From: Wolfgang Bangerth <bangerth@colostate.edu>
Date: Tue, 8 Dec 2020 21:32:54 +0000 (-0700)
Subject: Better document step-22 solvers.
X-Git-Tag: v9.3.0-rc1~779^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F11340%2Fhead;p=dealii.git

Better document step-22 solvers.
---

diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib
index 5ed6044eb3..aad5f5dc3d 100644
--- a/doc/doxygen/references.bib
+++ b/doc/doxygen/references.bib
@@ -276,6 +276,20 @@
 }
 
 
+% ------------------------------------
+% Step 22
+% ------------------------------------
+
+@Article{SW94,
+  author = 	 {D. Silvester and A. Wathen},
+  title = 	 {Fast iterative solution of stabilised {S}tokes systems. {P}art
+                  {II}: {U}sing general block preconditioners},
+  journal = 	 {SIAM J. Numer. Anal.},
+  year = 	 1994,
+  volume = 	 31,
+  pages = 	 {1352--1367}}
+
+
 % ------------------------------------
 % Step 47
 % ------------------------------------
diff --git a/examples/step-22/doc/intro.dox b/examples/step-22/doc/intro.dox
index 32da0ccba4..002bc0dd5c 100644
--- a/examples/step-22/doc/intro.dox
+++ b/examples/step-22/doc/intro.dox
@@ -588,6 +588,25 @@ classes, we will be able to use them interchangeably using the same syntax in
 all places.
 
 
+<h4> Is this how one should solve the Stokes equations? </h4>
+
+The discussions above showed *one* way in which the linear system that
+results from the Stokes equations can be solved, and because the
+tutorial programs are teaching tools that makes sense. But is this the
+way this system of equations *should* be solved?
+
+The answer to this is no. The primary bottleneck with the approach,
+already identified above, is that we have to repeatedly solve linear
+systems with $A$ inside the Schur complement, and because we don't
+have a good preconditioner for the Schur complement, these solves just
+have to happen too often. A better approach is to use a block
+decomposition, which is based on an observation of Silvester and
+Wathen @cite SW94 and explained in much greater detail in
+@cite elman2005 . An implementation of this alternative approach is
+discussed below, in the section on a <a href="#block-schur">block Schur
+complementation preconditioner</a> in the results section of this program.
+
+
 <h4> A note on the structure of the linear system </h4>
 
 Above, we have claimed that the linear system has the form
diff --git a/examples/step-22/doc/results.dox b/examples/step-22/doc/results.dox
index 7c3a6c7512..c9fc2037b0 100644
--- a/examples/step-22/doc/results.dox
+++ b/examples/step-22/doc/results.dox
@@ -346,6 +346,7 @@ mesh-independent number of iterations, say 10 to 30. We have seen such a
 candidate in step-16: multigrid.
 
 <h5>Block Schur complement preconditioner</h5>
+<a name="block-schur"></a>
 Even with a good preconditioner for $A$, we still
 need to solve of the same linear system repeatedly (with different
 right hand sides, though) in order to make the Schur complement solve