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IMA Journal of Numerical Analysis Advance Access originally published online on November 19, 2007
IMA Journal of Numerical Analysis 2008 28(3):440-468; doi:10.1093/imanum/drm038
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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.

Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates

Marcus J. Grote{dagger} and Anna Schneebeli{ddagger}

Department of Mathematics, University of Basel, Rheinsprung 21,4051 Basel, Switzerland

Dominik Schötzau§

Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

{dagger} Email: marcus.grote{at}unibas.ch.

{ddagger} Email: anna.schneebeli{at}unibas.ch

§ Corresponding author. Email: schoetzau{at}math.ubc.ca

Received on 13 December 2006. Revised on 10 September 2007.


  Abstract

We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375–386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetics.

Key Words: Maxwell's equations; discontinuous Galerkin methods; a priori error estimates


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