A Posteriori Error Analysis for a Discontinuous Galerkin Method Applied to Hyperbolic Problems on Tetrahedral Meshes

In this thesis, we present a simple and efficient \emph{a posteriori} error estimation procedure for a discontinuous finite element method applied to scalar first-order hyperbolic problems on structured and unstructured tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and find basis functions spanning the error for several finite element spaces. We describe an implicit error estimation procedure for the leading term of the discretization error by solving a local problem on each tetrahedron. Numerical computations show that the implicit \emph{a posteriori} error estimation procedure yields accurate estimates for linear and nonlinear problems with smooth solutions. Furthermore, we show the performance of our error estimates on problems with discontinuous solutions. We investigate pointwise superconvergence properties of the discontinuous Galerkin (DG) method using enriched polynomial spaces. We study the effect of finite element spaces on the superconvergence properties of DG solutions on each class and type of tetrahedral elements. We show that, using enriched polynomial spaces, the discretization error on tetrahedral elements having one inflow face, is O(h^{p+2}) superconvergent on the three edges of the inflow face, while on elements with one inflow and one outflow faces the DG solution is O(h^{p+2}) superconvergent on the outflow face in addition to the three edges of the inflow face. Furthermore, we show that, on tetrahedral elements with two inflow faces, the DG solution is O(h^{p+2}) superconvergent on the edge shared by two of the inflow faces. On elements with two inflow and one outflow faces and on elements with three inflow faces, the DG solution is O(h^{p+2}) superconvergent on two edges of the inflow faces. We also show that using enriched polynomial spaces lead to a simpler{a posterior error estimation procedure. Finally, we extend our error analysis for the discontinuous Galerkin method applied to linear three-dimensional hyperbolic systems of conservation laws with smooth solutions. We perform a local error analysis by expanding the local error as a series and showing that its leading term is O( h^{p+1}). We further simplify the leading term and express it in terms of an optimal set of polynomials which can be used to estimate the error.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2012

Author dc:creator

• Mechaii, Idir

Chair dc:contributor.committeechair

• Adjerid, Slimane

Committee members dc:contributor.committeemember

• Hagedorn, George A.
• Lin, Tao
• Renardy, Yuriko Y.

• a posteriori error estimation
• Discontinuous Galerkin method
• hyperbolic problems
• superconvergence
• tetrahedral meshes

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en_US

Dc Identifier Other

etd-04032012-131655

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/77344