[petsc-dev] DD FAQ

[Jed Brown]

On Sun, 2 May 2010 17:46:16 -0500, Barry Smith <bsmith at mcs.anl.gov> wrote:
>     I assume you mean where you use "optimized" linear combinations of
> solutions in overlapped regions? I am only vaguely aware as well. I've
> attempted to ignore them all these years. I don't know how to tune
> them. We currently use VecScatter to put in the overlapped values; now
> we'd need to multiple each value by some scale factor before putting
> in. I hate to increase the complexity of VecScatter even more by
> adding this support but we could.

An asymmetric balancing operation would be useful for Neumann-Neumann
methods, but I don't think it's sufficient for optimized Schwarz.  In
the most common form, these use asymmetric Robin conditions on subdomain
boundaries (which may or may not be overlapping).  Much like Neumann
conditions, these either require bookkeeping to deal with partially
unassembled subdomains or user-provided modification of boundary
conditions.  It's not just a weighted combination of solutions from
Dirichlet problems so I don't think adding complexity to VecScatter
would do any good.

There seem to be fairly good heuristics (on the model problems anyway, I
don't understand them well enough to generalize) for good asymmetric
subdomain boundary conditions, but of coures the availability of a
robust tuning procedure is important to the success of the method.  Note
that these essentially decay to Neumann-Dirichlet methods in the limit
of large coefficient jumps, so convergence rates often improve as the
jump size increases.

Jed