Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T18:31:10.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Element Analysis of Maxwell’s Equations in Dispersive Lossy Bi-Isotropic Media

Published online by Cambridge University Press:  03 June 2015

Yunqing Huang ,
Jichun Li  and
Yanping Lin
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China
Jichun Li*
Affiliation:
Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada 89154-4020, USA
Yanping Lin*
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, and Department of Mathematical and Statistics Science, University of Alberta, Edmonton AB, Alberta T6G 2G1, Canada
*
Email: [email protected]
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

In this paper, the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated. Existence and uniqueness of the modeling equations are proved. Two fully discrete finite element schemes are proposed, and their practical implementation and stability are discussed.

Keywords

65N3035L1578M10Maxwell’s equationsbi-isotropic mediafinite element method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 494 - 509
Copyright
Copyright © Global-Science Press 2013

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]Banks, H. T., Bokil, V. A. and Gibson, N. L., Analysis of stability and dispersion in a finite element method for Debye and Lorentz media, Numer. Methods Partial Differential Equations, 25 (2009), pp. 885–917.Google Scholar
[2]Demkowicz, L., Computing with hp-Adaptive Finite Elements I, One and Two-Dimensional Elliptic and Maxwell Problems, CRC Press, Taylor and Francis, 2006.Google Scholar
[3]Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W. and Zdunek, A., Computing with hp-Adaptive Finite Elements, Vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, CRC Press, Taylor and Francis, 2006.Google Scholar
[4]Efendiev, Y. and Hou, T. Y., Multiscale Finite Element Methods: Theory and Applications, Springer, New York, 2009.Google Scholar
[5]Fairweather, G., Finite Element Galerkin Methods for Differential Equations, M. Dekker, New York, 1978.Google Scholar
[6]Fernandes, P. and Raffetto, M., Well posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials, Math. Models Methods Appl. Sci., 19 (2009), pp. 2299–2335.Google Scholar
[7]Grande, A., Barba, I., Cabeceira, A. C. L., Represa, J., Karkkainen, K. and Sihvola, A. H., Two-dimensional extension of a novel FDTD technique for modeling dispersive lossy bi-isotropic media using the auxiliary differential equation method, IEEE Microwave Wireless Components Lett., 15(5) (2005), pp. 375–377.Google Scholar
[8]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, New York, 2008.Google Scholar
[9]Huang, Y., Li, J. and Yang, W., Interior penalty DG methods for Maxwell’s equations in dispersive media, J. Comput. Phys., 230 (2011), pp. 4559–4570.Google Scholar
[10]Li, J., Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell’s equations in dispersive media, Comput. Methods Appl. Mech. Eng., 196 (2007), pp. 3081–3094.CrossRefGoogle Scholar
[11]Li, J., Posteriori error estimation for an interiori penalty discontinuous Galerkin method for Maxwell’s equations in cold plasma, Adv. Appl. Math. Mech., 1 (2009), pp. 107–124.Google Scholar
[12]Li, J., Finite element study of the Lorentz model in metamaterials, Comput. Methods Appl. Mech. Eng., 200 (2011), pp. 626–637.CrossRefGoogle Scholar
[13]Li, J. and Huang, Y., Mathematical simulation of cloaking metamaterial structures, Adv. Appl. Math. Mech., 4 (2012), pp. 93–101.CrossRefGoogle Scholar
[14]Li, J., Huang, Y. and Lin, Y., Developing finite element methods for Maxwell’s equations in a Cole-Cole dispersive medium, SIAM J. Sci. Comput., 33(6) (2011), pp. 3153–3174.Google Scholar
[15]Lindell, I., Sihvola, A., Tretyakov, S. and Viitanen, A., Electromagnetic Waves on Chiral and Bi-Isotropic Media, Artech House, Boston, 1994.Google Scholar
[16]Lu, T., Zhang, P. and Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, J. Comput. Phys., 200 (2004), pp. 549–580.CrossRefGoogle Scholar
[17]Monk, P., Finite Element Methods for Maxwell’s Equations, Oxford University Press, 2003.Google Scholar
[18]Nédélec, J.-C., Mixed finite elements in R3, Numer. Math., 35 (1980), pp. 315–341.CrossRefGoogle Scholar
[19]Scheid, C. and Lanteri, S., Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell’s equations in dispersive media, IMA Journal of Numerical Analysis, 33 (2013), pp. 432–459.Google Scholar
[20]Shaw, S., Finite element approximation of Maxwell’s equations with Debye memory, Adv. Numer. Anal., Vol. 2010, Article ID 923832, 28 pages, doi:10.1155/2010/923832.Google Scholar
[21]Wang, B., Xie, Z. and Zhang, Z., Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys., 229 (2010), pp. 8552–8563.CrossRefGoogle Scholar
[22]Zhang, Y., Cao, L.-Q., Allegretto, W. and Lin, Y., Multiscale numerical algorithm for 3D Maxwell’s equations with memory effects in composite materials, Int. J. Numer. Anal. Model. Series B, 1 (2011), pp. 41–57.Google Scholar