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Fully Discrete A-ø Finite Element Method forMaxwell’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. Emailaddress: [email protected] (H.Zhang)
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Abstract

In this paper we present a fully discrete A-ø finite element method to solve Maxwell’sequations with a nonlinear degenerate boundary condition, which represents ageneralization of the classical Silver-Müller condition for anon-perfect conductor. The relationship between the normal components of theelectric 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 anduniqueness of the solutions of the proposed A-ø scheme and derive the error estimates. Finally, wepresent 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 

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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