From: kronbichler Date: Tue, 11 Nov 2008 14:28:51 +0000 (+0000) Subject: Corrected an error in derivation of block preconditioners. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f8c0de5a8c6909a623151c9fb0565692b78f0ed2;p=dealii-svn.git Corrected an error in derivation of block preconditioners. git-svn-id: https://svn.dealii.org/trunk@17543 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 c1854abea1..048f89ee34 100644 --- a/deal.II/examples/step-22/doc/results.dox +++ b/deal.II/examples/step-22/doc/results.dox @@ -416,16 +416,17 @@ would appear to be a good choice since \end{array}\right) = \left(\begin{array}{cc} - I & A^{-1} B^T \\ 0 & 0 + I & A^{-1} B^T \\ 0 & I \end{array}\right). @f} -This is the approach taken by the paper by Silvester and Wathen referenced to -in the introduction (with the exception that Silvester and Wathen use right -preconditioning). In this case, a Krylov-based iterative method would -converge in two steps if exact inverses of $A$ and $S$ were applied, since -there are only two distinct eigenvalues 0 and 1 of the matrix. Below, we will -discuss the choice of an adequate solver for this problem. First, we are going -to have a closer look at the implementation of the preconditioner. +This is the approach taken by the paper by Silvester and Wathen referenced +to in the introduction (with the exception that Silvester and Wathen use +right preconditioning). In this case, a Krylov-based iterative method would +converge in one step only if exact inverses of $A$ and $S$ were applied, +since all the eigenvalues are one (and the number of iterations in such a +method is bounded by the number of distinct eigenvalues). Below, we will +discuss the choice of an adequate solver for this problem. First, we are +going to have a closer look at the implementation of the preconditioner. Since $P$ is aimed to be a preconditioner only, we shall use approximations to the inverse of the Schur complement $S$ and the matrix $A$. Hence, the Schur diff --git a/deal.II/examples/step-31/step-31.cc b/deal.II/examples/step-31/step-31.cc index 9e40d25403..369abc7b6f 100644 --- a/deal.II/examples/step-31/step-31.cc +++ b/deal.II/examples/step-31/step-31.cc @@ -388,20 +388,21 @@ namespace LinearSolvers // & B^T \\ B & 0 // \end{array}\right) = // \left(\begin{array}{cc} I & - // A^{-1} B^T \\ 0 & 0 + // A^{-1} B^T \\ 0 & I // \end{array}\right), - // @f} - // which indeed is very simple. A - // GMRES solver based on exact - // matrices would converge in two - // iterations, since there are only - // two distinct eigenvalues. Such - // a preconditioner for the blocked - // Stokes system has been proposed - // by Silvester and Wathen ("Fast - // iterative solution of stabilised - // Stokes systems part II. Using - // general block preconditioners", + // @f} + // which indeed is very simple. A GMRES + // solver based on exact matrices would + // converge in one iteration, since all + // eigenvalues are equal (any Krylov + // method takes at most as many + // iterations as there are distinct + // eigenvalues). Such a preconditioner + // for the blocked Stokes system has been + // proposed by Silvester and Wathen + // ("Fast iterative solution of + // stabilised Stokes systems part II. + // Using general block preconditioners", // SIAM J. Numer. Anal., 31 (1994), // pp. 1352-1367). //