Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law
The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 53-89.

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.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.20
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__53_0,
     author = {Martin Campos Pinto and Eric Sonnendr\"ucker},
     title = {Compatible {Maxwell} solvers with particles {I:} conforming and non-conforming {2D} schemes with a strong {Ampere} law},
     journal = {The SMAI journal of computational mathematics},
     pages = {53--89},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {3},
     year = {2017},
     doi = {10.5802/smai-jcm.20},
     mrnumber = {3695788},
     zbl = {1416.78028},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.20/}
}
Martin Campos Pinto; Eric Sonnendrücker. Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law. The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 53-89. doi : 10.5802/smai-jcm.20. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.20/

[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, Volume 22 (1980), pp. 327-346 | Article | MR 579675 | Zbl 0433.73067

[2] D.N. Arnold; R.S. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numerica (2006) | Article | 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, Volume 198 (2009) no. 21, pp. 1660-1672 | Article | MR 2517938

[4] D.N. Arnold; R.S. Falk; R. Winther 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

[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, pp. 7-28 | Article | Zbl 1210.65179

[6] D. Boffi A note on the deRham complex and a discrete compactness property, Applied Mathematics Letters, Volume 14 (2001) no. 1, pp. 33-38 | Article | MR 1793699 | Zbl 0983.65125

[7] D. Boffi Compatible Discretizations for Eigenvalue Problems, Compatible Spatial Discretizations, Springer New York, New York, NY, 2006, pp. 121-142 | Article | MR 2249348 | Zbl 1110.65104

[8] D. Boffi; F. Brezzi; M. Fortin Mixed finite element methods and applications, Springer Series in Computational Mathematics, Volume 44, Springer, 2013 | MR 3097958 | Zbl 1277.65092

[9] J.P. Boris Relativistic plasma simulations - Optimization of a hybrid code, Proc. 4th Conf. Num. Sim. of Plasmas, (NRL Washington, Washington DC) (1970), pp. 3-67

[10] A. Bossavit Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, Physical Science, Measurement and Instrumentation, Management and Education - Reviews, IEE Proceedings A (1988), pp. 493-500 | Article

[11] A. Bossavit Computational electromagnetism: variational formulations, complementarity, edge elements, Academic Press, 1998 | Zbl 0945.78001

[12] H. Brezis Functional analysis, Sobolev spaces and partial differential equations, Springer, 2010 | Article

[13] A. Buffa; I. Perugia 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

[14] A. Buffa; G. Sangalli; R. Vázquez Isogeometric analysis in electromagnetics: B-splines approximation, Computer Methods in Applied Mechanics and Engineering, Volume 199 (2010) no. 17, pp. 1143-1152 | Article | MR 2594830 | Zbl 1227.78026

[15] M. Campos Pinto 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

[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, Volume 342 (2014) no. 10-11, pp. 570-582 | Article

[18] M. Campos Pinto; M. Lutz; M. Mounier Electromagnetic PIC simulations with smooth particles: a numerical study, ESAIM: Proc., Volume 53 (2016), pp. 133-148 | Article | MR 3517453 | Zbl 1355.78039

[19] M. Campos Pinto; M. Mounier; E. Sonnendrücker Handling the divergence constraints in Maxwell and Vlasov–Maxwell simulations, Applied Mathematics and Computation, Volume 272 (2016), pp. 403-419 | Article | MR 3421114 | Zbl 1410.82027

[20] M. Campos Pinto; E. Sonnendrücker Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampère law (2016) (HAL preprint, hal-01303852v1) | Zbl 1416.78028

[21] M. Campos Pinto; E. Sonnendrücker Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law (2016) (HAL preprint hal-01303861) | Zbl 1416.78029

[22] M. Campos Pinto; E. Sonnendrücker Gauss-compatible Galerkin schemes for time-dependent Maxwell equations, Mathematics of Computation (2016) | Article | MR 3522966 | Zbl 1344.65092

[23] M. Cessenat Mathematical methods in electromagnetism, Series on Advances in Mathematics for Applied Sciences, Volume 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996 | MR 1409140 | Zbl 0917.65099

[24] S.H. Christiansen Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension, Numerische Mathematik, Volume 107 (2007) no. 1, pp. 87-106 | Article | MR 2317829

[25] S.H. Christiansen; R. Winther Smoothed projections in finite element exterior calculus, Mathematics of Computation, Volume 77 (2008) no. 262, pp. 813-829 | Article | MR 2373181 | Zbl 1140.65081

[26] A. Crestetto; Ph. Helluy Resolution of the Vlasov-Maxwell system by PIC Discontinuous Galerkin method on GPU with OpenCL, ESAIM: Proceedings, Volume 38 (2012), pp. 257-274 | Article | MR 3006546 | Zbl 1337.78005

[27] L. Demkowicz Polynomial exact sequences and projection-based interpolation with application to Maxwell equations, Mixed finite elements, compatibility conditions, and applications, Springer, 2008, pp. 101-158 | Article | Zbl 1143.78366

[28] L. Demkowicz; A. Buffa H 1 , H( curl ) and H( div )-conforming projection-based interpolation in three dimensions: Quasi-optimal p-interpolation estimates, Computer Methods in Applied Mechanics and Engineering, Volume 194 (2005) no. 2, pp. 267-296 | Article | MR 2105164 | Zbl 1143.78365

[29] S. Depeyre; D. Issautier 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

[30] J.W. Eastwood The virtual particle electromagnetic particle-mesh method, Computer Physics Communications, Volume 64 (1991) no. 2, pp. 252-266 | Article

[31] A. Ern; J.-L. Guermond Finite Element Quasi-Interpolation and Best Approximation (2015) (hal-01155412v2)

[32] T.Z. Esirkepov Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor, Computer Physics Communications, Volume 135 (2001) no. 2, pp. 144-153 | Article | Zbl 0981.78014

[33] R. Falk; R. Winther Local bounded cochain projections, Mathematics of Computation, Volume 83 (2014) no. 290, pp. 2631-2656 | Article | MR 3246803 | Zbl 1300.65085

[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, Volume 39 (2005) no. 6, pp. 1149-1176 | Article | Numdam | MR 2195908 | Zbl 1094.78008

[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, 2007 International Conference on Electromagnetics in Advanced Applications (2007), pp. 356-359 | Article

[37] J.S. Hesthaven; T. Warburton Nodal High-Order Methods on Unstructured Grids, Journal of Computational Physics, Volume 181 (2002) no. 1, pp. 186-221 | Article | Zbl 1014.78016

[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, Volume 362 (2004) no. 1816, pp. 493-524 | Article | MR 2075904 | Zbl 1078.78014

[39] R. Hiptmair Canonical construction of finite elements, Mathematics of Computation, Volume 68 (1999) no. 228, pp. 1325-1346 | Article | MR 1665954 | Zbl 0938.65132

[40] R. Hiptmair Finite elements in computational electromagnetism, Acta Numerica, Volume 11 (2002), pp. 237-339 | Article | MR 2009375 | Zbl 1123.78320

[41] R. Hiptmair Maxwell’s Equations: Continuous and Discrete, Computational Electromagnetism, Lecture Notes in Math., Vol. 2148 (A Bermúdez de Castro; A Valli, eds.), Springer International Publishing, Switzerland, 2015, pp. 1-58 | MR 3382059 | Zbl 1331.78035

[42] D. Issautier; F. Poupaud; J.-P. Cioni; L. Fezoui 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

[43] G.B. Jacobs; J.S. Hesthaven 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

[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, Volume 180 (2009) no. 10, pp. 1760-1767 | Article | MR 2678449 | Zbl 1197.76081

[45] P. Joly Variational methods for time-dependent wave propagation problems, Topics in computational wave propagation (Lect. Notes Comput. Sci. Eng.) Volume 31 (2003), pp. 201-264 | Article | MR 2032871 | Zbl 1049.78028

[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., Volume 70 (1992), pp. 447-450 | Article

[48] T. Lau; E. Gjonaj; T. Weiland The Construction of Discrete Gauss Laws for Time Domain Schemes, Magnetics, IEEE Transactions on, Volume 44 (2008) no. 6, pp. 1294-1297 | Article

[49] R. Leis Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986 | Zbl 0599.35001

[50] C.G. Makridakis; P. Monk 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

[51] B. Marder A method for incorporating Gauss’s law into electromagnetic PIC codes, J. Comput. Phys., Volume 68 (1987), pp. 48-55 | Article | Zbl 0603.65079

[52] P. Monk A mixed method for approximating Maxwell’s equations, SIAM Journal on Numerical Analysis (1991), pp. 1610-1634 | Article | MR 1135758 | Zbl 0742.65091

[53] P. Monk Analysis of a Finite Element Method for Maxwell’s Equations, SIAM Journal on Numerical Analysis, Volume 29 (1992) no. 3, pp. 714-729 | Article | MR 1163353 | Zbl 0761.65097

[54] P. Monk 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

[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 | Zbl 1024.78009

[56] P. Monk; L Demkowicz Discrete compactness and the approximation of Maxwell’s equations in R3, Mathematics of Computation, Volume 70 (2001), pp. 507-523 | Article | Zbl 1035.65131

[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, Volume 194 (2015), pp. 43-53 | Article | MR 3349372 | Zbl 1344.65096

[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, Volume 161 (2000) no. 2, pp. 484-511 | Article | MR 1764247 | Zbl 0970.78010

[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, Volume 328 (1999) no. 5, pp. 431-436 | Article | MR 1678143 | Zbl 0937.78005

[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) | Zbl 1378.78038

[61] J.-C. Nédélec Mixed finite elements in R 3 , Numerische Mathematik, Volume 35 (1980) no. 3, pp. 315-341 | Article

[62] A. Pazy Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983 | Zbl 0516.47023

[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, pp. 292-315 | Article | Zbl 0362.65089

[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, Volume 204 (2005) no. 2, pp. 490-519 | Article

[65] J. Schöberl A posteriori error estimates for Maxwell equations, Mathematics of Computation, Volume 77 (2008) no. 262, pp. 633-649 | Article | MR 2373173 | Zbl 1136.78016

[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), Volume 19 (2012) no. 8, 084501 pages | Article

[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, Volume 40 (2012) no. 7, pp. 1860-1870 | Article

[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, 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, ICAP 2009 (2009)

[70] J. Villasenor; O. Buneman Rigorous charge conservation for local electromagnetic field solvers, Computer Physics Communications, Volume 69 (1992) no. 2-3, pp. 306-316 | Article

[71] T. Weiland Finite Integration Method and Discrete Electromagnetism, Computational Electromagnetics, Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, pp. 183-198 | Article | MR 1986137 | Zbl 1030.78012

[72] D.A. White; J.M. Koning; R.N. Rieben Development and application of compatible discretizations of Maxwell’s equations, Compatible Spatial Discretizations, Springer, New York, 2006, pp. 209-234 | Article

[73] F.S. Yee Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag., Volume 14 (1966), pp. 302-307 | Article

[74] K. Yosida Functional analysis, Classics in Mathematics, Volume 123, Springer-Verlag, Berlin, 1995

[75] S. Zaglmayr High order finite element methods for electromagnetic field computation (2006) (Ph. D. Thesis)

[76] J. Zhao 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