Preconditioning for Boundary Integral Equations (Preliminary Version)

We propose new classes of preconditioners 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. Our preconditioners, which are based on solving small linear systems at each node, reduce the number of iterations in some cases by a factor of 20. Two iterative methods are considered: conjugate gradient on the normal equations and GMREES of Saad and Shultz. For a simple model problem, we demonstrate the exact relationship between the preconditioners and the resulting condition number of the preconditioned system is decreased by a factor asymptotically greater than any constant.