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Schwartz type domain decomposition techniques require a transmission condition which can be hard to come by. Mortar type techniques enforce continuity with a Lagrange multiplier across domains. Are there limitations to this approach?

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    $\begingroup$ We don't know how to precondition the saddle point problem that results from the Lagrange multipliers? $\endgroup$ Oct 5, 2023 at 21:31
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    $\begingroup$ FETI-DP uses Lagrange multipliers to enforce continuity across domains and is (as far as I know) pretty popular. I always thought mortar methods meant that you might not have conforming meshes across the interface. So I wonder if you just have a terminology problem and need to be searching for slightly different things. $\endgroup$ Oct 5, 2023 at 23:35
  • $\begingroup$ @WolfgangBangerth There's a chapter on preconditioning saddle-point problems in the book by Toselli and Widlund. $\endgroup$
    – lightxbulb
    Oct 6, 2023 at 7:53
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    $\begingroup$ @WolfgangBangerth The best thing I have done in practice is DD + full multigrid (though in a simpler fdm setting with just schwarz DD used as a smoother). DD helped mainly with reducing the memory access overhead since I could work with local memory longer on a GPU core rather than accessing global memory often - it was several times faster than simple multigrid (asymptotically they were equivalent though). I am definitely not the best implementation person around, but dismissing DD outright sounds off to me considering the experience I have had with it. $\endgroup$
    – lightxbulb
    Oct 6, 2023 at 17:37
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    $\begingroup$ @lightxbulb My impression is that DD is a fine strategy if you want to partition into 100 subdomains, but that it just does not work very well with 10,000 or 100,000. I take your point of experience, that's the kind of thing I enjoy learning about. $\endgroup$ Oct 7, 2023 at 2:19

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