Exact solution of Poisson’s equation with an elliptical boundary

An exact Green’s function of the 2D Poisson equation for an elliptical boundary is derived in terms of elementary functions which can be readily implemented and efficiently evaluated. In addition the corresponding closed-form solution is derived and used for studying the convergence, the accuracy and the numerical stability of both expressions. The Green’s function for the elliptic hole is also considered by using the Joukowsky mapping in a modified form. In addition, the Green’s functions of the 2D Poisson equation in the rectangular domain are presented in elementary functions.