Smoothness-relaxation strategies for singular and hypersingular integral equations

Abstract

Three stages are involved in the formulation of a typical direct boundary element method: derivation of an integral representation; taking a Limit To the Boundary (LTB) so as to obtain an integral equation; and discretization. We examine the second and third stages, focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions. Two such strategies are indicated. The first is the introduction of various apparent or 'pseudo-LTBs'. The second is 'relaxed regularization', in which a regularized integral equation, derived rigorously under certain smoothness assumptions, is used when less smoothness is available. Both strategies are shown to be based on inconsistent reasoning. Nevertheless, reasons are offered for having some confidence in numerical results obtained with certain strategies. Our work is done in two physical contexts, namely two-dimensional potential theory (using functions of a complex variable) and three-dimensional elastostatics. © 1998 John Wiley & Sons, Ltd.