Optimized double sweep Schwarz method by complete radiation boundary conditions

Abstract

We present an optimized double sweep nonoverlapping Schwarz method for solving the Helmholtz equation in semi-infinite waveguides. The domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high-order absorbing boundary conditions are solved forward and backward sequentially. For communication between subdomains, Neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and Dirichlet data are exploited in the backward direction. The complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find Neumann data without introducing errors that would be produced if finite difference formulas were used for computing Neumann data. The convergence of the double sweep Schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory.