[petsc-users] boomerAmg scalability

[Jed Brown]

On Mon, Dec 19, 2011 at 20:57, Barry Smith <bsmith at mcs.anl.gov> wrote:

> So please tell use how we SHOULD use AMG with those "indefinite problem
> produced by most discretizations of incompressible flow" dear teacher :-)


If only there was a nice complete answer...

We can do block preconditioners advocated by Elman and others. These are
the most flexible and the simplest for code reuse. For low Reynolds number,
they can also have optimal complexity, although the constants are usually
not the best. Most variants are well-supported by PCFieldSplit (e.g. with
PCLSC), but some need the user to provide auxiliary operators (e.g. the
"pressure convection-diffusion" variant). We could improve support for
these cases, but it's a delicate balance and I don't know any way to avoid
asking the user to understand a reasonable amount about the method and
usually to provide auxiliary information.

We can do coupled multigrid with fieldsplit or "distributed relaxation" as
a smoother. These can often be made more robust, but they tend to be more
intrusive to implement. These are not usually purely algebraic due to
inf-sup issues when coarsening the dual variables (pressure), though Mark
Adams' work on this for contact mechanics could be used to coarsen pressure
algebraically. I would like to experiment with this in PCGAMG.

We can do coupled multigrid with compatible Vanka-type smoothers. Whether
these are algorithmically effective and/or efficient is quite dependent on
the discretization. These methods are also usually geometric, though it's
possible to algebraically define a Vanka-smoother (though not necessarily
efficient). This is straightforward for MAC finite differences on
structured grids. For continuous finite elements, the "rotated Q1"
Rannacher-Turek elements are most attractive for these smoothers, but
Rannacher-Turek elements do not satisfy a discrete Korn's inequality, so
they are unusable for many problems. Some variants of DG for incompressible
flow seem to be the most interesting for this approach in general domains.
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