Direct Solution of 2D Poisson's Equation Using Wavelet Scaling Functions

Hélvio de Farias Costa Peixoto¹; Rodrigo Bird Burgos²

¹ PUC-Rio; ² UERJ

doi:10.20906/CPS/COB-2015-2160

Resumo

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 wavelets as basis functions in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties. Daubechies and Deslauriers-Dubuc wavelets 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 is the fact that the calculation of integrals of inner products of wavelet basis 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.

Palavras-chave: Wavelets; Poisson's Equation; Wavelet-Galerkin Method