Preconditioning for Boundary Integral Equations

Abstract
New classes of preconditioners are proposed for the linear systems arising from a boundary integral equation method. The problem under consideration is Laplace’s equation in three dimensions. The system arising in this context is dense and unsymmetric. These preconditioners, which are based on solving small linear systems at each node, reduce the number of iterations in some cases by a factor of 8. Three iterative methods are considered: conjugate gradient on the normal equations, CGS of Sonneveld, and GMRES of Saad and Schultz. For a simple model problem, the exact relationship between the preconditioners and the resulting condition number of the system is investigated. This analysis proves that the condition number of the preconditioned system is decreased by a factor asymptotically greater than any constant.

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