Approximate moment matrix decomposition in wavelet Galerkin BEM

Abstract

We present an improvement to the wavelet Galerkin BEM in [J. Tausch, A variable order wavelet method for the sparse representation of layer potentials in the non-standard form, J. Numer. Math. 12 (3) (2004) 233–254]. In the non-standard form representation of integral operators the number of wavelets in every level determines the efficiency of the underlying Galerkin scheme. In order to increase this number, the partial singular value decomposition (PSVD) of the moment matrix is employed. Since the resulting wavelets do not exactly satisfy the conventional vanishing moment condition, i.e. a certain number of polynomial moments of the wavelets are zero, we introduce the concept of “quasi-vanishing moment”. For integral equations of the second kind we prove that the asymptotic convergence of the full Galerkin scheme can be retained by controlling the cut-off for singular values in the PSVD appropriately. Numerical results show that our improvement can largely decrease the time and memory usage while preserving the optimal convergence rate of the method.