Solution to the Dirichlet Problem on Irregular Domains Using Wavelet-Based Approach

We present a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) with penalty for solving two dimensional (2D) Dirichlet problem for linear elliptic PDE on irregular geometric domains. In this method, the 2-D Dirichlet problem is discretized along one of the spatial variables, reducing it to a 1-D problem. The problem and the boundaries of the irregular domain are approximated using compactly supported wavelets. Results from the numerical analysis indicate that, our method performs better in terms of accuracy and convergence of the approximate solution compared with finite element method.