From: Martin Kronbichler Date: Mon, 3 Aug 2009 07:44:39 +0000 (+0000) Subject: Fix language. X-Git-Tag: v8.0.0~7398 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e3cd523c2932721e68c51b4e901b9a3c0f5ee6a6;p=dealii.git Fix language. git-svn-id: https://svn.dealii.org/trunk@19167 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-22/doc/results.dox b/deal.II/examples/step-22/doc/results.dox index 00a5e21fc6..b28426bd54 100644 --- a/deal.II/examples/step-22/doc/results.dox +++ b/deal.II/examples/step-22/doc/results.dox @@ -360,14 +360,15 @@ mesh-independent number of iterations, say 10 to 30. We have seen such a canditate in @ref step_16 "step-16": multigrid.
Block Schur complement preconditioner
-But even with a good preconditioner for $A$ at hand, we still +Even with a good preconditioner for $A$, we still need to solve of the same linear system repeatedly (with different -right hand sides, though). The approach we are going to discuss here is how this -can be avoided. If we persist in calculating the Schur complement, there is no -other possibility. +right hand sides, though) in order to make the Schur complement solve +converge. The approach we are going to discuss here is how inner iteration +and outer iteration can be combined. If we persist in calculating the Schur +complement, there is no other possibility. The alternative is to attack the block system at once and use an approximate -Schur complement as efficient preconditioner. The basic idea is as +Schur complement as efficient preconditioner. The idea is as follows: If we find a block preconditioner $P$ such that the matrix @f{eqnarray*} P^{-1}\left(\begin{array}{cc} @@ -730,8 +731,9 @@ because GMRES(k) scales worse with the problem size than CG, as we discussed above. Nonetheless, the improvement by a factor of 3-5 for moderate problem sizes is quite impressive. -
Combining block preconditioner and multigrid
An ultimate linear -solver for this problem could be imagined as a combination of an optimal +
Combining block preconditioner and multigrid
+An ultimate linear solver for this problem could be imagined as a +combination of an optimal preconditioner for $A$ (e.g. multigrid) and the block preconditioner described above, which is the approach taken in the @ref step_31 "step-31" tutorial program.