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.

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/

[1] Finite element exterior calculus, homological techniques, and applications, Acta Numerica (2006) | Article | MR 2269741

[2] Geometric decompositions and local bases for spaces of finite element differential forms, Computer Methods in Applied Mechanics and Engineering, Volume 198 (2009) no. 21, pp. 1660-1672 | Article | MR 2517938

[3] Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc.(NS), Volume 47 (2010) no. 2, pp. 281-354 | Article | MR 2594630

[4] Mixed finite element methods and applications, Springer Series in Computational Mathematics, Volume 44, Springer, 2013 | MR 3097958 | Zbl 1277.65092

[5] Discrete Compactness for the hp Version of Rectangular Edge Finite Elements., SIAM Journal on Numerical Analysis, Volume 44 (2006) no. 3, pp. 979-1004 | Article | MR 2231852 | Zbl 1122.65110

[6] Discontinuous Galerkin Approximation of the Maxwell Eigenproblem, SIAM Journal on Numerical Analysis, Volume 44 (2006) no. 5, pp. 2198-2226 | Article | MR 2263045 | Zbl 1344.65110

[7] Constructing exact sequences on non-conforming discrete spaces, Comptes Rendus Mathematique, Volume 354 (2016) no. 7, pp. 691-696 | Article | MR 3508565 | Zbl 1338.65241

[8] Structure-preserving conforming and nonconforming discretizations of mixed problems, hal.archives-ouvertes.fr (2017)

[9] Charge conserving FEM-PIC schemes on general grids, C.R. Mecanique, Volume 342 (2014) no. 10-11, pp. 570-582 | Article

[10] Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampère law (2016) (HAL preprint, $\langle $hal-01303852$\rangle $) | Zbl 1416.78028

[11] Gauss-compatible Galerkin schemes for time-dependent Maxwell equations, Mathematics of Computation (2016) | Article | MR 3522966 | Zbl 1344.65092

[12] A new constrained formulation of the Maxwell system, Rairo-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique, Volume 31 (1997) no. 3, pp. 327-357 | Article | Numdam | MR 1451346 | Zbl 0874.65097

[13] Finite Element Quasi-Interpolation and Best Approximation (2015) ($\langle $hal-01155412v2$\rangle $)

[14] Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 39 (2005) no. 6, pp. 1149-1176 | Article | Numdam | MR 2195908 | Zbl 1094.78008

[15] Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986

[16] High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 362 (2004) no. 1816, pp. 493-524 | Article | MR 2075904 | Zbl 1078.78014

[17] A 2-D Vlasov-Maxwell solver on unstructured meshes, Third international conference on mathematical and numerical aspects of wave propagation (1995), pp. 355-371 | Zbl 0874.76061

[18] High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids, Journal of Computational Physics, Volume 214 (2006) no. 1, pp. 96-121 | Article | MR 2208672

[19] Time-discrete finite element schemes for Maxwell’s equations, RAIRO Modél Math Anal Numér, Volume 29 (1995) no. 2, pp. 171-197 | Article | Numdam | MR 1332480 | Zbl 0834.65120

[20] An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations, Journal of Computational and Applied Mathematics, Volume 47 (1993) no. 1, pp. 101-121 | Article | Zbl 0784.65091

[21] Mixed finite elements in ${\mathbf{R}}^{\mathbf{3}}$, Numerische Mathematik, Volume 35 (1980) no. 3, pp. 315-341 | Article

[22] A new family of mixed finite elements in ${\mathbf{R}}^{\mathbf{3}}$, Numerische Mathematik, Volume 50 (1986) no. 1, pp. 57-81 | Article | MR 864305 | Zbl 0625.65107

[23] A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods, Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292-315 | Article | Zbl 0362.65089

[24] Investigation of the Purely Hyperbolic Maxwell System for Divergence Cleaning in Discontinuous Galerkin based Particle-In-Cell Methods, COUPLED PROBLEMS 2011 IV International Conference on Computational Methods for Coupled Problems in Science and Engineering (2011)

[25] Analysis of finite element approximation for time-dependent Maxwell problems, Mathematics of Computation, Volume 73 (2004) no. 247, pp. 1089-1106 | Article | MR 2047079 | Zbl 1119.65392