Title page for ETD etd-04032012-131655


Type of Document Dissertation
Author Mechaii, Idir
Author's Email Address mechaii@vt.edu
URN etd-04032012-131655
Title A Posteriori Error Analysis for a Discontinuous Galerkin Method Applied to Hyperbolic Problems on Tetrahedral Meshes
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Adjerid, Slimane Committee Chair
Hagedorn, George A. Committee Member
Lin, Tao Committee Member
Renardy, Yuriko Y. Committee Member
Keywords
  • a posteriori error estimation
  • Discontinuous Galerkin method
  • hyperbolic problems
  • superconvergence
  • tetrahedral meshes
Date of Defense 2012-03-19
Availability unrestricted
Abstract
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 \emph{{\it{inflow}}} face, is $ O(h^{p+2})$ superconvergent on the three edges of the \emph{{\it{inflow}}} face, while on elements with one \emph{{\it{inflow}}} and one \emph{{\it{outflow}}} faces the DG solution is $ O(h^{p+2})$ superconvergent on the \emph{{\it{outflow}}} face in addition to the three edges of the \emph{{\it{inflow}}} face. Furthermore, we show that, on tetrahedral elements with two \emph{{\it{inflow}}} faces, the DG solution is $ O(h^{p+2})$ superconvergent on the edge shared by two of the \emph{{\it{inflow}}} faces. On elements with two \emph{{\it{inflow}}} and one \emph{{\it{outflow}}} faces and on elements with three \emph{{\it{inflow}}} faces, the DG solution is $ O(h^{p+2})$ superconvergent on two edges of the \emph{{\it{inflow}}} faces. We also show that using enriched polynomial spaces lead to a simpler {\it{a posteriori}} 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 $O\left( h^{p+1}\right)$. 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.

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