Finite Element Method on Riemann Surfaces and Applications to the Laplacian Spectrum

Abstract

Abstract The objective of this PhD thesis is the approximate computation of thesolutions of the Spectral Problem associated with the Laplace operator ona compact Riemann surface without boundaries. A Riemann surface canbe seen as a gluing of portions of the Hyperbolic Plane made with suitableconditions to obtain a 2 dimensional manifold. The solutions of the SpectralProblem associated with the Laplace operator are to be understood as theeigenfunctions de ned on the surface and their corresponding eigenvalues.This work is separated into two parts: the rst part describes the methodused to approximate the eigenvalues and eigenfunctions, the second focuseson the design of a program to compute these approximations.The approximation method is inspired by the Finite Element Method(FEM), in that it relies on the variational expression of the Spectral Problemand the de nition of a nite subspace of functions in which the approximatedeigenvalues and eigenfunctions are computed. However, it di ers from theFEM in that it removes the euclidian basis of the FEM and is invariant underthe isometries of the Hyperbolic Plane.To ful ll this objective, we begin by geodesically triangulating the surfaceas regularly as possible. This hyperbolic triangulation allows us to de ne the nite subspace of functions by using the concept of barycentric coordinatesassociated with each vertex of the triangulation (idea introduced by Whitneyand taken up by Dodziuk). We then prove that the approximated solutionsconvergence to the exact ones when the diameter of the triangulation de-creases, as well as the order of convergence.The program is a practical application of the theoretical work and allowsthe computation of the approximated eigenfunctions and eigenvalues.KeywordsHyperbolic Geometry, Riemann Surfaces, Laplace Operator, Hyperbolic Tri-angulation, Finite Elements Method, Eigenvalues, Eigenfunctions.vii