Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

This article is the first of a series where we develop and analyze finite element space discretizations for the time-dependent 2D Maxwell and Vlasov-Maxwell systems with long-time stability properties, and propose a structure-preserving (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 explicit 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