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IMA Journal of Numerical Analysis Advance Access originally published online on September 29, 2007
IMA Journal of Numerical Analysis 2008 28(3):496-521; doi:10.1093/imanum/drm023
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An optimal L{infty}(L2)-error estimate for the discontinuous Galerkin approximation of a nonlinear non-stationary convection–diffusion problem

Vít Dolejsí, Miloslav Feistauer{dagger} and Václav Kucera

Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic

Veronika Sobotíková{ddagger}

Faculty of Electrical Engineering, Czech Technical University Prague, Technická 2, 166 27 Praha 6, Czech Republic

{dagger} Corresponding author. Email: feist{at}karlin.mff.cuni.cz

{ddagger} Email: sobotik{at}math.feld.cvut.cz

Received on 21 February 2007. Revised on 27 June 2007. Accepted for publication 22 July 2007.


  Abstract

This paper is concerned with the analysis of the discontinuous Galerkin finite-element method applied to the space semi-discretization of a nonlinear non-stationary convection–diffusion problem. Attention is paid on the derivation of an L{infty}(L2)-optimal error estimate for the symmetric interior penalty Galerkin scheme. The error analysis is performed for standard simplicial meshes under the assumption that the exact solution of the problem and the solution of an elliptic dual problem are sufficiently regular. The theoretical results are illustrated by numerical experiments.

Key Words: nonlinear convection–diffusion equation; discontinuous Galerkin finite-element method; symmetric formulation of diffusion terms; interior and boundary penalty; method of lines; optimal error estimates; experimental order of convergence


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