The direct discontinuous Galerkin method with symmetric structure for diffusion problems

dc.contributor.advisor Jue Yan Vidden, Chad
dc.contributor.department Mathematics 2018-08-11T10:03:37.000 2020-06-30T02:43:06Z 2020-06-30T02:43:06Z Sun Jan 01 00:00:00 UTC 2012 2013-06-05 2012-01-01
dc.description.abstract <p>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.</p> <p>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.</p>
dc.format.mimetype application/pdf
dc.identifier archive/
dc.identifier.articleid 3505
dc.identifier.contextkey 3437867
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath etd/12498
dc.language.iso en
dc.source.bitstream archive/|||Fri Jan 14 19:22:44 UTC 2022
dc.subject.disciplines Applied Mathematics
dc.subject.disciplines Mathematics
dc.subject.keywords Diffusion
dc.subject.keywords Discontinuous Galerkin
dc.subject.keywords Numerical Analysis
dc.title The direct discontinuous Galerkin method with symmetric structure for diffusion problems
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48 dissertation Doctor of Philosophy
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
859.68 KB
Adobe Portable Document Format