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: 8
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 CG iterations: 9
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 CG iterations: 10
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 CG iterations: 10
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 CG iterations: 10
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 CG iterations: 11
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 CG iterations: 10
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 CG iterations: 11
</pre>
That's almost perfect multigrid performance: the linear residual gets reduced by 12 orders of
<h3> Possible extensions </h3>
-We encourage you to switch generate timings for the solve() call and compare to
+We encourage you to generate timings for the solve() call and compare to
step 6. You will see that the multigrid method has quite an overhead
on coarse meshes, but that it always beats other methods on fine
meshes because of its optimal complexity.
but can figure out level matrices and similar things by itself. Algebraic
multigrid methods do exactly this, and we will use them in step-31 for the
solution of a Stokes problem and in step-32 and step-40 for a parallel
-variation. That said, a parallel version of this example program with MPI is found
-as step-50.
+variation. That said, a parallel version of this example program with MPI can be
+found in step-50.
Finally, one may want to think how to use geometric multigrid for other kinds of
problems, specifically @ref vector_valued "vector valued problems". This is the
{
// @sect3{The Scratch and Copy objects}
//
- // We use MeshWorker::mesh_loop() to assemble our matrices. For this, we need
- // a ScratchData object to store temporary data on each cell (this is just the
- // FEValues object) and a CopyData object that will contain the output of each
- // cell assembly.
+ // We use MeshWorker::mesh_loop() to assemble our matrices. For this, we
+ // need a ScratchData object to store temporary data on each cell (this is
+ // just the FEValues object) and a CopyData object that will contain the
+ // output of each cell assembly. For more details about the usage of scratch
+ // and copy objects, see the WorkStream namespace.
template <int dim>
struct ScratchData
{
// us, and thus the difference between this function and the previous lies
// only in the setup of the assembler and the different iterators in the loop.
//
- // We generate an AffineConstraints<> object
+ // We generate an AffineConstraints<> object for each level containing the
+ // boundary and interface dofs as constrained entries. The corresponding
+ // object is then used to generate the level matrices.
template <int dim>
void LaplaceProblem<dim>::assemble_multigrid()
{
solution = 0;
solver.solve(system_matrix, solution, system_rhs, preconditioner);
- std::cout << " Number of CG iterations: " << solver_control.last_step()
+ std::cout << " Number of CG iterations: " << solver_control.last_step()
<< "\n"
<< std::endl;
constraints.distribute(solution);
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