Is there a reference on the effect of subdomain topology on performance of the overlapping additive Schwarz method for (high order) finite elements? For example, taking subdomains to be vertex patches (elements directly connected through a vertex) differs from taking a subdomain to be an element + all its neighbors connected through faces (the former I believe gave me iteration counts independent of subdomain size, the former did not for high order).
For your example, it is the ratio of the overlap size and the subdomain size that matters. With coarse grid, the condition number scales like H/d with H the subdomain size and d the overlap size.
It does depend on the geometry of the subdomains but it seems unaffected from high-order finite elements because this is a property holding already at the continuous level and h or p refinement just makes it closer to the continuous case. You can see Continuous Analysis of the Additive Schwarz Method: a Stable Decomposition in H^1.