Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:47:39.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Fully Discrete A-ø Finite Element Method for Maxwell’s Equations with a Nonlinear Boundary Condition

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems

Published online by Cambridge University Press:  10 November 2015

Tong Kang ,
Ran Wang ,
Tao Chen  and
Huai Zhang
Tong Kang
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Ran Wang
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Tao Chen
Affiliation:
Department of Applied Mathematics, School of Sciences, Communication University of China, Beijing, 100024, China.
Huai Zhang*
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, China.
*
*Corresponding author. Email address: huaizhang@gmail.com (H. Zhang)
Get access

Abstract

In this paper we present a fully discrete A-ø finite element method to solve Maxwell’s equations with a nonlinear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of the electric field E and the magnetic field H obeys a power-law nonlinearity of the type H x n = n x (|E x n|α-1E x n) with α ∈ (0,1]. We prove the existence and uniqueness of the solutions of the proposed A-ø scheme and derive the error estimates. Finally, we present some numerical experiments to verify the theoretical result.

Keywords

Maxwell’s equationsnonlinear Silver-Müller boundary conditionA-ø methodfull discretizationnodal elementserror estimates

MSC classification

Secondary: 35Q60: PDEs in connection with optics and electromagnetic theory 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 605 - 633
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albanese, R., Rubinacci, G., Fotmulation of the eddy-current problem, IEEE Proc., vol. 137 (1990), pp. 16–22.Google Scholar
[2]Amrouche, C., Bernardi, C., Dauge, M., and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math Meth. Appl. Sci., vol. 21 (1998), pp. 823–864.3.0.CO;2-B>CrossRefGoogle Scholar
[3]B’iró, O., Preis, K., On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents, IEEE Tran. Magn., vol. 25 (1989), pp. 3145–3159.CrossRefGoogle Scholar
[4]B’iró, O., Preis, K., and Buchgraber, G., Voltage-driven coils in finite-element formulations using a current vector and a magnetic scalar potential, IEEE Trans. Magn., vol. 40 (2004), pp. 1286–1289.CrossRefGoogle Scholar
[5]Buffa, A., Ciarlet, P., On trace for functional spaces related to Maxwell’s equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Meth. Appl. Sci., vol. 24 (2001), pp. 9–30.3.0.CO;2-2>CrossRefGoogle Scholar
[6]Chen, Z., Du, Q. and Zou, J., Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal., vol. 37 (2000), pp. 1542–1570.CrossRefGoogle Scholar
[7]Ciarlet, P., The finite element method for elliptic problems, North-Holland, Amsterdam, 1978.Google Scholar
[8]Ciarlet, P., Augmented formulations for solving Maxwell equations, Comput. Methods Appl. Mech. Engrg., vol. 194 (2005), pp. 559–586.CrossRefGoogle Scholar
[9]Ciarlet, P., Jamelot, E., Continuous Galerkin methods for solving Maxwell equations in 3D geometries, J. Comput. Phys., vol. 226 (2007), pp. 1122–1135.CrossRefGoogle Scholar
[10]Ciarlet, P. Jr., Wu, H. and Zou, J., Edge element methods for Maxwell’s equations with strong convergence for Gauss’ laws, SIAM J. Numer. Anal., vol. 52 (2014), pp. 779–807.CrossRefGoogle Scholar
[11]Ciarlet, P., Zou, J., Fully discrete finite element approaches for time-dependent Maxwell's equations, Numerische Mathematik, vol. 82 (1999), pp. 193–219.CrossRefGoogle Scholar
[12]Durand, S., Slodička, M., Fully discrete finite element method for Maxwells equations with nonlinear conductivity, IMA J. Numer. Anal., vol. 31 (2011), pp. 1713–1733.CrossRefGoogle Scholar
[13]Fabrizio, M., Morro, A., Electromagnetism of Continuous Media. Mathematical Modelling and Applications, Oxford University Press, Oxford, 2003.Google Scholar
[14]Fernandes, P., Raffetto, M., Existence, uniqueness and finite element approximation of the solution of time-harmonic electromagnetic boundary value problems involving meta-materials, COMPEL, vol. 24 (2005), pp. 1450–1469.CrossRefGoogle Scholar
[15]Kang, T., Chen, T., Zhang, H. and Kim, K. I., Fully discrete A-ø finite element method for Maxwell's equations with nonlinear conductivity, Numer. Meth. Part. D. E., vol. 30 (2014), pp. 2083–2108.CrossRefGoogle Scholar
[16]Kang, T., Kim, K. I., Fully discrete potential-based finite element methods for a transient eddy current problem, Computing, vol. 85 (2009), pp. 339–362.CrossRefGoogle Scholar
[17]Kim, K. I., Kang, T. , A potential-based finite element method of time-dependent Maxwell's equations, Int. J. Computer Math., vol. 83 (2006), pp. 107–122.CrossRefGoogle Scholar
[18]Kress, R., Uniqueness in inverse obstacle scattering for electromagnetic waves, in: Proceedings of the URSI General Assembly 2002, Maastricht.Google Scholar
[19]Monk, P., Finite Element Methods for Maxwells quations, Oxford University Press, USA, 2003.CrossRefGoogle Scholar
[20]Rod'iguez, A., Valli, A., Eddy Current Approximation of Maxwell Equations, Springer-Verlag, Italia, 2010.CrossRefGoogle Scholar
[21]Slodička, M., Zemanová, V., Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition, J. Comput. Appl. Math., vol. 216 (2008), pp. 514–522.CrossRefGoogle Scholar
[22]Slodička, M., Durand, S., Fully discrete finite element scheme for Maxwells equations with non-linear boundary condition, J. Math. Anal. Appl., vol. 375 (2011), pp. 230–244.CrossRefGoogle Scholar
[23]Vainberg, M., Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. New York: John Wiley, 1973.Google Scholar
[24]Vrábel, V., Slodička, M., An eddy current problem with a nonlinear evolution boundary condition. J. Math. Anal. Appl., vol. 387 (2012), pp. 267–283.CrossRefGoogle Scholar
[25]Zemanová, V., Slodička, M., Continuous Galerkin methods for solving Maxwell equations in 3D geometries, Numer. Anal. Appl. Math., vol. 1048 (2008), pp. 621–624.Google Scholar
[26]Zienkiewicz, O.C., Finite element-the basic concepts and the application to 3D magneto-static problems, John Wiley & Sons, INC., London, 1980.Google Scholar