Convergence of overlapping domain decomposition methods for the Helmholtz equation


The impedance boundary condition has been widely used in designing domain decomposition methods for the Helmholtz equation. The first algorithm with a rigorous convergence theory (Bennamou and Despres, 1997) was based on a nonoverlapping domain decomposition with general subdomains and swapped impedance data of neighbouring subdomains at each iteration. The convergence proof (but without a rate of convergence) was carried out using a “pseudo-energy” norm constructed from the sum of the $L^2$ norms of the impedance data on subdomain boundaries. This is a norm on the space of local solutions to the homogeneous Helmholtz equation and the convergence proof used the fact that the norms of the inward and outward impedance data are equal for solutions of the homogenous Helmholtz equation on each subdomain. In this talk, we will present some corresponding results for overlapping domain decomposition methods. We will show how convergence of the parallel Schwarz method (as an iterative method) depends on properties of the ‘impedance map’ which takes impedance data on the boundary to impedance data on an interior interface, via the solution of a homogeneous Helmholtz problem. Based on these properties, the convergence (and rate of convergence) of some overlapping domain decomposition methods can be theoretically guaranteed. We also present numerical results illustrating the theory.