Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law
The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 91-116.

This article is the second 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 second article we study two schemes which include a strong discretization of the Faraday law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by the natural ${L}^{2}$ projection for the current, also standard. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. In this 2D setting it is shown that the associated discrete curl operator coincides with that of a classical DG formulation with centered fluxes, and our analysis shows that a non-standard current approximation operator must be used to yield a charge-conserving scheme with long-time stability properties, while retaining the local nature of ${L}^{2}$ projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.21
Classification: 35Q61,  65M12,  65M60,  65M75
Keywords: Maxwell equations, Gauss laws, structure-preserving, PIC, charge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga method.
@article{SMAI-JCM_2017__3__91_0,
author = {Martin Campos Pinto and Eric Sonnendr\"ucker},
title = {Compatible {Maxwell} solvers with particles {II:} conforming and non-conforming {2D} schemes with a strong {Faraday} law},
journal = {The SMAI journal of computational mathematics},
pages = {91--116},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {3},
year = {2017},
doi = {10.5802/smai-jcm.21},
mrnumber = {3695789},
zbl = {1416.78029},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/}
}
Martin Campos Pinto; Eric Sonnendrücker. Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law. The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 91-116. doi : 10.5802/smai-jcm.21. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/

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