On boundary integral operators for the Laplace and the Helmholtz equations and their discretisations

Boundary integral operators for the solutions of the Laplace and the Helmholtz equations are considered. These an classical strongly elliptic pseudodifferential operators of integer orders alpha, mapping the Sobolev space H-r to H{r-alpha}. We study the spectral properties of the single layer Laplacian potential operator and its tangential derivative, and also the double layer Laplacian potential operator and its normal derivative, the hypersingular operator, over a circle boundary. We extend the analysis to the Helmholtz potential operators. We derive important analytical results for the elements of the discrete operators and their eigenvalues and eigenvectors. (C) 1999 Elsevier Science Ltd. All rights reserved.