Wavelet - Based Finitte Element Method

INTRODUCTION The finite element method (FEM) is a numerical technique to obtain an approximate solution of partial differential equations. Mathematical models of many engineering problems can be expressed as partial differential equations that do not have a closed formed solution and therefore require a numerical solution procedure. The FEM approximation requires the discretization of the problem domain into regular sub-domains (elements) in which the solution field is modeled and interpolated such that the problem domain boundary conditions and the inter-element continuity conditions are satisfied. As the discretization is refined the FEM approximation approaches the exact solution. Wavelet theory provides a powerful mathematical tool for function approximation and multiresolution analysis. The general procedure in wavelet analysis is to transform a problem into its wavelet domain using an appropriate basis. The transformation is done in such a way that the resulting representation is amenable to the desired processing. The problem is solved in the wavelet domain and then transformed back to obtain the desired results. An important property of wavelet analysis is the capability to represent information in a hierarchical manner. Typical applications of wavelet analysis include data compression, signal and image de-noising, data communication, and function approximation. 2 A wavelet-based approach can also be used for the numerical solution of partial differential equations. This approach proceeds in a manner similar to the traditional Galerkin method except that the problem is solved in the wavelet domain. Solution in the wavelet domain enables a hierarchical approximation to the exact solution. Two challenges prevent the wider practical use of wavelet-based partial differential equation solution techniques. The first challenge is the representation of general boundary conditions in the wavelet domain. Wavelet representation of periodic boundary conditions is straightforward. But periodic boundary conditions are inappropriate for many practical problems. The second challenge is the development of efficient computational techniques that are comparable to the traditional solution approaches for partial differential equations. This paper presents the motivation for wavelet-based FEM and the general procedure for wavelet-based solutions of partial differential equations by the detailed description of a 2-D elliptic Dirichlet boundary value problem.