Numerical Solution of a Generalized Elliptic Partial Differential Eigenvalue Problem

In this article we discuss a method for the solution of non-separable eigenvalue problems. These problems are taken to be elliptic and linear and arise in a whole host of physically interesting problems. The approach exploits finite differences and a pseudo-spectral scheme. We elect to normalise at a single point, which is usually internal to the domain, and exploit the fact that the partial differential equation has not been satisfied at this point to determine whether we have an eigenvalue of the system. The eigenvalue solver is of a local nature and is consequently relatively inexpensive to run.