Explain the benefit of the fast diagonalization method in terms of iteration counts.

diff --git a/examples/step-59/doc/intro.dox b/examples/step-59/doc/intro.dox
index 158b645c4ae162bbe08413082ada26caa69dbda8..d723fcc96b52d3fff709ed55a6b10af231b569cb 100644 (file)
--- a/examples/step-59/doc/intro.dox
+++ b/examples/step-59/doc/intro.dox
@@ -277,3 +277,15 @@ relaxation parameter (which would be around 0.7 in 2D and 0.5 in 3D for the
 block-Jacobi smoother), and often increases smoothing efficiency a bit over
 plain Jacobi smoothing in that it enables lower the time to solution when
 setting the degree of the Chebyshev polynomial to one or two.
+
+Note that the block-Jacobi smoother has an additional benefit: The fast
+diagonalization method can also be interpreted as a change from the
+Hermite-like polynomials underlying FE_DGQHermite to a basis where the cell
+Laplacian is diagonal. Thus, it cancels the effect of the basis, and we get
+the same iteration counts irrespective of whether we use FE_DGQHermite or
+FE_DGQ. This is in contrast to using the PreconditionChebyshev class with only
+the diagonal (a point-Jacobi scheme), where FE_DGQ and FE_DGQHermite do indeed
+behave differently and FE_DGQ needs 2-5 less iterations than FE_DGQHermite,
+despite the modification made to the Hermite-like shape functions to ensure a
+good conditioning.
+

diff --git a/examples/step-59/doc/results.dox b/examples/step-59/doc/results.dox
index 031ac9f383cd930814481b6ace164eba0c2ae391..2c6f975c76649ea3cf16a12690b22505efe9309c 100644 (file)
--- a/examples/step-59/doc/results.dox
+++ b/examples/step-59/doc/results.dox
@@ -338,4 +338,5 @@ design of the smoother parameters by selecting better values.
 
 Another way of extending the program would be to include support for adaptive
 meshes, for which an interface operation at edges of different refinement
-level becomes necessary, as discussed in step-39.
\ No newline at end of file
+level becomes necessary, as discussed in step-39.
+