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

  • $\begingroup$ Not directly related to your question and your particular problem: be careful with triangles like the top left one. scicomp.stackexchange.com/a/25568/21916 $\endgroup$ – 56th Jul 6 '17 at 21:18
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    $\begingroup$ @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. $\endgroup$ – C. E. Jul 6 '17 at 23:37
  • $\begingroup$ So, you are setting dirichlet bc on the nodes at the original domain boundary and alternating schwarz inside and it doesn't converge? $\endgroup$ – VorKir Jul 8 '17 at 23:10
  • $\begingroup$ @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. $\endgroup$ – C. E. Jul 8 '17 at 23:31
  • $\begingroup$ @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? $\endgroup$ – VorKir Jul 8 '17 at 23:37

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