calls from multiple
1215 * threads at the same time. Most implementations do support this, but some
1216 * libraries need to be built in a specific way to avoid problems. For
1217 * example, OpenBLAS compiled without multithreading inside the BLAS/LAPACK
1218 * calls needs to built with a flag called `USE_LOCKING` set to true.
1221 * template <
int dim>
1222 *
void HDG<dim>::assemble_system(const
bool trace_reconstruct)
1224 *
const QGauss<dim> quadrature_formula(fe.degree + 1);
1225 *
const QGauss<dim - 1> face_quadrature_formula(fe.degree + 1);
1235 * PerTaskData task_data(fe.n_dofs_per_cell(), trace_reconstruct);
1236 * ScratchData scratch(fe,
1238 * quadrature_formula,
1239 * face_quadrature_formula,
1245 * dof_handler.end(),
1247 * &HDG<dim>::assemble_system_one_cell,
1248 * &HDG<dim>::copy_local_to_global,
1258 * <a name=
"step_51-HDGassemble_system_one_cell"></a>
1259 * <h4>HDG::assemble_system_one_cell</h4>
1260 * The real work of the HDG program is done by @p assemble_system_one_cell.
1261 * Assembling the local matrices @f$A, B,
C@f$ is done here, along with the
1262 * local contributions of the global
matrix @f$D@f$.
1265 *
template <
int dim>
1266 *
void HDG<dim>::assemble_system_one_cell(
1268 * ScratchData &scratch,
1269 * PerTaskData &task_data)
1273 * Construct iterator
for dof_handler_local
for FEValues reinit function.
1277 * cell->as_dof_handler_iterator(dof_handler_local);
1279 *
const unsigned int n_q_points =
1280 * scratch.fe_values_local.get_quadrature().size();
1281 *
const unsigned int n_face_q_points =
1282 * scratch.fe_face_values_local.get_quadrature().size();
1284 *
const unsigned int loc_dofs_per_cell =
1285 * scratch.fe_values_local.get_fe().n_dofs_per_cell();
1290 * scratch.ll_matrix = 0;
1291 * scratch.l_rhs = 0;
1292 *
if (!task_data.trace_reconstruct)
1294 * scratch.lf_matrix = 0;
1295 * scratch.fl_matrix = 0;
1296 * task_data.cell_matrix = 0;
1297 * task_data.cell_vector = 0;
1299 * scratch.fe_values_local.reinit(loc_cell);
1303 * We
first compute the cell-interior contribution to @p ll_matrix
matrix
1304 * (referred to as
matrix @f$A@f$ in the introduction) corresponding to
1305 * local-local coupling, as well as the local right-hand-side vector. We
1306 * store the values at each quadrature point
for the basis functions, the
1307 * right-hand-side value, and the convection velocity, in order to have
1308 * quick access to these fields.
1311 * for (
unsigned int q = 0; q < n_q_points; ++q)
1313 *
const double rhs_value = scratch.right_hand_side.value(
1314 * scratch.fe_values_local.quadrature_point(q));
1315 *
const Tensor<1, dim> convection = scratch.convection_velocity.value(
1316 * scratch.fe_values_local.quadrature_point(q));
1317 *
const double JxW = scratch.fe_values_local.JxW(q);
1318 *
for (
unsigned int k = 0; k < loc_dofs_per_cell; ++k)
1320 * scratch.q_phi[k] = scratch.fe_values_local[fluxes].value(k, q);
1321 * scratch.q_phi_div[k] =
1322 * scratch.fe_values_local[fluxes].divergence(k, q);
1323 * scratch.u_phi[k] = scratch.fe_values_local[
scalar].value(k, q);
1324 * scratch.u_phi_grad[k] =
1325 * scratch.fe_values_local[
scalar].gradient(k, q);
1327 *
for (
unsigned int i = 0; i < loc_dofs_per_cell; ++i)
1329 *
for (
unsigned int j = 0; j < loc_dofs_per_cell; ++j)
1330 * scratch.ll_matrix(i, j) +=
1331 * (scratch.q_phi[i] * scratch.q_phi[j] -
1332 * scratch.q_phi_div[i] * scratch.u_phi[j] +
1333 * scratch.u_phi[i] * scratch.q_phi_div[j] -
1334 * (scratch.u_phi_grad[i] * convection) * scratch.u_phi[j]) *
1336 * scratch.l_rhs(i) += scratch.u_phi[i] * rhs_value * JxW;
1342 * Face terms are assembled on all faces of all elements. This is in
1343 * contrast to more traditional DG methods, where each face is only visited
1344 * once in the assembly procedure.
1347 *
for (
const auto face_no : cell->face_indices())
1349 * scratch.fe_face_values_local.
reinit(loc_cell, face_no);
1350 * scratch.fe_face_values.reinit(cell, face_no);
1354 * The already obtained @f$\hat{u}@f$
values are needed when solving
for the
1358 *
if (task_data.trace_reconstruct)
1359 * scratch.fe_face_values.get_function_values(solution,
1360 * scratch.trace_values);
1362 *
for (
unsigned int q = 0; q < n_face_q_points; ++q)
1364 *
const double JxW = scratch.fe_face_values.JxW(q);
1366 * scratch.fe_face_values.quadrature_point(q);
1368 * scratch.fe_face_values.normal_vector(q);
1370 * scratch.convection_velocity.value(quadrature_point);
1374 * Here we compute the stabilization parameter discussed in the
1375 * introduction: since the diffusion is one and the diffusion
1376 * length
scale is set to 1/5, it simply results in a contribution
1377 * of 5
for the diffusion part and the magnitude of convection
1378 * through the element boundary in a centered scheme
for the
1382 *
const double tau_stab = (5. +
std::abs(convection * normal));
1386 * We store the non-zero flux and
scalar values, making use of the
1387 * support_on_face information we created in @p ScratchData.
1390 *
for (
unsigned int k = 0;
1391 * k < scratch.fe_local_support_on_face[face_no].size();
1394 *
const unsigned int kk =
1395 * scratch.fe_local_support_on_face[face_no][k];
1396 * scratch.q_phi[k] =
1397 * scratch.fe_face_values_local[fluxes].value(kk, q);
1398 * scratch.u_phi[k] =
1399 * scratch.fe_face_values_local[
scalar].value(kk, q);
1404 * When @p trace_reconstruct=
false, we are preparing to
assemble the
1405 * system
for the skeleton variable @f$\hat{u}@f$. If
this is the
case,
1406 * we must
assemble all local matrices associated with the problem:
1407 * local-local, local-face, face-local, and face-face. The
1408 * face-face
matrix is stored as @p TaskData::cell_matrix, so that
1409 * it can be assembled into the global system by @p
1410 * copy_local_to_global.
1413 *
if (!task_data.trace_reconstruct)
1415 *
for (
unsigned int k = 0;
1416 * k < scratch.fe_support_on_face[face_no].size();
1418 * scratch.tr_phi[k] = scratch.fe_face_values.shape_value(
1419 * scratch.fe_support_on_face[face_no][k], q);
1420 *
for (
unsigned int i = 0;
1421 * i < scratch.fe_local_support_on_face[face_no].size();
1423 *
for (
unsigned int j = 0;
1424 * j < scratch.fe_support_on_face[face_no].size();
1427 *
const unsigned int ii =
1428 * scratch.fe_local_support_on_face[face_no][i];
1429 *
const unsigned int jj =
1430 * scratch.fe_support_on_face[face_no][j];
1431 * scratch.lf_matrix(ii, jj) +=
1432 * ((scratch.q_phi[i] * normal +
1433 * (convection * normal - tau_stab) * scratch.u_phi[i]) *
1434 * scratch.tr_phi[j]) *
1439 * Note the
sign of the face_no-local
matrix. We negate
1440 * the
sign during assembly here so that we can use the
1445 * scratch.fl_matrix(jj, ii) -=
1446 * ((scratch.q_phi[i] * normal +
1447 * tau_stab * scratch.u_phi[i]) *
1448 * scratch.tr_phi[j]) *
1452 *
for (
unsigned int i = 0;
1453 * i < scratch.fe_support_on_face[face_no].size();
1455 *
for (
unsigned int j = 0;
1456 * j < scratch.fe_support_on_face[face_no].size();
1459 *
const unsigned int ii =
1460 * scratch.fe_support_on_face[face_no][i];
1461 *
const unsigned int jj =
1462 * scratch.fe_support_on_face[face_no][j];
1463 * task_data.cell_matrix(ii, jj) +=
1464 * ((convection * normal - tau_stab) * scratch.tr_phi[i] *
1465 * scratch.tr_phi[j]) *
1469 *
if (cell->face(face_no)->at_boundary() &&
1470 * (cell->face(face_no)->boundary_id() == 1))
1472 *
const double neumann_value =
1473 * -scratch.exact_solution.gradient(quadrature_point) *
1475 * convection * normal *
1476 * scratch.exact_solution.value(quadrature_point);
1477 *
for (
unsigned int i = 0;
1478 * i < scratch.fe_support_on_face[face_no].size();
1481 *
const unsigned int ii =
1482 * scratch.fe_support_on_face[face_no][i];
1483 * task_data.cell_vector(ii) +=
1484 * scratch.tr_phi[i] * neumann_value * JxW;
1491 * This last term adds the contribution of the term @f$\left<
w,\tau
1492 * u_h\right>_{\partial \mathcal T}@f$ to the local
matrix. As opposed
1493 * to the face matrices above, we need it in both assembly stages.
1496 *
for (
unsigned int i = 0;
1497 * i < scratch.fe_local_support_on_face[face_no].size();
1499 *
for (
unsigned int j = 0;
1500 * j < scratch.fe_local_support_on_face[face_no].size();
1503 *
const unsigned int ii =
1504 * scratch.fe_local_support_on_face[face_no][i];
1505 *
const unsigned int jj =
1506 * scratch.fe_local_support_on_face[face_no][j];
1507 * scratch.ll_matrix(ii, jj) +=
1508 * tau_stab * scratch.u_phi[i] * scratch.u_phi[j] * JxW;
1513 * When @p trace_reconstruct=
true, we are solving
for the local
1514 * solutions on an element by element basis. The local
1515 * right-hand-side is calculated by replacing the basis
functions @p
1516 * tr_phi in the @p lf_matrix computation by the computed
values @p
1517 * trace_values. Of course, the
sign of the
matrix is now minus
1518 * since we have moved everything to the other side of the equation.
1521 *
if (task_data.trace_reconstruct)
1522 *
for (
unsigned int i = 0;
1523 * i < scratch.fe_local_support_on_face[face_no].size();
1526 *
const unsigned int ii =
1527 * scratch.fe_local_support_on_face[face_no][i];
1528 * scratch.l_rhs(ii) -=
1529 * (scratch.q_phi[i] * normal +
1530 * scratch.u_phi[i] * (convection * normal - tau_stab)) *
1531 * scratch.trace_values[q] * JxW;
1538 * Once assembly of all of the local contributions is complete, we must
1539 * either: (1) assemble the global system, or (2) compute the local solution
1544 * scratch.ll_matrix.gauss_jordan();
1548 * For (1), we compute the Schur complement and add it to the @p
1552 *
if (task_data.trace_reconstruct ==
false)
1554 * scratch.fl_matrix.mmult(scratch.tmp_matrix, scratch.ll_matrix);
1555 * scratch.tmp_matrix.vmult_add(task_data.cell_vector, scratch.l_rhs);
1556 * scratch.tmp_matrix.mmult(task_data.cell_matrix,
1557 * scratch.lf_matrix,
1559 * cell->get_dof_indices(task_data.dof_indices);
1563 * For (2), we are simply solving (ll_matrix).(solution_local) = (l_rhs).
1564 * Hence, we multiply @p l_rhs by our already inverted local-local
matrix
1565 * and store the result
using the <code>
set_dof_values</code> function.
1570 * scratch.ll_matrix.vmult(scratch.tmp_rhs, scratch.l_rhs);
1571 * loc_cell->set_dof_values(scratch.tmp_rhs, solution_local);
1580 * <a name=
"step_51-HDGcopy_local_to_global"></a>
1581 * <h4>HDG::copy_local_to_global</h4>
1582 * If we are in the
first step of the solution, i.e. @p trace_reconstruct=
false,
1583 * then we
assemble the local matrices into the global system.
1586 *
template <
int dim>
1587 *
void HDG<dim>::copy_local_to_global(
const PerTaskData &data)
1589 *
if (data.trace_reconstruct ==
false)
1590 * constraints.distribute_local_to_global(data.cell_matrix,
1602 * <a name=
"step_51-HDGsolve"></a>
1603 * <h4>HDG::solve</h4>
1604 * The skeleton solution is solved
for by
using a BiCGStab solver with
1608 *
template <
int dim>
1609 *
void HDG<dim>::solve()
1612 * 1e-11 * system_rhs.l2_norm());
1616 * std::cout <<
" Number of BiCGStab iterations: "
1617 * << solver_control.last_step() << std::endl;
1619 * system_matrix.clear();
1620 * sparsity_pattern.reinit(0, 0, 0, 1);
1622 * constraints.distribute(solution);
1626 * Once we have solved
for the skeleton solution,
1627 * we can solve
for the local solutions in an element-by-element
1628 * fashion. We
do this by re-
using the same @p assemble_system function
1629 * but switching @p trace_reconstruct to
true.
1632 * assemble_system(
true);
1640 * <a name=
"step_51-HDGpostprocess"></a>
1641 * <h4>HDG::postprocess</h4>
1645 * The postprocess method serves two purposes. First, we want to construct a
1646 * post-processed
scalar variables in the element space of degree @f$p+1@f$ that
1647 * we hope will converge at order @f$p+2@f$. This is again an element-by-element
1648 * process and only involves the
scalar solution as well as the
gradient on
1649 * the local cell. To
do this, we introduce the already defined scratch data
1650 * together with some update flags and
run the work stream to
do this in
1655 * Secondly, we want to compute discretization errors just as we did in
1656 * @ref step_7
"step-7". The overall procedure is similar with calls to
1658 * the errors
for the
scalar variable and the
gradient variable. In @ref step_7
"step-7",
1660 * contributions. Here, we have a
DoFHandler with these two contributions
1661 * computed and
sorted by their vector component, <code>[0, dim)</code>
for
1663 * gradient and @p dim
for the
scalar. To compute their value, we hence use
1665 * SolutionAndGradient
class introduced above that contains the analytic
1666 * parts of either of them. Eventually, we also compute the L2-error of the
1667 * post-processed solution and add the results into the convergence table.
1670 * template <int dim>
1671 *
void HDG<dim>::postprocess()
1674 *
const QGauss<dim> quadrature_formula(fe_u_post.degree + 1);
1679 * PostProcessScratchData scratch(
1680 * fe_u_post, fe_local, quadrature_formula, local_flags, flags);
1683 * dof_handler_u_post.begin_active(),
1684 * dof_handler_u_post.end(),
1686 * PostProcessScratchData &scratch,
1687 *
unsigned int &data) {
1688 * this->postprocess_one_cell(cell, scratch, data);
1690 * std::function<
void(
const unsigned int &)>(),
1700 * SolutionAndGradient<dim>(),
1701 * difference_per_cell,
1705 *
const double L2_error =
1707 * difference_per_cell,
1711 * std::pair<unsigned int, unsigned int>(0, dim), dim + 1);
1714 * SolutionAndGradient<dim>(),
1715 * difference_per_cell,
1718 * &gradient_select);
1719 *
const double grad_error =
1721 * difference_per_cell,
1727 * difference_per_cell,
1730 *
const double post_error =
1732 * difference_per_cell,
1735 * convergence_table.add_value(
"cells",
triangulation.n_active_cells());
1736 * convergence_table.add_value(
"dofs", dof_handler.n_dofs());
1738 * convergence_table.add_value(
"val L2", L2_error);
1739 * convergence_table.set_scientific(
"val L2",
true);
1740 * convergence_table.set_precision(
"val L2", 3);
1742 * convergence_table.add_value(
"grad L2", grad_error);
1743 * convergence_table.set_scientific(
"grad L2",
true);
1744 * convergence_table.set_precision(
"grad L2", 3);
1746 * convergence_table.add_value(
"val L2-post", post_error);
1747 * convergence_table.set_scientific(
"val L2-post",
true);
1748 * convergence_table.set_precision(
"val L2-post", 3);
1756 * <a name=
"step_51-HDGpostprocess_one_cell"></a>
1757 * <h4>HDG::postprocess_one_cell</h4>
1761 * This is the actual work done
for the postprocessing. According to the
1762 * discussion in the introduction, we need to set up a system that projects
1764 * post-processed variable. Moreover, we need to set the average of the
new
1765 * post-processed variable to
equal the average of the
scalar DG solution
1770 * More technically speaking, the projection of the
gradient is a system
1771 * that would potentially fills our @p dofs_per_cell times @p dofs_per_cell
1772 *
matrix but is singular (the sum of all rows would be zero because the
1773 * constant function has zero gradient). Therefore, we take one row away and
1775 * row
for the
scalar part, even though we could pick any row
for @f$\mathcal
1776 * Q_{-p}@f$ elements. However, had we used
FE_DGP elements instead, the
first
1777 * row would correspond to the
constant part already and deleting
e.g. the
1778 * last row would give us a singular system. This way, our program can also
1779 * be used
for those elements.
1782 *
template <
int dim>
1783 *
void HDG<dim>::postprocess_one_cell(
1785 * PostProcessScratchData &scratch,
1789 * cell->as_dof_handler_iterator(dof_handler_local);
1791 * scratch.fe_values_local.reinit(loc_cell);
1792 * scratch.fe_values.reinit(cell);
1797 *
const unsigned int n_q_points = scratch.fe_values.get_quadrature().size();
1798 *
const unsigned int dofs_per_cell = scratch.fe_values.dofs_per_cell;
1800 * scratch.fe_values_local[
scalar].get_function_values(solution_local,
1801 * scratch.u_values);
1802 * scratch.fe_values_local[fluxes].get_function_values(solution_local,
1803 * scratch.u_gradients);
1806 *
for (
unsigned int i = 1; i < dofs_per_cell; ++i)
1808 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1811 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1812 * sum += (scratch.fe_values.shape_grad(i, q) *
1813 * scratch.fe_values.shape_grad(j, q)) *
1814 * scratch.fe_values.JxW(q);
1815 * scratch.cell_matrix(i, j) =
sum;
1819 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1820 * sum -= (scratch.fe_values.shape_grad(i, q) * scratch.u_gradients[q]) *
1821 * scratch.fe_values.JxW(q);
1822 * scratch.cell_rhs(i) =
sum;
1824 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1827 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1828 * sum += scratch.fe_values.shape_value(j, q) * scratch.fe_values.JxW(q);
1829 * scratch.cell_matrix(0, j) =
sum;
1833 *
for (
unsigned int q = 0; q < n_q_points; ++q)
1834 * sum += scratch.u_values[q] * scratch.fe_values.JxW(q);
1835 * scratch.cell_rhs(0) =
sum;
1840 * Having assembled all terms, we can again go on and solve the linear
1841 * system. We
invert the
matrix and then multiply the inverse by the
1842 * right hand side. An alternative (and more numerically stable) method
1843 * would have been to only factorize the
matrix and
apply the factorization.
1846 * scratch.cell_matrix.gauss_jordan();
1847 * scratch.cell_matrix.vmult(scratch.cell_sol, scratch.cell_rhs);
1848 * cell->distribute_local_to_global(scratch.cell_sol, solution_u_post);
1856 * <a name=
"step_51-HDGoutput_results"></a>
1857 * <h4>HDG::output_results</h4>
1858 * We have 3 sets of results that we would like to output: the local
1859 * solution, the post-processed local solution, and the skeleton solution. The
1860 * former 2 both
'live' on element volumes, whereas the latter lives on
1861 * codimension-1 surfaces
1862 * of the
triangulation. Our @p output_results function writes all local solutions
1863 * to the same
vtk file, even though they correspond to different
1864 *
DoFHandler objects. The graphical output
for the skeleton
1865 * variable is done through use of the
DataOutFaces class.
1868 *
template <
int dim>
1869 *
void HDG<dim>::output_results(
const unsigned int cycle)
1871 * std::string filename;
1872 *
switch (refinement_mode)
1874 *
case global_refinement:
1875 * filename =
"solution-global";
1877 *
case adaptive_refinement:
1878 * filename =
"solution-adaptive";
1884 * std::string face_out(filename);
1885 * face_out +=
"-face";
1889 * filename +=
".vtk";
1890 * std::ofstream output(filename);
1896 * We
first define the names and
types of the local solution,
1897 * and add the data to @p data_out.
1900 * std::vector<std::string> names(dim,
"gradient");
1901 * names.emplace_back(
"solution");
1902 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
1903 * component_interpretation(
1905 * component_interpretation[dim] =
1907 * data_out.add_data_vector(dof_handler_local,
1910 * component_interpretation);
1914 * The
second data item we add is the post-processed solution.
1915 * In
this case, it is a single
scalar variable belonging to
1919 * std::vector<std::string> post_name(1,
"u_post");
1920 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
1922 * data_out.add_data_vector(dof_handler_u_post,
1927 * data_out.build_patches(fe.degree);
1928 * data_out.write_vtk(output);
1932 * face_out +=
".vtk";
1933 * std::ofstream face_output(face_out);
1937 * The <code>
DataOutFaces</code>
class works analogously to the
1939 * defines the solution on the skeleton of the
triangulation. We treat it
1940 * as such here, and the code is similar to that above.
1944 * std::vector<std::string> face_name(1,
"u_hat");
1945 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
1948 * data_out_face.add_data_vector(dof_handler,
1951 * face_component_type);
1953 * data_out_face.build_patches(fe.degree);
1954 * data_out_face.write_vtk(face_output);
1960 * <a name=
"step_51-HDGrefine_grid"></a>
1961 * <h4>HDG::refine_grid</h4>
1965 * We implement two different refinement cases
for HDG, just as in
1966 * <code>@ref step_7
"step-7"</code>: adaptive_refinement and global_refinement. The
1967 * global_refinement option recreates the entire
triangulation every
1968 * time. This is because we want to use a finer sequence of meshes than what
1969 * we would get with one refinement step, namely 2, 3, 4, 6, 8, 12, 16, ...
1970 * elements per direction.
1975 * give a decent indication of the non-regular regions in the
scalar local
1979 *
template <
int dim>
1980 *
void HDG<dim>::refine_grid(
const unsigned int cycle)
1988 *
switch (refinement_mode)
1990 *
case global_refinement:
2001 *
case adaptive_refinement:
2007 * std::map<types::boundary_id, const Function<dim> *>
2013 * estimated_error_per_cell,
2014 * fe_local.component_mask(
2033 * Just as in @ref step_7
"step-7", we set the boundary indicator of two of the faces to 1
2034 * where we want to specify Neumann boundary conditions instead of Dirichlet
2035 * conditions. Since we re-create the
triangulation every time
for global
2036 * refinement, the flags are set in every refinement step, not just at the
2041 * for (const auto &face : cell->face_iterators())
2042 * if (face->at_boundary())
2045 * face->set_boundary_id(1);
2051 * <a name=
"step_51-HDGrun"></a>
2053 * The functionality here is basically the same as <code>@ref step_7
"step-7"</code>.
2054 * We
loop over 10 cycles, refining the grid on each one. At the
end,
2055 * convergence tables are created.
2058 *
template <
int dim>
2059 *
void HDG<dim>::run()
2061 *
for (
unsigned int cycle = 0; cycle < 10; ++cycle)
2063 * std::cout <<
"Cycle " << cycle <<
':' << std::endl;
2065 * refine_grid(cycle);
2067 * assemble_system(
false);
2070 * output_results(cycle);
2075 * There is one minor change
for the convergence table compared to @ref step_7
"step-7":
2076 * Since we did not
refine our mesh by a factor two in each cycle (but
2077 * rather used the sequence 2, 3, 4, 6, 8, 12, ...), we need to tell the
2078 * convergence rate evaluation about this. We
do this by setting the
2079 * number of cells as a reference column and additionally specifying the
2080 * dimension of the problem, which gives the necessary information
for the
2081 * relation between number of cells and mesh size.
2084 *
if (refinement_mode == global_refinement)
2086 * convergence_table.evaluate_convergence_rates(
2088 * convergence_table.evaluate_convergence_rates(
2090 * convergence_table.evaluate_convergence_rates(
2093 * convergence_table.write_text(std::cout);
2102 *
const unsigned int dim = 2;
2108 * Now
for the three calls to the main
class in complete analogy to
2109 * @ref step_7
"step-7".
2113 * std::cout <<
"Solving with Q1 elements, adaptive refinement"
2115 * <<
"============================================="
2119 * Step51::HDG<dim> hdg_problem(1, Step51::HDG<dim>::adaptive_refinement);
2120 * hdg_problem.run();
2122 * std::cout << std::endl;
2126 * std::cout <<
"Solving with Q1 elements, global refinement" << std::endl
2127 * <<
"===========================================" << std::endl
2130 * Step51::HDG<dim> hdg_problem(1, Step51::HDG<dim>::global_refinement);
2131 * hdg_problem.run();
2133 * std::cout << std::endl;
2137 * std::cout <<
"Solving with Q3 elements, global refinement" << std::endl
2138 * <<
"===========================================" << std::endl
2141 * Step51::HDG<dim> hdg_problem(3, Step51::HDG<dim>::global_refinement);
2142 * hdg_problem.run();
2144 * std::cout << std::endl;
2147 *
catch (std::exception &exc)
2149 * std::cerr << std::endl
2151 * <<
"----------------------------------------------------"
2153 * std::cerr <<
"Exception on processing: " << std::endl
2154 * << exc.what() << std::endl
2155 * <<
"Aborting!" << std::endl
2156 * <<
"----------------------------------------------------"
2162 * std::cerr << std::endl
2164 * <<
"----------------------------------------------------"
2166 * std::cerr <<
"Unknown exception!" << std::endl
2167 * <<
"Aborting!" << std::endl
2168 * <<
"----------------------------------------------------"
2176<a name=
"step_51-Results"></a><h1>Results</h1>
2179<a name=
"step_51-Programoutput"></a><h3>Program output</h3>
2182We
first have a look at the output generated by the program when
run in 2D. In
2183the four images below, we show the solution
for polynomial degree @f$p=1@f$
2184and cycles 2, 3, 4, and 8 of the program. In the plots, we overlay the data
2185generated from the
internal data (DG part) with the skeleton part (@f$\hat{u}@f$)
2186into the same plot. We had to generate two different data sets because cells
2187and faces represent different geometric entities, the combination of which (in
2188the same file) is not supported in the VTK output of deal.II.
2190The images show the distinctive features of HDG: The cell solution (colored
2191surfaces) is discontinuous between the cells. The solution on the skeleton
2192variable sits on the faces and ties together the local parts. The skeleton
2193solution is not continuous on the
vertices where the faces meet, even though
2194its
values are quite close along lines in the same coordinate direction. The
2195skeleton solution can be interpreted as a rubber spring between the two sides
2196that balances the jumps in the solution (or rather, the flux @f$\kappa \nabla u
2197+ \mathbf{c} u@f$). From the picture at the top left, it is clear that
2198the bulk solution frequently over- and undershoots and that the
2199skeleton variable in indeed a better approximation to the exact
2200solution;
this explains why we can get a better solution
using a
2203As the mesh is refined, the jumps between the cells get
2204small (we represent a smooth solution), and the skeleton solution approaches
2205the interior parts. For cycle 8, there is no visible difference in the two
2206variables. We also see how boundary conditions are implemented weakly and that
2207the interior variables
do not exactly satisfy boundary conditions. On the
2208lower and left boundaries, we set Neumann boundary conditions, whereas we set
2209Dirichlet conditions on the right and top boundaries.
2211<table align=
"center">
2213 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.sol_2.png" alt=
""></td>
2214 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.sol_3.png" alt=
""></td>
2217 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.sol_4.png" alt=
""></td>
2218 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.sol_8.png" alt=
""></td>
2222Next, we have a look at the post-processed solution, again at cycles 2, 3, 4,
2223and 8. This is a discontinuous solution that is locally described by
second
2224order polynomials. While the solution does not look very good on the mesh of
2225cycle two, it looks much better
for cycles three and four. As shown by the
2226convergence table below, we find that is also converges more quickly to the
2229<table align=
"center">
2231 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.post_2.png" alt=
""></td>
2232 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.post_3.png" alt=
""></td>
2235 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.post_4.png" alt=
""></td>
2236 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.post_8.png" alt=
""></td>
2240Finally, we look at the solution
for @f$p=3@f$ at cycle 2. Despite the coarse
2241mesh with only 64 cells, the post-processed solution is similar in quality
2242to the linear solution (not post-processed) at cycle 8 with 4,096
2243cells. This clearly shows the superiority of high order methods
for smooth
2246<table align=
"center">
2248 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.sol_q3_2.png" alt=
""></td>
2249 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.post_q3_2.png" alt=
""></td>
2253<a name=
"step_51-Convergencetables"></a><h4>Convergence tables</h4>
2256When the program is
run, it also outputs information about the respective
2257steps and convergence tables with errors in the various components in the
2258end. In 2D, the convergence tables look the following:
2261Q1 elements, adaptive refinement:
2262cells dofs val
L2 grad
L2 val
L2-post
2263 16 80 1.804e+01 2.207e+01 1.798e+01
2264 31 170 9.874e+00 1.322e+01 9.798e+00
2265 61 314 7.452e-01 3.793e+00 4.891e-01
2266 121 634 3.240e-01 1.511e+00 2.616e-01
2267 238 1198 8.585e-02 8.212e-01 1.808e-02
2268 454 2290 4.802e-02 5.178e-01 2.195e-02
2269 898 4378 2.561e-02 2.947e-01 4.318e-03
2270 1720 7864 1.306e-02 1.664e-01 2.978e-03
2271 3271 14638 7.025e-03 9.815e-02 1.075e-03
2272 6217 27214 4.119e-03 6.407e-02 9.975e-04
2274Q1 elements, global refinement:
2275cells dofs val
L2 grad
L2 val
L2-post
2276 16 80 1.804e+01 - 2.207e+01 - 1.798e+01 -
2277 36 168 6.125e+00 2.66 9.472e+00 2.09 6.084e+00 2.67
2278 64 288 9.785e-01 6.38 4.260e+00 2.78 7.102e-01 7.47
2279 144 624 2.730e-01 3.15 1.866e+00 2.04 6.115e-02 6.05
2280 256 1088 1.493e-01 2.10 1.046e+00 2.01 2.880e-02 2.62
2281 576 2400 6.965e-02 1.88 4.846e-01 1.90 9.204e-03 2.81
2282 1024 4224 4.018e-02 1.91 2.784e-01 1.93 4.027e-03 2.87
2283 2304 9408 1.831e-02 1.94 1.264e-01 1.95 1.236e-03 2.91
2284 4096 16640 1.043e-02 1.96 7.185e-02 1.96 5.306e-04 2.94
2285 9216 37248 4.690e-03 1.97 3.228e-02 1.97 1.599e-04 2.96
2287Q3 elements, global refinement:
2288cells dofs val
L2 grad
L2 val
L2-post
2289 16 160 3.613e-01 - 1.891e+00 - 3.020e-01 -
2290 36 336 6.411e-02 4.26 5.081e-01 3.24 3.238e-02 5.51
2291 64 576 3.480e-02 2.12 2.533e-01 2.42 5.277e-03 6.31
2292 144 1248 8.297e-03 3.54 5.924e-02 3.58 6.330e-04 5.23
2293 256 2176 2.254e-03 4.53 1.636e-02 4.47 1.403e-04 5.24
2294 576 4800 4.558e-04 3.94 3.277e-03 3.96 1.844e-05 5.01
2295 1024 8448 1.471e-04 3.93 1.052e-03 3.95 4.378e-06 5.00
2296 2304 18816 2.956e-05 3.96 2.104e-04 3.97 5.750e-07 5.01
2297 4096 33280 9.428e-06 3.97 6.697e-05 3.98 1.362e-07 5.01
2298 9216 74496 1.876e-06 3.98 1.330e-05 3.99 1.788e-08 5.01
2302One can see the error reduction upon grid refinement, and
for the cases where
2303global refinement was performed, also the convergence rates. The quadratic
2304convergence rates of Q1 elements in the @f$L_2@f$
norm for both the
scalar
2305variable and the
gradient variable is apparent, as is the cubic rate
for the
2306postprocessed
scalar variable in the @f$L_2@f$
norm. Note
this distinctive
2307feature of an HDG solution. In typical continuous finite elements, the
2308gradient of the solution of order @f$p@f$ converges at rate @f$p@f$ only, as
2309opposed to @f$p+1@f$
for the actual solution. Even though superconvergence
2310results
for finite elements are also available (
e.g. superconvergent patch
2311recovery
first introduced by Zienkiewicz and Zhu), these are typically limited
2312to structured meshes and other special cases. For Q3 HDG variables, the
scalar
2313variable and
gradient converge at fourth order and the postprocessed
scalar
2314variable at fifth order.
2316The same convergence rates are observed in 3
d.
2318Q1 elements, adaptive refinement:
2319cells dofs val
L2 grad
L2 val
L2-post
2320 8 144 7.122e+00 1.941e+01 6.102e+00
2321 29 500 3.309e+00 1.023e+01 2.145e+00
2322 113 1792 2.204e+00 1.023e+01 1.912e+00
2323 379 5732 6.085e-01 5.008e+00 2.233e-01
2324 1317 19412 1.543e-01 1.464e+00 4.196e-02
2325 4579 64768 5.058e-02 5.611e-01 9.521e-03
2326 14596 199552 2.129e-02 3.122e-01 4.569e-03
2327 46180 611400 1.033e-02 1.622e-01 1.684e-03
2328144859 1864212 5.007e-03 8.371e-02 7.364e-04
2329451060 5684508 2.518e-03 4.562e-02 3.070e-04
2331Q1 elements, global refinement:
2332cells dofs val
L2 grad
L2 val
L2-post
2333 8 144 7.122e+00 - 1.941e+01 - 6.102e+00 -
2334 27 432 5.491e+00 0.64 2.184e+01 -0.29 4.448e+00 0.78
2335 64 960 3.646e+00 1.42 1.299e+01 1.81 3.306e+00 1.03
2336 216 3024 1.595e+00 2.04 8.550e+00 1.03 1.441e+00 2.05
2337 512 6912 6.922e-01 2.90 5.306e+00 1.66 2.511e-01 6.07
2338 1728 22464 2.915e-01 2.13 2.490e+00 1.87 8.588e-02 2.65
2339 4096 52224 1.684e-01 1.91 1.453e+00 1.87 4.055e-02 2.61
2340 13824 172800 7.972e-02 1.84 6.861e-01 1.85 1.335e-02 2.74
2341 32768 405504 4.637e-02 1.88 3.984e-01 1.89 5.932e-03 2.82
2342110592 1354752 2.133e-02 1.92 1.830e-01 1.92 1.851e-03 2.87
2344Q3 elements, global refinement:
2345cells dofs val
L2 grad
L2 val
L2-post
2346 8 576 5.670e+00 - 1.868e+01 - 5.462e+00 -
2347 27 1728 1.048e+00 4.16 6.988e+00 2.42 8.011e-01 4.73
2348 64 3840 2.831e-01 4.55 2.710e+00 3.29 1.363e-01 6.16
2349 216 12096 7.883e-02 3.15 7.721e-01 3.10 2.158e-02 4.55
2350 512 27648 3.642e-02 2.68 3.305e-01 2.95 5.231e-03 4.93
2351 1728 89856 8.546e-03 3.58 7.581e-02 3.63 7.640e-04 4.74
2352 4096 208896 2.598e-03 4.14 2.313e-02 4.13 1.783e-04 5.06
2353 13824 691200 5.314e-04 3.91 4.697e-03 3.93 2.355e-05 4.99
2354 32768 1622016 1.723e-04 3.91 1.517e-03 3.93 5.602e-06 4.99
2355110592 5419008 3.482e-05 3.94 3.055e-04 3.95 7.374e-07 5.00
2358<a name=
"step_51-Comparisonwithcontinuousfiniteelements"></a><h3>Comparison with continuous finite elements</h3>
2361<a name=
"step_51-Resultsfor2D"></a><h4>Results
for 2D</h4>
2364The convergence tables verify the expected convergence rates stated in the
2365introduction. Now, we want to show a quick comparison of the computational
2366efficiency of the HDG method compared to a usual finite element (continuous
2367Galkerin) method on the problem of
this tutorial. Of course, stability aspects
2368of the HDG method compared to continuous finite elements
for
2369transport-dominated problems are also important in practice, which is an
2370aspect not seen on a problem with smooth analytic solution. In the picture
2371below, we compare the @f$L_2@f$ error as a function of the number of degrees of
2372freedom (left) and of the computing time spent in the linear solver (right)
2373for two space dimensions of continuous finite elements (CG) and the hybridized
2374discontinuous Galerkin method presented in
this tutorial. As opposed to the
2375tutorial where we only use unpreconditioned BiCGStab, the times shown in the
2376figures below use the Trilinos algebraic multigrid preconditioner in
2381<table align=
"center">
2383 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2d_plain.png" width=
"400" alt=
""></td>
2384 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2dt_plain.png" width=
"400" alt=
""></td>
2388The results in the graphs show that the HDG method is slower than continuous
2389finite elements at @f$p=1@f$, about equally fast
for cubic elements and
2390faster
for sixth order elements. However, we have seen above that the HDG
2391method actually produces solutions which are more accurate than what is
2392represented in the original variables. Therefore, in the next two plots below
2393we instead display the error of the post-processed solution
for HDG (denoted
2394by @f$p=1^*@f$
for example). We now see a clear advantage of HDG
for the same
2395amount of work
for both @f$p=3@f$ and @f$p=6@f$, and about the same quality
2398<table align=
"center">
2400 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2d_post.png" width=
"400" alt=
""></td>
2401 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2dt_post.png" width=
"400" alt=
""></td>
2405Since the HDG method actually produces results converging as
2406@f$h^{p+2}@f$, we should compare it to a continuous Galerkin
2407solution with the same asymptotic convergence behavior, i.e.,
FE_Q with degree
2408@f$p+1@f$. If we
do this, we get the convergence curves below. We see that
2409CG with
second order polynomials is again clearly better than HDG with
2410linears. However, the advantage of HDG
for higher orders remains.
2412<table align=
"center">
2414 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2d_postb.png" width=
"400" alt=
""></td>
2415 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.2dt_postb.png" width=
"400" alt=
""></td>
2419The results are in line with properties of DG methods in
general: Best
2420performance is typically not achieved
for linear elements, but rather at
2421somewhat higher order, usually around @f$p=3@f$. This is because of a
2422volume-to-surface effect
for discontinuous solutions with too much of the
2423solution living on the surfaces and hence duplicating work when the elements
2424are linear. Put in other words, DG methods are often most efficient when used
2425at relatively high order, despite their focus on a discontinuous (and hence,
2426seemingly low accurate) representation of solutions.
2428<a name=
"step_51-Resultsfor3D"></a><h4>Results
for 3D</h4>
2431We now show the same figures in 3D: The
first row shows the number of degrees
2432of freedom and computing time versus the @f$L_2@f$ error in the
scalar variable
2433@f$u@f$
for CG and HDG at order @f$p@f$, the
second row shows the
2434post-processed HDG solution instead of the original one, and the third row
2435compares the post-processed HDG solution with CG at order @f$p+1@f$. In 3D,
2436the
volume-to-surface effect makes the cost of HDG somewhat higher and the CG
2437solution is clearly better than HDG
for linears by any metric. For cubics, HDG
2438and CG are of similar quality, whereas HDG is again more efficient
for sixth
2439order polynomials. One can alternatively also use the combination of
FE_DGP
2441polynomials of degree @f$p@f$ but Legendre polynomials of <i>complete</i>
2442degree @f$p@f$. There are fewer degrees of freedom on the skeleton variable
2443for FE_FaceP for a given mesh size, but the solution quality (error vs. number
2444of DoFs) is very similar to the results
for FE_FaceQ.
2446<table align=
"center">
2448 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3d_plain.png" width=
"400" alt=
""></td>
2449 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3dt_plain.png" width=
"400" alt=
""></td>
2452 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3d_post.png" width=
"400" alt=
""></td>
2453 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3dt_post.png" width=
"400" alt=
""></td>
2456 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3d_postb.png" width=
"400" alt=
""></td>
2457 <td><img src=
"https://www.dealii.org/images/steps/developer/step-51.3dt_postb.png" width=
"400" alt=
""></td>
2461One
final note on the efficiency comparison: We tried to use
general-purpose
2462sparse
matrix structures and similar solvers (optimal AMG preconditioners
for
2463both without particular tuning of the AMG parameters on any of them) to give a
2464fair picture of the cost versus accuracy of two methods, on a toy example. It
2465should be noted however that geometric multigrid (GMG)
for continuous finite
2466elements is about a factor four to five faster
for @f$p=3@f$ and @f$p=6@f$. As of
24672019, optimal-complexity iterative solvers
for HDG are still under development
2468in the research community. Also, there are other implementation aspects
for CG
2469available such as fast
matrix-free approaches as shown in @ref step_37
"step-37" that make
2470higher order continuous elements more competitive. Again, it is not clear to
2471the authors of the tutorial whether similar improvements could be made
for
2472HDG. We refer to <a href=
"https://dx.doi.org/10.1137/16M110455X">Kronbichler
2473and Wall (2018)</a>
for a recent efficiency evaluation.
2476<a name=
"step_51-Possibilitiesforimprovements"></a><h3>Possibilities
for improvements</h3>
2479As already mentioned in the introduction, one possibility is to implement
2480another post-processing technique as discussed in the literature.
2482A
second item that is not done optimally relates to the performance of
this
2483program, which is of course an issue in practical applications (weighing in
2484also the better solution quality of (H)DG methods
for transport-dominated
2485problems). Let us look at
2486the computing time of the tutorial program and the share of the individual
2489<table align=
"center" class=
"doxtable">
2496 <th>Trace reconstruct</th>
2497 <th>Post-processing</th>
2503 <th colspan=
"6">Relative share</th>
2506 <td align=
"left">2D, Q1, cycle 9, 37,248 dofs</td>
2507 <td align=
"center">5.34s</td>
2508 <td align=
"center">0.7%</td>
2509 <td align=
"center">1.2%</td>
2510 <td align=
"center">89.5%</td>
2511 <td align=
"center">0.9%</td>
2512 <td align=
"center">2.3%</td>
2513 <td align=
"center">5.4%</td>
2516 <td align=
"left">2D, Q3, cycle 9, 74,496 dofs</td>
2517 <td align=
"center">22.2s</td>
2518 <td align=
"center">0.4%</td>
2519 <td align=
"center">4.3%</td>
2520 <td align=
"center">84.1%</td>
2521 <td align=
"center">4.1%</td>
2522 <td align=
"center">3.5%</td>
2523 <td align=
"center">3.6%</td>
2526 <td align=
"left">3D, Q1, cycle 7, 172,800 dofs</td>
2527 <td align=
"center">9.06s</td>
2528 <td align=
"center">3.1%</td>
2529 <td align=
"center">8.9%</td>
2530 <td align=
"center">42.7%</td>
2531 <td align=
"center">7.0%</td>
2532 <td align=
"center">20.6%</td>
2533 <td align=
"center">17.7%</td>
2536 <td align=
"left">3D, Q3, cycle 7, 691,200 dofs</td>
2537 <td align=
"center">516s</td>
2538 <td align=
"center">0.6%</td>
2539 <td align=
"center">34.5%</td>
2540 <td align=
"center">13.4%</td>
2541 <td align=
"center">32.8%</td>
2542 <td align=
"center">17.1%</td>
2543 <td align=
"center">1.5%</td>
2547As can be seen from the table, the solver and assembly calls dominate the
2548runtime of the program. This also gives a clear indication of where
2549improvements would make the most sense:
2552 <li> Better linear solvers: We use a BiCGStab iterative solver without
2553 preconditioner, where the number of iteration increases with increasing
2554 problem size (the number of iterations
for Q1 elements and global
2555 refinements starts at 35
for the small sizes but increase up to 701
for the
2556 largest size). To
do better, one could
for example use an algebraic
2557 multigrid preconditioner from Trilinos, or some more advanced variants as
2558 the one discussed in <a
2559 href=
"https://dx.doi.org/10.1137/16M110455X">Kronbichler and Wall
2560 (2018)</a>. For diffusion-dominated problems such as the problem at hand
2561 with finer meshes, such a solver can be designed that uses the
matrix-vector
2563 long as we are not working in
parallel with
MPI. For
MPI-parallelized
2566 <li> Speed up assembly by pre-assembling parts that
do not change from one
2567 cell to another (those that
do neither contain variable coefficients nor
2568 mapping-dependent terms).
2572<a name=
"step_51-PlainProg"></a>
2573<h1> The plain program</h1>
2574@include
"step-51.cc"
void mmult(FullMatrix< number2 > &C, const FullMatrix< number2 > &B, const bool adding=false) const
virtual RangeNumberType value(const Point< dim > &p, const unsigned int component=0) const
virtual void vector_value(const Point< dim > &p, Vector< RangeNumberType > &values) const
static void estimate(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim - 1 > &quadrature, const std::map< types::boundary_id, const Function< spacedim, Number > * > &neumann_bc, const ReadVector< Number > &solution, Vector< float > &error, const ComponentMask &component_mask={}, const Function< spacedim > *coefficients=nullptr, const unsigned int n_threads=numbers::invalid_unsigned_int, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id, const types::material_id material_id=numbers::invalid_material_id, const Strategy strategy=cell_diameter_over_24)
virtual value_type value(const Point< dim > &p) const
#define AssertDimension(dim1, dim2)
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())
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
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.
#define DEAL_II_NOT_IMPLEMENTED()
@ component_is_part_of_vector
Expression fabs(const Expression &x)
Expression sign(const Expression &x)
void subdivided_hyper_cube(Triangulation< dim, spacedim > &tria, const unsigned int repetitions, 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)
void refine_and_coarsen_fixed_number(Triangulation< dim, spacedim > &triangulation, const Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max())
@ matrix
Contents is actually a matrix.
@ general
No special properties.
void cell_matrix(FullMatrix< double > &M, const FEValuesBase< dim > &fe, const FEValuesBase< dim > &fetest, const ArrayView< const std::vector< double > > &velocity, const double factor=1.)
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
void L2(Vector< number > &result, const FEValuesBase< dim > &fe, const std::vector< double > &input, const double factor=1.)
SymmetricTensor< 2, dim, Number > C(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > w(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void apply(const Kokkos::TeamPolicy< MemorySpace::Default::kokkos_space::execution_space >::member_type &team_member, const Kokkos::View< Number *, MemorySpace::Default::kokkos_space > shape_data, const ViewTypeIn in, ViewTypeOut out)
constexpr ReturnType< rank, T >::value_type & extract(T &t, const ArrayType &indices)
VectorType::value_type * end(VectorType &V)
T sum(const T &t, const MPI_Comm mpi_communicator)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
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 run(const std::vector< std::vector< Iterator > > &colored_iterators, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length=2 *MultithreadInfo::n_threads(), const unsigned int chunk_size=8)
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)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
void set_dof_values(const DoFCellAccessor< dim, spacedim, lda > &cell, const Vector< number > &local_values, OutputVector &values, const bool perform_check)
static constexpr double PI
::VectorizedArray< Number, width > exp(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)
DEAL_II_HOST constexpr Number trace(const SymmetricTensor< 2, dim2, Number > &)