Degree Type

Dissertation

Date of Award

2012

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Jue Yan

Abstract

In this thesis, a discontinuous Galerkin (DG) finite element method for nonlinear diffusion equations named the symmetric direct discontinuous Galerkin (DDG) method is studied. The scheme is first developed for the one dimensional heat equation using the DG approach. To define a numerical flux for the numerical solution derivative, the solution derivative trace formula of the heat equation with discontinuous initial data is used. A numerical flux for the test function is introduced in order to arrive at a symmetric scheme.

Having a symmetric scheme is the key to proving an optimal $L^2(L^2)$ error estimate. In addition, stability results and an optimal energy error estimate are proven. In order to ensure stability of the scheme, a notion of flux admissibility is defined. Flux admissibility is analyzed resulting in explicit guidelines for choosing free coefficients in the numerical flux formula. The scheme is extended to one dimensional nonlinear diffusion, nonlinear convection diffusion, as well as two dimensional linear and nonlinear diffusion problems. Numerical examples are carried out to demonstrate the optimal $(k + 1)$th order of accuracy for the method with degree $k$ polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings. In addition, admissibility analysis results are explored numerically.

Copyright Owner

Chad Nathaniel Vidden

Language

en

Date Available

2012-10-31

File Format

application/pdf

File Size

85 pages

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