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Solving Maxwell's Equation in Meta-Materials by a CG-DG Method

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

Published online by Cambridge University Press:  17 May 2016

Ziqing Xie ,
Jiangxing Wang ,
Bo Wang  and
Chuanmiao Chen
Ziqing Xie*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
Jiangxing Wang*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China Beijing Computational Science Research Center, Beijing 100094, China
Bo Wang*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
Chuanmiao Chen*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
*
*Corresponding author. Email addresses:ziqingxie@foxmail.com (Z. Xie), wangjianghai3871@163.com (J. Wang), bowanghn@gmail.com (B. Wang), cmchen@hunnu.edu.cn (C. Chen)
*Corresponding author. Email addresses:ziqingxie@foxmail.com (Z. Xie), wangjianghai3871@163.com (J. Wang), bowanghn@gmail.com (B. Wang), cmchen@hunnu.edu.cn (C. Chen)
*Corresponding author. Email addresses:ziqingxie@foxmail.com (Z. Xie), wangjianghai3871@163.com (J. Wang), bowanghn@gmail.com (B. Wang), cmchen@hunnu.edu.cn (C. Chen)
*Corresponding author. Email addresses:ziqingxie@foxmail.com (Z. Xie), wangjianghai3871@163.com (J. Wang), bowanghn@gmail.com (B. Wang), cmchen@hunnu.edu.cn (C. Chen)
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Abstract

In this paper, an approach combining the DG method in space with CG method in time (CG-DG method) is developed to solve time-dependent Maxwell's equations when meta-materials are involved. Both the unconditional L2-stability and error estimate of order are obtained when polynomials of degree at most r is used for the temporal discretization and at most k for the spatial discretization. Numerical results in 3D are given to validate the theoretical results.

Keywords

Maxwell's equationsmeta-materialsfull-discretizationCG-DG methodL2-stabilityL2-error estimate

MSC classification

Secondary: 35Q61: Maxwell equations 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1242 - 1264
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D.. Unified Analysis Of Discontinuous Galerkin Methods For Elliptic Problems. SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779.Google Scholar
[2]Banks, H., Bokil, V. and Gibson, N.. Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media. Numer. Methods Partial Differ. Equ., 25 (2009), pp. 885–917.Google Scholar
[3]Chen, C.. Structure theory of superconvergence of finite elements, Hunan Science and Technology Press, Changsha, 2001.Google Scholar
[4]Chung, E. T., Ciarlet, P. and Yu, F.. Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids, J. Comp. Phys., 235 (2013), pp. 14–31.CrossRefGoogle Scholar
[5]Cockburn, B., Karniadakis, G. E. and Shu, C.-W.. Discontinuous Galerkin Methods: Theory, Computation and Applications, Brown University, Providence, RI (US), 2000.Google Scholar
[6]Cockburn, B., Li, F., and Shu, C. W.-. Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comp. Phys., 194 (2004), pp. 488–610.Google Scholar
[7]Cockburn, B. and Shu, C. W.-. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comp., 16 (2001), pp. 173–261.CrossRefGoogle Scholar
[8]Cui, T. J., Smith, D. R. and Liu, R.. Metamaterials: Theory, Design, and Applications, Springer, 2010.Google Scholar
[9]Di Pietro, D. A. and Ern, A.. Mathematical Aspects of Discontinuous Galerkin Methods, Springer, 2011.Google Scholar
[10]Dolean, V., Fans, H., Fezoui, L and Lanteri, S.. Locally implicit discontinuous Galerkin method for time domain electromagnetics, J. Comp. Phys., 229 (2010), pp. 512–526.Google Scholar
[11]Douglas, T., Dupont, T. and Wheeler, M. F.. An L ∞ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials, RAIRO Anal. Numer., 4 (1974), pp. 61–66.Google Scholar
[12]Fezoui, L., Lanteri, S., Lohrengel, S. and Piperno, S.. Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Model. Math. Anal. Numer., 39 (2005), pp. 1149–1176.Google Scholar
[13]Hao, Y. and Mittra, R.. FDTD Modeling of Metamaterials: Theory and Applications, Artech house, 2008.Google Scholar
[14]Hesthaven, J. S. and Warburton, T.. Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations, J. Comp. Phys., 181 (2002), PP. 186–221.Google Scholar
[15]Hesthaven, J. S. and Warburton, T.. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008.Google Scholar
[16]Huang, Y., Li, J. and Yang, W.. Interior penalty DG methods for Maxwells equations in dispersive media, J. Comp. Phys., 230 (2011), pp. 4559–4570.Google Scholar
[17]Lanteri, S. and Scheid, C.. Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell's equations in dispersive media, IMA J. Numer. Anal., 33 (2013), pp. 432–459.Google Scholar
[18]Li, J.. Development of discontinuous Galerkin methods for Maxwells equations in metamaterials and perfectly matched layers, J. Comput. Appl. Math., 47 (2011), pp. 950–961.Google Scholar
[19]Li, J.. Finite element study of the Lorentz model in metamaterials, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 626–637.Google Scholar
[20]Li, J. and Hesthaven, J. S.. Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comp. Phys,. 258 (2014), pp. 915–930.Google Scholar
[21]Li, J. and Huang, Y.. Time-domain Finite Element Methods for Maxwell's equations in Metamaterials, Springer, 2013.Google Scholar
[22]Li, J., Huang, Y. and Yang, W.. Numerical study of the Plasma-Lorentz model in metamaterials, J. Sci. Comput., 54 (2013), pp. 121–144.Google Scholar
[23]Li, J., Waters, J. W. and Mchorro, E. A.. An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwells equations in metamaterials, Comput. Methods Appl. Mech. Eng., 223 (2012), pp. 43–54.Google Scholar
[24]Lu, T., Zhang, P. and Cai, W.. Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions, J. Comp. Phys., 200 (2004), pp. 549–580.Google Scholar
[25]Pendry, J. B.. Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), pp. 3966.Google Scholar
[26]Reed, W. H. and Hill, T.. Triangular Mesh Methods for the Neutron Transport Equation, Los Alamos Report LA-UR-73-479, (1973).Google Scholar
[27]Smith, D. R. and Kroll, N.. Negative refractive index in left-handed materials, Phys. Rev. Lett., 85 (2000), pp. 2933.Google Scholar
[28]Tavlove, A. and Hagness, S. C.. Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, 1995.Google Scholar
[29]Veselago, V. G.. Electrodynamics of substance with simultaneously negative values of sigma and mu, Sov. Phys. usp., 10 (1968), pp. 509–514.Google Scholar
[30]Wang, B., Xie, Z. and Zhang, Z.. Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comp. Phys., 229 (2010), pp. 8552–8563.Google Scholar
[31]Wang, B., Xie, Z. and Zhang, Z.. Space-time discontinuous galerkin method for maxwell equations in dispersive media, Acta Math. Sci., 34 (2014), pp. 1357–1376.Google Scholar
[32]Wang, J., Xie, Z. and Chen, C.. Implicit DG method for time domain maxwell's equations involving metamaterials, Adv. Appl. Math. Mech., 7 (2015), pp. 796–817.Google Scholar
[33]Xie, Z., Wang, B. and Zhang, Z.. Space-Time Discontinuous Galerkin Method for Maxwell's Equations, Commun. Comput. Phys., 14 (2013), pp. 916–939.Google Scholar
[34]Ziolkowski, R. W.. Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs, Opt. Exp., 11 (2003), pp. 662–681.Google Scholar