Solution of 1d and 2d Poisson's Equation by Using Wavelet Scaling Functions

Abstract

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelet scaling functions as a basis in computational analysis holds some promise due to their compact support, orthogonality and localization properties. Daubechies and Deslauriers-Dubuc functions have been successfully used as basis functions in several schemes like the Wavelet- Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Another possible advantage of their use is the fact that the calculation of integrals of inner products of wavelet scaling functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of using approximations by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness matrices and load vectors. In this work, some mathematical foundations regarding wavelet scaling functions, their derivatives and connection coefficients are reviewed. A scheme based on the Galerkin Method is proposed for the direct solution of Poisson's equation (potential problems) in a meshless formulation using interpolating wavelet scaling functions (Interpolets). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.

Use of Daubechies Wavelets in the Representation of Analytical Functions

“…Since M k m the moment of the wavelet scales concerning the x k monomial, where k is the degree of the polynomial, m and j are the translation and resolution of the ϕ wavelet. The justification for the construction of the equation is found in the work of [8][9][10], in which the author concludes that the c j m coefficients for approximating a monomial of the x k form, using a Daubechies wavelet base ϕ, looks like this:…”