Conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary

Abstract

The conjugate gradient methods (CGMs) have been successfully applied to solve the complex matrix equations arising from discretization of boundary integral equations. If the underlying integral operator is compact, its eigenvalue clustering property ensures the fast convergence of these methods. Such an integral operator is usually compact if the integral boundary is globally smooth. In this paper however, we consider the numerical solution of the boundary integral equation with a non-compact operator where the non-compactness is due to the non-smoothness of a pieceiwe smooth boundary. Two particular algorithms are presented and tested for a model problem. We show that such non-compact integral equations can be solved efficiently by the preconditioned conjugate gradient method and that the algorithm using the normal equation appears to be particularly efficient.