## The direct discontinuous Galerkin method with symmetric structure for diffusion problems

 dc.contributor.advisor Jue Yan dc.contributor.author Vidden, Chad dc.contributor.department Mathematics dc.date 2018-08-11T10:03:37.000 dc.date.accessioned 2020-06-30T02:43:06Z dc.date.available 2020-06-30T02:43:06Z dc.date.copyright Sun Jan 01 00:00:00 UTC 2012 dc.date.embargo 2013-06-05 dc.date.issued 2012-01-01 dc.description.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.

dc.format.mimetype application/pdf dc.identifier archive/lib.dr.iastate.edu/etd/12498/ dc.identifier.articleid 3505 dc.identifier.contextkey 3437867 dc.identifier.doi https://doi.org/10.31274/etd-180810-75 dc.identifier.s3bucket isulib-bepress-aws-west dc.identifier.submissionpath etd/12498 dc.identifier.uri https://dr.lib.iastate.edu/handle/20.500.12876/26687 dc.language.iso en dc.source.bitstream archive/lib.dr.iastate.edu/etd/12498/Vidden_iastate_0097E_12773.pdf|||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 thesis.degree.level dissertation thesis.degree.name Doctor of Philosophy
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