<h3> Stokes Problem </h3>
-The purpose of this tutorial is to create an efficient linear solver
-for the Stokes equation and compare it to alternative
-approaches. Using FGMRES with geometric multigrid as a preconditioner
-for the velocity block, we see that the linear solvers used in step-22
-cannot keep up since multigrid is the only way to get $O(n)$ solve
-time. Using the Timer class, we collect some statistics to compare
-setup times, solve times, and number of iterations. We also compute
+The purpose of this tutorial is to create an efficient linear solver for the
+Stokes equation and compare it to alternative approaches. Here, we will use
+FGMRES with geometric multigrid as a preconditioner velocity block, and we
+will show in the results section that this is a fundamentally better approach
+than the linear solvers used in step-22 (including the scheme described in
+"Possible Extensions"). Fundamentally, this is because only with multigrid it
+is possible to get $O(n)$ solve time, where $n$ is the number of unknowns of
+the linear system. Using the Timer class, we collect some statistics to
+compare setup times, solve times, and number of iterations. We also compute
errors to make sure that what we have implemented is correct.
Let $u \in H_0^1 = \{ u \in H^1(\Omega), u|_{\partial \Omega} = 0 \}$
solves the linear system using a "Schur complement approach" in two
separate steps, we instead attack the
block system at once using FMGRES with an efficient
-preconditioner. The idea is as follows: if we find a block
+preconditioner, in the spirit of the approach outlined in the "Results"
+section of step-22. The idea is as follows: if we find a block
preconditioner $P$ such that the matrix
@f{eqnarray*}
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