I am solving 1D laplace problem discretized with finite differences (3-point stencil). I would like to use additive Schwarz method in classical form:

$U_{k+1}=U_{k}+M^{−1} r_k,$

where $r_k=F−A U_k$ and $ M^{−1} = \sum_{i=1}^N R_i^T(R_i A R_i^T)^{−1} R_i$.

Restriction operators $R_i$ are defined as binary matrices, with ones marking existing parts of the domain.

Method converges nicely for non-overlapping case (block Jacobi). However, I am not able to obtain convergence in case of any overlap.

If I use restricted additive Schwarz, method is convergent again. What is reason for such a behaviour?

  • 1
    $\begingroup$ There is not enough information in your question. It may be that your formulas are wrong; that your implementation is wrong (in one of many ways); that your postprocessing step is wrong. You need to be far more specific in explaining what you are doing, how exactly the solution is wrong, what you've already tried to debug the problem, etc. $\endgroup$ – Wolfgang Bangerth Feb 28 '18 at 17:02
  • $\begingroup$ Apparently the spectral radius of the iteration matrix is 1. Therefore the additive Schwarz (AS) method does not converge. This is not uncommon for the AS method, see Why Restricted Additive Schwarz Converges Faster than Additive Schwarz, Evridiki Efstathiou, Martin J. Gander. $\endgroup$ – wim Mar 1 '18 at 21:47
  • $\begingroup$ @WolfgangBangerth It would be nice to elaborate a bit further on my previous comment. Unfortunately the opportunity to write an answer has been closed. Can you please reopen the possibility to answer the question? The question is quite clear to me. I don't think that the formula's, implementation or postprocessing is wrong. $\endgroup$ – wim Mar 1 '18 at 21:53
  • $\begingroup$ Just start a new question that has all of the information necessary. That said, this question here was missing a lot of information. $\endgroup$ – Wolfgang Bangerth Mar 1 '18 at 23:29
  • $\begingroup$ @WolfgangBangerth I think that formulas, implementation and post/pre-processing are correct, since I am using solver to solve other problems and it works. This was more question about math properties of given method. $\endgroup$ – SmallElephant Mar 2 '18 at 22:50