The discrete layer potentials

Abstract

In this chapter we explore fully discrete CQ-BEM methods (Convolution Quadrature in time, Galerkin Boundary Elements in space) for the exterior Dirichlet and Neumann problems associated with the wave equation. For the first one we will use a retarded single layer potential representation, and for the second one, a retarded double layer potential representation. We will detail the analysis of Galerkin semidiscretization error and some stability estimates that are needed to apply the results on Convolution Quadrature to the fully discrete scheme. It has already been mentioned, but let us repeat it here: the analysis of full discretizations for time domain integral equations in acoustics seems to be restricted to equations of the first kind, since the analysis is based on variational (energy) estimates. Not much is known about the behavior of the integral equations of the second kind, especially once they are discretized. The theory of full Galerkin discretizations is based on the weak weighted coercivity estimates given in Section 3.7 In the way the method is used, space and time Galerkin can be understood as a Galerkin-in-time discretization of a semidiscrete Galerkin problem. From that point of view, all the results given here about semidiscretization in space could eventually be used for the fully discrete method.