Multiscale compression of BEM equations for electrostatic systems

Abstract

This paper describes the use of wavelet bases to create a sparse approximation of the fully populated matrix that one obtains using an integral formulation like charge simulation or surface charge simulation for numerically solving Laplace's equation with mixed boundary conditions. The sparse approximation is formed by a similarity transform of the N/spl times/N coefficient matrix, and the cost of the one employed here is of optimal order N/sup 2/. We must emphasize that benefits of computing with a sparse matrix typically do not justify the costs of the transformation, unless the problem has multiple right hand sides, i.e. one wants to simulate multiple excitation modes. The special orthogonal matrices we need for the similarity transform are built from wavelet bases. Wavelets are a well studied and mature topic in pure and applied mathematics, however, the fundamental ideas are probably new to many researchers interested in electrostatic field computation. Towards this end an important purpose of this paper is to describe some of the basic concepts of multiresolutional analysis using wavelet bases.