Tuong Ha-Duong (Compiègne. France)

On boundary retarded potential equations in scattering problems

Abstract

This paper deals with the technique of boundary element methods in transient scattering problems. We begin with a review of some basic scattering problems for transient waves (acoustic, electromagnetic and elastic) and the different boundary retarded potential equations (BRPE) that can be used for solving them. Then, we will survey the literature on numerical methods for discretising these equations and computing their solutions. We notice that unstable solutions are very commonly observed. In an attempt to understand these equations, a section will be devoted to the functional analysis of the involved retarded boundary integral operators. The results from the author's work in transient acoustic problems are reported, as well as the generalisation s to electromagnetic and elastic waves by other researchers. Beside a functional framework that can be used for the analysis of theBRPE, this analysis also suggests the use of the so-called first kind BRPE in most transient scattering problems. Actually, space-time variational formulations can be set up for them, with natural links to the energy of the waves. Galerkin methods for their discretisation are then described with some details, as well as many numerical experiments that confirm the stability obtained from the energy estimates.