Conjugate gradient methods for three-dimensional BEM systems of equations

Abstract

A fundamental advantage of the boundary element method (BEM) is that the dimensionality of the problems is reduced by one. However, this advantage has to be weighted against the difficulty in solving the resulting systems of algebraic linear equations whose matrices are dense, non-symmetric and sometimes ill conditioned. For large three-dimensional problems the application of the classical direct methods becomes too expensive. This paper studies the comparative performance of iterative techniques based on conjugate gradient solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residuals (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for potential and exterior problems. Preconditioning is also considered and assessed. Two examples, one from electrostatics and other from fluid mechanics, were employed to test these methods, which proved to be effective and competitive as solvers for BEM linear algebraic systems of equations.