Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation

Based on a mathematical model of laser beams, we present a spectral Galerkin method for solving a Cauchy problem of the Helmholtz equation in a rectangle, where the Cauchy data pairs are given at y = 0 and boundary data are for x = 0 and x = π. The solution is sought in the interval 0 < y < 1. The spectral Galerkin method is considered as a regularization method. We then perform an analysis on the error bound for this method. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.