]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Corrected an error in derivation of block preconditioners.
authorkronbichler <kronbichler@0785d39b-7218-0410-832d-ea1e28bc413d>
Tue, 11 Nov 2008 14:28:51 +0000 (14:28 +0000)
committerkronbichler <kronbichler@0785d39b-7218-0410-832d-ea1e28bc413d>
Tue, 11 Nov 2008 14:28:51 +0000 (14:28 +0000)
git-svn-id: https://svn.dealii.org/trunk@17543 0785d39b-7218-0410-832d-ea1e28bc413d

deal.II/examples/step-22/doc/results.dox
deal.II/examples/step-31/step-31.cc

index c1854abea15196b600ce84066d41501bc540cf08..048f89ee34de6c85349d4e627c0a454e4a807be6 100644 (file)
@@ -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
index 9e40d25403488497e11029b90912c45e4f73db63..369abc7b6f0631e22ef457650c9c768c7f0ac248 100644 (file)
@@ -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).
                                   // 

In the beginning the Universe was created. This has made a lot of people very angry and has been widely regarded as a bad move.

Douglas Adams


Typeset in Trocchi and Trocchi Bold Sans Serif.