Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions

Abstract

This dissertation has been duly read, reviewed, and critiqued by the Committee listed below, which hereby approves the manuscript of Vladimir Delengov as fulfilling the scope and quality requirements for meriting the degree of Doctor of Philosophy.In this work, numerical approaches based on meshless methods are proposed to obtain eigenmodes of elliptic operators on manifolds, and their performance is compared against existing alternative methods. Radial Basis Function (RBF)based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly's formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces. Prospective extensions of the methods include application to problems with boundary conditions and incorporating a multi-layer approach in order to improve accuracy. The latter is justified by an asymptotic expansion of eigenvalues of Laplace operator on a thin ring and a thin shell. This work is dedicated to my beloved fiancée Sara, who always supported me unconditionally and believed in me no matter what. v