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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
SMAI-Journal of computational mathematics, 3 (2017), p. 53-89, doi: 10.5802/smai-jcm.20
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Class. Math.: 35Q61, 65M12, 65M60, 65M75
Keywords: Maxwell equations, Gauss laws, structure-preserving, PIC, charge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga method.


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.


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