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IMA Journal of Numerical Analysis Advance Access originally published online on March 20, 2008
IMA Journal of Numerical Analysis 2009 29(2):235-256; doi:10.1093/imanum/drm050
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity

Alexandre Ern

Cermics, Ecole des Ponts, University Paris-Est, 6 et 8 avenue Blaise Pascal, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France

Annette F. Stephansen{dagger}

Cermics, Ecole des Ponts, University Paris-Est, 6 et 8 avenue Blaise Pascal, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France and Andra, Parc de la Croix-Blanche, 1-7 rue Jean Monnet, 92298 Châtenay-Malabry Cedex, France

Paolo Zunino

MOX, Dipartimento di Matematica ‘F. Brioschi’, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy

{dagger} Email: Stephansen{at}cermics.enpc.fr

Received on 5 December 2006. Revised on 5 September 2007.


  Abstract

We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection–diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Péclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

Key Words: discontinuous Galerkin; weighted averages; locally small diffusion with advection; anisotropic diffusion


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