Is there a reference on the effect of subdomain topology on performance of the overlapping additive Schwarz method for (high order) finite elements? For example, taking subdomains to be vertex patches (elements directly connected through a vertex) differs from taking a subdomain to be an element + all its neighbors connected through faces (the former I believe gave me iteration counts independent of subdomain size, the former did not for high order).


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For your example, it is the ratio of the overlap size and the subdomain size that matters. With coarse grid, the condition number scales like H/d with H the subdomain size and d the overlap size.

It does depend on the geometry of the subdomains but it seems unaffected from high-order finite elements because this is a property holding already at the continuous level and h or p refinement just makes it closer to the continuous case. You can see Continuous Analysis of the Additive Schwarz Method: a Stable Decomposition in H^1.

  • $\begingroup$ Hi Hui - thanks for the response, I'll take a look. I thought that Luca Pavarino showed for higher $p$ that convergence depended on the number of total number of subdomains contributing to an overlap region. Is that result consistent with Martin's work? $\endgroup$
    – Jesse Chan
    Feb 16, 2015 at 15:24
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    $\begingroup$ Yes, you are right that the number of intersecting subdomains also matters. It is classical see Toselli--Widlund's book and must also be in Martin's work. $\endgroup$
    – Hui Zhang
    Feb 22, 2015 at 12:10

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