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Have FETI-DP or BDDC methods been applied to alternative FEM discretizations - for example, least squares finite elements?

My Google searching doesn't seem to yield many results, so I'm wondering if I've missed any field-specific papers. If not, is there a reason for this (beyond "lack of interest or funding")?

Edit: I meant FETI-DP or BDDC as preconditioners, primarily.

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I don't see any reason why they wouldn't work. That said, domain decomposition as a discretization technique is dead -- it was used for a while in the 1990s and early 2000s when we didn't know how to solve very large linear systems that are distributed across processors. However, with good partitioners and libraries such as PETSc and Trilinos that can store linear systems and provide ghost elements on vectors, everybody has moved to solving PDEs as one big system across all processors, rather than one small system per processor.

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    $\begingroup$ "Domain decomposition" is a family of algorithms and is still very much alive. Many libraries provide DD methods and users customize those for their needs. You can think of modern DD as being much like a multigrid method that has been modified to accommodate extremely rapid coarsening in exchange for more expensive local solves. (Many DD methods are only formulated as 2-level methods, which restricts their "dynamic range" in problem sizes where neither the local or coarse (global) problem is too big, but can still scale.) The favorite SC13 Algorithms paper is a DD paper. $\endgroup$ – Jed Brown Oct 30 '13 at 16:14
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    $\begingroup$ Agree with Jed - I think this answer overlooks the beneficial preconditioning/deflation effects that DDM can have on tough systems. IMO DDM + parallel direct solver for subdomains is a reasonable way to approach e.g. helmholtz systems (resistant to multigrid because the nyquist criteria enforces a "floor"/"point of no return" on the V-cycle, and resistant to minimization over Krylov spaces because wave transport can be so non-local). Being able to deflate the 3D problem to a coupled set of 2D interface problems can help. Am far from an expert in DDM, but "dead" feels like an exaggeration. $\endgroup$ – rchilton1980 Oct 30 '13 at 22:01
  • $\begingroup$ Wolfgang, I meant using DD as preconditioning. Dead as a solver technique, maybe, but not as a preonditioner, surely? $\endgroup$ – Jesse Chan Oct 31 '13 at 1:33
  • $\begingroup$ I can agree to that statement. DD was big in the 1990s as a solver technique, but I think that's a direction no longer promising. I will agree that as a preconditioner, it's got its place. Of course, it shifts the problem to a different place -- how do you precondition the interface problems? $\endgroup$ – Wolfgang Bangerth Oct 31 '13 at 3:25
  • $\begingroup$ That's worth an entirely new question on its own! I only know of ones proposed by BDDC/FETI-DP, where a second level splitting decomposes basis functions into ones that are vertex continuous vs edge-average continuous. It appears to have been used by the HDG folks (Schoberl has some HDG results) with some success. This is also a question I'm wondering about (related: can you precondition a statically condensed system?) $\endgroup$ – Jesse Chan Oct 31 '13 at 4:18

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