Is there a standard approach to testing codes with refined regions? Specifically, I am interested in testing whether the refinement is working correctly.
For the sake of simplicity, let's consider a (time-independent) boundary value problem in a domain with only one refined area, and a case with a known exact solution. Let's consider a finite difference method with a uniform grid size. Assume I've already verified that the code obtains the correct convergence rate for a uniform grid with the exact solution I have.
Now consider a uniform grid of size $h$ in the fine region and $H$ in the coarse region. I can compare the error (however it might be defined) between that case and the case where the grid is uniform with resolution $h$. Hopefully, the accuracy degradation is not "too bad". I don't know precisely what "too bad" in this case might mean, however, and hence my question. There may be a better approach altogether than what I am thinking here.