On 9/9/25 14:07, Simon wrote:
I am aware that choosing an unstable element is generally not recommended.
However, despite not being LBB stable, the Q1-P0 element is still commonly used
in solid mechanics and included in commercial codes. In practice, it performs
quite well for my benchmark problems and the computational savings for not
choosing quadratic displacement elements are significant.
Yes, that's true. As mentioned in the paper you referenced, the element is
also widely used in the geosciences. Of course, the issue with the small
eigenvalues is a problem for solvers everywhere else too.
Besides the condition number, I am wondering whether the increased outer
iterations I presented
could stem from a poor approximation of the Schur complement, S = B*A^-1*B^T?
Following one of the approachs PETSc uses for saddle point systems (see here
<https://nam10.safelinks.protection.outlook.com/?
url=https%3A%2F%2Fpetsc.org%2Fmain%2Fmanualpages%2FPC%2FPCFieldSplitSetSchurPre%2F&data=05%7C02%7CWolfgang.Bangerth%40colostate.edu%7Cb0a3dfe376814b2d690608ddefdc8187%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C638930452796650916%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=4u4HhaAE03L2Xf95FFuICzDzVgqh2MndaRdZyugWM00%3D&reserved=0>),
I tried approximating the Schur complement as
S = B*diag(A)^-1*B^T
and obtained the following results:
* Cycle 2: 41(Outer), 434 Inner (A), 3379 Inner (S)
* Cycle 3: 82 (Outer), 1669 Inner (A), 13308 Inner (S)
That's much better than what you had before. What option specifically did you
try?
The outer FMGRES iterations are quite reasonable, but the number of inner
iterations for solving
with the Schur complement clearly increased significantly.
Sloppy speaking, did I simply shift the conditioning issue from the outer
solve to the inner Schur solve?
Using AMG for the Schur solve helped to reduce the inner iteration counts for
S to
530 (Cycle 2) and 2746 (Cycle 3). However, the total cpu time remained similar
to that of using sparse ilu.
Given all this, do you have any recommendations for a more effective Schur
complement approximation
and/or preconditioner? Ideally, this can be generalized to incompressible
elasticity as well.
I don't have any good suggestions. You probably already saw that, but it's
worth reading through the "Possibilities for extensions" section of step-22.
I had a graduate student who spent 3 years working on finding preconditioners
for a block system, with only moderate success. I'm currently working on a
problem where I'm also having great trouble finding decent preconditioners,
despite trying the best ideas I have on that topic. Preconditioning block
systems is hard :-( That's why people still write papers about it, and why the
issue is mentioned in many tutorials.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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