Fast wavelet transform for BIEM matrix

Abstract

We consider a matrix arising from a Nystrom method for the numerical solution of the boundary integral equation of the second kind for the planar harmonic Dirichlet problem in domains with a corner. The Nystrom method is based on the trapezoidal rule with a graded mesh at near corner. We concentrate our efforts on expressing this matrix in terms of wavelet bases with compact support via a fast wavelet transform on the purpose of obtaining sparse matrix. Upper bounds on the size of the wavelet transform elements are obtained. These bounds are then used to show that the resulting transformed matrix is sparse, having only O(NlogN) significant entries. Some numerical results are presented.