481 *
const unsigned int = 0)
const override
487 *
template <
typename number>
490 *
const unsigned int = 0) const
499 * This next
class represents the diffusion coefficient. We use a variable
500 * coefficient which is 100.0 at any
point where at least one coordinate is
501 * less than -0.5, and 1.0 at all other points. As above, a separate
value()
503 * average() function computes the arithmetic average for a set of points.
507 * class Coefficient : public
Function<dim>
511 *
const unsigned int = 0)
const override;
513 *
template <
typename number>
515 *
const unsigned int = 0)
const;
517 *
template <
typename number>
522 * When
using a coefficient in the
MatrixFree framework, we also
523 * need a function that creates a
Table of coefficient
values for a
524 * set of cells provided by the
MatrixFree operator argument here.
527 *
template <
typename number>
528 * std::shared_ptr<Table<2, VectorizedArray<number>>> make_coefficient_table(
535 *
double Coefficient<dim>::value(
const Point<dim> &p,
const unsigned int)
const
537 *
for (
int d = 0;
d < dim; ++
d)
548 *
template <
typename number>
551 *
const unsigned int)
const
554 *
for (
unsigned int i = 0; i < VectorizedArray<number>::size(); ++i)
556 *
for (
int d = 0;
d < dim; ++
d)
557 *
if (p[d][i] < -0.5)
559 * return_value[i] = 100.0;
564 *
return return_value;
570 *
template <
typename number>
571 * number Coefficient<dim>::average_value(
575 *
for (
unsigned int i = 0; i < points.size(); ++i)
576 * average +=
value(points[i]);
577 * average /= points.size();
585 *
template <
typename number>
586 * std::shared_ptr<Table<2, VectorizedArray<number>>>
587 * Coefficient<dim>::make_coefficient_table(
590 *
auto coefficient_table =
591 * std::make_shared<Table<2, VectorizedArray<number>>>();
593 *
FEEvaluation<dim, -1, 0, 1, number> fe_eval(mf_storage);
595 *
const unsigned int n_cells = mf_storage.n_cell_batches();
597 * coefficient_table->reinit(n_cells, 1);
599 *
for (
unsigned int cell = 0; cell <
n_cells; ++cell)
601 * fe_eval.reinit(cell);
604 *
for (
const unsigned int q : fe_eval.quadrature_point_indices())
605 * average_value +=
value(fe_eval.quadrature_point(q));
606 * average_value /= fe_eval.n_q_points;
608 * (*coefficient_table)(cell, 0) = average_value;
611 *
return coefficient_table;
619 * <a name=
"step_50-Runtimeparameters"></a>
620 * <h3>Run time parameters</h3>
625 * structure @p
Settings parses and stores these parameters to be queried
626 * throughout the program.
631 *
bool try_parse(
const std::string &prm_filename);
643 *
double smoother_dampen;
644 *
unsigned int smoother_steps;
645 *
unsigned int n_steps;
651 *
bool Settings::try_parse(
const std::string &prm_filename)
657 *
"The problem dimension.");
658 * prm.declare_entry(
"n_steps",
661 *
"Number of adaptive refinement steps.");
662 * prm.declare_entry(
"smoother dampen",
665 *
"Dampen factor for the smoother.");
666 * prm.declare_entry(
"smoother steps",
669 *
"Number of smoother steps.");
670 * prm.declare_entry(
"solver",
673 *
"Switch between matrix-free GMG, "
674 *
"matrix-based GMG, and AMG.");
675 * prm.declare_entry(
"output",
678 *
"Output graphical results.");
680 *
if (prm_filename.empty())
683 * <<
"**** Error: No input file provided!\n"
684 * <<
"**** Error: Call this program as './step-50 input.prm\n"
686 * <<
"**** You may want to use one of the input files in this\n"
687 * <<
"**** directory, or use the following default values\n"
688 * <<
"**** to create an input file:\n";
696 * prm.parse_input(prm_filename);
698 *
catch (std::exception &e)
701 * std::cerr <<
e.what() << std::endl;
705 *
if (prm.get(
"solver") ==
"MF")
706 * this->solver = gmg_mf;
707 *
else if (prm.get(
"solver") ==
"MB")
708 * this->solver = gmg_mb;
709 *
else if (prm.get(
"solver") ==
"AMG")
710 * this->solver = amg;
714 * this->dimension = prm.get_integer(
"dim");
715 * this->n_steps = prm.get_integer(
"n_steps");
716 * this->smoother_dampen = prm.get_double(
"smoother dampen");
717 * this->smoother_steps = prm.get_integer(
"smoother steps");
718 * this->output = prm.get_bool(
"output");
728 * <a name=
"step_50-LaplaceProblemclass"></a>
729 * <h3>LaplaceProblem
class</h3>
733 * This is the main
class of the program. It looks very similar to
734 * @ref step_16
"step-16", @ref step_37
"step-37", and @ref step_40
"step-40". For the
MatrixFree setup, we use the
737 * that the polynomial degree is a
template parameter of
this class. This is
738 * necessary
for the
matrix-free code.
741 *
template <
int dim,
int degree>
742 *
class LaplaceProblem
745 * LaplaceProblem(
const Settings &settings);
751 * We will use the following
types throughout the program. First the
753 *
matrix-free implementation, we use @p
float for the
level operators.
776 *
void setup_system();
777 *
void setup_multigrid();
778 *
void assemble_system();
779 *
void assemble_multigrid();
780 *
void assemble_rhs();
783 *
void refine_grid();
784 *
void output_results(
const unsigned int cycle);
802 * MatrixFreeActiveMatrix mf_system_matrix;
819 * The only interesting part about the constructor is that we construct the
820 * multigrid hierarchy unless we use AMG. For that, we need to parse the
821 *
run time parameters before
this constructor completes.
824 *
template <
int dim,
int degree>
825 * LaplaceProblem<dim, degree>::LaplaceProblem(
const Settings &settings)
826 * : settings(settings)
827 * , mpi_communicator(MPI_COMM_WORLD)
833 * (settings.solver == Settings::amg) ?
851 * <a name=
"step_50-LaplaceProblemsetup_system"></a>
852 * <h4>LaplaceProblem::setup_system()</h4>
856 * Unlike @ref step_16
"step-16" and @ref step_37
"step-37", we
split the set up into two parts,
857 * setup_system() and setup_multigrid(). Here is the typical setup_system()
858 * function for the active mesh found in most tutorials. For matrix-free, the
859 * active mesh set up is similar to @ref step_37 "step-37"; for matrix-based (GMG and AMG
860 * solvers), the setup is similar to @ref step_40 "step-40".
863 * template <
int dim,
int degree>
864 *
void LaplaceProblem<dim, degree>::setup_system()
868 * dof_handler.distribute_dofs(fe);
870 * locally_relevant_dofs =
872 * locally_owned_dofs = dof_handler.locally_owned_dofs();
874 * solution.reinit(locally_owned_dofs, mpi_communicator);
875 * right_hand_side.reinit(locally_owned_dofs, mpi_communicator);
876 * constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
881 * constraints.close();
883 *
switch (settings.solver)
885 *
case Settings::gmg_mf:
890 * additional_data.mapping_update_flags =
892 * std::shared_ptr<MatrixFree<dim, double>> mf_storage =
893 * std::make_shared<MatrixFree<dim, double>>();
894 * mf_storage->reinit(mapping,
900 * mf_system_matrix.initialize(mf_storage);
902 *
const Coefficient<dim> coefficient;
903 * mf_system_matrix.set_coefficient(
904 * coefficient.make_coefficient_table(*mf_storage));
909 *
case Settings::gmg_mb:
910 *
case Settings::amg:
912 * #ifdef USE_PETSC_LA
917 * locally_owned_dofs,
919 * locally_relevant_dofs);
921 * system_matrix.reinit(locally_owned_dofs,
922 * locally_owned_dofs,
927 * locally_owned_dofs,
928 * locally_relevant_dofs,
932 * system_matrix.reinit(dsp);
946 * <a name=
"step_50-LaplaceProblemsetup_multigrid"></a>
947 * <h4>LaplaceProblem::setup_multigrid()</h4>
951 * This function does the multilevel setup
for both
matrix-free and
952 *
matrix-based GMG. The
matrix-free setup is similar to that of @ref step_37
"step-37", and
953 * the
matrix-based is similar to @ref step_16
"step-16", except we must use appropriate
954 * distributed sparsity patterns.
958 * The function is not called
for the AMG approach, but to err on the
959 * safe side, the main `
switch` statement of
this function
960 * nevertheless makes sure that the function only operates on known
961 * multigrid settings by throwing an assertion
if the function were
962 * called
for anything other than the two geometric multigrid methods.
965 *
template <
int dim,
int degree>
966 *
void LaplaceProblem<dim, degree>::setup_multigrid()
970 * dof_handler.distribute_mg_dofs();
972 * mg_constrained_dofs.clear();
973 * mg_constrained_dofs.initialize(dof_handler);
976 * mg_constrained_dofs.make_zero_boundary_constraints(dof_handler,
979 *
const unsigned int n_levels =
triangulation.n_global_levels();
981 *
switch (settings.solver)
983 *
case Settings::gmg_mf:
985 * mf_mg_matrix.resize(0, n_levels - 1);
990 * dof_handler.locally_owned_mg_dofs(
level),
994 * mg_constrained_dofs.get_boundary_indices(
level))
995 * level_constraints.constrain_dof_to_zero(dof_index);
996 * level_constraints.close();
1001 * additional_data.mapping_update_flags =
1004 * additional_data.mg_level =
level;
1005 * std::shared_ptr<MatrixFree<dim, float>> mf_storage_level(
1007 * mf_storage_level->reinit(mapping,
1009 * level_constraints,
1013 * mf_mg_matrix[
level].initialize(mf_storage_level,
1014 * mg_constrained_dofs,
1017 *
const Coefficient<dim> coefficient;
1018 * mf_mg_matrix[
level].set_coefficient(
1019 * coefficient.make_coefficient_table(*mf_storage_level));
1021 * mf_mg_matrix[
level].compute_diagonal();
1027 *
case Settings::gmg_mb:
1029 * mg_matrix.resize(0, n_levels - 1);
1030 * mg_matrix.clear_elements();
1031 * mg_interface_in.resize(0, n_levels - 1);
1032 * mg_interface_in.clear_elements();
1041 * #ifdef USE_PETSC_LA
1047 * dof_handler.locally_owned_mg_dofs(
level),
1051 * mg_matrix[
level].reinit(
1052 * dof_handler.locally_owned_mg_dofs(
level),
1053 * dof_handler.locally_owned_mg_dofs(
level),
1055 * mpi_communicator);
1058 * dof_handler.locally_owned_mg_dofs(
level),
1059 * dof_handler.locally_owned_mg_dofs(
level),
1061 * mpi_communicator);
1065 * mg_matrix[
level].reinit(dsp);
1070 * #ifdef USE_PETSC_LA
1073 * mg_constrained_dofs,
1079 * dof_handler.locally_owned_mg_dofs(
level),
1083 * mg_interface_in[
level].reinit(
1084 * dof_handler.locally_owned_mg_dofs(
level),
1085 * dof_handler.locally_owned_mg_dofs(
level),
1087 * mpi_communicator);
1090 * dof_handler.locally_owned_mg_dofs(
level),
1091 * dof_handler.locally_owned_mg_dofs(
level),
1093 * mpi_communicator);
1096 * mg_constrained_dofs,
1100 * mg_interface_in[
level].reinit(dsp);
1116 * <a name=
"step_50-LaplaceProblemassemble_system"></a>
1117 * <h4>LaplaceProblem::assemble_system()</h4>
1121 * The assembly is
split into three parts: `assemble_system()`,
1122 * `assemble_multigrid()`, and `assemble_rhs()`. The
1123 * `assemble_system()` function here assembles and stores the (global)
1124 * system
matrix and the right-hand side
for the
matrix-based
1125 * methods. It is similar to the assembly in @ref step_40
"step-40".
1129 * Note that the
matrix-free method does not execute
this function as it does
1131 * side in assemble_rhs().
1134 *
template <
int dim,
int degree>
1135 *
void LaplaceProblem<dim, degree>::assemble_system()
1139 *
const QGauss<dim> quadrature_formula(degree + 1);
1142 * quadrature_formula,
1146 *
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1147 *
const unsigned int n_q_points = quadrature_formula.size();
1152 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1154 *
const Coefficient<dim> coefficient;
1155 * RightHandSide<dim> rhs;
1156 * std::vector<double> rhs_values(n_q_points);
1158 *
for (
const auto &cell : dof_handler.active_cell_iterators())
1159 * if (cell->is_locally_owned())
1164 * fe_values.reinit(cell);
1166 *
const double coefficient_value =
1167 * coefficient.average_value(fe_values.get_quadrature_points());
1168 * rhs.value_list(fe_values.get_quadrature_points(), rhs_values);
1170 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1171 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1173 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1175 * coefficient_value *
1176 * fe_values.shape_grad(i, q_point) *
1177 * fe_values.shape_grad(j, q_point) *
1178 * fe_values.JxW(q_point);
1181 * fe_values.shape_value(i, q_point) *
1182 * rhs_values[q_point] *
1183 * fe_values.JxW(q_point);
1186 * cell->get_dof_indices(local_dof_indices);
1187 * constraints.distribute_local_to_global(cell_matrix,
1189 * local_dof_indices,
1202 * <a name=
"step_50-LaplaceProblemassemble_multigrid"></a>
1203 * <h4>LaplaceProblem::assemble_multigrid()</h4>
1207 * The following function assembles and stores the multilevel matrices
for the
1208 *
matrix-based GMG method. This function is similar to the one found in
1209 * @ref step_16
"step-16", only here it works
for distributed meshes. This difference amounts
1210 * to adding a condition that we only
assemble on locally owned
level cells
1211 * and a call to
compress() for each matrix that is built.
1214 * template <
int dim,
int degree>
1215 *
void LaplaceProblem<dim, degree>::assemble_multigrid()
1219 *
const QGauss<dim> quadrature_formula(degree + 1);
1222 * quadrature_formula,
1226 *
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1227 *
const unsigned int n_q_points = quadrature_formula.size();
1231 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1233 *
const Coefficient<dim> coefficient;
1235 * std::vector<AffineConstraints<double>> boundary_constraints(
1240 * boundary_constraints[
level].reinit(
1241 * dof_handler.locally_owned_mg_dofs(
level),
1245 * mg_constrained_dofs.get_refinement_edge_indices(
level))
1246 * boundary_constraints[
level].constrain_dof_to_zero(dof_index);
1248 * mg_constrained_dofs.get_boundary_indices(
level))
1249 * boundary_constraints[
level].constrain_dof_to_zero(dof_index);
1250 * boundary_constraints[
level].close();
1253 *
for (
const auto &cell : dof_handler.cell_iterators())
1254 * if (cell->level_subdomain_id() ==
triangulation.locally_owned_subdomain())
1257 * fe_values.reinit(cell);
1259 *
const double coefficient_value =
1260 * coefficient.average_value(fe_values.get_quadrature_points());
1262 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1263 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1264 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1266 * coefficient_value * fe_values.shape_grad(i, q_point) *
1267 * fe_values.shape_grad(j, q_point) * fe_values.JxW(q_point);
1269 * cell->get_mg_dof_indices(local_dof_indices);
1271 * boundary_constraints[cell->level()].distribute_local_to_global(
1272 * cell_matrix, local_dof_indices, mg_matrix[cell->level()]);
1274 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1275 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1276 *
if (mg_constrained_dofs.is_interface_matrix_entry(
1277 * cell->level(), local_dof_indices[i], local_dof_indices[j]))
1278 * mg_interface_in[cell->level()].add(local_dof_indices[i],
1279 * local_dof_indices[j],
1283 *
for (
unsigned int i = 0; i <
triangulation.n_global_levels(); ++i)
1295 * <a name=
"step_50-LaplaceProblemassemble_rhs"></a>
1296 * <h4>LaplaceProblem::assemble_rhs()</h4>
1300 * The
final function in
this triptych assembles the right-hand side
1301 * vector
for the
matrix-free method -- because in the
matrix-free
1302 * framework, we don
't have to assemble the matrix and can get away
1303 * with only assembling the right hand side. We could do this by extracting
1304 * the code from the `assemble_system()` function above that deals with the
1305 * right hand side, but we decide instead to go all in on the matrix-free
1306 * approach and do the assembly using that way as well.
1310 * The result is a function that is similar
1311 * to the one found in the "Use FEEvaluation::read_dof_values_plain()
1312 * to avoid resolving constraints" subsection in the "Possibilities
1313 * for extensions" section of @ref step_37 "step-37".
1317 * The reason for this function is that the MatrixFree operators do not take
1318 * into account non-homogeneous Dirichlet constraints, instead treating all
1319 * Dirichlet constraints as homogeneous. To account for this, the right-hand
1320 * side here is assembled as the residual @f$r_0 = f-Au_0@f$, where @f$u_0@f$ is a
1321 * zero vector except in the Dirichlet values. Then when solving, we have that
1322 * the solution is @f$u = u_0 + A^{-1}r_0@f$. This can be seen as a Newton
1323 * iteration on a linear system with initial guess @f$u_0@f$. The CG solve in the
1324 * `solve()` function below computes @f$A^{-1}r_0@f$ and the call to
1325 * `constraints.distribute()` (which directly follows) adds the @f$u_0@f$.
1329 * Obviously, since we are considering a problem with zero Dirichlet boundary,
1330 * we could have taken a similar approach to @ref step_37 "step-37" `assemble_rhs()`, but
1331 * this additional work allows us to change the problem declaration if we so
1336 * This function has two parts in the integration loop: applying the negative
1337 * of matrix @f$A@f$ to @f$u_0@f$ by submitting the negative of the gradient, and
1338 * adding the right-hand side contribution by submitting the value @f$f@f$. We
1339 * must be sure to use `read_dof_values_plain()` for evaluating @f$u_0@f$ as
1340 * `read_dof_values()` would set all Dirichlet values to zero.
1344 * Finally, the system_rhs vector is of type LA::MPI::Vector, but the
1345 * MatrixFree class only work for
1346 * LinearAlgebra::distributed::Vector. Therefore we must
1347 * compute the right-hand side using MatrixFree functionality and then
1348 * use the functions in the `ChangeVectorType` namespace to copy it to
1352 * template <int dim, int degree>
1353 * void LaplaceProblem<dim, degree>::assemble_rhs()
1355 * TimerOutput::Scope timing(computing_timer, "Assemble right-hand side");
1357 * MatrixFreeActiveVector solution_copy;
1358 * MatrixFreeActiveVector right_hand_side_copy;
1359 * mf_system_matrix.initialize_dof_vector(solution_copy);
1360 * mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
1362 * solution_copy = 0.;
1363 * constraints.distribute(solution_copy);
1364 * solution_copy.update_ghost_values();
1365 * right_hand_side_copy = 0;
1366 * const Table<2, VectorizedArray<double>> &coefficient =
1367 * *(mf_system_matrix.get_coefficient());
1369 * RightHandSide<dim> right_hand_side_function;
1371 * FEEvaluation<dim, degree, degree + 1, 1, double> phi(
1372 * *mf_system_matrix.get_matrix_free());
1374 * for (unsigned int cell = 0;
1375 * cell < mf_system_matrix.get_matrix_free()->n_cell_batches();
1379 * phi.read_dof_values_plain(solution_copy);
1380 * phi.evaluate(EvaluationFlags::gradients);
1382 * for (const unsigned int q : phi.quadrature_point_indices())
1384 * phi.submit_gradient(-1.0 *
1385 * (coefficient(cell, 0) * phi.get_gradient(q)),
1388 * right_hand_side_function.value(phi.quadrature_point(q)), q);
1391 * phi.integrate_scatter(EvaluationFlags::values |
1392 * EvaluationFlags::gradients,
1393 * right_hand_side_copy);
1396 * right_hand_side_copy.compress(VectorOperation::add);
1398 * ChangeVectorTypes::copy(right_hand_side, right_hand_side_copy);
1406 * <a name="step_50-LaplaceProblemsolve"></a>
1407 * <h4>LaplaceProblem::solve()</h4>
1411 * Here we set up the multigrid preconditioner, test the timing of a single
1412 * V-cycle, and solve the linear system. Unsurprisingly, this is one of the
1413 * places where the three methods differ the most.
1416 * template <int dim, int degree>
1417 * void LaplaceProblem<dim, degree>::solve()
1419 * TimerOutput::Scope timing(computing_timer, "Solve");
1421 * SolverControl solver_control(1000, 1.e-10 * right_hand_side.l2_norm());
1422 * solver_control.enable_history_data();
1428 * The solver for the matrix-free GMG method is similar to @ref step_37 "step-37", apart
1429 * from adding some interface matrices in complete analogy to @ref step_16 "step-16".
1432 * switch (settings.solver)
1434 * case Settings::gmg_mf:
1436 * computing_timer.enter_subsection("Solve: Preconditioner setup");
1438 * MGTransferMatrixFree<dim, float> mg_transfer(mg_constrained_dofs);
1439 * mg_transfer.build(dof_handler);
1441 * SolverControl coarse_solver_control(1000, 1e-12, false, false);
1442 * SolverCG<MatrixFreeLevelVector> coarse_solver(
1443 * coarse_solver_control);
1444 * PreconditionIdentity identity;
1445 * MGCoarseGridIterativeSolver<MatrixFreeLevelVector,
1446 * SolverCG<MatrixFreeLevelVector>,
1447 * MatrixFreeLevelMatrix,
1448 * PreconditionIdentity>
1449 * coarse_grid_solver(coarse_solver, mf_mg_matrix[0], identity);
1451 * using Smoother = PreconditionJacobi<MatrixFreeLevelMatrix>;
1452 * MGSmootherPrecondition<MatrixFreeLevelMatrix,
1454 * MatrixFreeLevelVector>
1456 * smoother.initialize(mf_mg_matrix,
1457 * typename Smoother::AdditionalData(
1458 * settings.smoother_dampen));
1459 * smoother.set_steps(settings.smoother_steps);
1461 * mg::Matrix<MatrixFreeLevelVector> mg_m(mf_mg_matrix);
1464 * MatrixFreeOperators::MGInterfaceOperator<MatrixFreeLevelMatrix>>
1465 * mg_interface_matrices;
1466 * mg_interface_matrices.resize(0,
1467 * triangulation.n_global_levels() - 1);
1468 * for (unsigned int level = 0;
1469 * level < triangulation.n_global_levels();
1471 * mg_interface_matrices[level].initialize(mf_mg_matrix[level]);
1472 * mg::Matrix<MatrixFreeLevelVector> mg_interface(
1473 * mg_interface_matrices);
1475 * Multigrid<MatrixFreeLevelVector> mg(
1476 * mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
1477 * mg.set_edge_matrices(mg_interface, mg_interface);
1479 * PreconditionMG<dim,
1480 * MatrixFreeLevelVector,
1481 * MGTransferMatrixFree<dim, float>>
1482 * preconditioner(dof_handler, mg, mg_transfer);
1486 * Copy the solution vector and right-hand side from LA::MPI::Vector
1487 * to LinearAlgebra::distributed::Vector so that we can solve.
1490 * MatrixFreeActiveVector solution_copy;
1491 * MatrixFreeActiveVector right_hand_side_copy;
1492 * mf_system_matrix.initialize_dof_vector(solution_copy);
1493 * mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
1495 * ChangeVectorTypes::copy(solution_copy, solution);
1496 * ChangeVectorTypes::copy(right_hand_side_copy, right_hand_side);
1497 * computing_timer.leave_subsection("Solve: Preconditioner setup");
1501 * Timing for 1 V-cycle.
1505 * TimerOutput::Scope timing(computing_timer,
1506 * "Solve: 1 multigrid V-cycle");
1507 * preconditioner.vmult(solution_copy, right_hand_side_copy);
1509 * solution_copy = 0.;
1513 * Solve the linear system, update the ghost values of the solution,
1514 * copy back to LA::MPI::Vector and distribute constraints.
1518 * SolverCG<MatrixFreeActiveVector> solver(solver_control);
1520 * TimerOutput::Scope timing(computing_timer, "Solve: CG");
1521 * solver.solve(mf_system_matrix,
1523 * right_hand_side_copy,
1527 * solution_copy.update_ghost_values();
1528 * ChangeVectorTypes::copy(solution, solution_copy);
1529 * constraints.distribute(solution);
1536 * Solver for the matrix-based GMG method, similar to @ref step_16 "step-16", only
1537 * using a Jacobi smoother instead of a SOR smoother (which is not
1538 * implemented in parallel).
1541 * case Settings::gmg_mb:
1543 * computing_timer.enter_subsection("Solve: Preconditioner setup");
1545 * MGTransferPrebuilt<VectorType> mg_transfer(mg_constrained_dofs);
1546 * mg_transfer.build(dof_handler);
1548 * SolverControl coarse_solver_control(1000, 1e-12, false, false);
1549 * SolverCG<VectorType> coarse_solver(coarse_solver_control);
1550 * PreconditionIdentity identity;
1551 * MGCoarseGridIterativeSolver<VectorType,
1552 * SolverCG<VectorType>,
1554 * PreconditionIdentity>
1555 * coarse_grid_solver(coarse_solver, mg_matrix[0], identity);
1557 * using Smoother = LA::MPI::PreconditionJacobi;
1558 * MGSmootherPrecondition<MatrixType, Smoother, VectorType> smoother;
1560 * #ifdef USE_PETSC_LA
1561 * smoother.initialize(mg_matrix);
1563 * settings.smoother_dampen == 1.0,
1564 * ExcNotImplemented(
1567 * smoother.initialize(mg_matrix, settings.smoother_dampen);
1570 * smoother.set_steps(settings.smoother_steps);
1572 * mg::Matrix<VectorType> mg_m(mg_matrix);
1573 * mg::Matrix<VectorType> mg_in(mg_interface_in);
1574 * mg::Matrix<VectorType> mg_out(mg_interface_in);
1576 * Multigrid<VectorType> mg(
1577 * mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
1578 * mg.set_edge_matrices(mg_out, mg_in);
1581 * PreconditionMG<dim, VectorType, MGTransferPrebuilt<VectorType>>
1582 * preconditioner(dof_handler, mg, mg_transfer);
1584 * computing_timer.leave_subsection("Solve: Preconditioner setup
");
1588 * Timing for 1 V-cycle.
1592 * TimerOutput::Scope timing(computing_timer,
1593 * "Solve: 1 multigrid V-cycle
");
1594 * preconditioner.vmult(solution, right_hand_side);
1600 * Solve the linear system and distribute constraints.
1604 * SolverCG<VectorType> solver(solver_control);
1606 * TimerOutput::Scope timing(computing_timer, "Solve: CG
");
1607 * solver.solve(system_matrix,
1613 * constraints.distribute(solution);
1620 * Solver for the AMG method, similar to @ref step_40 "step-40
".
1623 * case Settings::amg:
1625 * computing_timer.enter_subsection("Solve: Preconditioner setup
");
1627 * PreconditionAMG preconditioner;
1628 * PreconditionAMG::AdditionalData Amg_data;
1630 * #ifdef USE_PETSC_LA
1631 * Amg_data.symmetric_operator = true;
1633 * Amg_data.elliptic = true;
1634 * Amg_data.smoother_type = "Jacobi
";
1635 * Amg_data.higher_order_elements = true;
1636 * Amg_data.smoother_sweeps = settings.smoother_steps;
1637 * Amg_data.aggregation_threshold = 0.02;
1640 * Amg_data.output_details = false;
1642 * preconditioner.initialize(system_matrix, Amg_data);
1643 * computing_timer.leave_subsection("Solve: Preconditioner setup
");
1647 * Timing for 1 V-cycle.
1651 * TimerOutput::Scope timing(computing_timer,
1652 * "Solve: 1 multigrid V-cycle
");
1653 * preconditioner.vmult(solution, right_hand_side);
1659 * Solve the linear system and distribute constraints.
1663 * SolverCG<VectorType> solver(solver_control);
1665 * TimerOutput::Scope timing(computing_timer, "Solve: CG
");
1666 * solver.solve(system_matrix,
1671 * constraints.distribute(solution);
1677 * DEAL_II_ASSERT_UNREACHABLE();
1680 * pcout << " Number of CG iterations:
" << solver_control.last_step()
1688 * <a name="step_50-Theerrorestimator
"></a>
1689 * <h3>The error estimator</h3>
1693 * We use the FEInterfaceValues class to assemble an error estimator to decide
1694 * which cells to refine. See the exact definition of the cell and face
1695 * integrals in the introduction. To use the method, we define Scratch and
1696 * Copy objects for the MeshWorker::mesh_loop() with much of the following
1697 * code being in essence as was set up in @ref step_12 "step-12
" already (or at least similar
1701 * template <int dim>
1702 * struct ScratchData
1704 * ScratchData(const Mapping<dim> &mapping,
1705 * const FiniteElement<dim> &fe,
1706 * const unsigned int quadrature_degree,
1707 * const UpdateFlags update_flags,
1708 * const UpdateFlags interface_update_flags)
1709 * : fe_values(mapping, fe, QGauss<dim>(quadrature_degree), update_flags)
1710 * , fe_interface_values(mapping,
1712 * QGauss<dim - 1>(quadrature_degree),
1713 * interface_update_flags)
1717 * ScratchData(const ScratchData<dim> &scratch_data)
1718 * : fe_values(scratch_data.fe_values.get_mapping(),
1719 * scratch_data.fe_values.get_fe(),
1720 * scratch_data.fe_values.get_quadrature(),
1721 * scratch_data.fe_values.get_update_flags())
1722 * , fe_interface_values(scratch_data.fe_values.get_mapping(),
1723 * scratch_data.fe_values.get_fe(),
1724 * scratch_data.fe_interface_values.get_quadrature(),
1725 * scratch_data.fe_interface_values.get_update_flags())
1728 * FEValues<dim> fe_values;
1729 * FEInterfaceValues<dim> fe_interface_values;
1737 * : cell_index(numbers::invalid_unsigned_int)
1743 * unsigned int cell_indices[2];
1747 * unsigned int cell_index;
1749 * std::vector<FaceData> face_data;
1753 * template <int dim, int degree>
1754 * void LaplaceProblem<dim, degree>::estimate()
1756 * TimerOutput::Scope timing(computing_timer, "Estimate
");
1758 * VectorType temp_solution;
1759 * temp_solution.reinit(locally_owned_dofs,
1760 * locally_relevant_dofs,
1761 * mpi_communicator);
1762 * temp_solution = solution;
1764 * const Coefficient<dim> coefficient;
1766 * estimated_error_square_per_cell.reinit(triangulation.n_active_cells());
1768 * using Iterator = typename DoFHandler<dim>::active_cell_iterator;
1772 * Assembler for cell residual @f$h^2 \| f + \epsilon \triangle u \|_K^2@f$
1775 * auto cell_worker = [&](const Iterator &cell,
1776 * ScratchData<dim> &scratch_data,
1777 * CopyData ©_data) {
1778 * FEValues<dim> &fe_values = scratch_data.fe_values;
1779 * fe_values.reinit(cell);
1781 * RightHandSide<dim> rhs;
1782 * const double rhs_value = rhs.value(cell->center());
1784 * const double nu = coefficient.value(cell->center());
1786 * std::vector<Tensor<2, dim>> hessians(fe_values.n_quadrature_points);
1787 * fe_values.get_function_hessians(temp_solution, hessians);
1789 * copy_data.cell_index = cell->active_cell_index();
1791 * double residual_norm_square = 0.;
1792 * for (unsigned k = 0; k < fe_values.n_quadrature_points; ++k)
1794 * const double residual = (rhs_value + nu * trace(hessians[k]));
1795 * residual_norm_square += residual * residual * fe_values.JxW(k);
1799 * cell->diameter() * cell->diameter() * residual_norm_square;
1804 * Assembler for face term @f$\sum_F h_F \| \jump{\epsilon \nabla u \cdot n}
1808 * auto face_worker = [&](const Iterator &cell,
1809 * const unsigned int &f,
1810 * const unsigned int &sf,
1811 * const Iterator &ncell,
1812 * const unsigned int &nf,
1813 * const unsigned int &nsf,
1814 * ScratchData<dim> &scratch_data,
1815 * CopyData ©_data) {
1816 * FEInterfaceValues<dim> &fe_interface_values =
1817 * scratch_data.fe_interface_values;
1818 * fe_interface_values.reinit(cell, f, sf, ncell, nf, nsf);
1820 * copy_data.face_data.emplace_back();
1821 * CopyData::FaceData ©_data_face = copy_data.face_data.back();
1823 * copy_data_face.cell_indices[0] = cell->active_cell_index();
1824 * copy_data_face.cell_indices[1] = ncell->active_cell_index();
1826 * const double coeff1 = coefficient.value(cell->center());
1827 * const double coeff2 = coefficient.value(ncell->center());
1829 * std::vector<Tensor<1, dim>> grad_u[2];
1831 * for (unsigned int i = 0; i < 2; ++i)
1833 * grad_u[i].resize(fe_interface_values.n_quadrature_points);
1834 * fe_interface_values.get_fe_face_values(i).get_function_gradients(
1835 * temp_solution, grad_u[i]);
1838 * double jump_norm_square = 0.;
1840 * for (unsigned int qpoint = 0;
1841 * qpoint < fe_interface_values.n_quadrature_points;
1844 * const double jump =
1845 * coeff1 * grad_u[0][qpoint] * fe_interface_values.normal(qpoint) -
1846 * coeff2 * grad_u[1][qpoint] * fe_interface_values.normal(qpoint);
1848 * jump_norm_square += jump * jump * fe_interface_values.JxW(qpoint);
1851 * const double h = cell->face(f)->measure();
1852 * copy_data_face.values[0] = 0.5 * h * jump_norm_square;
1853 * copy_data_face.values[1] = copy_data_face.values[0];
1856 * auto copier = [&](const CopyData ©_data) {
1857 * if (copy_data.cell_index != numbers::invalid_unsigned_int)
1858 * estimated_error_square_per_cell[copy_data.cell_index] +=
1861 * for (const auto &cdf : copy_data.face_data)
1862 * for (unsigned int j = 0; j < 2; ++j)
1863 * estimated_error_square_per_cell[cdf.cell_indices[j]] += cdf.values[j];
1866 * const unsigned int n_gauss_points = degree + 1;
1867 * ScratchData<dim> scratch_data(mapping,
1870 * update_hessians | update_quadrature_points |
1871 * update_JxW_values,
1872 * update_values | update_gradients |
1873 * update_JxW_values | update_normal_vectors);
1874 * CopyData copy_data;
1878 * We need to assemble each interior face once but we need to make sure that
1879 * both processes assemble the face term between a locally owned and a ghost
1880 * cell. This is achieved by setting the
1881 * MeshWorker::assemble_ghost_faces_both flag. We need to do this, because
1882 * we do not communicate the error estimator contributions here.
1885 * MeshWorker::mesh_loop(dof_handler.begin_active(),
1886 * dof_handler.end(),
1891 * MeshWorker::assemble_own_cells |
1892 * MeshWorker::assemble_ghost_faces_both |
1893 * MeshWorker::assemble_own_interior_faces_once,
1894 * /*boundary_worker=*/nullptr,
1897 * const double global_error_estimate =
1898 * std::sqrt(Utilities::MPI::sum(estimated_error_square_per_cell.l1_norm(),
1899 * mpi_communicator));
1900 * pcout << " Global error estimate:
" << global_error_estimate
1908 * <a name="step_50-LaplaceProblemrefine_grid
"></a>
1909 * <h4>LaplaceProblem::refine_grid()</h4>
1913 * We use the cell-wise estimator stored in the vector @p estimate_vector and
1914 * refine a fixed number of cells (chosen here to roughly double the number of
1915 * DoFs in each step).
1918 * template <int dim, int degree>
1919 * void LaplaceProblem<dim, degree>::refine_grid()
1921 * TimerOutput::Scope timing(computing_timer, "Refine grid
");
1923 * const double refinement_fraction = 1. / (std::pow(2.0, dim) - 1.);
1924 * parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
1925 * triangulation, estimated_error_square_per_cell, refinement_fraction, 0.0);
1927 * triangulation.execute_coarsening_and_refinement();
1934 * <a name="step_50-LaplaceProblemoutput_results
"></a>
1935 * <h4>LaplaceProblem::output_results()</h4>
1939 * The output_results() function is similar to the ones found in many of the
1940 * tutorials (see @ref step_40 "step-40
" for example).
1943 * template <int dim, int degree>
1944 * void LaplaceProblem<dim, degree>::output_results(const unsigned int cycle)
1946 * TimerOutput::Scope timing(computing_timer, "Output results
");
1948 * VectorType temp_solution;
1949 * temp_solution.reinit(locally_owned_dofs,
1950 * locally_relevant_dofs,
1951 * mpi_communicator);
1952 * temp_solution = solution;
1954 * DataOut<dim> data_out;
1955 * data_out.attach_dof_handler(dof_handler);
1956 * data_out.add_data_vector(temp_solution, "solution
");
1958 * Vector<float> subdomain(triangulation.n_active_cells());
1959 * for (unsigned int i = 0; i < subdomain.size(); ++i)
1960 * subdomain(i) = triangulation.locally_owned_subdomain();
1961 * data_out.add_data_vector(subdomain, "subdomain
");
1963 * Vector<float> level(triangulation.n_active_cells());
1964 * for (const auto &cell : triangulation.active_cell_iterators())
1965 * level(cell->active_cell_index()) = cell->level();
1966 * data_out.add_data_vector(level, "level");
1968 * if (estimated_error_square_per_cell.size() > 0)
1969 * data_out.add_data_vector(estimated_error_square_per_cell,
1970 * "estimated_error_square_per_cell
");
1972 * data_out.build_patches();
1974 * const std::string pvtu_filename = data_out.write_vtu_with_pvtu_record(
1975 * "", "solution
", cycle, mpi_communicator, 2 /*n_digits*/, 1 /*n_groups*/);
1977 * pcout << " Wrote
" << pvtu_filename << std::endl;
1984 * <a name="step_50-LaplaceProblemrun
"></a>
1985 * <h4>LaplaceProblem::run()</h4>
1989 * As in most tutorials, this function calls the various functions defined
1990 * above to set up, assemble, solve, and output the results.
1993 * template <int dim, int degree>
1994 * void LaplaceProblem<dim, degree>::run()
1996 * for (unsigned int cycle = 0; cycle < settings.n_steps; ++cycle)
1998 * pcout << "Cycle
" << cycle << ':' << std::endl;
2002 * pcout << " Number of active cells:
"
2003 * << triangulation.n_global_active_cells();
2007 * We only output level cell data for the GMG methods (same with DoF
2008 * data below). Note that the partition efficiency is irrelevant for AMG
2009 * since the level hierarchy is not distributed or used during the
2013 * if (settings.solver == Settings::gmg_mf ||
2014 * settings.solver == Settings::gmg_mb)
2015 * pcout << " (
" << triangulation.n_global_levels() << " global levels)
"
2017 * << " Partition efficiency:
"
2018 * << 1.0 / MGTools::workload_imbalance(triangulation);
2019 * pcout << std::endl;
2025 * Only set up the multilevel hierarchy for GMG.
2028 * if (settings.solver == Settings::gmg_mf ||
2029 * settings.solver == Settings::gmg_mb)
2030 * setup_multigrid();
2032 * pcout << " Number of degrees of freedom:
" << dof_handler.n_dofs();
2033 * if (settings.solver == Settings::gmg_mf ||
2034 * settings.solver == Settings::gmg_mb)
2036 * pcout << " (by
level:
";
2037 * for (unsigned int level = 0;
2038 * level < triangulation.n_global_levels();
2040 * pcout << dof_handler.n_dofs(level)
2041 * << (level == triangulation.n_global_levels() - 1 ? ")
" :
2044 * pcout << std::endl;
2048 * For the matrix-free method, we only assemble the right-hand side.
2049 * For both matrix-based methods, we assemble both active matrix and
2050 * right-hand side, and only assemble the multigrid matrices for
2054 * if (settings.solver == Settings::gmg_mf)
2056 * else /*gmg_mb or amg*/
2058 * assemble_system();
2059 * if (settings.solver == Settings::gmg_mb)
2060 * assemble_multigrid();
2066 * if (settings.output)
2067 * output_results(cycle);
2069 * computing_timer.print_summary();
2070 * computing_timer.reset();
2073 * } // namespace Step50
2078 * <a name="step_50-Themainfunction
"></a>
2079 * <h3>The main() function</h3>
2083 * This is a similar main function to @ref step_40 "step-40
", with the exception that
2084 * we require the user to pass a .prm file as a sole command line
2085 * argument (see @ref step_29 "step-29
" and the documentation of the ParameterHandler
2086 * class for a complete discussion of parameter files).
2089 * int main(int argc, char *argv[])
2091 * using namespace dealii;
2092 * using namespace Step50;
2093 * Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
2095 * Settings settings;
2096 * if (!settings.try_parse((argc > 1) ? (argv[1]) : ""))
2101 * constexpr unsigned int fe_degree = 2;
2103 * switch (settings.dimension)
2107 * LaplaceProblem<2, fe_degree> test(settings);
2115 * LaplaceProblem<3, fe_degree> test(settings);
2122 * DEAL_II_NOT_IMPLEMENTED();
2125 * catch (std::exception &exc)
2127 * std::cerr << std::endl
2129 * << "----------------------------------------------------
"
2131 * std::cerr << "Exception on processing:
" << std::endl
2132 * << exc.what() << std::endl
2133 * << "Aborting!
" << std::endl
2134 * << "----------------------------------------------------
"
2136 * MPI_Abort(MPI_COMM_WORLD, 1);
2141 * std::cerr << std::endl
2143 * << "----------------------------------------------------
"
2145 * std::cerr << "Unknown exception!
" << std::endl
2146 * << "Aborting!
" << std::endl
2147 * << "----------------------------------------------------
"
2149 * MPI_Abort(MPI_COMM_WORLD, 2);
2156<a name="step_50-Results
"></a><h1>Results</h1>
2159When you run the program using the following command
2161mpirun -np 16 ./step-50 gmg_mf_2d.prm
2163the screen output should look like the following:
2166 Number of active cells: 12 (2 global levels)
2167 Partition efficiency: 0.1875
2168 Number of degrees of freedom: 65 (by level: 21, 65)
2169 Number of CG iterations: 10
2170 Global error estimate: 0.355373
2171 Wrote solution_00.pvtu
2174+---------------------------------------------+------------+------------+
2175| Total wallclock time elapsed since start | 0.0163s | |
2177| Section | no. calls | wall time | % of total |
2178+---------------------------------+-----------+------------+------------+
2179| Assemble right-hand side | 1 | 0.000374s | 2.3% |
2180| Estimate | 1 | 0.000724s | 4.4% |
2181| Output results | 1 | 0.00277s | 17% |
2182| Setup | 1 | 0.00225s | 14% |
2183| Setup multigrid | 1 | 0.00181s | 11% |
2184| Solve | 1 | 0.00364s | 22% |
2185| Solve: 1 multigrid V-cycle | 1 | 0.000354s | 2.2% |
2186| Solve: CG | 1 | 0.00151s | 9.3% |
2187| Solve: Preconditioner setup | 1 | 0.00125s | 7.7% |
2188+---------------------------------+-----------+------------+------------+
2191 Number of active cells: 24 (3 global levels)
2192 Partition efficiency: 0.276786
2193 Number of degrees of freedom: 139 (by level: 21, 65, 99)
2194 Number of CG iterations: 10
2195 Global error estimate: 0.216726
2196 Wrote solution_01.pvtu
2199+---------------------------------------------+------------+------------+
2200| Total wallclock time elapsed since start | 0.0169s | |
2202| Section | no. calls | wall time | % of total |
2203+---------------------------------+-----------+------------+------------+
2204| Assemble right-hand side | 1 | 0.000309s | 1.8% |
2205| Estimate | 1 | 0.00156s | 9.2% |
2206| Output results | 1 | 0.00222s | 13% |
2207| Refine grid | 1 | 0.00278s | 16% |
2208| Setup | 1 | 0.00196s | 12% |
2209| Setup multigrid | 1 | 0.0023s | 14% |
2210| Solve | 1 | 0.00565s | 33% |
2211| Solve: 1 multigrid V-cycle | 1 | 0.000349s | 2.1% |
2212| Solve: CG | 1 | 0.00285s | 17% |
2213| Solve: Preconditioner setup | 1 | 0.00195s | 12% |
2214+---------------------------------+-----------+------------+------------+
2217 Number of active cells: 51 (4 global levels)
2218 Partition efficiency: 0.41875
2219 Number of degrees of freedom: 245 (by level: 21, 65, 225, 25)
2220 Number of CG iterations: 11
2221 Global error estimate: 0.112098
2222 Wrote solution_02.pvtu
2225+---------------------------------------------+------------+------------+
2226| Total wallclock time elapsed since start | 0.0183s | |
2228| Section | no. calls | wall time | % of total |
2229+---------------------------------+-----------+------------+------------+
2230| Assemble right-hand side | 1 | 0.000274s | 1.5% |
2231| Estimate | 1 | 0.00127s | 6.9% |
2232| Output results | 1 | 0.00227s | 12% |
2233| Refine grid | 1 | 0.0024s | 13% |
2234| Setup | 1 | 0.00191s | 10% |
2235| Setup multigrid | 1 | 0.00295s | 16% |
2236| Solve | 1 | 0.00702s | 38% |
2237| Solve: 1 multigrid V-cycle | 1 | 0.000398s | 2.2% |
2238| Solve: CG | 1 | 0.00376s | 21% |
2239| Solve: Preconditioner setup | 1 | 0.00238s | 13% |
2240+---------------------------------+-----------+------------+------------+
2245Here, the timing of the `solve()` function is split up in 3 parts: setting
2246up the multigrid preconditioner, execution of a single multigrid V-cycle, and
2247the CG solver. The V-cycle that is timed is unnecessary for the overall solve
2248and only meant to give an insight at the different costs for AMG and GMG.
2249Also it should be noted that when using the AMG solver, "Workload imbalance
"
2250is not included in the output since the hierarchy of coarse meshes is not
2253All results in this section are gathered on Intel Xeon Platinum 8280 (Cascade
2254Lake) nodes which have 56 cores and 192GB per node and support AVX-512 instructions,
2255allowing for vectorization over 8 doubles (vectorization used only in the matrix-free
2256computations). The code is compiled using gcc 7.1.0 with intel-mpi 17.0.3. Trilinos
225712.10.1 is used for the matrix-based GMG/AMG computations.
2259We can then gather a variety of information by calling the program
2260with the input files that are provided in the directory in which
2261@ref step_50 "step-50
" is located. Using these, and adjusting the number of mesh
2262refinement steps, we can produce information about how well the
2265The following table gives weak scaling timings for this program on up to 256M DoFs
2266and 7,168 processors. (Recall that weak scaling keeps the number of
2267degrees of freedom per processor constant while increasing the number of
2268processors; i.e., it considers larger and larger problems.)
2269Here, @f$\mathbb{E}@f$ is the partition efficiency from the
2270 introduction (also equal to 1.0/workload imbalance), "Setup
" is a combination
2271of setup, setup multigrid, assemble, and assemble multigrid from the timing blocks,
2272and "Prec
" is the preconditioner setup. Ideally all times would stay constant
2273over each problem size for the individual solvers, but since the partition
2274efficiency decreases from 0.371 to 0.161 from largest to smallest problem size,
2275we expect to see an approximately @f$0.371/0.161=2.3@f$ times increase in timings
2276for GMG. This is, in fact, pretty close to what we really get:
2278<table align="center" class="doxtable
">
2280 <th colspan="4
"></th>
2282 <th colspan="4
">MF-GMG</th>
2284 <th colspan="4
">MB-GMG</th>
2286 <th colspan="4
">AMG</th>
2289 <th align="right
">Procs</th>
2290 <th align="right
">Cycle</th>
2291 <th align="right
">DoFs</th>
2292 <th align="right
">@f$\mathbb{E}@f$</th>
2294 <th align="right
">Setup</th>
2295 <th align="right
">Prec</th>
2296 <th align="right
">Solve</th>
2297 <th align="right
">Total</th>
2299 <th align="right
">Setup</th>
2300 <th align="right
">Prec</th>
2301 <th align="right
">Solve</th>
2302 <th align="right
">Total</th>
2304 <th align="right
">Setup</th>
2305 <th align="right
">Prec</th>
2306 <th align="right
">Solve</th>
2307 <th align="right
">Total</th>
2310 <td align="right
">112</th>
2311 <td align="right
">13</th>
2312 <td align="right
">4M</th>
2313 <td align="right
">0.37</th>
2315 <td align="right
">0.742</th>
2316 <td align="right
">0.393</th>
2317 <td align="right
">0.200</th>
2318 <td align="right
">1.335</th>
2320 <td align="right
">1.714</th>
2321 <td align="right
">2.934</th>
2322 <td align="right
">0.716</th>
2323 <td align="right
">5.364</th>
2325 <td align="right
">1.544</th>
2326 <td align="right
">0.456</th>
2327 <td align="right
">1.150</th>
2328 <td align="right
">3.150</th>
2331 <td align="right
">448</th>
2332 <td align="right
">15</th>
2333 <td align="right
">16M</th>
2334 <td align="right
">0.29</th>
2336 <td align="right
">0.884</th>
2337 <td align="right
">0.535</th>
2338 <td align="right
">0.253</th>
2339 <td align="right
">1.672</th>
2341 <td align="right
">1.927</th>
2342 <td align="right
">3.776</th>
2343 <td align="right
">1.190</th>
2344 <td align="right
">6.893</th>
2346 <td align="right
">1.544</th>
2347 <td align="right
">0.456</th>
2348 <td align="right
">1.150</th>
2349 <td align="right
">3.150</th>
2352 <td align="right
">1,792</th>
2353 <td align="right
">17</th>
2354 <td align="right
">65M</th>
2355 <td align="right
">0.22</th>
2357 <td align="right
">1.122</th>
2358 <td align="right
">0.686</th>
2359 <td align="right
">0.309</th>
2360 <td align="right
">2.117</th>
2362 <td align="right
">2.171</th>
2363 <td align="right
">4.862</th>
2364 <td align="right
">1.660</th>
2365 <td align="right
">8.693</th>
2367 <td align="right
">1.654</th>
2368 <td align="right
">0.546</th>
2369 <td align="right
">1.460</th>
2370 <td align="right
">3.660</th>
2373 <td align="right
">7,168</th>
2374 <td align="right
">19</th>
2375 <td align="right
">256M</th>
2376 <td align="right
">0.16</th>
2378 <td align="right
">1.214</th>
2379 <td align="right
">0.893</th>
2380 <td align="right
">0.521</th>
2381 <td align="right
">2.628</th>
2383 <td align="right
">2.386</th>
2384 <td align="right
">7.260</th>
2385 <td align="right
">2.560</th>
2386 <td align="right
">12.206</th>
2388 <td align="right
">1.844</th>
2389 <td align="right
">1.010</th>
2390 <td align="right
">1.890</th>
2391 <td align="right
">4.744</th>
2395On the other hand, the algebraic multigrid in the last set of columns
2396is relatively unaffected by the increasing imbalance of the mesh
2397hierarchy (because it doesn't use the mesh hierarchy) and the growth
2398in time is rather driven by other factors that are well documented in
2399the literature (most notably that the algorithmic complexity of
2400some parts of algebraic multigrid methods appears to be @f${\cal O}(N
2401\log N)@f$ instead of @f${\cal O}(N)@f$ for geometric multigrid).
2403The upshort of the table above is that the matrix-free geometric multigrid
2404method appears to be the fastest approach to solving this equation if
2405not by a huge margin. Matrix-based methods, on the other hand, are
2406consistently the worst.
2408The following figure provides strong scaling results for each method, i.e.,
2409we solve the same problem on more and more processors. Specifically,
2410we consider the problems after 16 mesh refinement cycles
2411(32M DoFs) and 19 cycles (256M DoFs), on between 56 to 28,672 processors:
2413<img width="600px
" src="https:
2415While the
matrix-based GMG solver and AMG
scale similarly and have a
2416similar time to solution (at least as long as there is a substantial
2417number of unknowns per processor -- say, several 10,000), the
2418matrix-free GMG solver scales much better and solves the finer problem
2419in roughly the same time as the AMG solver for the coarser mesh with
2420only an eighth of the number of processors. Conversely, it can solve the
2421same problem on the same number of processors in about one eighth the
2425<a name=
"step_50-Possibilitiesforextensions"></a><h3> Possibilities for extensions </h3>
2428<a name=
"step_50-Testingconvergenceandhigherorderelements"></a><h4> Testing convergence and higher order elements </h4>
2431The finite element degree is currently hard-coded as 2, see the template
2432arguments of the main class. It is easy to change. To test, it would be
2433interesting to switch to a test problem with a reference solution. This way,
2434you can compare error rates.
2436<a name=
"step_50-Coarsesolver"></a><h4> Coarse solver </h4>
2439A more interesting example would involve a more complicated coarse mesh (see
2440@ref step_49
"step-49" for inspiration). The issue in that case is that the coarsest
2441level of the mesh hierarchy is actually quite large, and one would
2442have to think about ways to solve the coarse
level problem
2443efficiently. (This is not an issue for algebraic multigrid methods
2444because they would just continue to build coarser and coarser levels
2445of the
matrix, regardless of their geometric origin.)
2447In the program here, we simply solve the coarse
level problem with a
2448Conjugate Gradient method without any preconditioner. That is acceptable
2449if the coarse problem is really small -- for example, if the coarse
2450mesh had a single cell, then the coarse mesh problems has a @f$9\times 9@f$
2451matrix in 2
d, and a @f$27\times 27@f$
matrix in 3
d;
for the coarse mesh we
2452use on the @f$L@f$-shaped domain of the current program, these sizes are
2453@f$21\times 21@f$ in 2
d and @f$117\times 117@f$ in 3
d. But
if the coarse mesh
2454consists of hundreds or thousands of cells,
this approach will no
2455longer work and might start to dominate the overall
run-time of each V-cycle.
2456A common approach is then to solve the coarse mesh problem
using an
2457algebraic multigrid preconditioner;
this would then, however, require
2458assembling the coarse
matrix (even
for the matrix-free version) as
2459input to the AMG implementation.
2462<a name=
"step_50-PlainProg"></a>
2463<h1> The plain program</h1>
2464@include
"step-50.cc"
void declare_entry(const std::string &entry, const std::string &default_value, const Patterns::PatternBase &pattern=Patterns::Anything(), const std::string &documentation="", const bool has_to_be_set=false)
#define AssertThrow(cond, exc)
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_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
#define DEAL_II_NOT_IMPLEMENTED()
std::vector< value_type > split(const typename ::Triangulation< dim, spacedim >::cell_iterator &parent, const value_type parent_value)
void hyper_L(Triangulation< dim > &tria, const double left=-1., const double right=1., const bool colorize=false)
@ matrix
Contents is actually a matrix.
PETScWrappers::PreconditionBoomerAMG PreconditionAMG
Tpetra::Vector< Number, LO, GO, NodeType< MemorySpace > > VectorType
Tpetra::CrsMatrix< Number, LO, GO, NodeType< MemorySpace > > MatrixType
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.)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
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)
@ construct_multigrid_hierarchy
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
std::string compress(const std::string &input)
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)
unsigned int n_cells(const internal::TriangulationImplementation::NumberCache< 1 > &c)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
TasksParallelScheme tasks_parallel_scheme