480 *
const unsigned int = 0)
const override
486 *
template <
typename number>
489 *
const unsigned int = 0) const
498 * This next
class represents the diffusion coefficient. We use a variable
499 * coefficient which is 100.0 at any
point where at least
one coordinate is
500 * less than -0.5, and 1.0 at all other points. As above, a separate
value()
502 * average() function computes the arithmetic average for a
set of points.
506 * class Coefficient : public
Function<dim>
510 *
const unsigned int = 0)
const override;
512 *
template <
typename number>
514 *
const unsigned int = 0)
const;
516 *
template <
typename number>
521 * When
using a coefficient in the
MatrixFree framework, we also
522 * need a function that creates a
Table of coefficient
values for a
523 *
set of cells provided by the
MatrixFree operator argument here.
526 *
template <
typename number>
527 * std::shared_ptr<Table<2, VectorizedArray<number>>> make_coefficient_table(
534 *
double Coefficient<dim>::value(
const Point<dim> &p,
const unsigned int)
const
536 *
for (
int d = 0;
d < dim; ++
d)
547 *
template <
typename number>
550 *
const unsigned int)
const
553 *
for (
unsigned int i = 0; i < VectorizedArray<number>::size(); ++i)
555 *
for (
int d = 0;
d < dim; ++
d)
556 *
if (p[
d][i] < -0.5)
558 * return_value[i] = 100.0;
563 *
return return_value;
569 *
template <
typename number>
570 * number Coefficient<dim>::average_value(
574 *
for (
unsigned int i = 0; i < points.size(); ++i)
575 * average +=
value(points[i]);
576 * average /= points.size();
584 *
template <
typename number>
585 * std::shared_ptr<Table<2, VectorizedArray<number>>>
586 * Coefficient<dim>::make_coefficient_table(
589 *
auto coefficient_table =
590 * std::make_shared<Table<2, VectorizedArray<number>>>();
592 *
FEEvaluation<dim, -1, 0, 1, number> fe_eval(mf_storage);
594 *
const unsigned int n_cells = mf_storage.n_cell_batches();
595 *
const unsigned int n_q_points = fe_eval.n_q_points;
597 * coefficient_table->reinit(
n_cells, 1);
599 *
for (
unsigned int cell = 0; cell <
n_cells; ++cell)
601 * fe_eval.reinit(cell);
604 *
for (
unsigned int q = 0; q < n_q_points; ++q)
605 * average_value +=
value(fe_eval.quadrature_point(q));
606 * average_value /= n_q_points;
608 * (*coefficient_table)(cell, 0) = average_value;
611 *
return coefficient_table;
619 * <a name=
"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)
658 *
"Number of adaptive refinement steps.");
662 *
"Dampen factor for the smoother.");
666 *
"Number of smoother steps.");
670 *
"Switch between matrix-free GMG, "
671 *
"matrix-based GMG, and AMG.");
675 *
"Output graphical results.");
677 *
if (prm_filename.size() == 0)
679 * std::cout <<
"**** Error: No input file provided!\n"
680 * <<
"**** Error: Call this program as './step-50 input.prm\n"
682 * <<
"**** You may want to use one of the input files in this\n"
683 * <<
"**** directory, or use the following default values\n"
684 * <<
"**** to create an input file:\n";
694 *
catch (std::exception &
e)
697 * std::cerr <<
e.what() << std::endl;
701 *
if (prm.
get(
"solver") ==
"MF")
702 * this->solver = gmg_mf;
703 *
else if (prm.
get(
"solver") ==
"MB")
704 * this->solver = gmg_mb;
705 *
else if (prm.
get(
"solver") ==
"AMG")
706 * this->solver = amg;
712 * this->smoother_dampen = prm.
get_double(
"smoother dampen");
713 * this->smoother_steps = prm.
get_integer(
"smoother steps");
714 * this->output = prm.
get_bool(
"output");
724 * <a name=
"LaplaceProblemclass"></a>
725 * <h3>LaplaceProblem
class</h3>
729 * This is the main
class of the program. It looks very similar to
730 * @ref step_16
"step-16", @ref step_37
"step-37", and @ref step_40
"step-40". For the
MatrixFree setup, we use the
733 * the polynomial degree is a
template parameter of
this class. This is
737 *
template <
int dim,
int degree>
738 *
class LaplaceProblem
741 * LaplaceProblem(
const Settings &settings);
747 * We will use the following
types throughout the program. First the
773 *
void setup_system();
774 *
void setup_multigrid();
775 *
void assemble_system();
776 *
void assemble_multigrid();
777 *
void assemble_rhs();
780 *
void refine_grid();
781 *
void output_results(
const unsigned int cycle);
798 * MatrixType system_matrix;
799 * MatrixFreeActiveMatrix mf_system_matrix;
800 * VectorType solution;
801 * VectorType right_hand_side;
816 * The only interesting part about the constructor is that we construct the
817 * multigrid hierarchy unless we use AMG. For that, we need to parse the
818 *
run time parameters before
this constructor completes.
821 *
template <
int dim,
int degree>
822 * LaplaceProblem<dim, degree>::LaplaceProblem(
const Settings &settings)
823 * : settings(settings)
824 * , mpi_communicator(MPI_COMM_WORLD)
828 * (settings.solver == Settings::amg) ?
846 * <a name=
"LaplaceProblemsetup_system"></a>
847 * <h4>LaplaceProblem::setup_system()</h4>
851 * Unlike @ref step_16
"step-16" and @ref step_37
"step-37", we
split the
set up into two parts,
852 * setup_system() and setup_multigrid(). Here is the typical setup_system()
853 * function for the active mesh found in most tutorials. For
matrix-
free, the
854 * active mesh
set up is similar to @ref step_37 "step-37"; for
matrix-based (GMG and AMG
855 * solvers), the setup is similar to @ref step_40 "step-40".
858 * template <
int dim,
int degree>
859 *
void LaplaceProblem<dim, degree>::setup_system()
863 * dof_handler.distribute_dofs(fe);
866 * locally_owned_dofs = dof_handler.locally_owned_dofs();
868 * solution.reinit(locally_owned_dofs, mpi_communicator);
869 * right_hand_side.reinit(locally_owned_dofs, mpi_communicator);
870 * constraints.reinit(locally_relevant_dofs);
875 * constraints.close();
877 *
switch (settings.solver)
879 *
case Settings::gmg_mf:
886 * std::shared_ptr<MatrixFree<dim, double>> mf_storage =
887 * std::make_shared<MatrixFree<dim, double>>();
888 * mf_storage->reinit(mapping,
894 * mf_system_matrix.initialize(mf_storage);
896 *
const Coefficient<dim> coefficient;
897 * mf_system_matrix.set_coefficient(
898 * coefficient.make_coefficient_table(*mf_storage));
903 *
case Settings::gmg_mb:
904 *
case Settings::amg:
906 * #ifdef USE_PETSC_LA
911 * locally_owned_dofs,
913 * locally_relevant_dofs);
915 * system_matrix.reinit(locally_owned_dofs,
916 * locally_owned_dofs,
921 * locally_owned_dofs,
922 * locally_relevant_dofs,
926 * system_matrix.reinit(dsp);
940 * <a name=
"LaplaceProblemsetup_multigrid"></a>
941 * <h4>LaplaceProblem::setup_multigrid()</h4>
945 * This function does the multilevel setup
for both
matrix-
free and
946 *
matrix-based GMG. The
matrix-
free setup is similar to that of @ref step_37
"step-37", and
947 * the
matrix-based is similar to @ref step_16
"step-16", except we must use appropriate
948 * distributed sparsity patterns.
952 * The function is not called
for the AMG approach, but to err on the
953 * safe side, the main `
switch` statement of
this function
954 * nevertheless makes sure that the function only operates on known
955 * multigrid settings by throwing an assertion
if the function were
956 * called
for anything other than the two geometric multigrid methods.
959 *
template <
int dim,
int degree>
960 *
void LaplaceProblem<dim, degree>::setup_multigrid()
964 * dof_handler.distribute_mg_dofs();
966 * mg_constrained_dofs.clear();
967 * mg_constrained_dofs.initialize(dof_handler);
970 * mg_constrained_dofs.make_zero_boundary_constraints(dof_handler, boundary_ids);
972 *
const unsigned int n_levels =
triangulation.n_global_levels();
974 *
switch (settings.solver)
976 *
case Settings::gmg_mf:
978 * mf_mg_matrix.resize(0, n_levels - 1);
987 * level_constraints.
reinit(relevant_dofs);
989 * mg_constrained_dofs.get_boundary_indices(
level));
990 * level_constraints.
close();
999 * std::shared_ptr<MatrixFree<dim, float>> mf_storage_level(
1001 * mf_storage_level->reinit(mapping,
1003 * level_constraints,
1007 * mf_mg_matrix[
level].initialize(mf_storage_level,
1008 * mg_constrained_dofs,
1011 *
const Coefficient<dim> coefficient;
1012 * mf_mg_matrix[
level].set_coefficient(
1013 * coefficient.make_coefficient_table(*mf_storage_level));
1015 * mf_mg_matrix[
level].compute_diagonal();
1021 *
case Settings::gmg_mb:
1023 * mg_matrix.resize(0, n_levels - 1);
1024 * mg_matrix.clear_elements();
1025 * mg_interface_in.resize(0, n_levels - 1);
1026 * mg_interface_in.clear_elements();
1036 * #ifdef USE_PETSC_LA
1042 * dof_handler.locally_owned_mg_dofs(
level),
1046 * mg_matrix[
level].reinit(
1047 * dof_handler.locally_owned_mg_dofs(
level),
1048 * dof_handler.locally_owned_mg_dofs(
level),
1050 * mpi_communicator);
1053 * dof_handler.locally_owned_mg_dofs(
level),
1054 * dof_handler.locally_owned_mg_dofs(
level),
1056 * mpi_communicator);
1060 * mg_matrix[
level].reinit(dsp);
1065 * #ifdef USE_PETSC_LA
1068 * mg_constrained_dofs,
1074 * dof_handler.locally_owned_mg_dofs(
level),
1078 * mg_interface_in[
level].reinit(
1079 * dof_handler.locally_owned_mg_dofs(
level),
1080 * dof_handler.locally_owned_mg_dofs(
level),
1082 * mpi_communicator);
1085 * dof_handler.locally_owned_mg_dofs(
level),
1086 * dof_handler.locally_owned_mg_dofs(
level),
1088 * mpi_communicator);
1091 * mg_constrained_dofs,
1095 * mg_interface_in[
level].reinit(dsp);
1111 * <a name=
"LaplaceProblemassemble_system"></a>
1112 * <h4>LaplaceProblem::assemble_system()</h4>
1116 * The assembly is
split into three parts: `assemble_system()`,
1117 * `assemble_multigrid()`, and `assemble_rhs()`. The
1118 * `assemble_system()` function here assembles and stores the (global)
1119 * system
matrix and the right-hand side
for the
matrix-based
1120 * methods. It is similar to the assembly in @ref step_40
"step-40".
1124 * Note that the
matrix-
free method does not execute
this function as it does
1126 * side in assemble_rhs().
1129 *
template <
int dim,
int degree>
1130 *
void LaplaceProblem<dim, degree>::assemble_system()
1134 *
const QGauss<dim> quadrature_formula(degree + 1);
1137 * quadrature_formula,
1141 *
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1142 *
const unsigned int n_q_points = quadrature_formula.size();
1147 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1149 *
const Coefficient<dim> coefficient;
1150 * RightHandSide<dim> rhs;
1151 * std::vector<double> rhs_values(n_q_points);
1153 *
for (
const auto &cell : dof_handler.active_cell_iterators())
1154 *
if (cell->is_locally_owned())
1159 * fe_values.reinit(cell);
1161 *
const double coefficient_value =
1162 * coefficient.average_value(fe_values.get_quadrature_points());
1163 * rhs.value_list(fe_values.get_quadrature_points(), rhs_values);
1165 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1166 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1168 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1170 * coefficient_value *
1171 * fe_values.shape_grad(i, q_point) *
1172 * fe_values.shape_grad(j, q_point) *
1173 * fe_values.JxW(q_point);
1176 * fe_values.shape_value(i, q_point) *
1177 * rhs_values[q_point] *
1178 * fe_values.JxW(q_point);
1181 * cell->get_dof_indices(local_dof_indices);
1182 * constraints.distribute_local_to_global(
cell_matrix,
1184 * local_dof_indices,
1197 * <a name=
"LaplaceProblemassemble_multigrid"></a>
1198 * <h4>LaplaceProblem::assemble_multigrid()</h4>
1202 * The following function assembles and stores the multilevel matrices
for the
1203 *
matrix-based GMG method. This function is similar to the
one found in
1204 * @ref step_16
"step-16", only here it works
for distributed meshes. This difference amounts
1205 * to adding a condition that we only
assemble on locally owned
level cells and
1209 * template <
int dim,
int degree>
1210 *
void LaplaceProblem<dim, degree>::assemble_multigrid()
1217 * quadrature_formula,
1221 *
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
1222 *
const unsigned int n_q_points = quadrature_formula.size();
1226 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
1228 *
const Coefficient<dim> coefficient;
1230 * std::vector<AffineConstraints<double>> boundary_constraints(
1238 * boundary_constraints[
level].reinit(dof_set);
1239 * boundary_constraints[
level].add_lines(
1240 * mg_constrained_dofs.get_refinement_edge_indices(
level));
1241 * boundary_constraints[
level].add_lines(
1242 * mg_constrained_dofs.get_boundary_indices(
level));
1244 * boundary_constraints[
level].close();
1247 *
for (
const auto &cell : dof_handler.cell_iterators())
1248 *
if (cell->level_subdomain_id() ==
triangulation.locally_owned_subdomain())
1251 * fe_values.reinit(cell);
1253 *
const double coefficient_value =
1254 * coefficient.average_value(fe_values.get_quadrature_points());
1256 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
1257 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1258 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1260 * coefficient_value * fe_values.shape_grad(i, q_point) *
1261 * fe_values.shape_grad(j, q_point) * fe_values.JxW(q_point);
1263 * cell->get_mg_dof_indices(local_dof_indices);
1265 * boundary_constraints[cell->level()].distribute_local_to_global(
1266 *
cell_matrix, local_dof_indices, mg_matrix[cell->level()]);
1268 *
for (
unsigned int i = 0; i < dofs_per_cell; ++i)
1269 *
for (
unsigned int j = 0; j < dofs_per_cell; ++j)
1270 *
if (mg_constrained_dofs.is_interface_matrix_entry(
1271 * cell->level(), local_dof_indices[i], local_dof_indices[j]))
1272 * mg_interface_in[cell->level()].add(local_dof_indices[i],
1273 * local_dof_indices[j],
1277 *
for (
unsigned int i = 0; i <
triangulation.n_global_levels(); ++i)
1289 * <a name=
"LaplaceProblemassemble_rhs"></a>
1290 * <h4>LaplaceProblem::assemble_rhs()</h4>
1294 * The
final function in
this triptych assembles the right-hand side
1296 * framework, we don
't have to assemble the matrix and can get away
1297 * with only assembling the right hand side. We could do this by extracting the
1298 * code from the `assemble_system()` function above that deals with the right
1299 * hand side, but we decide instead to go all in on the matrix-free approach and
1300 * do the assembly using that way as well.
1304 * The result is a function that is similar
1305 * to the one found in the "Use FEEvaluation::read_dof_values_plain()
1306 * to avoid resolving constraints" subsection in the "Possibilities
1307 * for extensions" section of @ref step_37 "step-37".
1311 * The reason for this function is that the MatrixFree operators do not take
1312 * into account non-homogeneous Dirichlet constraints, instead treating all
1313 * Dirichlet constraints as homogeneous. To account for this, the right-hand
1314 * side here is assembled as the residual @f$r_0 = f-Au_0@f$, where @f$u_0@f$ is a
1315 * zero vector except in the Dirichlet values. Then when solving, we have that
1316 * the solution is @f$u = u_0 + A^{-1}r_0@f$. This can be seen as a Newton
1317 * iteration on a linear system with initial guess @f$u_0@f$. The CG solve in the
1318 * `solve()` function below computes @f$A^{-1}r_0@f$ and the call to
1319 * `constraints.distribute()` (which directly follows) adds the @f$u_0@f$.
1323 * Obviously, since we are considering a problem with zero Dirichlet boundary,
1324 * we could have taken a similar approach to @ref step_37 "step-37" `assemble_rhs()`, but this
1325 * additional work allows us to change the problem declaration if we so
1330 * This function has two parts in the integration loop: applying the negative
1331 * of matrix @f$A@f$ to @f$u_0@f$ by submitting the negative of the gradient, and adding
1332 * the right-hand side contribution by submitting the value @f$f@f$. We must be sure
1333 * to use `read_dof_values_plain()` for evaluating @f$u_0@f$ as `read_dof_vaues()`
1334 * would set all Dirichlet values to zero.
1338 * Finally, the system_rhs vector is of type LA::MPI::Vector, but the
1339 * MatrixFree class only work for
1340 * ::LinearAlgebra::distributed::Vector. Therefore we must
1341 * compute the right-hand side using MatrixFree functionality and then
1342 * use the functions in the `ChangeVectorType` namespace to copy it to
1346 * template <int dim, int degree>
1347 * void LaplaceProblem<dim, degree>::assemble_rhs()
1349 * TimerOutput::Scope timing(computing_timer, "Assemble right-hand side");
1351 * MatrixFreeActiveVector solution_copy;
1352 * MatrixFreeActiveVector right_hand_side_copy;
1353 * mf_system_matrix.initialize_dof_vector(solution_copy);
1354 * mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
1356 * solution_copy = 0.;
1357 * constraints.distribute(solution_copy);
1358 * solution_copy.update_ghost_values();
1359 * right_hand_side_copy = 0;
1360 * const Table<2, VectorizedArray<double>> &coefficient =
1361 * *(mf_system_matrix.get_coefficient());
1363 * RightHandSide<dim> right_hand_side_function;
1365 * FEEvaluation<dim, degree, degree + 1, 1, double> phi(
1366 * *mf_system_matrix.get_matrix_free());
1368 * for (unsigned int cell = 0;
1369 * cell < mf_system_matrix.get_matrix_free()->n_cell_batches();
1373 * phi.read_dof_values_plain(solution_copy);
1374 * phi.evaluate(EvaluationFlags::gradients);
1376 * for (unsigned int q = 0; q < phi.n_q_points; ++q)
1378 * phi.submit_gradient(-1.0 *
1379 * (coefficient(cell, 0) * phi.get_gradient(q)),
1382 * right_hand_side_function.value(phi.quadrature_point(q)), q);
1385 * phi.integrate_scatter(EvaluationFlags::values |
1386 * EvaluationFlags::gradients,
1387 * right_hand_side_copy);
1390 * right_hand_side_copy.compress(VectorOperation::add);
1392 * ChangeVectorTypes::copy(right_hand_side, right_hand_side_copy);
1400 * <a name="LaplaceProblemsolve"></a>
1401 * <h4>LaplaceProblem::solve()</h4>
1405 * Here we set up the multigrid preconditioner, test the timing of a single
1406 * V-cycle, and solve the linear system. Unsurprisingly, this is one of the
1407 * places where the three methods differ the most.
1410 * template <int dim, int degree>
1411 * void LaplaceProblem<dim, degree>::solve()
1413 * TimerOutput::Scope timing(computing_timer, "Solve");
1415 * SolverControl solver_control(1000, 1.e-10 * right_hand_side.l2_norm());
1416 * solver_control.enable_history_data();
1422 * The solver for the matrix-free GMG method is similar to @ref step_37 "step-37", apart
1423 * from adding some interface matrices in complete analogy to @ref step_16 "step-16".
1426 * switch (settings.solver)
1428 * case Settings::gmg_mf:
1430 * computing_timer.enter_subsection("Solve: Preconditioner setup");
1432 * MGTransferMatrixFree<dim, float> mg_transfer(mg_constrained_dofs);
1433 * mg_transfer.build(dof_handler);
1435 * SolverControl coarse_solver_control(1000, 1e-12, false, false);
1436 * SolverCG<MatrixFreeLevelVector> coarse_solver(coarse_solver_control);
1437 * PreconditionIdentity identity;
1438 * MGCoarseGridIterativeSolver<MatrixFreeLevelVector,
1439 * SolverCG<MatrixFreeLevelVector>,
1440 * MatrixFreeLevelMatrix,
1441 * PreconditionIdentity>
1442 * coarse_grid_solver(coarse_solver, mf_mg_matrix[0], identity);
1444 * using Smoother = ::PreconditionJacobi<MatrixFreeLevelMatrix>;
1445 * MGSmootherPrecondition<MatrixFreeLevelMatrix,
1447 * MatrixFreeLevelVector>
1449 * smoother.initialize(mf_mg_matrix,
1450 * typename Smoother::AdditionalData(
1451 * settings.smoother_dampen));
1452 * smoother.set_steps(settings.smoother_steps);
1454 * mg::Matrix<MatrixFreeLevelVector> mg_m(mf_mg_matrix);
1457 * MatrixFreeOperators::MGInterfaceOperator<MatrixFreeLevelMatrix>>
1458 * mg_interface_matrices;
1459 * mg_interface_matrices.resize(0, triangulation.n_global_levels() - 1);
1460 * for (unsigned int level = 0; level < triangulation.n_global_levels();
1462 * mg_interface_matrices[level].initialize(mf_mg_matrix[level]);
1463 * mg::Matrix<MatrixFreeLevelVector> mg_interface(mg_interface_matrices);
1465 * Multigrid<MatrixFreeLevelVector> mg(
1466 * mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
1467 * mg.set_edge_matrices(mg_interface, mg_interface);
1469 * PreconditionMG<dim,
1470 * MatrixFreeLevelVector,
1471 * MGTransferMatrixFree<dim, float>>
1472 * preconditioner(dof_handler, mg, mg_transfer);
1476 * Copy the solution vector and right-hand side from LA::MPI::Vector
1477 * to ::LinearAlgebra::distributed::Vector so that we can solve.
1480 * MatrixFreeActiveVector solution_copy;
1481 * MatrixFreeActiveVector right_hand_side_copy;
1482 * mf_system_matrix.initialize_dof_vector(solution_copy);
1483 * mf_system_matrix.initialize_dof_vector(right_hand_side_copy);
1485 * ChangeVectorTypes::copy(solution_copy, solution);
1486 * ChangeVectorTypes::copy(right_hand_side_copy, right_hand_side);
1487 * computing_timer.leave_subsection("Solve: Preconditioner setup");
1491 * Timing for 1 V-cycle.
1495 * TimerOutput::Scope timing(computing_timer,
1496 * "Solve: 1 multigrid V-cycle");
1497 * preconditioner.vmult(solution_copy, right_hand_side_copy);
1499 * solution_copy = 0.;
1503 * Solve the linear system, update the ghost values of the solution,
1504 * copy back to LA::MPI::Vector and distribute constraints.
1508 * SolverCG<MatrixFreeActiveVector> solver(solver_control);
1510 * TimerOutput::Scope timing(computing_timer, "Solve: CG");
1511 * solver.solve(mf_system_matrix,
1513 * right_hand_side_copy,
1517 * solution_copy.update_ghost_values();
1518 * ChangeVectorTypes::copy(solution, solution_copy);
1519 * constraints.distribute(solution);
1526 * Solver for the matrix-based GMG method, similar to @ref step_16 "step-16", only
1527 * using a Jacobi smoother instead of a SOR smoother (which is not
1528 * implemented in parallel).
1531 * case Settings::gmg_mb:
1533 * computing_timer.enter_subsection("Solve: Preconditioner setup");
1535 * MGTransferPrebuilt<VectorType> mg_transfer(mg_constrained_dofs);
1536 * mg_transfer.build(dof_handler);
1538 * SolverControl coarse_solver_control(1000, 1e-12, false, false);
1539 * SolverCG<VectorType> coarse_solver(coarse_solver_control);
1540 * PreconditionIdentity identity;
1541 * MGCoarseGridIterativeSolver<VectorType,
1542 * SolverCG<VectorType>,
1544 * PreconditionIdentity>
1545 * coarse_grid_solver(coarse_solver, mg_matrix[0], identity);
1547 * using Smoother = LA::MPI::PreconditionJacobi;
1548 * MGSmootherPrecondition<MatrixType, Smoother, VectorType> smoother;
1550 * #ifdef USE_PETSC_LA
1551 * smoother.initialize(mg_matrix);
1553 * settings.smoother_dampen == 1.0,
1554 * ExcNotImplemented(
1557 * smoother.initialize(mg_matrix, settings.smoother_dampen);
1560 * smoother.set_steps(settings.smoother_steps);
1562 * mg::Matrix<VectorType> mg_m(mg_matrix);
1563 * mg::Matrix<VectorType> mg_in(mg_interface_in);
1564 * mg::Matrix<VectorType> mg_out(mg_interface_in);
1566 * Multigrid<VectorType> mg(
1567 * mg_m, coarse_grid_solver, mg_transfer, smoother, smoother);
1568 * mg.set_edge_matrices(mg_out, mg_in);
1571 * PreconditionMG<dim, VectorType, MGTransferPrebuilt<VectorType>>
1572 * preconditioner(dof_handler, mg, mg_transfer);
1574 * computing_timer.leave_subsection("Solve: Preconditioner setup
");
1578 * Timing for 1 V-cycle.
1582 * TimerOutput::Scope timing(computing_timer,
1583 * "Solve: 1 multigrid
V-cycle
");
1584 * preconditioner.vmult(solution, right_hand_side);
1590 * Solve the linear system and distribute constraints.
1594 * SolverCG<VectorType> solver(solver_control);
1596 * TimerOutput::Scope timing(computing_timer, "Solve: CG
");
1597 * solver.solve(system_matrix,
1603 * constraints.distribute(solution);
1610 * Solver for the AMG method, similar to @ref step_40 "step-40
".
1613 * case Settings::amg:
1615 * computing_timer.enter_subsection("Solve: Preconditioner setup
");
1617 * PreconditionAMG preconditioner;
1618 * PreconditionAMG::AdditionalData Amg_data;
1620 * #ifdef USE_PETSC_LA
1621 * Amg_data.symmetric_operator = true;
1623 * Amg_data.elliptic = true;
1624 * Amg_data.smoother_type = "Jacobi
";
1625 * Amg_data.higher_order_elements = true;
1626 * Amg_data.smoother_sweeps = settings.smoother_steps;
1627 * Amg_data.aggregation_threshold = 0.02;
1630 * Amg_data.output_details = false;
1632 * preconditioner.initialize(system_matrix, Amg_data);
1633 * computing_timer.leave_subsection("Solve: Preconditioner setup
");
1637 * Timing for 1 V-cycle.
1641 * TimerOutput::Scope timing(computing_timer,
1642 * "Solve: 1 multigrid
V-cycle
");
1643 * preconditioner.vmult(solution, right_hand_side);
1649 * Solve the linear system and distribute constraints.
1653 * SolverCG<VectorType> solver(solver_control);
1655 * TimerOutput::Scope timing(computing_timer, "Solve: CG
");
1656 * solver.solve(system_matrix,
1661 * constraints.distribute(solution);
1667 * Assert(false, ExcInternalError());
1670 * pcout << " Number of CG iterations:
" << solver_control.last_step()
1678 * <a name="Theerrorestimator
"></a>
1679 * <h3>The error estimator</h3>
1683 * We use the FEInterfaceValues class to assemble an error estimator to decide
1684 * which cells to refine. See the exact definition of the cell and face
1685 * integrals in the introduction. To use the method, we define Scratch and
1686 * Copy objects for the MeshWorker::mesh_loop() with much of the following code
1687 * being in essence as was set up in @ref step_12 "step-12
" already (or at least similar in
1691 * template <int dim>
1692 * struct ScratchData
1694 * ScratchData(const Mapping<dim> & mapping,
1695 * const FiniteElement<dim> &fe,
1696 * const unsigned int quadrature_degree,
1697 * const UpdateFlags update_flags,
1698 * const UpdateFlags interface_update_flags)
1699 * : fe_values(mapping, fe, QGauss<dim>(quadrature_degree), update_flags)
1700 * , fe_interface_values(mapping,
1702 * QGauss<dim - 1>(quadrature_degree),
1703 * interface_update_flags)
1707 * ScratchData(const ScratchData<dim> &scratch_data)
1708 * : fe_values(scratch_data.fe_values.get_mapping(),
1709 * scratch_data.fe_values.get_fe(),
1710 * scratch_data.fe_values.get_quadrature(),
1711 * scratch_data.fe_values.get_update_flags())
1712 * , fe_interface_values(scratch_data.fe_values.get_mapping(),
1713 * scratch_data.fe_values.get_fe(),
1714 * scratch_data.fe_interface_values.get_quadrature(),
1715 * scratch_data.fe_interface_values.get_update_flags())
1718 * FEValues<dim> fe_values;
1719 * FEInterfaceValues<dim> fe_interface_values;
1727 * : cell_index(numbers::invalid_unsigned_int)
1731 * CopyData(const CopyData &) = default;
1735 * unsigned int cell_indices[2];
1739 * unsigned int cell_index;
1741 * std::vector<FaceData> face_data;
1745 * template <int dim, int degree>
1746 * void LaplaceProblem<dim, degree>::estimate()
1748 * TimerOutput::Scope timing(computing_timer, "Estimate
");
1750 * VectorType temp_solution;
1751 * temp_solution.reinit(locally_owned_dofs,
1752 * locally_relevant_dofs,
1753 * mpi_communicator);
1754 * temp_solution = solution;
1756 * const Coefficient<dim> coefficient;
1758 * estimated_error_square_per_cell.reinit(triangulation.n_active_cells());
1760 * using Iterator = typename DoFHandler<dim>::active_cell_iterator;
1764 * Assembler for cell residual @f$h^2 \| f + \epsilon \triangle u \|_K^2@f$
1767 * auto cell_worker = [&](const Iterator & cell,
1768 * ScratchData<dim> &scratch_data,
1769 * CopyData & copy_data) {
1770 * FEValues<dim> &fe_values = scratch_data.fe_values;
1771 * fe_values.reinit(cell);
1773 * RightHandSide<dim> rhs;
1774 * const double rhs_value = rhs.value(cell->center());
1776 * const double nu = coefficient.value(cell->center());
1778 * std::vector<Tensor<2, dim>> hessians(fe_values.n_quadrature_points);
1779 * fe_values.get_function_hessians(temp_solution, hessians);
1781 * copy_data.cell_index = cell->active_cell_index();
1783 * double residual_norm_square = 0.;
1784 * for (unsigned k = 0; k < fe_values.n_quadrature_points; ++k)
1786 * const double residual = (rhs_value + nu * trace(hessians[k]));
1787 * residual_norm_square += residual * residual * fe_values.JxW(k);
1791 * cell->diameter() * cell->diameter() * residual_norm_square;
1796 * Assembler for face term @f$\sum_F h_F \| \jump{\epsilon \nabla u \cdot n}
1800 * auto face_worker = [&](const Iterator & cell,
1801 * const unsigned int &f,
1802 * const unsigned int &sf,
1803 * const Iterator & ncell,
1804 * const unsigned int &nf,
1805 * const unsigned int &nsf,
1806 * ScratchData<dim> & scratch_data,
1807 * CopyData & copy_data) {
1808 * FEInterfaceValues<dim> &fe_interface_values =
1809 * scratch_data.fe_interface_values;
1810 * fe_interface_values.reinit(cell, f, sf, ncell, nf, nsf);
1812 * copy_data.face_data.emplace_back();
1813 * CopyData::FaceData ©_data_face = copy_data.face_data.back();
1815 * copy_data_face.cell_indices[0] = cell->active_cell_index();
1816 * copy_data_face.cell_indices[1] = ncell->active_cell_index();
1818 * const double coeff1 = coefficient.value(cell->center());
1819 * const double coeff2 = coefficient.value(ncell->center());
1821 * std::vector<Tensor<1, dim>> grad_u[2];
1823 * for (unsigned int i = 0; i < 2; ++i)
1825 * grad_u[i].resize(fe_interface_values.n_quadrature_points);
1826 * fe_interface_values.get_fe_face_values(i).get_function_gradients(
1827 * temp_solution, grad_u[i]);
1830 * double jump_norm_square = 0.;
1832 * for (unsigned int qpoint = 0;
1833 * qpoint < fe_interface_values.n_quadrature_points;
1836 * const double jump =
1837 * coeff1 * grad_u[0][qpoint] * fe_interface_values.normal(qpoint) -
1838 * coeff2 * grad_u[1][qpoint] * fe_interface_values.normal(qpoint);
1840 * jump_norm_square += jump * jump * fe_interface_values.JxW(qpoint);
1843 * const double h = cell->face(f)->measure();
1844 * copy_data_face.values[0] = 0.5 * h * jump_norm_square;
1845 * copy_data_face.values[1] = copy_data_face.values[0];
1848 * auto copier = [&](const CopyData ©_data) {
1849 * if (copy_data.cell_index != numbers::invalid_unsigned_int)
1850 * estimated_error_square_per_cell[copy_data.cell_index] += copy_data.value;
1852 * for (auto &cdf : copy_data.face_data)
1853 * for (unsigned int j = 0; j < 2; ++j)
1854 * estimated_error_square_per_cell[cdf.cell_indices[j]] += cdf.values[j];
1857 * const unsigned int n_gauss_points = degree + 1;
1858 * ScratchData<dim> scratch_data(mapping,
1861 * update_hessians | update_quadrature_points |
1862 * update_JxW_values,
1863 * update_values | update_gradients |
1864 * update_JxW_values | update_normal_vectors);
1865 * CopyData copy_data;
1869 * We need to assemble each interior face once but we need to make sure that
1870 * both processes assemble the face term between a locally owned and a ghost
1871 * cell. This is achieved by setting the
1872 * MeshWorker::assemble_ghost_faces_both flag. We need to do this, because
1873 * we do not communicate the error estimator contributions here.
1876 * MeshWorker::mesh_loop(dof_handler.begin_active(),
1877 * dof_handler.end(),
1882 * MeshWorker::assemble_own_cells |
1883 * MeshWorker::assemble_ghost_faces_both |
1884 * MeshWorker::assemble_own_interior_faces_once,
1885 * /*boundary_worker=*/nullptr,
1888 * const double global_error_estimate =
1889 * std::sqrt(Utilities::MPI::sum(estimated_error_square_per_cell.l1_norm(),
1890 * mpi_communicator));
1891 * pcout << " Global error estimate:
" << global_error_estimate
1899 * <a name="LaplaceProblemrefine_grid
"></a>
1900 * <h4>LaplaceProblem::refine_grid()</h4>
1904 * We use the cell-wise estimator stored in the vector @p estimate_vector and
1905 * refine a fixed number of cells (chosen here to roughly double the number of
1906 * DoFs in each step).
1909 * template <int dim, int degree>
1910 * void LaplaceProblem<dim, degree>::refine_grid()
1912 * TimerOutput::Scope timing(computing_timer, "Refine grid
");
1914 * const double refinement_fraction = 1. / (std::pow(2.0, dim) - 1.);
1915 * parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number(
1916 * triangulation, estimated_error_square_per_cell, refinement_fraction, 0.0);
1918 * triangulation.execute_coarsening_and_refinement();
1925 * <a name="LaplaceProblemoutput_results
"></a>
1926 * <h4>LaplaceProblem::output_results()</h4>
1930 * The output_results() function is similar to the ones found in many of the
1931 * tutorials (see @ref step_40 "step-40
" for example).
1934 * template <int dim, int degree>
1935 * void LaplaceProblem<dim, degree>::output_results(const unsigned int cycle)
1937 * TimerOutput::Scope timing(computing_timer, "Output results
");
1939 * VectorType temp_solution;
1940 * temp_solution.reinit(locally_owned_dofs,
1941 * locally_relevant_dofs,
1942 * mpi_communicator);
1943 * temp_solution = solution;
1945 * DataOut<dim> data_out;
1946 * data_out.attach_dof_handler(dof_handler);
1947 * data_out.add_data_vector(temp_solution, "solution
");
1949 * Vector<float> subdomain(triangulation.n_active_cells());
1950 * for (unsigned int i = 0; i < subdomain.size(); ++i)
1951 * subdomain(i) = triangulation.locally_owned_subdomain();
1952 * data_out.add_data_vector(subdomain, "subdomain
");
1954 * Vector<float> level(triangulation.n_active_cells());
1955 * for (const auto &cell : triangulation.active_cell_iterators())
1956 * level(cell->active_cell_index()) = cell->level();
1957 * data_out.add_data_vector(level, "level");
1959 * if (estimated_error_square_per_cell.size() > 0)
1960 * data_out.add_data_vector(estimated_error_square_per_cell,
1961 * "estimated_error_square_per_cell
");
1963 * data_out.build_patches();
1965 * const std::string pvtu_filename = data_out.write_vtu_with_pvtu_record(
1966 * "", "solution
", cycle, mpi_communicator, 2 /*n_digits*/, 1 /*n_groups*/);
1968 * pcout << " Wrote
" << pvtu_filename << std::endl;
1975 * <a name="LaplaceProblemrun
"></a>
1976 * <h4>LaplaceProblem::run()</h4>
1980 * As in most tutorials, this function calls the various functions defined
1981 * above to setup, assemble, solve, and output the results.
1984 * template <int dim, int degree>
1985 * void LaplaceProblem<dim, degree>::run()
1987 * for (unsigned int cycle = 0; cycle < settings.n_steps; ++cycle)
1989 * pcout << "Cycle
" << cycle << ':' << std::endl;
1993 * pcout << " Number of active cells:
"
1994 * << triangulation.n_global_active_cells();
1998 * We only output level cell data for the GMG methods (same with DoF
1999 * data below). Note that the partition efficiency is irrelevant for AMG
2000 * since the level hierarchy is not distributed or used during the
2004 * if (settings.solver == Settings::gmg_mf ||
2005 * settings.solver == Settings::gmg_mb)
2006 * pcout << " (
" << triangulation.n_global_levels() << " global levels)
"
2008 * << " Partition efficiency:
"
2009 * << 1.0 / MGTools::workload_imbalance(triangulation);
2010 * pcout << std::endl;
2016 * Only set up the multilevel hierarchy for GMG.
2019 * if (settings.solver == Settings::gmg_mf ||
2020 * settings.solver == Settings::gmg_mb)
2021 * setup_multigrid();
2023 * pcout << " Number of degrees of freedom:
" << dof_handler.n_dofs();
2024 * if (settings.solver == Settings::gmg_mf ||
2025 * settings.solver == Settings::gmg_mb)
2027 * pcout << " (by
level:
";
2028 * for (unsigned int level = 0; level < triangulation.n_global_levels();
2030 * pcout << dof_handler.n_dofs(level)
2031 * << (level == triangulation.n_global_levels() - 1 ? ")
" :
2034 * pcout << std::endl;
2038 * For the matrix-free method, we only assemble the right-hand side.
2039 * For both matrix-based methods, we assemble both active matrix and
2040 * right-hand side, and only assemble the multigrid matrices for
2044 * if (settings.solver == Settings::gmg_mf)
2046 * else /*gmg_mb or amg*/
2048 * assemble_system();
2049 * if (settings.solver == Settings::gmg_mb)
2050 * assemble_multigrid();
2056 * if (settings.output)
2057 * output_results(cycle);
2059 * computing_timer.print_summary();
2060 * computing_timer.reset();
2068 * <a name="Themainfunction
"></a>
2069 * <h3>The main() function</h3>
2073 * This is a similar main function to @ref step_40 "step-40
", with the exception that
2074 * we require the user to pass a .prm file as a sole command line
2075 * argument (see @ref step_29 "step-29
" and the documentation of the ParameterHandler
2076 * class for a complete discussion of parameter files).
2079 * int main(int argc, char *argv[])
2081 * using namespace dealii;
2082 * Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
2084 * Settings settings;
2085 * if (!settings.try_parse((argc > 1) ? (argv[1]) : ""))
2090 * constexpr unsigned int fe_degree = 2;
2092 * switch (settings.dimension)
2096 * LaplaceProblem<2, fe_degree> test(settings);
2104 * LaplaceProblem<3, fe_degree> test(settings);
2111 * Assert(false, ExcMessage("This program only works in 2
d and 3
d.
"));
2114 * catch (std::exception &exc)
2116 * std::cerr << std::endl
2118 * << "----------------------------------------------------
"
2120 * std::cerr << "Exception on processing:
" << std::endl
2121 * << exc.what() << std::endl
2122 * << "Aborting!
" << std::endl
2123 * << "----------------------------------------------------
"
2125 * MPI_Abort(MPI_COMM_WORLD, 1);
2130 * std::cerr << std::endl
2132 * << "----------------------------------------------------
"
2134 * std::cerr << "Unknown exception!
" << std::endl
2135 * << "Aborting!
" << std::endl
2136 * << "----------------------------------------------------
"
2138 * MPI_Abort(MPI_COMM_WORLD, 2);
2145<a name="Results
"></a><h1>Results</h1>
2148When you run the program using the following command
2150mpirun -np 16 ./step-50 gmg_mf_2d.prm
2152the screen output should look like the following:
2155 Number of active cells: 12 (2 global levels)
2156 Partition efficiency: 0.1875
2157 Number of degrees of freedom: 65 (by level: 21, 65)
2158 Number of CG iterations: 10
2159 Global error estimate: 0.355373
2160 Wrote solution_00.pvtu
2163+---------------------------------------------+------------+------------+
2164| Total wallclock time elapsed since start | 0.0163s | |
2166| Section | no. calls | wall time | % of total |
2167+---------------------------------+-----------+------------+------------+
2168| Assemble right-hand side | 1 | 0.000374s | 2.3% |
2169| Estimate | 1 | 0.000724s | 4.4% |
2170| Output results | 1 | 0.00277s | 17% |
2171| Setup | 1 | 0.00225s | 14% |
2172| Setup multigrid | 1 | 0.00181s | 11% |
2173| Solve | 1 | 0.00364s | 22% |
2174| Solve: 1 multigrid V-cycle | 1 | 0.000354s | 2.2% |
2175| Solve: CG | 1 | 0.00151s | 9.3% |
2176| Solve: Preconditioner setup | 1 | 0.00125s | 7.7% |
2177+---------------------------------+-----------+------------+------------+
2180 Number of active cells: 24 (3 global levels)
2181 Partition efficiency: 0.276786
2182 Number of degrees of freedom: 139 (by level: 21, 65, 99)
2183 Number of CG iterations: 10
2184 Global error estimate: 0.216726
2185 Wrote solution_01.pvtu
2188+---------------------------------------------+------------+------------+
2189| Total wallclock time elapsed since start | 0.0169s | |
2191| Section | no. calls | wall time | % of total |
2192+---------------------------------+-----------+------------+------------+
2193| Assemble right-hand side | 1 | 0.000309s | 1.8% |
2194| Estimate | 1 | 0.00156s | 9.2% |
2195| Output results | 1 | 0.00222s | 13% |
2196| Refine grid | 1 | 0.00278s | 16% |
2197| Setup | 1 | 0.00196s | 12% |
2198| Setup multigrid | 1 | 0.0023s | 14% |
2199| Solve | 1 | 0.00565s | 33% |
2200| Solve: 1 multigrid V-cycle | 1 | 0.000349s | 2.1% |
2201| Solve: CG | 1 | 0.00285s | 17% |
2202| Solve: Preconditioner setup | 1 | 0.00195s | 12% |
2203+---------------------------------+-----------+------------+------------+
2206 Number of active cells: 51 (4 global levels)
2207 Partition efficiency: 0.41875
2208 Number of degrees of freedom: 245 (by level: 21, 65, 225, 25)
2209 Number of CG iterations: 11
2210 Global error estimate: 0.112098
2211 Wrote solution_02.pvtu
2214+---------------------------------------------+------------+------------+
2215| Total wallclock time elapsed since start | 0.0183s | |
2217| Section | no. calls | wall time | % of total |
2218+---------------------------------+-----------+------------+------------+
2219| Assemble right-hand side | 1 | 0.000274s | 1.5% |
2220| Estimate | 1 | 0.00127s | 6.9% |
2221| Output results | 1 | 0.00227s | 12% |
2222| Refine grid | 1 | 0.0024s | 13% |
2223| Setup | 1 | 0.00191s | 10% |
2224| Setup multigrid | 1 | 0.00295s | 16% |
2225| Solve | 1 | 0.00702s | 38% |
2226| Solve: 1 multigrid V-cycle | 1 | 0.000398s | 2.2% |
2227| Solve: CG | 1 | 0.00376s | 21% |
2228| Solve: Preconditioner setup | 1 | 0.00238s | 13% |
2229+---------------------------------+-----------+------------+------------+
2234Here, the timing of the `solve()` function is split up in 3 parts: setting
2235up the multigrid preconditioner, execution of a single multigrid V-cycle, and
2236the CG solver. The V-cycle that is timed is unnecessary for the overall solve
2237and only meant to give an insight at the different costs for AMG and GMG.
2238Also it should be noted that when using the AMG solver, "Workload imbalance
"
2239is not included in the output since the hierarchy of coarse meshes is not
2242All results in this section are gathered on Intel Xeon Platinum 8280 (Cascade
2243Lake) nodes which have 56 cores and 192GB per node and support AVX-512 instructions,
2244allowing for vectorization over 8 doubles (vectorization used only in the matrix-free
2245computations). The code is compiled using gcc 7.1.0 with intel-mpi 17.0.3. Trilinos
224612.10.1 is used for the matrix-based GMG/AMG computations.
2248We can then gather a variety of information by calling the program
2249with the input files that are provided in the directory in which
2250@ref step_50 "step-50
" is located. Using these, and adjusting the number of mesh
2251refinement steps, we can produce information about how well the
2254The following table gives weak scaling timings for this program on up to 256M DoFs
2255and 7,168 processors. (Recall that weak scaling keeps the number of
2256degrees of freedom per processor constant while increasing the number of
2257processors; i.e., it considers larger and larger problems.)
2258Here, @f$\mathbb{E}@f$ is the partition efficiency from the
2259 introduction (also equal to 1.0/workload imbalance), "Setup
" is a combination
2260of setup, setup multigrid, assemble, and assemble multigrid from the timing blocks,
2261and "Prec
" is the preconditioner setup. Ideally all times would stay constant
2262over each problem size for the individual solvers, but since the partition
2263efficiency decreases from 0.371 to 0.161 from largest to smallest problem size,
2264we expect to see an approximately @f$0.371/0.161=2.3@f$ times increase in timings
2265for GMG. This is, in fact, pretty close to what we really get:
2267<table align="center" class="doxtable
">
2269 <th colspan="4
"></th>
2271 <th colspan="4
">MF-GMG</th>
2273 <th colspan="4
">MB-GMG</th>
2275 <th colspan="4
">AMG</th>
2278 <th align="right
">Procs</th>
2279 <th align="right
">Cycle</th>
2280 <th align="right
">DoFs</th>
2281 <th align="right
">@f$\mathbb{E}@f$</th>
2283 <th align="right
">Setup</th>
2284 <th align="right
">Prec</th>
2285 <th align="right
">Solve</th>
2286 <th align="right
">Total</th>
2288 <th align="right
">Setup</th>
2289 <th align="right
">Prec</th>
2290 <th align="right
">Solve</th>
2291 <th align="right
">Total</th>
2293 <th align="right
">Setup</th>
2294 <th align="right
">Prec</th>
2295 <th align="right
">Solve</th>
2296 <th align="right
">Total</th>
2299 <td align="right
">112</th>
2300 <td align="right
">13</th>
2301 <td align="right
">4M</th>
2302 <td align="right
">0.37</th>
2304 <td align="right
">0.742</th>
2305 <td align="right
">0.393</th>
2306 <td align="right
">0.200</th>
2307 <td align="right
">1.335</th>
2309 <td align="right
">1.714</th>
2310 <td align="right
">2.934</th>
2311 <td align="right
">0.716</th>
2312 <td align="right
">5.364</th>
2314 <td align="right
">1.544</th>
2315 <td align="right
">0.456</th>
2316 <td align="right
">1.150</th>
2317 <td align="right
">3.150</th>
2320 <td align="right
">448</th>
2321 <td align="right
">15</th>
2322 <td align="right
">16M</th>
2323 <td align="right
">0.29</th>
2325 <td align="right
">0.884</th>
2326 <td align="right
">0.535</th>
2327 <td align="right
">0.253</th>
2328 <td align="right
">1.672</th>
2330 <td align="right
">1.927</th>
2331 <td align="right
">3.776</th>
2332 <td align="right
">1.190</th>
2333 <td align="right
">6.893</th>
2335 <td align="right
">1.544</th>
2336 <td align="right
">0.456</th>
2337 <td align="right
">1.150</th>
2338 <td align="right
">3.150</th>
2341 <td align="right
">1,792</th>
2342 <td align="right
">17</th>
2343 <td align="right
">65M</th>
2344 <td align="right
">0.22</th>
2346 <td align="right
">1.122</th>
2347 <td align="right
">0.686</th>
2348 <td align="right
">0.309</th>
2349 <td align="right
">2.117</th>
2351 <td align="right
">2.171</th>
2352 <td align="right
">4.862</th>
2353 <td align="right
">1.660</th>
2354 <td align="right
">8.693</th>
2356 <td align="right
">1.654</th>
2357 <td align="right
">0.546</th>
2358 <td align="right
">1.460</th>
2359 <td align="right
">3.660</th>
2362 <td align="right
">7,168</th>
2363 <td align="right
">19</th>
2364 <td align="right
">256M</th>
2365 <td align="right
">0.16</th>
2367 <td align="right
">1.214</th>
2368 <td align="right
">0.893</th>
2369 <td align="right
">0.521</th>
2370 <td align="right
">2.628</th>
2372 <td align="right
">2.386</th>
2373 <td align="right
">7.260</th>
2374 <td align="right
">2.560</th>
2375 <td align="right
">12.206</th>
2377 <td align="right
">1.844</th>
2378 <td align="right
">1.010</th>
2379 <td align="right
">1.890</th>
2380 <td align="right
">4.744</th>
2384On the other hand, the algebraic multigrid in the last set of columns
2385is relatively unaffected by the increasing imbalance of the mesh
2386hierarchy (because it doesn't use the mesh hierarchy) and the growth
2387in time is rather driven by other factors that are well documented in
2388the literature (most notably that the algorithmic complexity of
2389some parts of algebraic multigrid methods appears to be @f${\cal O}(N
2390\log N)@f$ instead of @f${\cal O}(N)@f$ for geometric multigrid).
2392The upshort of the table above is that the matrix-free geometric multigrid
2393method appears to be the fastest approach to solving this equation if
2394not by a huge margin. Matrix-based methods, on the other hand, are
2395consistently the worst.
2397The following figure provides strong scaling results for each method, i.e.,
2398we solve the same problem on more and more processors. Specifically,
2399we consider the problems after 16 mesh refinement cycles
2400(32M DoFs) and 19 cycles (256M DoFs), on between 56 to 28,672 processors:
2402<img width="600px
" src="https:
2404While the
matrix-based GMG solver and AMG
scale similarly and have a
2405similar time to solution (at least as
long as there is a substantial
2406number of unknowns per processor -- say, several 10,000), the
2407matrix-
free GMG solver scales much better and solves the finer problem
2408in roughly the same time as the AMG solver
for the coarser mesh with
2409only an eighth of the number of processors. Conversely, it can solve the
2410same problem on the same number of processors in about
one eighth the
2414<a name=
"Possibilitiesforextensions"></a><h3> Possibilities
for extensions </h3>
2417<a name=
"Testingconvergenceandhigherorderelements"></a><h4> Testing convergence and higher order elements </h4>
2420The finite element degree is currently hard-coded as 2, see the
template
2421arguments of the main
class. It is easy to change. To test, it would be
2422interesting to
switch to a test problem with a reference solution. This way,
2423you can compare error rates.
2425<a name=
"Coarsesolver"></a><h4> Coarse solver </h4>
2428A more interesting example would involve a more complicated coarse mesh (see
2429@ref step_49
"step-49" for inspiration). The issue in that
case is that the coarsest
2430level of the mesh hierarchy is actually quite large, and
one would
2431have to think about ways to solve the coarse
level problem
2432efficiently. (This is not an issue
for algebraic multigrid methods
2433because they would just
continue to build coarser and coarser levels
2434of the
matrix, regardless of their geometric origin.)
2436In the program here, we simply solve the coarse
level problem with a
2437Conjugate Gradient method without any preconditioner. That is acceptable
2438if the coarse problem is really small --
for example,
if the coarse
2439mesh had a single cell, then the coarse mesh problems has a @f$9\times 9@f$
2440matrix in 2
d, and a @f$27\times 27@f$
matrix in 3
d;
for the coarse mesh we
2441use on the @f$L@f$-shaped domain of the current program, these sizes are
2442@f$21\times 21@f$ in 2
d and @f$117\times 117@f$ in 3
d. But
if the coarse mesh
2443consists of hundreds or thousands of cells,
this approach will no
2444longer work and might start to dominate the overall
run-time of each
V-cyle.
2445A common approach is then to solve the coarse mesh problem
using an
2446algebraic multigrid preconditioner;
this would then, however, require
2448input to the AMG implementation.
2451<a name=
"PlainProg"></a>
2452<h1> The plain program</h1>
2453@include
"step-50.cc"
void reinit(const IndexSet &local_constraints=IndexSet())
void add_lines(const std::vector< bool > &lines)
virtual void parse_input(std::istream &input, const std::string &filename="input file", const std::string &last_line="", const bool skip_undefined=false)
std::ostream & print_parameters(std::ostream &out, const OutputStyle style) const
long int get_integer(const std::string &entry_string) const
bool get_bool(const std::string &entry_name) const
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)
std::string get(const std::string &entry_string) const
double get_double(const std::string &entry_name) const
@ update_values
Shape function values.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
__global__ void set(Number *val, const Number s, const size_type N)
static ::ExceptionBase & ExcNotImplemented()
#define Assert(cond, exc)
#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, SparsityPatternType &sparsity_pattern, const AffineConstraints< number > &constraints=AffineConstraints< number >(), const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
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.
static const types::blas_int one
SparseMatrix< double > SparseMatrix
PETScWrappers::PreconditionBoomerAMG PreconditionAMG
PETScWrappers::PreconditionJacobi PreconditionJacobi
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)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
@ 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 typename identity< Iterator >::type &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
UpdateFlags mapping_update_flags