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Martin Campos Pinto; Eric Sonnendrücker
Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law
SMAI-Journal of computational mathematics, 3 (2017), p. 91-116, doi: 10.5802/smai-jcm.21
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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 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.


[1] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus, homological techniques, and applications”, Acta Numerica (2006) Article
[2] D.N. Arnold, R.S. Falk & R. Winther, “Geometric decompositions and local bases for spaces of finite element differential forms”, Computer Methods in Applied Mechanics and Engineering 198 (2009) no. 21, p. 1660-1672 Article
[3] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus: from Hodge theory to numerical stability”, Bull. Amer. Math. Soc.(NS) 47 (2010) no. 2, p. 281-354 Article
[4] D. Boffi, F. Brezzi & M. Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics 44, Springer, 2013
[5] Daniele Boffi, Martin Costabel, Monique Dauge & Leszek F Demkowicz, “Discrete Compactness for the hp Version of Rectangular Edge Finite Elements.”, SIAM Journal on Numerical Analysis 44 (2006) no. 3, p. 979-1004 Article
[6] A. Buffa & I. Perugia, “Discontinuous Galerkin Approximation of the Maxwell Eigenproblem”, SIAM Journal on Numerical Analysis 44 (2006) no. 5, p. 2198-2226 Article
[7] M. Campos Pinto, “Constructing exact sequences on non-conforming discrete spaces”, Comptes Rendus Mathematique 354 (2016) no. 7, p. 691-696 Article
[8] M. Campos Pinto, “Structure-preserving conforming and nonconforming discretizations of mixed problems”, hal.archives-ouvertes.fr (2017)
[9] M. Campos Pinto, S. Jund, S. Salmon & E. Sonnendrücker, “Charge conserving FEM-PIC schemes on general grids”, C.R. Mecanique 342 (2014) no. 10-11, p. 570-582 Article
[10] M. Campos Pinto & E. Sonnendrücker, “Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampère law”, HAL preprint, hal-01303852, 2016
[11] M. Campos Pinto & E. Sonnendrücker, “Gauss-compatible Galerkin schemes for time-dependent Maxwell equations”, Mathematics of Computation (2016) Article |  MR 3522966
[12] S. Depeyre & D. Issautier, “A new constrained formulation of the Maxwell system”, Rairo-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique 31 (1997) no. 3, p. 327-357 Article
[13] A. Ern & J.-L. Guermond, “Finite Element Quasi-Interpolation and Best Approximation ”, hal-01155412v2, 2015
[14] L. Fezoui, S. Lanteri, S. Lohrengel & S. Piperno, “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 39 (2005) no. 6, p. 1149-1176 Article |  MR 2195908
[15] V. Girault & P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986
[16] J.S. Hesthaven & T. Warburton, “High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 362 (2004) no. 1816, p. 493-524 Article |  MR 2075904
[17] D. Issautier, F. Poupaud, J.-P. Cioni & L. Fezoui, A 2-D Vlasov-Maxwell solver on unstructured meshes, in Third international conference on mathematical and numerical aspects of wave propagation, 1995, p. 355-371
[18] G.B. Jacobs & J.S. Hesthaven, “High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids”, Journal of Computational Physics 214 (2006) no. 1, p. 96-121 Article
[19] C.G. Makridakis & P. Monk, “Time-discrete finite element schemes for Maxwell’s equations”, RAIRO Modél Math Anal Numér 29 (1995) no. 2, p. 171-197 Article
[20] P. Monk, “An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations”, Journal of Computational and Applied Mathematics 47 (1993) no. 1, p. 101-121 Article
[21] J.-C. Nédélec, “Mixed finite elements in $\bf R^3$”, Numerische Mathematik 35 (1980) no. 3, p. 315-341 Article |  MR 593835
[22] J.-C. Nédélec, “A new family of mixed finite elements in $\bf R^3$”, Numerische Mathematik 50 (1986) no. 1, p. 57-81 Article
[23] P.-A. Raviart & J.-M. Thomas, 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, p. 292–315
[24] A. Stock, J. Neudorfer, R. Schneider, C. Altmann & C.-D. Munz, Investigation of the Purely Hyperbolic Maxwell System for Divergence Cleaning in Discontinuous Galerkin based Particle-In-Cell Methods, in COUPLED PROBLEMS 2011 IV International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2011
[25] J. Zhao, “Analysis of finite element approximation for time-dependent Maxwell problems”, Mathematics of Computation 73 (2004) no. 247, p. 1089-1106 Article