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Nonconforming Finite Element Methods for Wave Propagation in Metamaterials

Part of: Partial differential equations, boundary value problems Elliptic equations and systems

Published online by Cambridge University Press:  20 February 2017

Changhui Yao  and
Lixiu Wang
Changhui Yao*
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, P. R. China
Lixiu Wang*
Affiliation:
Beijing Computational Science Research Center, Beijing, 100193, P. R. China
*
*Corresponding author. Email addresses:chyao@lsec.cc.ac.cn (C.-H. Yao), lxwang@csrc.ac.cn (L.-X. Wang)
*Corresponding author. Email addresses:chyao@lsec.cc.ac.cn (C.-H. Yao), lxwang@csrc.ac.cn (L.-X. Wang)
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Abstract

In this paper, nonconforming mixed finite element method is proposed to simulate the wave propagation in metamaterials. The error estimate of the semi-discrete scheme is given by convergence order O(h2), which is less than 40 percent of the computational costs comparing with the same effect by using Nédélec-Raviart element. A Crank-Nicolson full discrete scheme is also presented with O(τ2 + h2) by traditional discrete formula without using penalty method. Numerical examples of 2D TE, TM cases and a famous re-focusing phenomena are shown to verify our theories.

Keywords

Wave PropagationMetamaterialsNonconforming Mixed Finite ElementError Estimates

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N15: Error bounds 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 145 - 166
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Hesthaven, J. S., Warburton, T., Nodal Discontinious Galerkin Methods:Algorithm, Analysis and Applications, Springer-verlag, Berlin Heidelbert, 2008.CrossRefGoogle Scholar
[2] Li, J., Huang, Y., Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterilas, Springer-Verlag, Berlin Heidelbert, 2013.Google Scholar
[3] Monk, P., Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.Google Scholar
[4] Monk, P., A mixed method for approximating Maxwell's equations, SIAM J. Numer. Anal., 28:6(1991), pp, 1610–1634.Google Scholar
[5] Monk, P., Analysis of a finite element method for Maxwell's equations, SIAM J. Numer. Anal., 29:3(1992), pp, 714–729.Google Scholar
[6] Monk, P., A comparison of three mixed methods for the time-dependent Maxwell's equations, SIAM J. Sci. Statist. Comput., 13:5(1992), pp, 1097–1122.Google Scholar
[7] Duan, H. Y., Jia, F., Lin, P. and Roger Tan, C. E., The Local L2 Projected C0 Finite Element Method for Maxwell Problem, SIAM J. Numer. Anal., 47:2(2009), pp, 1274–1303.Google Scholar
[8] Jian, Bo-Nan, Wu, Jie and Povinelli, L. A., The origin of spurious solutions in computational electromagnetics, NASA. Technical Memorandum 106921, (1995), pp, 1–44.Google Scholar
[9] Li, J.C. and Chen, Y., Analysis of a time-domain finite element method for 3-D Maxwell's equations in dispersive media, Comput. Methods Appl. Mech. Engrg., 195:33-36, (2006), pp, 4220–4229.CrossRefGoogle Scholar
[10] Li, J., Hesthaven, J.S., Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258(2014), pp, 915–930.Google Scholar
[11] Ziolkowski, R.W., Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs, Opt. Express 11.(2003), pp, 662–681.Google Scholar
[12] Lin, Q. and Li, J.C., uperconvergence analysis for Maxwell's equations in dispersive media, Math. Comp., 77:262, (2008), pp, 757–771.Google Scholar
[13] Li, J.C., Numerical convergence and physical fidelity analysis for Maxwell's equations in metamaterials, Comput. Methods Appl. Mech. Engrg., 198:37-40 (2009), pp. 3161–3172.Google Scholar
[14] Li, J.C. and Zhang, Z.M., Unified analysis of time domain mixed finite element methods for maxwell's equations in dispersive media, J. Sci. Comput., 28:5 (2010), pp. 693–710.Google Scholar
[15] Li, J.C., Unified analysis of leap-frog methods for solving time-domain Maxwell's equations in dispersive media, J. Sci. Comput., 47:1 (2011), pp. 1–26.CrossRefGoogle Scholar
[16] Huang, Y.Q., Li, J.C. and Yang, W., Interior penalty DG methods for Maxwell's equations in dispersive media, J. Comput. Phys., 230:12 (2011), pp. 4559–4570.Google Scholar
[17] Li, J.C., Huang, Y., 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
[18] Huang, Y., Li, J. and Yang, W., Modeling Backward wave propagation in meta- materials by the finite element time domain method, SIAM J. Sci. Comput., 35:1(2013), pp, 248–B274.Google Scholar
[19] Qiao, Z., Yao, C.H. and Jia, S.J., Superconvergence and extrapolation analysis of a non-conforming mixed finite element approximation for time-harmonic Maxwell's equations, J. Sci. Comput., 46:1 (2011), pp 1–19.Google Scholar
[20] Qiao, Z., Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Comm. Comput. Phys., 3 (2008), pp. 406–426.Google Scholar
[21] Qiao, Z., Zhang, Z. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), pp, 1395–1414.Google Scholar
[22] Shi, D. Y. and Yao, C. H., Nonconforming finite element approximation of time-dependeent Maxwell's equations in Debye medium, Numer. Methods PDEs., (2014). DOI:10.1002/num21843.Google Scholar
[23] Brenner, S. C., Li, F. and Sung, L.-Y., A local divergence-free interior penalty method for two-dimensional curl-curl problem, SIAM J. Numer. Anal., 46:3 (2008), pp. 1190–1211.Google Scholar
[24] Brenner, S. C., Cui, J. Li, F. and Sung, L.-Y., A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem, Numer. Math., 109 (2008), pp. 509–533.Google Scholar
[25] Brenner, S. C., Li, F. and Sung, L.-Y., A locally divergence-free nonconforming finite element method for the time-harmonic maxwell equations, Math. Comp., 76:258 (2007), pp. 573–595.Google Scholar