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  • $\begingroup$ Do you mean the same number of nodes for the inner and outer circles? $\endgroup$ Commented Jul 28, 2015 at 11:40
  • $\begingroup$ If F is a map from the inner circle to the outer circle, and x_i are the nodes on the inner circle, I need F[x_i] to be the nodes on the outer circle. This is more than just saying that the number of nodes are the same. $\endgroup$ Commented Jul 28, 2015 at 12:44
  • $\begingroup$ For your general problem -- be aware that it is easy to introduce an orientation non-preserving map between two boundary components. This will embed a Moebius strip in your region (if, for instance, the boundary condition is equality on the two components). As a consequence, odd modes along paths in the region starting at one associated point and ending in the others will be driven to zero by topology. Are you sure you can reliably make properly oriented maps when this will matter? $\endgroup$ Commented Jul 28, 2015 at 16:50
  • $\begingroup$ Will the maps between your boundary components have (approximately) unit magnitude of derivative? If not, then you may not be able to get a sufficiently uniform grid when a tiny piece of one boundary is mapped to a long piece of another. $\endgroup$ Commented Jul 28, 2015 at 17:16
  • $\begingroup$ Can your map between boundary components be "thickened" to include nonintersecting neighbourhoods of each boundary component? (If so, an abstract solution is to transport one neighbourhood along the map to the other component, grid the target and moved neighbourhoods, map back, and repeat for all the pairs. The result leaves only the whole region minus the neighbourhoods to grid compatibly with the "outer" boundaries of the neighbourhoods -- i.e., not compatibly requirements need be enforced. I'm sure I don't want to implement this method.) $\endgroup$ Commented Jul 28, 2015 at 17:22