ALGEBRAIC WAY TO DERIVE ABSORBING BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION

Abstract

This paper is dedicated to the optimal convergence properties of a domain decomposition method involving two-Lagrange multipliers at the interface between the subdomains and additional augmented interface operators. Most methods for optimizing these augmented interface operators are based on the discretization of continuous approximations of the optimal transparent operators.1–5 Such approach is strongly linked to the continuous equation, and to the discretization scheme. At the discrete level, the optimal transparent operator can be proved to be equal to the Schur complement of the outer subdomain. Our idea consists of approximating directly the Schur complement matrix with purely algebraic techniques involving local condensation of the subdomain degree of freedom on small patch defined on the interface between the subdomains. The main advantage of such approach is that it is much more easy to implement in existing code without any information on the geometry of the interface and of the finite element formulation used. Such technique leads to a so-called "black box" for the users. Convergence results and parallel efficiency of this new and original algebraic optimization technique of the interface operators are presented for acoustics applications.