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Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law
Martin Campos Pinto1; Eric Sonnendrücker2
1 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France
2 Max-Planck Institute for plasma physics, Boltzmannstr. 2, D-85748 Garching, Germany, Mathematics Center, TU Munich, Boltzmannstr. 3, D-85747 Garching, Germany
The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 53-89.

This article is the first of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific charge-conserving schemes are provided for the 2D case. In this article we study two schemes which include a strong discretization of the Ampere law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by a Raviart-Thomas finite element interpolation for the current source, thanks to its commuting diagram properties. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.

Published online:
DOI: 10.5802/smai-jcm.20
Classification: 35Q61, 65M12, 65M60, 65M75
Keywords: Maxwell equations, Gauss laws, structure-preserving, PIC, charge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga method.
Martin Campos Pinto 1; Eric Sonnendrücker 2

1 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France
2 Max-Planck Institute for plasma physics, Boltzmannstr. 2, D-85748 Garching, Germany, Mathematics Center, TU Munich, Boltzmannstr. 3, D-85747 Garching, Germany
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Compatible {Maxwell} solvers with particles {I:} conforming and non-conforming {2D} schemes with a strong {Ampere} law},
     journal = {The SMAI Journal of computational mathematics},
     pages = {53--89},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Martin Campos Pinto; Eric Sonnendrücker. Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law. The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 53-89. doi : 10.5802/smai-jcm.20. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.20/

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