From: Timo Heister Date: Fri, 17 May 2019 20:13:39 +0000 (-0600) Subject: step-63 documention follow-up X-Git-Tag: v9.1.0-rc3~2^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F8250%2Fhead;p=dealii.git step-63 documention follow-up see #8225 --- diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox index 8dd4b53f10..6cc9fc5c7d 100644 --- a/examples/step-63/doc/intro.dox +++ b/examples/step-63/doc/intro.dox @@ -125,20 +125,23 @@ One of the goals of this tutorial is to expand from using a simple (class PreconditionSOR) on each level of the multigrid hierarchy. The term "point-wise" is traditionally used in solvers to indicate that one solves at one "grid point" at a time; for scalar problems, this means -to use a solver or smoother that updates one unknown of the linear +to use a solver that updates one unknown of the linear system at a time, keeping all of the others fixed; one would then iterate over all unknowns in the problem and, once done, start over again from the first unknown until these "sweeps" converge. Jacobi, Gauss-Seidel, and SOR iterations can all be interpreted in this way. -(In the context of multigrid, one does not think of these methods as -"solvers", but as "smoothers", and only does one loop or "sweep" over all -unknowns. This does not solve the linear system, of course, but is -the kind of operation one needs in the multigrid context.) +In the context of multigrid, one does not think of these methods as +"solvers", but as "smoothers". As such, one is not interested in +actually solving the linear system. It is enough to remove the high-frequency +part of the residual for the multigrid method to work, because that allows +restricting the solution to a coarser mesh. Therefore, one only does a few, +fixed number of "sweeps" over all unknowns. In the code in this +tutorial this is controlled by the "Smoothing steps" parameter. But these methods are known to converge rather slowly when used as solvers. While as multigrid smoothers, they are surprisingly good, -they can also be improved upon. In particular, we here also consider -"cell-based" smoothers. These methods solve for all +they can also be improved upon. In particular, we consider +"cell-based" smoothers here as well. These methods solve for all unknowns on a cell at once, keeping all other unknowns fixed; they then move on to the next cell, and so on and so forth. One can think of them as "block" versions of Jacobi, Gauss-Seidel, or SOR, but