Schur complement domain decomposition methods for the solution of multiple scattering problems

Abstract

We present a Schur complement domain decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods, we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of non-overlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers and (3) we connect the solutions of the subproblems via Robin boundary conditions matching on the common interfaces between subdomains. We use subdomain Robin-To-Robin maps to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between subdomains-Two unknowns per interface. The Robin-To-Robin maps are computed in terms of well conditioned boundary integral operators, and thus the method of solution proposed in this paper can be viewed as a boundary integral equation (BIE)/BIE coupling via artificial subdomains. Unlike classical DD, we do not reformulate the DD problem in the form a fixed point iteration, but rather we solve the ensuing linear system by Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, we solve a much smaller linear system involving unknowns on the inner and outer artificial boundary. We present numerical evidence that our Schur complement DD algorithm can produce accurate solutions of very large multiple scattering problems that are out of reach for other existing approaches.