[petsc-users] boomerAmg scalability

[Matthew Knepley]

On Wed, Dec 21, 2011 at 6:40 PM, Mohamad M. Nasr-Azadani
<mmnasr at gmail.com>wrote:

> Thank you Jed for the detailed explanation. To tell you the truth, that
> was a bit overwhelming/scary for an engineer like me-that I have to be
> supercareful when using those solvers and preconditioners- :D
> For my case though, I decided to fix the pressure at one node in the
> domain so  I won't end up with a null-space case of the Pressure equation.
> I assume that way, I would not have to worry about the nullspace problems
> and the convergence issues you pointed out, right?
>

Nope. You have removed the null space from the operator on the full domain,
but not the subdomains that Jed
was talking about. I think this method only complicates things since now
one subdomain looks weird, and the
behavior of the coarse operator in unpredictable since i don't know where
this point is.

   Matt


> Best,
> Mohamad
>
>
>
> On Mon, Dec 19, 2011 at 9:09 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
>
>> 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.
>>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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