From: Timo Heister Date: Thu, 2 Jun 2016 11:25:23 +0000 (+0100) Subject: address comments X-Git-Tag: v8.5.0-rc1~990^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=c56f1a4e1e10d165db14c78cb31dfcc412f686f8;p=dealii.git address comments --- diff --git a/examples/step-56/doc/intro.dox b/examples/step-56/doc/intro.dox index d3eb87877a..1c16afcd36 100644 --- a/examples/step-56/doc/intro.dox +++ b/examples/step-56/doc/intro.dox @@ -5,13 +5,15 @@

Stokes Problem

-The purpose of this tutorial is to create an efficient linear solver -for the Stokes equation and compare it to alternative -approaches. Using FGMRES with geometric multigrid as a preconditioner -for the velocity block, we see that the linear solvers used in step-22 -cannot keep up since multigrid is the only way to get $O(n)$ solve -time. Using the Timer class, we collect some statistics to compare -setup times, solve times, and number of iterations. We also compute +The purpose of this tutorial is to create an efficient linear solver for the +Stokes equation and compare it to alternative approaches. Here, we will use +FGMRES with geometric multigrid as a preconditioner velocity block, and we +will show in the results section that this is a fundamentally better approach +than the linear solvers used in step-22 (including the scheme described in +"Possible Extensions"). Fundamentally, this is because only with multigrid it +is possible to get $O(n)$ solve time, where $n$ is the number of unknowns of +the linear system. Using the Timer class, we collect some statistics to +compare setup times, solve times, and number of iterations. We also compute errors to make sure that what we have implemented is correct. Let $u \in H_0^1 = \{ u \in H^1(\Omega), u|_{\partial \Omega} = 0 \}$ @@ -44,7 +46,8 @@ Our goal is to compare several solution approaches. While step-22 solves the linear system using a "Schur complement approach" in two separate steps, we instead attack the block system at once using FMGRES with an efficient -preconditioner. The idea is as follows: if we find a block +preconditioner, in the spirit of the approach outlined in the "Results" +section of step-22. The idea is as follows: if we find a block preconditioner $P$ such that the matrix @f{eqnarray*}