\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
// & 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).
//