Finite Element Approximation of Eigenvalues and Eigenfunctions of the Laplace-Beltrami Operator

The surface finite element method is an important tool for discretizing and solving elliptic partial differential equations on surfaces. Recently the surface finite element method has been used for computing approximate eigenvalues and eigenfunctions of the Laplace-Beltrami operator, but no theoretical analysis exists to offer computational guidance. In this dissertation we develop approximations of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator using the surface finite element method. We develop a priori estimates for the eigenvalues and eigenfunctions of the Laplace-Beltrami operator. We then use these a priori estimates to develop and analyze an optimal adaptive method for approximating eigenfunctions of the Laplace-Beltrami operator