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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
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Class. Math.: 35Q61, 65M12, 65M60, 65M75
Mots clés: 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.


[1] J.-C. Adam, A. Gourdin Serveniere, J.-C. Nédélec & P.-A. Raviart, “Study of an implicit scheme for integrating Maxwell’s equations”, Computer Methods in Applied Mechanics and Engineering 22 (1980), p. 327-346
[2] D.N. Arnold, R.S. Falk & R. Winther, “Finite element exterior calculus, homological techniques, and applications”, Acta Numerica (2006)  MR 2269741
[3] 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
[4] 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
[5] R. Barthelmé & C. Parzani, Numerical charge conservation in particle-in-cell codes, Numerical methods for hyperbolic and kinetic problems, Eur. Math. Soc., Zürich, 2005, p. 7–28
[6] D. Boffi, “A note on the deRham complex and a discrete compactness property”, Applied Mathematics Letters 14 (2001) no. 1, p. 33-38
[7] D. Boffi, Compatible Discretizations for Eigenvalue Problems, Compatible Spatial Discretizations, Springer New York, 2006, p. 121–142
[8] D. Boffi, F. Brezzi & M. Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics 44, Springer, 2013
[9] J.P. Boris, Relativistic plasma simulations - Optimization of a hybrid code, in Proc. 4th Conf. Num. Sim. of Plasmas, (NRL Washington, Washington DC), 1970, p. 3-67
[10] A. Bossavit, Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, in Physical Science, Measurement and Instrumentation, Management and Education - Reviews, IEE Proceedings A, 1988, p. 493-500
[11] A. Bossavit, Computational electromagnetism: variational formulations, complementarity, edge elements, Academic Press, 1998
[12] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, 2010
[13] A. Buffa & I. Perugia, “Discontinuous Galerkin Approximation of the Maxwell Eigenproblem”, SIAM Journal on Numerical Analysis 44 (2006) no. 5, p. 2198-2226
[14] A. Buffa, G. Sangalli & R. Vázquez, “Isogeometric analysis in electromagnetics: B-splines approximation”, Computer Methods in Applied Mechanics and Engineering 199 (2010) no. 17, p. 1143-1152
[15] M. Campos Pinto, “Constructing exact sequences on non-conforming discrete spaces”, Comptes Rendus Mathematique 354 (2016) no. 7, p. 691-696
[16] M. Campos Pinto, “Structure-preserving conforming and nonconforming discretizations of mixed problems”, hal.archives-ouvertes.fr (2017)
[17] 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
[18] M. Campos Pinto, M. Lutz & M. Mounier, “Electromagnetic PIC simulations with smooth particles: a numerical study”, ESAIM: Proc. 53 (2016), p. 133-148 Article
[19] M. Campos Pinto, M. Mounier & E. Sonnendrücker, “Handling the divergence constraints in Maxwell and Vlasov–Maxwell simulations”, Applied Mathematics and Computation 272 (2016), p. 403-419
[20] 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-01303852v1, 2016
[21] M. Campos Pinto & E. Sonnendrücker, “Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law”, HAL preprint hal-01303861, 2016
[22] M. Campos Pinto & E. Sonnendrücker, “Gauss-compatible Galerkin schemes for time-dependent Maxwell equations”, Mathematics of Computation (2016)  MR 3522966
[23] M. Cessenat, Mathematical methods in electromagnetism, Series on Advances in Mathematics for Applied Sciences 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996
[24] S.H. Christiansen, “Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension”, Numerische Mathematik 107 (2007) no. 1, p. 87-106
[25] S.H. Christiansen & R. Winther, “Smoothed projections in finite element exterior calculus”, Mathematics of Computation 77 (2008) no. 262, p. 813-829
[26] A. Crestetto & Ph. Helluy, “Resolution of the Vlasov-Maxwell system by PIC Discontinuous Galerkin method on GPU with OpenCL”, ESAIM: Proceedings 38 (2012), p. 257-274
[27] L. Demkowicz, Polynomial exact sequences and projection-based interpolation with application to Maxwell equations, Mixed finite elements, compatibility conditions, and applications, Springer, 2008, p. 101–158
[28] L. Demkowicz & A. Buffa, “${H}^1$, ${H}({\rm curl})$ and ${H}({\rm div})$-conforming projection-based interpolation in three dimensions: Quasi-optimal p-interpolation estimates”, Computer Methods in Applied Mechanics and Engineering 194 (2005) no. 2, p. 267-296
[29] 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
[30] J.W. Eastwood, “The virtual particle electromagnetic particle-mesh method”, Computer Physics Communications 64 (1991) no. 2, p. 252-266
[31] A. Ern & J.-L. Guermond, “Finite Element Quasi-Interpolation and Best Approximation ”, hal-01155412v2, 2015
[32] T.Z. Esirkepov, “Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor”, Computer Physics Communications 135 (2001) no. 2, p. 144-153
[33] R. Falk & R. Winther, “Local bounded cochain projections”, Mathematics of Computation 83 (2014) no. 290, p. 2631-2656
[34] 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
[35] V. Girault & P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986
[36] E. Gjonaj, T. Lau & T. Weiland, Conservation Properties of the Discontinuous Galerkin Method for Maxwell Equations, in 2007 International Conference on Electromagnetics in Advanced Applications, IEEE, 2007, p. 356-359
[37] J.S. Hesthaven & T. Warburton, “Nodal High-Order Methods on Unstructured Grids”, Journal of Computational Physics 181 (2002) no. 1, p. 186-221
[38] 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
[39] R. Hiptmair, “Canonical construction of finite elements”, Mathematics of Computation 68 (1999) no. 228, p. 1325-1346
[40] R. Hiptmair, “Finite elements in computational electromagnetism”, Acta Numerica 11 (2002), p. 237-339
[41] R. Hiptmair, Maxwell’s Equations: Continuous and Discrete, in A Bermúdez de Castro, A Valli, éd., Computational Electromagnetism, Lecture Notes in Math., Vol. 2148, Springer International Publishing, 2015, p. 1–58
[42] 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  MR 1328210
[43] 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
[44] G.B. Jacobs & J.S. Hesthaven, “Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning”, Computer Physics Communications 180 (2009) no. 10, p. 1760-1767
[45] P. Joly, Variational methods for time-dependent wave propagation problems, in Topics in computational wave propagation, Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2003, p. 201-264
[46] M. Kraus, K. Kormann, P.J Morrison & E. Sonnendrücker, “GEMPIC: Geometric electromagnetic particle-in-cell methods”, arXiv preprint arXiv:1609.03053 (2016)
[47] A.B. Langdon, “On enforcing Gauss’ law in electromagnetic particle-in-cell codes”, Comput. Phys. Comm. 70 (1992), p. 447-450
[48] T. Lau, E. Gjonaj & T. Weiland, “The Construction of Discrete Gauss Laws for Time Domain Schemes”, Magnetics, IEEE Transactions on 44 (2008) no. 6, p. 1294-1297
[49] R. Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986
[50] 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
[51] B. Marder, “A method for incorporating Gauss’s law into electromagnetic PIC codes”, J. Comput. Phys. 68 (1987), p. 48-55
[52] P. Monk, “A mixed method for approximating Maxwell’s equations”, SIAM Journal on Numerical Analysis (1991), p. 1610-1634
[53] P. Monk, “Analysis of a Finite Element Method for Maxwell’s Equations”, SIAM Journal on Numerical Analysis 29 (1992) no. 3, p. 714-729
[54] 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
[55] P. Monk, Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, University of Delaware, Newark, 2003
[56] P. Monk & L Demkowicz, “Discrete compactness and the approximation of Maxwell’s equations in R3”, Mathematics of Computation 70 (2001), p. 507-523
[57] H. Moon, F.L. Teixeira & Y.A. Omelchenko, “Exact charge-conserving scatter–gather algorithm for particle-in-cell simulations on unstructured grids: A geometric perspective”, Computer Physics Communications 194 (2015), p. 43-53
[58] C.-D. Munz, P. Omnes, R. Schneider, E. Sonnendrücker & U. Voß, “Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model”, Journal of Computational Physics 161 (2000) no. 2, p. 484-511  MR 1764247
[59] C.-D. Munz, R. Schneider, E. Sonnendrücker & U. Voß, “Maxwell’s equations when the charge conservation is not satisfied”, Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 328 (1999) no. 5, p. 431-436
[60] D.-Y. Na, Y.A. Omelchenko, H. Moon, B.-H. Borges & F.L. Teixeira, “Axisymmetric Charge-Conservative Electromagnetic Particle Simulation Algorithm on Unstructured Grids: Application to Vacuum Electronic Devices”, arXiv:1112.1859v1 [math.NA] (2017)  MR 3670639
[61] J.-C. Nédélec, “Mixed finite elements in $\bf R^3$”, Numerische Mathematik 35 (1980) no. 3, p. 315-341
[62] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983
[63] 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
[64] R.N. Rieben, G.H. Rodrigue & D.A. White, “A high order mixed vector finite element method for solving the time dependent Maxwell equations on unstructured grids”, Journal of Computational Physics 204 (2005) no. 2, p. 490-519
[65] J. Schöberl, “A posteriori error estimates for Maxwell equations”, Mathematics of Computation 77 (2008) no. 262, p. 633-649
[66] J. Squire, H. Qin & W.M. Tang, “Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme”, Physics of Plasmas (1994-present) 19 (2012) no. 8
[67] A. Stock, J. Neudorfer, M. Riedlinger, G. Pirrung, G. Gassner, R. Schneider, S. Roller & C.-D. Munz, “Three-Dimensional Numerical Simulation of a 30-GHz Gyrotron Resonator With an Explicit High-Order Discontinuous-Galerkin-Based Parallel Particle-In-Cell Method”, IEEE Transactions on Plasma Science 40 (2012) no. 7, p. 1860-1870
[68] 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
[69] M.L. Stowell & D.A. White, Discretizing Transient Current Densities in the Maxwell Equations, in ICAP 2009, 2009
[70] J. Villasenor & O. Buneman, “Rigorous charge conservation for local electromagnetic field solvers”, Computer Physics Communications 69 (1992) no. 2-3, p. 306-316
[71] T. Weiland, Finite Integration Method and Discrete Electromagnetism, Computational Electromagnetics, Springer Berlin Heidelberg, 2003, p. 183–198
[72] D.A. White, J.M. Koning & R.N. Rieben, Development and application of compatible discretizations of Maxwell’s equations, Compatible Spatial Discretizations, Springer, 2006, p. 209–234
[73] F.S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media”, IEEE Trans. Antennas Propag. 14 (1966), p. 302-307
[74] K. Yosida, Functional analysis, Classics in Mathematics 123, Springer-Verlag, Berlin, 1995
[75] S. Zaglmayr, High order finite element methods for electromagnetic field computation, Ph. D. Thesis, Universität Linz, Diss, 2006
[76] J. Zhao, “Analysis of finite element approximation for time-dependent Maxwell problems”, Mathematics of Computation 73 (2004) no. 247, p. 1089-1106