An operator splitting preconditioner for matrices arising from a wavelet boundary element method for the Helmholtz equation

Abstract

An operator splitting type preconditioner is presented for fast solution of linear systems obtained by Galerkin discretization of the Burton and Miller formulation for the Helmholtz equation. Our approach differs from usual boundary element treatments of the three-dimensional scattering problem because we use a basis of biorthogonal wavelets. Such wavelets result in a sparse linear system and that facilitates preconditioning and makes matrix vector products cheap to form. In this Part I of our work, we implement a biorthogonal wavelet transform on a closed surface in three dimensions. Numerical results demonstrate the gains in efficiency that are already achievable with this convenient but non-optimal implementation.