On fast matrix–vector multiplication in wavelet Galerkin BEM

So far, the wavelet boundary element method (BEM) has been recognized as a different kind of fast BEM from the methods based on low-rank approximation, such as fast multipole method, panel clustering method and H2H2-matrices.We consider the matrix–vector multiplication in the wavelet Galerkin BEM [Tausch J. A variable order wavelet method for the sparse representation of layer potentials in the non-standard form. J Numer Math 2004;12(3):233–54]. We show that the system matrix of the wavelet Galerkin BEM can be transformed into a matrix with hierarchical structure by combining the forward and inverse wavelet transform in matrix–vector multiplications phase. The new system matrix only involves the data about the scaling functions. Any operations concerning the wavelet functions are thus avoided. A new version of matrix–vector multiplication scheme is proposed. We prove that the complexity of the new scheme never exceeds that of the old scheme.This work: (1) simplifies the implementation wavelet Galerkin BEM; (2) bridges a link between the wavelet BEMs and the methods of low-rank approximation.