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Note: Thanks to comments, I realized that I have two problems, each which can be described more clearly on its own. This revised question covers the first.

I would like to solve $$ -\Delta u - 1=0,\quad u(0,y)= 0,\quad u(1,y)=0 $$ over a rectangle using alternating Schwarz. The following is my mesh, where the small disks indicate the internal boundary nodes of the subdomains (the nodes that have Dirichlet conditions that I update on them):

$x = 0$ and $x = 1$ are the physical boundaries of the full mesh. The local solutions do not converge to the global solution. The problem is mostly in the overlapping part of the mesh, here is the difference between the actual solution and the solution given by the local solutions:

Mathematica graphics

If I impose the Dirichlet conditions $u(x,0) = 0$ and $u(x,1/\phi) = 0$, where $1/\phi$ is the height of the mesh, then the local solutions converge to the global solutions.

Should it not also work with only the boundary conditions that I mentioned about earlier?

EDIT: I solved it by selecting the artifical boundary in this manner:

Mathematica graphics

C. E.
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  • Not directly related to your question and your particular problem: be careful with triangles like the top left one. https://scicomp.stackexchange.com/a/25568/21916 – 56th Jul 06 '17 at 21:18
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    @56th The answer suggests to "[add] a node in the middle of any edge having both ends on the boundary". I have that. The image where boundary nodes are marked with disks shows that it is the case for the boundary edges, and it's also true for all other edges. Thanks though, I wasn't aware of this problem. – C. E. Jul 06 '17 at 23:37
  • So, you are setting dirichlet bc on the nodes at the original domain boundary and alternating schwarz inside and it doesn't converge? – VorKir Jul 08 '17 at 23:10
  • @VorKir Yes, I have Dirichlet conditions on the outer boundary of the original domain. The inner rectangles don't have any boundary conditions on them at all. I would like to not have to specify boundary conditions except for at some boundaries. – C. E. Jul 08 '17 at 23:31
  • @C. E., it sounds like a contradiction to theory if you are solving nice elliptic pdes. What are your pdes? Have you tried a simple two domain model poisson problem in rectangle with vertical interface? – VorKir Jul 08 '17 at 23:37
  • @VorKir I'm using $-\Delta u - 1 = 0$. I have tried my code on several other geometries, and it seemingly converges very nicely even with many subdomains – despite the issue I mention about points that belong to all subdomains – but when I remove the Dirichlet condition from one of the boundaries of the original domain (f.e. one side of the square in my last example above) it stops converging to the real solution. I just noticed this, it also relates to the problem above where there are no Dirichlet conditions on the inner rectangles. Does this contradict theory? – C. E. Jul 09 '17 at 00:57
  • @VorKir I updated my question, addressing specifically the situation you mentioned. – C. E. Jul 10 '17 at 23:18
  • So if you impose Dirichlet BCs on the whole bndry everything works, right? In the case when your soln does not converge, what are BCs of your global problem on $y = 0$ and $y = 1$? What are BCs of local problems? Please add them explicitly. I may assume you use hom. Neumann. – 56th Jul 17 '17 at 16:23
  • @56th You are correct, with Dirichlet boundary conditions on all boundaries it does work. Yes, a Neumann value of zero can be assumed wherever boundary conditions have not been specified. Note that it says "does not converge to the global solution," rather than "does not converge." The boundary conditions of boundaries between subdomains in the alternating Schwarz algorithm are Dirichlet conditions determined by the neighboring subdomain. – C. E. Jul 17 '17 at 23:15

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