canditate in @ref step_16 "step-16": multigrid.
<h5>Block Schur complement preconditioner</h5>
-But even with a good preconditioner for $A$ at hand, we still
+Even with a good preconditioner for $A$, we still
need to solve of the same linear system repeatedly (with different
-right hand sides, though). The approach we are going to discuss here is how this
-can be avoided. If we persist in calculating the Schur complement, there is no
-other possibility.
+right hand sides, though) in order to make the Schur complement solve
+converge. The approach we are going to discuss here is how inner iteration
+and outer iteration can be combined. If we persist in calculating the Schur
+complement, there is no other possibility.
The alternative is to attack the block system at once and use an approximate
-Schur complement as efficient preconditioner. The basic idea is as
+Schur complement as efficient preconditioner. The idea is as
follows: If we find a block preconditioner $P$ such that the matrix
@f{eqnarray*}
P^{-1}\left(\begin{array}{cc}
above. Nonetheless, the improvement by a factor of 3-5 for moderate problem
sizes is quite impressive.
-<h5>Combining block preconditioner and multigrid</h5> An ultimate linear
-solver for this problem could be imagined as a combination of an optimal
+<h5>Combining block preconditioner and multigrid</h5>
+An ultimate linear solver for this problem could be imagined as a
+combination of an optimal
preconditioner for $A$ (e.g. multigrid) and the block preconditioner
described above, which is the approach taken in the @ref step_31 "step-31"
tutorial program.