Optimized Schwarz methods with overlap for the Helmholtz equation

Optimized Schwarz methods are based on optimized transmission conditions between subdomains and can have substantially improved convergence behavior compared to classical Schwarz methods. This is especially true when the method is applied to the Helmholtz equation, and better transmission conditions in the form of perfectly matched layers have, for example, led to the new class of sweeping preconditioners. We present here for the first time a complete analysis of optimized Schwarz methods with overlap for the Helmholtz equation. We obtain closed form asymptotically optimized transmission conditions for the case of two subdomains and study numerically the influence of the number of subdomains on this optimized choice.