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

Abstract

Abstract. 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 chargeconserving 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.Math. classification. 35Q61, 65M12, 65M60, 65M75.

Local, Explicit, and Charge-Conserving Electromagnetic Particle-In-Cell Algorithm on Unstructured Grids

“…In typical mixed finite-element time-domain schemes, E and B fields are assumed to be primal quantities [21,35,33,34,47,37,1,2]. However, this is not strictly necessarily and one can choose for D and H instead to be discretized on the primal mesh [48,49,31,50]. For example, for TE ϕ polarized fields in zρ plane, the D field is represented as a (twisted) 2-form with dergees of freedom associated with the area elements of the primal mesh and expanded using Whitney 2-forms.…”

Broken-FEEC approximations of Hodge Laplace problems

“…More recently, compatible discretizations have been extended to nonconforming (broken) finite element spaces associated to sequences of conforming subspaces via stable projection operators. For the modelling of time Maxwell [15] and Maxwell-Vlasov equations [16,17] posed in contractible domains where the de Rham sequence is exact, the resulting conforming/nonconforming Galerkin (CONGA) method has been shown to have long time stability, be spectrally correct and to preserve key physical invariants such as the Gauss laws, without requiring numerical stabilization mechanisms as commonly used in discontinuous Galerkin schemes.…”