This thesis considers the numerical solution to elliptic boundary value problems (BVPs) in convex domains. Specifically we look at the two-dimensional problem in a polygon, and the three dimensional problem in a polyhedron. The nature of elliptic equations means that, knowing the values of a solution on the boundary, one can reconstruct this function inside the domain. This amounts to finding a Dirichlet-to-Neumann (D2N) map, which reconstructs the unknown (Neumann) boundary data from the known (Dirichlet) boundary data. Much is known about the solution to elliptic equations, both theoretically and numerically, but we shall pursue a newer development called the unified approach of [Fok08], or “Fokas method”. It is hoped that the positive results we present here will motivate further inclusion of the Fokas method in numerical packages.