On Retarded Potential Boundary Integral Equations and their Discretisation

The paper deals with the retarded potential boundary integral equations (RPBIE) used in the numerical resolution of transient scattering problems (the so-called time domain boundary element methods). We propose here a review and update of the mathematical analysis of the involved RPBIE. Our approach, via Laplace transform, is described in some details for the classical acoustic scattering problems. The main results are: (i) existence and uniqueness theorems on a functional framework closely linked to the energy of the scattered waves; (ii) space-time variational formulations for the so-called “first kind” RPBIE, with coerciveness obtained by energy estimates. That leads us to advocate choosing these first kind RPBIE and their Galerkin approximations, instead of the second kind RPBIE and the collocation approximations. The actual space-time boundary elements are described in some detail. Examples of numerical experiments that confirm the unconditional stability of our schemes are reported, as well as references for similar results in other related work.