[petsc-users] Failure of MUMPS

[Dave May]

On Thu, 11 Oct 2018 at 20:26, Michael Wick <michael.wick.1980 at gmail.com>
wrote:

> Thanks for all the suggestions!
>
> Increasing the value of icntl_14 in MUMPS helps a lot for my case.
>
> Do you have any suggestions for higher-order methods in saddle-point
> problems?
>

If the saddle point system arises from Stokes or an incompressible
elasticity formulation, then the standard block factorizations of
Silvester, Elman, Wathan will work very well for high-order - assuming of
course you use inf-sup stable basis for u/p. For Stokes/elasticity, the
pressure mass matrix is a decent spectrally equivalent operator for the
Schur complement.

These preconditioners are discussed here:

* Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of
saddle point problems,
Acta Numerica, 14 (2005), pp. 1–137.

* Howard C. Elman, David J. Silvester, and Andrew J. Wathen, Finite
elements and fast iterative
solvers: with applications in incompressible fluid dynamics, Oxford
University Press, 2014.

High order examples can be found here:

* https://arxiv.org/abs/1607.03936

* Rudi, Johann, Georg Stadler, and Omar Ghattas. "Weighted BFBT
Preconditioner for Stokes Flow Problems with Highly Heterogeneous
Viscosity." SIAM Journal on Scientific Computing 39.5 (2017): S272-S297.

Note that in the Rudi et al papers, due the highly variable nature of the
viscosity, the authors advocate using a more complex definition of the
preconditioner for the Schur complement. Whether you need to use their
approach is dependent on the nature of the problem you are solving.

Thanks,
  Dave






>
> Mike
>
> Dave May <dave.mayhem23 at gmail.com> 于2018年10月11日周四 上午1:50写道：
>
>>
>>
>> On Sat, 6 Oct 2018 at 12:42, Matthew Knepley <knepley at gmail.com> wrote:
>>
>>> On Fri, Oct 5, 2018 at 9:08 PM Mike Wick <michael.wick.1980 at gmail.com>
>>> wrote:
>>>
>>>> Hello PETSc team:
>>>>
>>>> I am trying to solve a PDE problem with high-order finite elements. The
>>>> matrix is getting denser and my experience is that MUMPS just outperforms
>>>> iterative solvers.
>>>>
>>>
>>> If the problem is elliptic, there is a lot of evidence that the P1
>>> preconditioner is descent for the system. Some people
>>> just project the system to P1, invert that with multigrid, and use that
>>> as the PC for Krylov. It should be worth trying.
>>>
>>
>> Matt means project to P1 directly from your high order function space in
>> one step. It is definitely worth trying.
>> For those interested, this approach is first described and discussed (to
>> my knowledge) in this paper:
>>
>> Persson, Per-Olof, and Jaime Peraire. "An efficient low memory implicit
>> DG algorithm for time dependent problems." *44th AIAA Aerospace Sciences
>> Meeting and Exhibit*. 2006.
>>
>>
>>> Moreover, as Jed will tell you, forming matrices for higher order is
>>> counterproductive. You should apply those matrix-free.
>>>
>>
>> I definitely agree with that.
>>
>> Cheers,
>>   Dave
>>
>>
>>
>>>
>>>   Thanks,
>>>
>>>      Matt
>>>
>>>
>>>> For certain problems, MUMPS just fail in the middle for no clear
>>>> reason. I just wander if there is any suggestion to improve the robustness
>>>> of MUMPS? Or in general, any suggestion for interative solver with very
>>>> high-order finite elements?
>>>>
>>>> Thanks!
>>>>
>>>> Mike
>>>>
>>>
>>>
>>> --
>>> 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
>>>
>>> https://www.cse.buffalo.edu/~knepley/
>>> <http://www.cse.buffalo.edu/~knepley/>
>>>
>>
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