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Gauss-compatible Galerkin schemes for time-dependent Maxwell equations


Authors: Martin Campos Pinto and Eric Sonnendrücker
Journal: Math. Comp. 85 (2016), 2651-2685
MSC (2010): Primary 65M60, 35Q61, 65M12
DOI: https://doi.org/10.1090/mcom/3079
Published electronically: February 15, 2016
MathSciNet review: 3522966
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Abstract: In this article we propose a unified analysis for conforming and non-conforming finite element methods that provides a partial answer to the problem of preserving discrete divergence constraints when computing numerical solutions to the time-dependent Maxwell system. In particular, we formulate a compatibility condition relative to the preservation of genuinely oscillating modes that takes the form of a generalized commuting diagram, and we show that compatible schemes satisfy convergence estimates leading to long-time stability with respect to stationary solutions. These findings are applied by specifying compatible formulations for several classes of Galerkin methods, such as the usual curl-conforming finite elements and the centered discontinuous Galerkin (DG) scheme. We also propose a new conforming/non-conforming Galerkin (Conga) method where fully discontinuous solutions are computed by embedding the general structure of curl-conforming finite elements into larger DG spaces. In addition to naturally preserving one of the Gauss laws in a strong sense, the Conga method is both spectrally correct and energy conserving, unlike existing DG discretizations where the introduction of a dissipative penalty term is needed to avoid the presence of spurious modes.


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Additional Information

Martin Campos Pinto
Affiliation: CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France – and – UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
Email: campos@ljll.math.upmc.fr

Eric Sonnendrücker
Affiliation: Max Planck Institute for Plasma Physics and Center for Mathematics, TU Munich, 85748 Garching, Germany
Email: eric.sonnendruecker@ipp.mpg.de

DOI: https://doi.org/10.1090/mcom/3079
Received by editor(s): March 11, 2014
Received by editor(s) in revised form: April 2, 2014, and February 17, 2015
Published electronically: February 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society