Computing Solutions for Helmholtz Equation: Domain Versus Boundary Decomposition

When computing solutions of the Helmholtz equation, for a scatterer which boundary shows different geometric features at different locations, needing different kind of discretization or algorithms (integral equations or asymptotic formulas…) one is led to use domain decomposition techniques. This technique can be also useful when the scatterer is too huge, or when distributed computation is the aim. We give here a domain decomposition algorithm to compute acoustics in a non-dissipating bounded cavity. It relies on a Despres type condition on the fictitious interaction boundary between the subdomains. Its convergence does not depend on the (different) type of approximations used for solving the equation in each subdomain.But when computing acoustics outside a bounded scatterer (computing outgoing solutions), the domain decomposition technique leads to infinite fictitious boundaries between subdomains, showing specific features that are not part of the problem to solve, but come in because of the technique used. We introduce a new technique, we call boundary decomposition, and prove its convergence when the scatterer is a disjoint union of subscatterers. It relies on an homological decomposition of the solution, and makes it possible to analyze contribution of each subscatterer. It is specifically suited when the geometric features of the different subscatterers need different approximation algorithms. Tests were made using integral equation algorithm for one subscatterer and Kirchhoff high frequency approximation for another subscatterer: they proved efficency of the method.Proof for convergence of the two algorithms can be found in [2, 3, 4]. Numerical test for both techniques were carried out at EADS — Centre Commun de Recherche — Suresnes.