On Thu, 11 Oct 2018 at 20:26, Michael Wick <[email protected]> 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 <[email protected]> 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 <[email protected]> wrote: >> >>> On Fri, Oct 5, 2018 at 9:08 PM Mike Wick <[email protected]> >>> 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/> >>> >>
