Time domain analysis of the single layer potential

Abstract

In this chapter we develop a systematic approach to a pure time domain analysis of the single layer potential and operator, the Galerkin semidiscrete error operator and the Galerkin solver. Namely, we want to obtain estimates of

$$ \displaystyle{\mathcal{S}{\ast}\lambda,\qquad \mathcal{V}{\ast}\lambda,\qquad \mathcal{K}^{t}{\ast}\lambda,} $$

of its semidiscrete inverses

$$ \displaystyle{\mathcal{G}_{\lambda }^{h}{\ast}\beta,\qquad \mathcal{G}_{ u}^{h}{\ast}\beta = \mathcal{S}{\ast}\mathcal{G}_{\lambda }^{h}{\ast}\beta } $$

and of the error operators

$$ \displaystyle{\mathcal{E}_{\lambda }^{h}{\ast}\lambda = \mathcal{G}_{\lambda }^{h} {\ast}\mathcal{V}{\ast}\lambda -\lambda,\qquad \mathcal{E}_{ u}^{h}{\ast}\lambda = \mathcal{S}{\ast}\mathcal{E}_{\lambda }^{h} {\ast}\lambda.} $$