Direct Solution of 2D Helmholtz Equations Using the Wavelet-Galerkin Method
Rodrigo Bird Burgos1; Hélvio F. C. Peixoto2; Marco Antonio Cetale Santos3
1 UERJ; 2 PUC-Rio; 3 UFF
doi:10.20906/CPS/CILAMCE2015-0733
Resumo
In the last three decades, the solution of partial differential equations (PDEs) using wavelets has become increasingly popular. Therefore, the use of wavelet scaling functions as a basis in numerical methods holds some promise due to their compact support, orthogonality, localization and multiresolution properties. Daubechies functions have been successfully used as a basis in several schemes like the Wavelet-Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Later, Gilles Deslauriers and Serge Dubuc dyadic subdivision algorithm served as a foundation for the arrival of wavelet scaling functions with interpolating properties, later called Interpolets. In this work, some mathematical foundations and computer implementation aspects 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 Helmoltz and Poisson's equations in a meshless formulation using Deslauriers-Dubuc scaling functions (Interpolets). Provided that the analysis domain is rectangular, results can be obtained also for a randomly distributed data set (scattered points). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.
Palavras-chave: Wavelets; Interpolets; Helmholtz equation; Poisson's equation