When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields "better" convergence than other partitions?

For concreteness, let us consider the classical Poisson equation $-\Delta u = f(x)$ with homogeneous Dirichlet boundary conditions.


We use domain decomposition because we want to exploit the power of more than one processor. As a consequence, the right question to pose is: "How do we need to partition the domain so that we get the maximal speedup by using as many processors as subdomains?"

The answer to that question is "subdomains need to be chosen so that the work necessary on every subdomain is equal among all processors", because if it isn't, some processors will sit idle while others are still working. In practice, that typically means that for standard finite element discretizations, the (global) mesh needs to be partitioned in such a way that every processor has (roughly) the same number of cells. (I will note that this isn't true if you have, for example, different polynomial degrees on different cells, or are doing something different/additional on some cells than on others.)

  • $\begingroup$ While true, I think this answer would be more complete if it also mentioned the additional desire to minimize the contact area between domains. $\endgroup$ Sep 13 '21 at 13:08
  • $\begingroup$ Yes, excellent point. I should have added that to begin with, but am happy to give @rchilton1980 the credit via the comment! $\endgroup$ Sep 13 '21 at 19:40
  • $\begingroup$ @WolfgangBangerth Thank you very much. Do you have any references to theoretical results on this? $\endgroup$
    – user298455
    Oct 4 '21 at 16:07
  • $\begingroup$ @user298455 About what? $\endgroup$ Oct 4 '21 at 18:56
  • $\begingroup$ About the error estimate being "better" when the subdomains are chosen to minimize contact area and roughly the same size $\endgroup$
    – user298455
    Oct 4 '21 at 20:58

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