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

The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on 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 Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.