Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare–Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h1/4)1−O(h1/4) on a class of general domains, where hh is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.