<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><br><br><div class="gmail_quote"><div dir="ltr">On Thu, 11 Oct 2018 at 20:26, Michael Wick <<a href="mailto:michael.wick.1980@gmail.com">michael.wick.1980@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>Thanks for all the suggestions!<br></div><div><br></div><div>Increasing the value of icntl_14 in MUMPS helps a lot for my case.</div><div><br></div><div>Do you have any suggestions for higher-order methods in saddle-point problems?</div></div></blockquote><div><br></div><div>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.</div><div><br></div><div>These preconditioners are discussed here:</div><div><br></div><div>* Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems,</div><div>Acta Numerica, 14 (2005), pp. 1–137.</div><div><br></div><div>* Howard C. Elman, David J. Silvester, and Andrew J. Wathen, Finite elements and fast iterative</div><div>solvers: with applications in incompressible fluid dynamics, Oxford University Press, 2014.</div><div><br></div><div>High order examples can be found here:</div><div><br></div><div>* <a href="https://arxiv.org/abs/1607.03936">https://arxiv.org/abs/1607.03936</a><br></div><div><br></div><div><div>* 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.<br></div></div><div><br></div><div>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. </div><div><br></div><div>Thanks,</div><div> Dave</div><div><br></div><div><br></div><div><br></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div><br></div><div>Mike<br></div></div><br><div class="gmail_quote"><div dir="ltr">Dave May <<a href="mailto:dave.mayhem23@gmail.com" target="_blank">dave.mayhem23@gmail.com</a>> 于2018年10月11日周四 上午1:50写道：<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div dir="ltr"><div dir="ltr"><br><br><div class="gmail_quote"><div dir="ltr">On Sat, 6 Oct 2018 at 12:42, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div class="gmail_quote"><div dir="ltr">On Fri, Oct 5, 2018 at 9:08 PM Mike Wick <<a href="mailto:michael.wick.1980@gmail.com" target="_blank">michael.wick.1980@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>Hello PETSc team:<br></div><div><br></div><div>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.</div></div></blockquote><div><br></div><div>If the problem is elliptic, there is a lot of evidence that the P1 preconditioner is descent for the system. Some people</div><div>just project the system to P1, invert that with multigrid, and use that as the PC for Krylov. It should be worth trying.</div></div></div></blockquote><div><br></div><div>Matt means project to P1 directly from your high order function space in one step. It is definitely worth trying.</div><div>For those interested, this approach is first described and discussed (to my knowledge) in this paper:</div><div><br></div><div>Persson, Per-Olof, and Jaime Peraire. "An efficient low memory implicit DG algorithm for time dependent problems." <i>44th AIAA Aerospace Sciences Meeting and Exhibit</i>. 2006.<br></div><div> <br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div class="gmail_quote"><div>Moreover, as Jed will tell you, forming matrices for higher order is counterproductive. You should apply those matrix-free.</div></div></div></blockquote><div><br></div><div>I definitely agree with that.</div><div><br></div><div><div>Cheers,</div><div> Dave</div></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div class="gmail_quote"><div><br></div><div> Thanks,</div><div><br></div><div> Matt</div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div>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?</div><div><br></div><div>Thanks!</div><div><br></div><div>Mike<br></div></div>
</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail-m_-3086393893594107885m_7766853203029900727gmail-m_5560003937257014756gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener</div><div><br></div><div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br></div></div></div></div></div></div></div></div>
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