Numerical analysis of Fokas' unified method for linear elliptic PDEs

In this paper we examine a numerical implementation of Fokas' unified method for elliptic boundary value problems on convex polygons. Within this setting the unified method provides a reconstruction of the unknown boundary values from the known boundary data for an important class of elliptic boundary value problems. We demonstrate that this approach can be used to study the classical Dirichlet eigenvalue problem for the Laplacian on convex polygons. We show that this problem is equivalent to a nonlinear eigenvalue problem for a semi-Fredholm operator which has holomorphic dependence on the eigenvalue itself. We also study the classical Dirichlet problem for the Helmholtz operator. We provide a new Galerkin scheme for the underlying Dirichlet–Neumann map, and prove that for sufficiently regular Dirichlet data this scheme converges with spectral accuracy.