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
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
Recommended Citation
Delengov, Vladimir. (2018). Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions. CGU Theses & Dissertations, 113. https://scholarship.claremont.edu/cgu_etd/113. doi: 10.5642/cguetd/113
