A generalized SOR method for dense linear systems of boundary element equations

In this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form Hx = c where H is a dense square nonsingular matrix. The iterative scheme considered is (D + (ΩH)_sl)x^(k+1) = (D- (ΩH)_u)x^(k) + Ωc, where (ΩH)_u and (ΩH)_sl are defined as the upper triangular and strictly lower triangular terms of ΩH, respectively. The parameter matrix Ω is selected to minimize the Frobenius norm ∥D - (ΩH)u∥F. Mathematical arguments and numerical experiments are presented to show that minimizing ∥D - (ΩH)u∥F provides for faster convergence. Numerical tests are performed for systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Gaussian elimination and Gauss-Seidel iteration.