Date of Award

Fall 2018

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD

Program

School of Mathematical Sciences

Concentration

Computational Mathematics and Numerical Analysis

Advisor/Supervisor/Committee Chair

Chiu-Yen Kao

Dissertation or Thesis Committee Member

Chiu-Yen Kao

Dissertation or Thesis Committee Member

Marina Chugunova

Dissertation or Thesis Committee Member

Ali Nadim

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2018 Vladimir Delengov

Keywords

radial basis functions, Laplace-Beltrami operator, eigenproblem, meshfree, elliptic operators, manifold

Subject Categories

Numerical Analysis and Computation | Partial Differential Equations

Abstract

In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its 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.

DOI

10.5642/cguetd/113

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