Finite Difference Method on Flat Surfaces with a Flat Unitary Vector Bundle

Abstract

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a tileable surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with Neumann boundary conditions. Our proof is based on the precise study of the singularities of the eigenvectors near the conical points and corners of a half-translation surface, and on establishing Harnack-type estimates on “almost harmonic” discrete functions, defined on the graphs, which approximate the given surface.