From: bangerth Date: Sun, 20 Jun 2010 18:47:11 +0000 (+0000) Subject: Add a paragraph on the suboptimality of the solver. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a6d06a0c83b933731484f34743d0bbb873489c91;p=dealii-svn.git Add a paragraph on the suboptimality of the solver. git-svn-id: https://svn.dealii.org/trunk@21237 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-20/doc/intro.dox b/deal.II/examples/step-20/doc/intro.dox index f96eae3f87..0f992b7362 100644 --- a/deal.II/examples/step-20/doc/intro.dox +++ b/deal.II/examples/step-20/doc/intro.dox @@ -342,10 +342,19 @@ the next problem immediately surfaces: due to the zero block, there are zeros on the diagonal and none of the usual preconditioners (Jacobi, SSOR) will work as they require division by diagonal elements. +In the following, we will introduce some techniques that can be used in cases +like these. Namely, we will consider the linear system as not consisting of one +large matrix and vectors, but we will want to decompose matrices into vectors +into blocks that correspond to the individual operators that appear in +the system. We note that the resulting solver is not optimal -- there are much +better ways, for example those explained in the results section of step-22 -- +but that the goal is to introduce techniques rather than optimal solvers. +

Solving using the Schur complement

-In view of this, let us take another look at the matrix. If we sort our +In view of the difficulties using standard solvers and preconditioners +mentioned above, let us take another look at the matrix. If we sort our degrees of freedom so that all velocity come before all pressure variables, then we can subdivide the linear system $AX=B$ into the following blocks: @f{eqnarray*}