[petsc-dev] DD FAQ

[Barry Smith]

On May 3, 2010, at 4:23 AM, Jed Brown wrote:

> 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).

    Ok, we are talking about something different. I was thinking only  
of additive Schwarz but where the
"restriction" from the overlapped region and the "interpolation" back  
to the overlapped region was scaled
by some factors between 0 and 1 instead of being just 0 or 1 as is  
supported in PCASM.


    Barry

>  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