A Finite Element Method to a Periodic Steady-State Problem for an Electromagnetic Field System Using the Space-Time Finite-Element Exterior Calculus

Abstract

This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a new potential variable, we reformulate two systems composed of the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace problem for the 1-form in

{\mathbb{R}}^4

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^4$$\end{document}

using the standard de Rham complex. Consequently, we can directly apply the finite-element exterior calculus (FEEC) theory in

{\mathbb{R}}^4

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^4$$\end{document}

to deduce the well-posedness, stability, and convergence. Numerical examples using the cubical element are reported to validate the theoretical results.

Keywords

Finite-element exterior calculus (FEEC) / Maxwell equation / Periodic steady-state analysis / Hodge Laplacian / Cubical element / 65N12 / 65N30 / 35J25

References

[1] Arnold, D. N.: Finite element exterior calculus. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol.93. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2018). https://doi.org/10.1137/1.9781611975543.ch1

[2] Arnold, D.N., Boffi, D., Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129(1), 1–20 (2015). https://doi.org/10.1007/s00211-014-0631-3.

[3] Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006). https://doi.org/10.1017/S0962492906210018

[4] Arnold, D. N., Falk, R. S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2) 281–354 (2010). https://doi.org/10.1090/S0273-0979-10-01278-4

[5] Arrow, K. J., Azawa, H., Hurwicz, L., Uzawa, H.: Studies in linear and non-linear programming. In: Stanford Mathematical Studies in the Social Sciences, Vol. 2. Stanford University Press , Stanford, Calif. (1958)

[6] Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. In: Springer Series in Computational Mathematics, Vol. 44. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36519-5

[7] Brüning, J., Lesch, M.: Hilbert complexes. J. Funct. Anal. 108(1), 88–132 (1992). https://doi.org/10.1016/0022-1236(92)90147-B

[8] Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011). https://doi.org/10.1017/S096249291100002X.

[9] Christiansen, S.H., Winther, R.: Smoothed projections in finite element exterior calculus. Math. Comp. 77(262), 813–829 (2008). https://doi.org/10.1090/S0025-5718-07-02081-9.

[10] Hirani, A. N.: Discrete exterior calculus. Ph.D. thesis. California Institute of Technology (2003)

[11] Jackson, J. D.: Classical Electrodynamics, 3rd Edition. Wiley, New York (1999). http://cdsweb.cern.ch/record/490457

[12] Kraus, M., Kormann, K., Morrison, P. J., Sonnendrücker, E.: Gempic: geometric electromagnetic particle-in-cell methods. J. Plasma Phys. 83(4) (2017)

[13] Lieberman MA, Lichtenberg AJ. Principles of Plasma Discharges and Materials Processing, 2005, Newark, Wiley

[14] Marsden J. E., Ratiu T. S., Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems. In: Texts in Applied Mathematics, Vol. 17. Springer, New York (2013)

[15] Petersen, P.: Riemannian geometry, 3rd Edition. In: Graduate Texts in Mathematics, Vol.171. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-26654-1

[16] Quenneville-Belair, V.: A new approach to finite element simulations of general relativity. Ph.D. thesis. University of Minnesota (2015)

[17] Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd Edition. Society for Industrial and Applied Mathematics, Philadelphia, PA (2003). https://doi.org/10.1137/1.9780898718003

[18] Salamon, J., Moody, J., Leok, M.: Space-time finite-element exterior calculus and variational discretizations of gauge field theories. In: 21st International Symposium on Mathematical Theory of Networks and Systems. Citeseer, Groningen, the Netherlands (2014)

[19] Sánchez, M.A., Du, S., Cockburn, B., Nguyen, N.-C., Peraire, J.: Symplectic Hamiltonian finite element methods for electromagnetics. Comput. Methods Appl. Mech. Eng. 396, 114969 (2022)

[20] Squire, J., Qin, H., Tang, W.M.: Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme. Phys. Plasmas 19(8), 084501 (2012)

[21] Tu, L. W.: An introduction to manifolds, 2nd Edition. In: Universitext. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-7400-6

Funding

Japan Society for the Promotion of Science(21H0443)

RIGHTS & PERMISSIONS

Shanghai University