Wavelet Regularized Solution of Laplace Equation in an Arbitrary Shaped Domain

A numerical method for the solution of interior Dirichlet problem for the two-dimensional Laplace equation in an arbitrary shaped bounded domain is presented. This domain is embedded in a circular domain and, exploiting the idea of analytic continuation, an inverse problem is formulated and solved for the boundary values of the unknown function on the circular domain. The ill-conditioning, usually associated with such inverse problems, is easily handled by a wavelet regularization. Numerical results are presented for an ellipse and a hexagon.