A wavelet based approach for solving Poisson´s equation

This paper presents a new approach for solving elliptic PDEs using wavelets. In this paper scalets and wavelets are used as basis functions for solving Poissons equation. The scalets are constructed using the Lagrangian interpolating functions (linear polynomials) which are C0. The corresponding wavelets are chosen to be Hierarchical basis functions. The basis functions are tailored for the PDE and their boundary conditions so that the resulting discretization matrix is block diagonal and permits optimal O(N) solving speed. The solution to the problem is obtained in multiresolutions and can be improved in a systematic manner by adding detail functions obtained from wavelets.