Type
Text
Type
Dissertation
Advisor
Samulyak, Roman | Jiao, Xiangmin, Glimm, James | Qin, Hong
Date
2012-05-01
Keywords
discrete mesh, finite element method, general finite difference method, mean curvature flow, surface diffusion, surface laplacian | Applied mathematics Ð Computer science Ð Mathematics
Department
Department of Applied Mathematics and Statistics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/71019
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
Geometric partial differential equations, such as mean-curvature flow and surface diffusion, are challenging to solve numerically due to their strong non-linearity and stiffness, when solved explicitly. Solving these high-order PDEs using explicit methods would require very small time steps to achieve stability, whereas using implicit methods would result in complex nonlinear systems of equations that are expensive to solve. In addition, accurate spatial discretizations of these equations pose challenges in their own rights, especially on triangulated surfaces. We propose new methods for mean curvature flow and surface diffusion using triangulated surfaces. Our method uses a weighted least-squares approximation for improved accuracy and stability, and semi-implicit schemes for time integration for larger time steps and higher efficiency. If mesh element quality is initially poor, or becomes poor through evolution under mean curvature flow or surface diffusion, we utilize mesh adaptivity to improve mesh quality and proceed further in evolution. Numerical experiments and comparisons demonstrate that our method can achieve second-order accuracy for both mean-curvature flow and surface diffusion, while being much more accurate and stable than using explicit schemes or alternative methods. | 84 pages
Recommended Citation
Clark, Bryan L., "Accurate, Semi-Implicit Methods with Mesh Adaptivity for Mean Curvature and Surface Diffusion Flows Using Triangulated Surfaces" (2012). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 226.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/226