From: bangerth Date: Thu, 11 Feb 2010 16:42:57 +0000 (+0000) Subject: More text, more outlook. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d8177de9d12d5af3f775f0845434e78316456b39;p=dealii-svn.git More text, more outlook. git-svn-id: https://svn.dealii.org/trunk@20554 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-16/doc/results.dox b/deal.II/examples/step-16/doc/results.dox index f0893878ba..94a2117995 100644 --- a/deal.II/examples/step-16/doc/results.dox +++ b/deal.II/examples/step-16/doc/results.dox @@ -56,7 +56,13 @@ finest level. This theory is easily tested by comparing results when we use a constant coefficient: in that case, the number of iterations remains constant at 9 after the first three or four refinement steps. -Let us compare the iteration steps needed to obtain convergence for the two methods: +We can also compare what this program produces with how +@ref step_5 "step-5" performed. To solve the same problem as in +step-5, the only change that is necessary is to replace the body of +the function LaplaceProblem::refine_grid by a call to +triangulation.refine_global(1) — the rest of the +program remains unchanged. In that case, here is how the solvers used +in step-5 and the multigrid solver used in the current program compare: @@ -78,5 +84,16 @@ for second order problems.

Possible extensions

A close inspection of this program's performance shows that it is mostly -dominated by matrix-vector operations. @ref step_37 "Step-37" shows one way +dominated by matrix-vector operations. @ref step_37 "step-37" shows one way how this can be avoided by working with matrix-free methods. + +Another avenue would be to use algebraic multigrid methods. The +geometric multigrid method used here can at times be a bit awkward to +implement because it needs all those additional data structures, and +it becomes even more difficult if the program is to run in parallel on +machines coupled through MPI, for example. In that case, it would be +simpler if one could use a black-box preconditioner that uses some +sort of multigrid hierarchy for good performance but can figure out +level matrices and similar things out by itself. Algebraic multigrid +methods do exactly this, and we will use them in +@ref step_31 "step-31" for the solution of a Stokes problem.
cellsstep-5step-16
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