OPTIMAL CONVERGENCE OF NON-OVERLAPPING SCHWARZ METHODS FOR THE HELMHOLTZ EQUATION

Abstract

The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.