Cycle 0
Number of active cells: 80
Number of degrees of freedom: 89 (by level: 8, 25, 89)
- Number of CG iterations: 8
+ Number of CG iterations: 5
Cycle 1
- Number of active cells: 158
- Number of degrees of freedom: 183 (by level: 8, 25, 89, 138)
- Number of CG iterations: 9
+ Number of active cells: 152
+ Number of degrees of freedom: 174 (by level: 8, 25, 89, 123)
+ Number of CG iterations: 5
Cycle 2
- Number of active cells: 302
- Number of degrees of freedom: 352 (by level: 8, 25, 89, 223, 160)
- Number of CG iterations: 10
+ Number of active cells: 287
+ Number of degrees of freedom: 331 (by level: 8, 25, 89, 223, 127)
+ Number of CG iterations: 6
Cycle 3
- Number of active cells: 578
- Number of degrees of freedom: 649 (by level: 8, 25, 89, 231, 494, 66)
- Number of CG iterations: 10
+ Number of active cells: 545
+ Number of degrees of freedom: 608 (by level: 8, 25, 89, 231, 476, 24)
+ Number of CG iterations: 6
Cycle 4
- Number of active cells: 1100
- Number of degrees of freedom: 1218 (by level: 8, 25, 89, 274, 764, 417, 126)
- Number of CG iterations: 10
+ Number of active cells: 1034
+ Number of degrees of freedom: 1137 (by level: 8, 25, 89, 274, 756, 417, 15)
+ Number of CG iterations: 6
Cycle 5
- Number of active cells: 2096
- Number of degrees of freedom: 2317 (by level: 8, 25, 89, 304, 779, 1214, 817)
- Number of CG iterations: 11
+ Number of active cells: 1964
+ Number of degrees of freedom: 2181 (by level: 8, 25, 89, 304, 779, 1030, 817)
+ Number of CG iterations: 6
Cycle 6
- Number of active cells: 3986
- Number of degrees of freedom: 4366 (by level: 8, 25, 89, 337, 836, 2270, 897, 1617)
- Number of CG iterations: 10
+ Number of active cells: 3734
+ Number of degrees of freedom: 4101 (by level: 8, 25, 89, 337, 779, 2046, 885, 1545)
+ Number of CG iterations: 6
Cycle 7
- Number of active cells: 7574
- Number of degrees of freedom: 8350 (by level: 8, 25, 89, 337, 1086, 2835, 2268, 1789, 3217)
- Number of CG iterations: 11
+ Number of active cells: 7094
+ Number of degrees of freedom: 7833 (by level: 8, 25, 89, 337, 1056, 2835, 1740, 1765, 3085)
+ Number of CG iterations: 6
+
</pre>
-That's almost perfect multigrid performance: the linear residual gets reduced by 12 orders of
-magnitude in 10 iteration steps, and the results are almost independent of the mesh size. That's
-obviously in part due to the simple nature of the problem solved, but
-it shows the power of multigrid methods.
+That's almost perfect multigrid performance: the linear residual gets
+reduced by 6 orders of magnitude in 5 or 6 iterations, and the results
+are almost independent of the mesh size. That's obviously in part due
+to the simple nature of the problem solved, but it also shows the
+power of multigrid methods.
<h3> Possibilities for extensions </h3>
// With all this together, we can finally get about solving the linear
// system in the usual way:
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> solver(solver_control);
solution = 0;
Number of active cells: 256
Total number of cells: 341
Number of degrees of freedom: 289
- 26 CG iterations needed to obtain convergence.
+ 19 CG iterations needed to obtain convergence.
Solving problem in 3 space dimensions.
Number of active cells: 4096
Total number of cells: 4681
Number of degrees of freedom: 4913
- 30 CG iterations needed to obtain convergence.
+ 20 CG iterations needed to obtain convergence.
@endcode
It is obvious that in three spatial dimensions the number of cells and
therefore also the number of degrees of freedom is
template <int dim>
void Step4<dim>::solve()
{
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> solver(solver_control);
solver.solve(system_matrix, solution, system_rhs, PreconditionIdentity());
When you run the program, on a single processor or with your local MPI
installation on a few, you should get output like this:
@code
+Running with PETSc on 1 MPI rank(s)...
Cycle 0:
Number of active cells: 1024
Number of degrees of freedom: 4225
- Solved in 10 iterations.
+ Solved in 7 iterations.
+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start | 0.176s | |
+| Total wallclock time elapsed since start | 0.548s | |
| | | |
| Section | no. calls | wall time | % of total |
+---------------------------------+-----------+------------+------------+
-| assembly | 1 | 0.0209s | 12% |
-| output | 1 | 0.0189s | 11% |
-| setup | 1 | 0.0299s | 17% |
-| solve | 1 | 0.0419s | 24% |
+| assembly | 1 | 0.242s | 44% |
+| output | 1 | 0.0495s | 9% |
+| setup | 1 | 0.102s | 19% |
+| solve | 1 | 0.0283s | 5.2% |
+---------------------------------+-----------+------------+------------+
Cycle 1:
- Number of active cells: 1954
- Number of degrees of freedom: 8399
- Solved in 10 iterations.
+ Number of active cells: 1963
+ Number of degrees of freedom: 8437
+ Solved in 7 iterations.
+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start | 0.327s | |
+| Total wallclock time elapsed since start | 1.19s | |
| | | |
| Section | no. calls | wall time | % of total |
+---------------------------------+-----------+------------+------------+
-| assembly | 1 | 0.0368s | 11% |
-| output | 1 | 0.0208s | 6.4% |
-| refine | 1 | 0.157s | 48% |
-| setup | 1 | 0.0452s | 14% |
-| solve | 1 | 0.0668s | 20% |
+| assembly | 1 | 0.469s | 40% |
+| output | 1 | 0.0899s | 7.6% |
+| refine | 1 | 0.429s | 36% |
+| setup | 1 | 0.177s | 15% |
+| solve | 1 | 0.0204s | 1.7% |
+---------------------------------+-----------+------------+------------+
Cycle 2:
- Number of active cells: 3664
- Number of degrees of freedom: 16183
- Solved in 11 iterations.
+ Number of active cells: 3670
+ Number of degrees of freedom: 16175
+ Solved in 7 iterations.
...
@endcode
LA::MPI::Vector completely_distributed_solution(locally_owned_dofs,
mpi_communicator);
- SolverControl solver_control(dof_handler.n_dofs(), 1e-12);
+ SolverControl solver_control(dof_handler.n_dofs(),
+ 1e-6 * system_rhs.l2_norm());
LA::SolverCG solver(solver_control);
Number of active cells: 20
Total number of cells: 20
Number of degrees of freedom: 25
- 13 CG iterations needed to obtain convergence.
+ 8 CG iterations needed to obtain convergence.
Cycle 1:
Number of active cells: 80
Total number of cells: 100
Number of degrees of freedom: 89
- 18 CG iterations needed to obtain convergence.
+ 12 CG iterations needed to obtain convergence.
Cycle 2:
Number of active cells: 320
Total number of cells: 420
Number of degrees of freedom: 337
- 29 CG iterations needed to obtain convergence.
+ 21 CG iterations needed to obtain convergence.
Cycle 3:
Number of active cells: 1280
Total number of cells: 1700
Number of degrees of freedom: 1313
- 52 CG iterations needed to obtain convergence.
+ 38 CG iterations needed to obtain convergence.
Cycle 4:
Number of active cells: 5120
Total number of cells: 6820
Number of degrees of freedom: 5185
- 95 CG iterations needed to obtain convergence.
+ 70 CG iterations needed to obtain convergence.
Cycle 5:
Number of active cells: 20480
Total number of cells: 27300
Number of degrees of freedom: 20609
- 182 CG iterations needed to obtain convergence.
+ 136 CG iterations needed to obtain convergence.
@endcode
template <int dim>
void Step5<dim>::solve()
{
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> solver(solver_control);
PreconditionSSOR<SparseMatrix<double>> preconditioner;
Number of active cells: 20
Number of degrees of freedom: 89
Cycle 1:
- Number of active cells: 44
- Number of degrees of freedom: 209
+ Number of active cells: 38
+ Number of degrees of freedom: 183
Cycle 2:
- Number of active cells: 92
- Number of degrees of freedom: 449
+ Number of active cells: 71
+ Number of degrees of freedom: 325
Cycle 3:
- Number of active cells: 200
- Number of degrees of freedom: 921
+ Number of active cells: 146
+ Number of degrees of freedom: 689
Cycle 4:
- Number of active cells: 440
- Number of degrees of freedom: 2017
+ Number of active cells: 284
+ Number of degrees of freedom: 1373
Cycle 5:
- Number of active cells: 956
- Number of degrees of freedom: 4425
+ Number of active cells: 599
+ Number of degrees of freedom: 2769
Cycle 6:
- Number of active cells: 1916
- Number of degrees of freedom: 8993
+ Number of active cells: 1241
+ Number of degrees of freedom: 5833
Cycle 7:
- Number of active cells: 3860
- Number of degrees of freedom: 18353
+ Number of active cells: 2507
+ Number of degrees of freedom: 11916
@endcode
template <int dim>
void Step6<dim>::solve()
{
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> solver(solver_control);
PreconditionSSOR<SparseMatrix<double>> preconditioner;
Number of active cells: 64
Number of degrees of freedom: 81
Cycle 1:
- Number of active cells: 124
- Number of degrees of freedom: 157
+ Number of active cells: 121
+ Number of degrees of freedom: 154
Cycle 2:
Number of active cells: 280
Number of degrees of freedom: 341
Cycle 3:
- Number of active cells: 577
- Number of degrees of freedom: 690
+ Number of active cells: 565
+ Number of degrees of freedom: 678
Cycle 4:
- Number of active cells: 1099
- Number of degrees of freedom: 1264
+ Number of active cells: 1075
+ Number of degrees of freedom: 1240
Cycle 5:
- Number of active cells: 2191
- Number of degrees of freedom: 2452
+ Number of active cells: 2041
+ Number of degrees of freedom: 2306
Cycle 6:
- Number of active cells: 4165
- Number of degrees of freedom: 4510
+ Number of active cells: 3913
+ Number of degrees of freedom: 4216
Cycle 7:
- Number of active cells: 7915
- Number of degrees of freedom: 8440
+ Number of active cells: 7432
+ Number of degrees of freedom: 7909
Cycle 8:
- Number of active cells: 15196
- Number of degrees of freedom: 15912
+ Number of active cells: 14203
+ Number of degrees of freedom: 14870
cycle cells dofs L2 H1 Linfty
0 64 81 1.840e+00 2.858e+00 1.835e+00
- 1 124 157 5.190e-02 1.200e+00 1.344e-01
+ 1 121 154 5.336e-02 1.200e+00 1.354e-01
2 280 341 1.439e-02 7.892e-01 7.554e-02
- 3 577 690 8.627e-03 5.061e-01 2.805e-02
- 4 1099 1264 3.217e-03 3.030e-01 1.073e-02
- 5 2191 2452 1.445e-03 2.097e-01 5.073e-03
- 6 4165 4510 8.387e-04 1.460e-01 2.013e-03
- 7 7915 8440 7.051e-04 1.053e-01 1.804e-03
- 8 15196 15912 2.774e-04 7.463e-02 6.911e-04
+ 3 565 678 8.696e-03 5.086e-01 2.843e-02
+ 4 1075 1240 3.245e-03 3.059e-01 1.072e-02
+ 5 2041 2306 2.407e-03 2.147e-01 5.156e-03
+ 6 3913 4216 8.501e-04 1.503e-01 2.033e-03
+ 7 7432 7909 7.113e-04 1.086e-01 1.808e-03
+ 8 14203 14870 3.140e-04 7.671e-02 7.181e-04
Solving with Q1 elements, global refinement
===========================================
0 64 289 1.606e-01 1.278e+00 3.029e-01
1 256 1089 7.638e-03 5.248e-01 4.816e-02
2 1024 4225 8.601e-04 1.086e-01 4.827e-03
- 3 4096 16641 1.107e-04 2.756e-02 7.802e-04
- 4 16384 66049 1.393e-05 6.915e-03 9.971e-05
+ 3 4096 16641 1.107e-04 2.756e-02 7.804e-04
+ 4 16384 66049 1.394e-05 6.915e-03 9.991e-05
n cells H1 L2
0 64 1.278e+00 - - 1.606e-01 - -
1 256 5.248e-01 2.43 1.28 7.638e-03 21.03 4.39
2 1024 1.086e-01 4.83 2.27 8.601e-04 8.88 3.15
3 4096 2.756e-02 3.94 1.98 1.107e-04 7.77 2.96
-4 16384 6.915e-03 3.99 1.99 1.393e-05 7.94 2.99
+4 16384 6.915e-03 3.99 1.99 1.394e-05 7.94 2.99
Solving with Q2 elements, adaptive refinement
===========================================
Number of active cells: 64
Number of degrees of freedom: 289
Cycle 1:
- Number of active cells: 124
- Number of degrees of freedom: 577
+ Number of active cells: 121
+ Number of degrees of freedom: 569
Cycle 2:
- Number of active cells: 289
- Number of degrees of freedom: 1353
+ Number of active cells: 280
+ Number of degrees of freedom: 1317
Cycle 3:
- Number of active cells: 547
- Number of degrees of freedom: 2531
+ Number of active cells: 529
+ Number of degrees of freedom: 2459
Cycle 4:
- Number of active cells: 1057
- Number of degrees of freedom: 4919
+ Number of active cells: 1015
+ Number of degrees of freedom: 4719
Cycle 5:
- Number of active cells: 2059
- Number of degrees of freedom: 9223
+ Number of active cells: 1963
+ Number of degrees of freedom: 9039
Cycle 6:
- Number of active cells: 3913
- Number of degrees of freedom: 17887
+ Number of active cells: 3727
+ Number of degrees of freedom: 17143
Cycle 7:
- Number of active cells: 7441
- Number of degrees of freedom: 33807
+ Number of active cells: 7081
+ Number of degrees of freedom: 32343
Cycle 8:
- Number of active cells: 14212
- Number of degrees of freedom: 64731
+ Number of active cells: 13525
+ Number of degrees of freedom: 60895
cycle cells dofs L2 H1 Linfty
0 64 289 1.606e-01 1.278e+00 3.029e-01
- 1 124 577 7.891e-03 5.256e-01 4.852e-02
- 2 289 1353 1.070e-03 1.155e-01 4.868e-03
- 3 547 2531 5.962e-04 5.101e-02 1.876e-03
- 4 1057 4919 1.977e-04 3.094e-02 7.923e-04
- 5 2059 9223 7.738e-05 1.974e-02 7.270e-04
- 6 3913 17887 2.925e-05 8.772e-03 1.463e-04
- 7 7441 33807 1.024e-05 4.121e-03 8.567e-05
- 8 14212 64731 3.761e-06 2.108e-03 2.167e-05
+ 1 121 569 7.916e-03 5.257e-01 4.857e-02
+ 2 280 1317 1.092e-03 1.165e-01 4.832e-03
+ 3 529 2459 5.999e-04 5.177e-02 1.873e-03
+ 4 1015 4719 2.100e-04 3.245e-02 7.938e-04
+ 5 1963 9039 7.821e-05 1.990e-02 7.261e-04
+ 6 3727 17143 2.868e-05 8.498e-03 1.462e-04
+ 7 7081 32343 1.146e-05 4.360e-03 8.576e-05
+ 8 13525 60895 3.747e-06 2.123e-03 2.174e-05
@endcode
But from an engineering perspective, the question is often more
practical: How fine do I have to make my mesh so that the error is
"small enough"? In other words, if in the table above the $H^1$
-semi-norm has been reduced to `4.121e-03`, is this good enough for me
+semi-norm has been reduced to `2.123e-03`, is this good enough for me
to sign the blueprint and declare that our numerical simulation showed
that the bridge is strong enough?
typically compare the numerical solution $u_h$ against the exact
solution $u$ in situations that matter -- if I knew $u$, I would not
have to compute $u_h$. But even if I could, the question to ask in
-general is then: `4.121e-03` *what*? The solution will have physical
+general is then: `2.123e-03` *what*? The solution will have physical
units, say kg-times-meter-squared, and I'm integrating a function with
units square of the above over the domain, and then take the square
root. So if the domain is two-dimensional, the units of
$\|u-u_h\|_{L_2}$ are kg-times-meter-cubed. The question is then: Is
-$4.121\times 10^{-3}$ kg-times-meter-cubed small? That depends on what
+$2.123\times 10^{-3}$ kg-times-meter-cubed small? That depends on what
you're trying to simulate: If you're an astronomer used to masses
measured in solar masses and distances in light years, then yes, this
is a fantastically small number. But if you're doing atomic physics,
template <int dim>
void HelmholtzProblem<dim>::solve()
{
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> cg(solver_control);
PreconditionSSOR<SparseMatrix<double>> preconditioner;
// @sect4{ElasticProblem::solve}
- // The solver does not care about where the system of equations comes, as
- // long as it stays positive definite and symmetric (which are the
+ // The solver does not care about where the system of equations comes from, as
+ // long as it is positive definite and symmetric (which are the
// requirements for the use of the CG solver), which the system indeed
// is. Therefore, we need not change anything.
template <int dim>
void ElasticProblem<dim>::solve()
{
- SolverControl solver_control(1000, 1e-12);
+ SolverControl solver_control(1000, 1e-6 * system_rhs.l2_norm());
SolverCG<Vector<double>> cg(solver_control);
PreconditionSSOR<SparseMatrix<double>> preconditioner;
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