On the use of conjugate gradient-type methods for boundary integral equations

Abstract

This paper deals with a number of iterative methods for solving matrix equations that result from boundary integral equations. The matrices are non-sparse, and in general neither positive definite nor symmetric. Traditional methods like Gauss-Seidel do not give satisfactory results, therefore the use of conjugate gradient- and Krylov-type methods is investigated based on work of Kleinman and Van den Berg, who presented a general framework for these methods. Eleven of these algorithms are given and their performance (without preconditioning) is compared in a test case involving four different integral operators arising in potential theory. For all four matrix equations the Generalized Minimal Residual method (GMRES) outperforms all other iterative methods in both computation time per iteration and total computation time. For the Fredholm equations of the first kind this method also is the fastest with respect to the number of iterations. The Bi-conjugate gradient method (Bi-CG) and the Quasi-minimal residual method (QMR) are the best alternatives. For the Fredholm equations of the second kind more methods can be used efficiently besides GMRES. The Conjugate Gradient Squared method (CGS), the Bi-conjugate gradient method (Bi-CG), its stabilized version (Bi-CGSTAB) and the Quasi Minimal Residual method (QMR) are efficient alternatives.