Potential Integral Equations of the 2 D Laplace Operator in Wavelet

In this paper we solve the Laplace equation by solving a boundary integral equation in a wavelet basis. The idea of solving integral equations using a hierarchical approach is used in the Multipole method and Wavelet-Galerkin methods in the non-standard form. Due to the orthogonality of the wavelet transform, the discrete linear systems preserve their good conditioning while their sparse structures speed up iterative solvers. We investigate the link between the two multi-scale methods and the convergence of Wavelet Galerkin discretizations. The behaviour of GCR algorithms under wavelet transformations and in non-standard form is investigated. Finaly we analyze the optimal choice of Daubechies lters in the non-standard form and describe some numerical experiments.