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 L2(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.
|Commitee:||Hou, L. Steven, Liu, Hailiang, Sacks, Paul, Wang, Zhi Jian|
|School:||Iowa State University|
|School Location:||United States -- Iowa|
|Source:||DAI-B 73/10(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics|
|Keywords:||Diffusion, Discontinuous galerkin, Numerical analysis, Symmetric structures|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be