A preconditioning strategy for the solution of linear boundary element systems using the GMRES method

Abstract

In this paper a novel preconditioning strategy is presented that is designed to improve the convergence rates of the Generalized Minimal Residual (GMRES) method when applied to dense linear systems of boundary element equations of the form Hx = c. The GMRES method is applied to the preconditioned system (D+ L) −1ΩHx = (D+ L) −1Ωc , where D = diag( H), L is the strictly lower triangular part of ΩH and Ω is a sparsely populated upper triangular matrix. The coefficients in Ω are determined via the minimization of the square of the Frobenius norm map; U+ D−Dmap; F , where U is the strictly upper triangular part of ΩH and D= diag(ΩH) . Several proofs are given to demonstrate that minimizing map; U+ D−Dmap; F 2 provides for improved conditioning and consequently faster convergence rates. Numerical experiments are performed on systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Jacobi preconditioned GMRES, preconditioned Gauss-Seidel and Gaussian elimination. Moreover, condition numbers are noted and up to 100-fold reductions are observed for the systems tested.