In this paper we present a novel approach for solving linear elliptic PDEs in regular convex polygons. The proposed algorithm relies on the so-called unified transform, or Fokas method. The basic step of this method involves the formulation of an equation coupling the finite Fourier transforms of the given boundary data and of the unknown boundary values, which is called the global relation. Herewith, a numerical scheme is proposed which computes the solution in the interior of a regular convex polygon using only the associated global relation. In particular, an adaptive complex collocation method is presented in order to solve numerically the global relation, using discrete boundary data. Additionally, the solution of a given PDE is computed in the entire computational domain, using a spatial-stepping scheme in conjunction with an adaptive complex collocation method. Moreover, a polynomial interpolation scheme is used near the center of the domain, and this increases the accuracy of the proposed method. We provide numerical results illustrating the applicability of the method as well as a comparison to a finite element formulation.