[petsc-users] Iterative solution for schur complement

[Matthew Knepley]

On Wed, Aug 14, 2013 at 3:56 PM, Umut Tabak <u.tabak at tudelft.nl> wrote:

> On 08/14/2013 10:37 PM, Jed Brown wrote:
>
>> Umut Tabak <u.tabak at tudelft.nl> writes:
>>
>>  Dear all,
>>>
>>> I am looking at a system where I am trying to investigate this
>>> ill-conditioned problem with some iterative tricks or not. Namely, the
>>> system that I try to solve is
>>>
>>> (B - C^T A^{-1}C) x2 = b2
>>>
>>> which results from block symmetric representation
>>>
>>> A  C
>>> C^T B
>>>
>> What physics do you have here?
>>
> Hi Jed,
>
> 'A' results from the discretization of structural field equations which is
> also ill-conditioned. More specifically, it is (Ks-a*Ms) where Ks and Ms
> are stiffness and mass matrices of the structural domain.
> However, 'B' results from the discretization of the Helmholtz operator for
> the fluid domain. It is also similarly represented as (Kf-a*Mf) as above.
>

Okay, this is fluid-structure interaction. Why not start with the
multiplicative combination in PCFIELDSPLIT first? Then you
can move to Schur complement with just an option if you figure out a good
preconditioner?

   Matt


>
>> Both CG and MINRES require an SPD preconditioner.  It sounds like B is a
>> poor approximation to the Schur complement S = B - C^T A^{-1} C.
>> Depending on your application area, there are a few classes of
>> preconditioners that you might consider.  These include the
>> least-squares commutator, physics-based approximate commutator,
>> SIMPLE(R), and DD and multigrid methods applied directly to the
>> indefinite problem.
>>
> Unfortunaltely, yes, even if I have the complete factor for B and even if
> C is a pretty sparse matrix, this is not a good preconditioner eventually,
> that is clear to me as well.
>
> Before leaving these ideas, I am trying to convince myselft that this idea
> is not useful and cannot be improved further. But, as a poor engineer ;), I
> had the feeling that since the fluid part only includes one variable which
> is the pressure and the domain is homogeneous, I would expect some better
> ways to exist in order to solve this problem.
>
> Since the domain is homogeneous, at least the fluid domain, and it is
>  modelled with a scalar variable, I was thinking that scaling should not be
> a problem.
>
> But, there is another important point, due to the modelling approach used
> Kf is a singular matrix with one zero eigenvalue(and this is always the
> case for a specific type of boundary condition which is the hard wall
> condition) and Mf is pretty well conditioned as a standalone matrix. The
> source of the problem is writing representing B as (Kf-a*Mf) or as
> (Kf/a-Mf) in the original block diagonal representation.
>
> Can you figure out something more after these explanations? What would you
> suggest as a first try and, maybe, a couple of more?
> Thanks,
> Umut
>



-- 
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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