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Time-Domain Finite Element Methods for Metamaterials

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Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 43))

Abstract

In this chapter, we present several fully discrete mixed finite element methods for solving Maxwell’s equations in metamaterials described by the Drude model and the Lorentz model. In Sects. 3.1 and 3.2, we respectively discuss the constructions of divergence and curl conforming finite elements, and the corresponding interpolation error estimates. These two sections are quite important, since we will use both the divergence and curl conforming finite elements for solving Maxwell’s equations in the rest of the book. The material for Sects. 3.1 and 3.2 is quite classic, and we mainly follow the book by Monk (Finite element methods for Maxwell’s equations. Oxford Science Publications, New York, 2003). After introducing the basic theory of divergence and curl conforming finite elements, we focus our discussion on developing some finite element methods for solving the time-dependent Maxwell’s equations when metamaterials are involved. More specifically, in Sect. 3.3, we discuss the well posedness of the Drude model. Then in Sects. 3.4 and 3.5, we present detailed stability and error analysis for the Crank-Nicolson scheme and the leap-frog scheme, respectively. Finally, we extend our discussion on the well posedness, scheme development and analysis to the Lorentz model and the Drude-Lorentz model in Sects. 3.6 and 3.7, respectively.

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References

  1. Abarbanel, S., Gottlieb, D., Hesthaven, J.S.: Long time behavior of the perfectly matched layer equations in computational electromagnetics. J. Sci. Comput. 17, 405–421 (2002)

    MathSciNet  MATH  Google Scholar 

  2. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  3. Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190, 6709–6733 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience, New York (2000)

    MATH  Google Scholar 

  5. Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68, 607–631 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Alu, A., Engheta, N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005)

    Google Scholar 

  7. Alu, A., Bilotti, E., Engheta, N., Vegni, L.: Sub-wavelength, compact, resonant patch antennas loaded with metamaterials. IEEE Trans. Antennas Propag. AP-55(1), 13–25 (2007)

    Google Scholar 

  8. Alu, A., Bilotti, E., Engheta, N., Vegni, L.: A conformal omni-directional sub-wavelength metamaterial leaky-wave antenna. IEEE Trans. Antennas Propag. AP-55(6), 1698–1708 (2007)

    Google Scholar 

  9. Ammari, H., Garnier, J., Jugnon, V., Kang, H., Lee, H., Lim, M.: Enhancement of near-cloaking. Part III: numerical simulations, statistical stability, and related questions. Contemporary Mathematics 577, 1–24 (2012)

    Google Scholar 

  10. Appelo, D., Hagstrom, T., Kreiss, G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J. Appl. Math. 67, 1–23 (2006)

    MathSciNet  Google Scholar 

  11. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85, 175–195 (2000)

    MathSciNet  Google Scholar 

  13. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Avitzour, Y., Urzhumov, Y.A., Shvets, G.: Wide-angle infrared absorber based on negative index plasmonic metamaterial. Phys. Rev. B 79, 045131 (2008)

    Google Scholar 

  15. Aydin, K., Bulu, I., Guven, K., Kafesaki, M., Soukoulis, C.M., Ozbay, E.: Investigation of magnetic resonances for different split-ring resonator parameters and designs. New J. Phys. 7, 168 (2005)

    Google Scholar 

  16. Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    MathSciNet  MATH  Google Scholar 

  17. Babuška, I., Suri, M.: The p and h − p versions of the finite element method, basic principles and properties. SIAM Rev. 36, 578–632 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Baena, J.D., Jelinek, L., Marques, R., Mock, J.J., Gollub, J., Smith, D.R.: Isotropic frequency selective surfaces made of cubic resonators. Appl. Phys. Lett. 91, 191105 (2007)

    Google Scholar 

  19. Banerjee, B.: An Introduction to Metamaterials and Waves in Composites. CRC, Boca Raton (2011)

    Google Scholar 

  20. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Solving Differential Equations. Birkhäuser, Basel (2003)

    Google Scholar 

  21. Bank, R.E.: PLTMG: A Software Package for Solving Elliptic Partial Differential Equations: Users’ Guide 8.0. SIAM, Philadelphia (1998)

    Google Scholar 

  22. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, part I: grids with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2004)

    MathSciNet  Google Scholar 

  23. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, part II: general unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2004)

    MathSciNet  Google Scholar 

  24. Banks, H.T., Bokil, V.A., Cioranescu, D., Gibson, N.L., Griso, G., Miara, B.: Homogenization of periodically varying coefficients in electromagnetic materials. J. Sci. Comput. 28, 191–221 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Banks, H.T., Bokil, V.A., Gibson, N.L.: Analysis of stability and dispersion in a finite element method for Debye and Lorentz media. Numer. Methods Partial Differ. Equ. 25, 885–917 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Bao, G., Li, P., Wu, H.: An adaptive edge element with perfectly matched absorbing layers for wave scattering by biperiodic structures. Math. Comput. 79, 1–34 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Barbatis, G., Stratis, I.G.: Homogenization of Maxwells equations in dissipative bianisotropic media. Math. Methods Appl. Sci. 26, 1241–1253 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Barrett, R., Berry, M.W., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia (1993)

    MATH  Google Scholar 

  29. Becache, E., Joly, P.: On the analysis of Berenger’s perfectly matched layers for Maxwell’s equations. Math. Model. Numer. Anal. 36, 87–119 (2002)

    MathSciNet  MATH  Google Scholar 

  30. Becache, E., Petropoulos, P., Gedney, S.: On the long-time behavior of unsplit Perfectly Matched Layers. IEEE Trans. Antennas Propag. 54, 1335–1342 (2004)

    MathSciNet  Google Scholar 

  31. Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. M2AN Math. Model. Numer. Anal. 34, 159–182 (2000)

    Google Scholar 

  32. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 11, 1–102 (2001)

    MathSciNet  Google Scholar 

  33. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, New York (1978)

    MATH  Google Scholar 

  34. Berenger, J.P.: A perfectly matched layer for the absorbing EM waves. J. Comput. Phys. 114, 185–200 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Berenger, J.P.: Three-dimensional perfectly matched layer for the absorbtion of electromagnetic waves. J. Comput. Phys. 127, 363–379 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Berenger, J.P.: Numerical reflection from FDTD-PMLs: a comparison of the split PML with the unsplit and CFS PMLs. IEEE Trans. Antennas Propag. 50, 258–265 (2002)

    Google Scholar 

  37. Beruete, M., Falcone, F., Freire, M.J., Marques, R., Baena, J.D.: Electromagnetic waves in chains of complementary metamaterial elements. Appl. Phys. Lett. 88, 083503 (2006)

    Google Scholar 

  38. Bilotti, E., Alu, A., Vegni, L.: Design of miniaturized patch antennas with μ-negative loading. IEEE Trans. Antennas Propag. AP-56(6), 1640–1647 (2008)

    Google Scholar 

  39. Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer, New York (2009)

    MATH  Google Scholar 

  40. Boffi, D., Fernandes, P., Gastaldi, L., Perudia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36, 1264–1290 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Boffi, D., Costabel, M., Dauge, M., Demkowicz, L., Hiptmair, R.: Discrete compactness for the p-version of discrete differential forms. SIAM J. Numer. Anal. 49, 135–158 (2011)

    MathSciNet  MATH  Google Scholar 

  42. Bondeson, A., Rylander, T., Ingelstrom, P.: Computational Electromagnetics. Springer, New York (2010)

    Google Scholar 

  43. Bonnet-Ben Dhia, A.S., Ciarlet, P., Zwölf, C.M.: Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants. J. Comput. Appl. Math. 204, 408–417 (2007)

    MathSciNet  MATH  Google Scholar 

  44. Bonnet-Ben Dhia, A.S., Ciarlet, P., Zwölf, C.M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912–1919 (2010). Corrigendum 234, 2616 (2010)

    Google Scholar 

  45. Bossavit, A.: Computational Electromagnetism. Academic, San Diego (1998)

    MATH  Google Scholar 

  46. Bossavit, A., Griso, G., Miara, B.: Modelling of periodic electromagnetic structures bianisotropic materials with memory effects. J. Math. Pures Appl. 84, 819–850 (2005)

    MathSciNet  MATH  Google Scholar 

  47. Bouchitté, G., Schweizer, B.: Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8, 717–750 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77, 651–672 (2008)

    MATH  Google Scholar 

  49. Bramble, J.H., Pasciak, J.E.: Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem. Math. Comput. 77, 1–10 (2008)

    MathSciNet  MATH  Google Scholar 

  50. Brandts, J.: Superconvergence of mixed finite element semi-discretization of two time-dependent problems. Appl. Math. 44, 43–53 (1999)

    MathSciNet  MATH  Google Scholar 

  51. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin/Heidelberg (1994)

    MATH  Google Scholar 

  52. Brenner, S.C., Li, F., Sung, L.-Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comput. 76, 573–595 (2007)

    MathSciNet  MATH  Google Scholar 

  53. Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  54. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin/Heidelberg (1991)

    MATH  Google Scholar 

  55. Buffa, A., Costabel, M., Schwab, C.: Boundary element methods for maxwell’s equations on non-smooth domains. Numer. Math. 92, 679–710 (2002)

    MathSciNet  MATH  Google Scholar 

  56. Buffa, A., Hiptmair, R., von Petersdorff, T., Schwab, C.: Boundary element methods for maxwell’s equations on Lipschitz domains. Numer. Math. 95, 459–485 (2003)

    MathSciNet  MATH  Google Scholar 

  57. Cai, W., Shalaev, V.: Optical Metamaterials: Fundamentals and Applications. Springer, New York (2009)

    Google Scholar 

  58. Caloz, C., Itoh, T.: Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Wiley, Hoboken (2005)

    Google Scholar 

  59. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2010)

    Google Scholar 

  60. Cao, L., Zhang, Y., Allegretto, W., Lin, Y.: Multiscale asymptotic method for Maxwell’s equations in composite materials. SIAM J. Numer. Anal. 47, 4257–4289 (2010)

    MathSciNet  MATH  Google Scholar 

  61. Capolino, F. (ed.): Metamaterials Handbook – Two Volume Slipcase Set: Theory and Phenomena of Metamaterials. CRC, Boca Raton (2009)

    Google Scholar 

  62. Carey, G.F., Oden, J.T.: Finite Elements: Computational Aspects. Prentice-Hall, Englewood Cliffs (1983)

    Google Scholar 

  63. Carstensen, C., Hu, J.: A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, 473–502 (2007)

    MathSciNet  MATH  Google Scholar 

  64. Carstensen, C., Eigel, M., Löbhard, C., Hoppe, R.H.W.: A review of unified a posteriori finite element error control. IMA Preprint Series # 2338, University of Minnesota, Oct. 2010

    Google Scholar 

  65. Chen, Z.: Finite Element Methods and Their Applications. Springer, Berlin (2005)

    MATH  Google Scholar 

  66. Chen, H., Chen, M.: Flipping photons backward: reversed Cherenkov radiation. Materialstoday 14, 34–41 (2011)

    Google Scholar 

  67. Chen, C.M., Huang, Y.: High Accuracy Theory of Finite Element Methods (in Chinese). Hunan Science Press, China (1995)

    Google Scholar 

  68. Chen, Q., Monk, P.: Introduction to applications of numerical analysis in time domain computational electromagnetism. In: Blowey, J., Jensen, M. (eds.) Frontiers in Numerical Analysis – Durham 2010, pp. 149–225. Springer, Berlin (2012)

    Google Scholar 

  69. Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41, 799–826 (2003)

    MathSciNet  MATH  Google Scholar 

  70. Chen, M.-H., Cockburn, B., Reitich, F.: High-order RKDG methods for computational electromagnetics. J. Sci. Comput. 22, 205–226 (2005)

    MathSciNet  Google Scholar 

  71. Chen, X., Wu, B.-I., Kong, J.-A., Grzegorezyk, T.: Retrieval of the effective constitutive parameters of bianisotropic metamaterials. Phys. Rev. E 71, 046610 (2005)

    Google Scholar 

  72. Chen, J., Xu, Y., Zou, J.: Convergence analysis of an adaptive edge element method for Maxwell’s equations. Appl. Numer. Math. 59, 2950–2969 (2009)

    MathSciNet  MATH  Google Scholar 

  73. Chen, H., Chan, C.T., Sheng, P.: Transformation optics and metamaterials. Nat. Mater. 9, 387–396 (2010)

    Google Scholar 

  74. Chew, W.C., Weedon, W.H.: A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw. Opt. Technol. Lett. 7, 599–604 (1994).

    Google Scholar 

  75. Christiansen, S.H.: Foundations of finite element methods for wave equations of Maxwell type. In: Quak, E., Soomere, T. (eds.) Applied Wave Mathematics, pp. 335–393. Springer, Berlin (2009)

    Google Scholar 

  76. Chung, E.T., Engquist, B.: Convergene analysis of fully discrete finite volume methods for Maxwell’s equations in nonhomogenous media. SIAM J. Numer. Anal. 43, 303–317 (2005)

    MathSciNet  MATH  Google Scholar 

  77. Chung, E.T., Du, Q., Zou, J.: Convergence analysis on a finite volume method for Maxwell’s equations in non-homogeneous media. SIAM J. Numer. Anal. 41, 37–63 (2003)

    MathSciNet  MATH  Google Scholar 

  78. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  79. Ciarlet, P. Jr., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999)

    MathSciNet  MATH  Google Scholar 

  80. Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris 335, 99–104 (2002)

    MathSciNet  MATH  Google Scholar 

  81. Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  82. Cochez-Dhondt, S., Nicaise, S.: Robust a posteriori error estimation for the Maxwell equations. Comput. Methods Appl. Mech. Eng. 196, 2583–2595 (2007)

    MathSciNet  MATH  Google Scholar 

  83. Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 3–50. Springer, Berlin (2000)

    Google Scholar 

  84. Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004)

    MathSciNet  MATH  Google Scholar 

  85. Cohen, G.C.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2001)

    Google Scholar 

  86. Cohen, G.C., Monk, P.: Gauss point mass lumping schemes for Maxwell’s equations. Numer. Methods Partial Diff. Equ. 14, 63–88 (1998)

    MathSciNet  MATH  Google Scholar 

  87. Cohen, G.C., Monk, P.: Mur-Nédélec finite element schemes for Maxwell’s equations. Comput. Methods Appl. Mech. Eng. 169, 197–217 (1999)

    MathSciNet  MATH  Google Scholar 

  88. Correia, D., Jin, J.-M.: 3D-FDTD-PML analysis of left-handed metamaterials. Microw. Opt. Technol. Lett. 40, 201–205 (2004)

    Google Scholar 

  89. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151(3), 221–276 (2000)

    MathSciNet  MATH  Google Scholar 

  90. Costabel, M., Dauge, M., Nicaise, S.: Singularities of eddy current problems. M2AN Math. Model. Numer. Anal. 37, 807–831 (2003)

    Google Scholar 

  91. Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci. 15, 575–622 (2005)

    MathSciNet  MATH  Google Scholar 

  92. Coutts, T.J.: A review of progress in thermophotovoltaic generation of electricity. Renew. Sustain. Energy Rev. 3, 77–184 (1999)

    Google Scholar 

  93. Craster, R.V., Guenneau, S. (eds.): Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. Springer, New York (2013)

    Google Scholar 

  94. Cui, T.J., Smith, D., Liu, R. (eds.): Metamaterials: Theory, Design, and Applications. Springer, New York (2009)

    Google Scholar 

  95. Cummer, S.A.: Perfectly matched layer behavior in negative refractive index materials. IEEE Antennas Wirel. Propag. Lett. 3, 172–175 (2004)

    Google Scholar 

  96. Cummer, S.A., Popa, B.-I., Schurig, D., Smith, D.R., Pendry, J.: Full-wave simulations of electromagnetic cloaking structures. Phys. Rev. E 74, 036621 (2006)

    Google Scholar 

  97. Demkowicz, L.: Computing with hp-Adaptive Finite Elements I. One and Two-Dimensional Elliptic and Maxwell Problems. CRC, Boca Raton (2006)

    Google Scholar 

  98. Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with Hp-Adaptive Finite Elements, Vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications. CRC, Boca Raton (2007)

    Google Scholar 

  99. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)

    MATH  Google Scholar 

  100. Dolean, V., Fahs, H., Fezoui, L., Lanteri, S.: Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. 229, 512–526 (2010)

    MathSciNet  MATH  Google Scholar 

  101. Dolling, G., Enkrich, C., Wegener, M., Soukoulis, C.M., Linden, S.: Simultaneous negative phase and group velocity of light in a metamaterial. Science 312, 892–894 (2006)

    Google Scholar 

  102. Dong, X.T., Rao, X.S., Gan, Y.B., Guo, B., Yin, W.Y.: Perfectly matched layer-absorbing boundary condition for left-handed materials. IEEE Microw. Wirel. Compon. Lett. 14, 301–303 (2004)

    Google Scholar 

  103. Douglas, J. Jr., Santos, J.E., Sheen, D.: A nonconforming mixed finite element method for Maxwell’s equations. Math. Models Methods Appl. Sci. 10, 593–613 (2000)

    MathSciNet  MATH  Google Scholar 

  104. Duan, Z.Y., Wu, B.-I., Chen, H.-S., Xi, S., Chen, M.: Research progress in reversed Cherenkov radiation in double-negative metamaterials. Prog. Electromagn. Res. 90, 75–87 (2009)

    Google Scholar 

  105. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications. Springer, New York (2009)

    MATH  Google Scholar 

  106. Eleftheriades, G.V., Balmain, K.G. (eds.): Negative Refraction Metamaterials: Fundamental Principles and Applications. Wiley, Hoboken (2005)

    Google Scholar 

  107. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  108. Elmkies, A., Joly, P.: Elements finis d’arete et condesation de masse pour les equations de Maxwell: le cas 3D. C. R. Acad. Sci. Paris Serie 1 325, 1217–1222 (1997)

    MathSciNet  Google Scholar 

  109. Engheta, N., Ziolkowski, R.W. (eds.): Electromagnetic Metamaterials: Physics and Engineering Explorations. Wiley, Hoboken (2006)

    Google Scholar 

  110. Engquist, B., Runborg, O., Tsai, Y.-H.R. (eds.): Numerical Analysis of Multiscale Computations: Proceedings of a Winter Workshop at the Banff International Research Station 2009. Springer, New York (2011)

    Google Scholar 

  111. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 4, 105–158 (1995)

    MathSciNet  Google Scholar 

  112. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    MATH  Google Scholar 

  113. Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp L2-error estimates and super-convergence of mixed finite element methods for non-Fickian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002)

    MathSciNet  MATH  Google Scholar 

  114. Fairweather, G.: Finite Element Galerkin Methods for Differential Equations. Marcel Dekker, New York-Basel (1978)

    MATH  Google Scholar 

  115. Fang, J., Wu, Z.: Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media. IEEE Trans. Microw. Theory Tech. 44, 2216–2222 (1996)

    Google Scholar 

  116. Fang, N., Lee, H., Sun, C., Zhang, X.: Sub-diffraction-limited optical imaging with a silver superlens. Science 308, 534–537 (2005)

    Google Scholar 

  117. Fernandes, P., Raffetto, M.: Existence, uniqueness and finite element approximation of the solution of time-harmonic electromagnetic boundary value problems involving metamaterials. COMPEL 24, 1450–1469 (2005)

    MathSciNet  MATH  Google Scholar 

  118. Fernandes, P., 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, 2299–2335 (2009)

    MathSciNet  MATH  Google Scholar 

  119. Fernandes, P., Raffetto, M.: Realistic and correct models of impressed sources for time-harmonic electromagnetic boundary value problems involving metamaterials. Preprint, Oct. 2011

    Google Scholar 

  120. Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerkin time-domain methods for the 3D heterogeneous Maxwell equations on unstructured meshes. Model. Math. Anal. Numer. 39(6), 1149–1176 (2005)

    MathSciNet  MATH  Google Scholar 

  121. Fisher, A., Rieben, R.N., Rodrigue, G.H., White, D.A.: A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations. IEEE Trans. Antennas Propag. 53(9), 2900–2910 (2005)

    MathSciNet  Google Scholar 

  122. Frantzeskakis, D.J., Ioannidis, A., Roach, G.F., Stratis, I.G., Yannacopoulos, A.N.: On the error of the optical response approximation in chiral media. Appl. Anal. 82, 839–856 (2003)

    MathSciNet  MATH  Google Scholar 

  123. Galyamin, S.N., Tyukhtin, A.V.: Electromagnetic field of a moving charge in the presence of a left-handed medium. Phys. Rev. B 81(23), 235134 (2010)

    Google Scholar 

  124. Gay-Balmaz, P., Martin, O.J.F.: Efficient isotropic magnetic resonators. Appl. Phys. Lett. 81, 939–941 (2002)

    Google Scholar 

  125. Gedney, S.D.: An anisotropic PML absorbing medium for the FDTD simulation of fields in lossy and dispersive media. Electromagnetics 16, 399–415 (1996)

    Google Scholar 

  126. Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)

    MathSciNet  MATH  Google Scholar 

  127. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms. Springer, Berlin (1986)

    MATH  Google Scholar 

  128. Goodsell, G., Whiteman, J.R.: Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part II: L ∞ -error estimates. Numer. Methods PDEs 7, 85–99 (1991)

    MathSciNet  MATH  Google Scholar 

  129. Gopalakrishnan, J., Pasciak, J.E., Demkowicz, L.F.: Analysis of a multigrid algorithm for time harmonic Maxwell equations. SIAM J. Numer. Anal. 42, 90–108 (2004)

    MathSciNet  MATH  Google Scholar 

  130. Greenleaf, A., Lassas, M., Uhlmann, G.: On non-uniqueness for Calderón’s inverse problem. Math. Res. Lett. 10, 685–693 (2003)

    MathSciNet  MATH  Google Scholar 

  131. Greenleaf, A., Lassas, M., Uhlmann, G.: Anisotropic conductivities that cannot be detected by EIT. Physiol. Meas. 24, 413–419 (2003)

    Google Scholar 

  132. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Cloaking devices, electromagnetics wormholes and transformation optics. SIAM Rev. 51, 3–33 (2009)

    MathSciNet  MATH  Google Scholar 

  133. Grote, M.J., Schneebeli, A., Schötzau, D.: Interior penalty discontinuous Galerkin method for Maxwell’s equations: energy norm error estimates. J. Comput. Appl. Math. 204, 375–386 (2007)

    MathSciNet  MATH  Google Scholar 

  134. Grote, M.J., Schneebeli, A., Schötzau, D.: Interior penalty discontinuous Galerkin method for Maxwell’s equations: optimal L 2-norm error estimates. IMA J. Numer. Anal. 28, 440–468 (2008)

    MathSciNet  MATH  Google Scholar 

  135. Guenneau, S., McPhedran, R.C., Enoch, S., Movchan, A.B., Farhat, M., Nicorovici, N.-A.P.: The colours of cloaks. J. Opt. 13, 024014 (2011)

    Google Scholar 

  136. Hackbusch, W.: Multi-Grid Methods and Applications. Springer, New York (1985)

    MATH  Google Scholar 

  137. Hao, Y., Mittra, R.: FDTD Modeling of Metamaterials: Theory and Applications. Artech House Publishers, Boston (2008)

    Google Scholar 

  138. Harrington, R.F.: Field Computation by Moment Methods. Wiley-IEEE, Hoboken (1993)

    Google Scholar 

  139. Harutyunyan, D., Izsak, F., van der Vegt, J.J.W., Botchev, M.A.: Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates. Comput. Methods Appl. Mech. Eng. 197, 1620–1638 (2008)

    MATH  Google Scholar 

  140. Hesthaven, J.S., Warburton, T.: High-order nodal methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)

    MathSciNet  MATH  Google Scholar 

  141. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)

    MATH  Google Scholar 

  142. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  143. Hetmaniuk, U., Liu, H.Y., Uhlmann, G.: On three dimensional active acoustic cloaking devices and their simulation. Preprint, University of Washington (2009)

    Google Scholar 

  144. Hiptmair, R.: Multigrid method for Maxwells equations. SIAM J. Numer. Anal. 36, 204–225 (1998)

    MathSciNet  MATH  Google Scholar 

  145. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    MathSciNet  MATH  Google Scholar 

  146. Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)

    MathSciNet  MATH  Google Scholar 

  147. Houston, P., Perugia, I., Schötzau, D.: Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator. Comput. Methods Appl. Mech. Eng. 194, 499–510 (2005)

    MATH  Google Scholar 

  148. Houston, P., Perugia, I., Schötzau, D.: An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations. IMA J. Numer. Anal. 27, 122–150 (2007)

    MathSciNet  MATH  Google Scholar 

  149. Hu, Q., Zou, J.: A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal. 41, 1682–1708 (2003)

    MathSciNet  MATH  Google Scholar 

  150. Huang, J., Zhang, S.: A divergence-free finite element method for a type of 3D Maxwell equations. Appl. Numer. Math. 62, 802–813 (2012)

    MathSciNet  MATH  Google Scholar 

  151. Huang, Y., Li, J.: Interior penalty discontinuous Galerkin method for Maxwell’s equation in cold plasma. J. Sci. Comput. 41, 321–340 (2009)

    MathSciNet  MATH  Google Scholar 

  152. Huang, Y., Li, J.: Numerical analysis of a PML model for time-dependent Maxwell’s equations. J. Comput. Appl. Math. 235, 3932–3942 (2011)

    MathSciNet  MATH  Google Scholar 

  153. Huang, Y., Li, J., Lin, Q.: Superconvergence analysis for time-dependent Maxwell’s equations in metamaterials. Numer. Methods Partial Differ. Equ. 28, 1794–1816 (2012)

    MATH  Google Scholar 

  154. Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2D edge elements to Maxwell’s equations in metamaterials. Preprint, Oct. 2011

    Google Scholar 

  155. Huang, Y., Li, J., Yang, W.: Interior penalty DG methods for Maxwell’s equations in dispersive media. J. Comput. Phys. 230, 4559–4570 (2011)

    MathSciNet  MATH  Google Scholar 

  156. Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials. J. Comput. Phys. 230, 8275–8289 (2011)

    MathSciNet  MATH  Google Scholar 

  157. Huang, Y., Li, J., Yang, W.: Modeling backward wave propagation in metamaterials by a finite element time domain method. SIAM J. Sci. Comput. (in press)

    Google Scholar 

  158. Hughes, T.J.R.: Finite Element Method – Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  159. Izsak, F., Harutyunyan, D., van der Vegt, J.J.W.: Implicit a posteriori error estimates for the Maxwell equations. Math. Comput. 77, 1355–1386 (2008)

    MATH  Google Scholar 

  160. Jiao, D., Jin, J.-M.: Time-domain finite-element modeling of dispersive media. IEEE Microw. Wirel. Compon. Lett. 11, 220–222 (2001)

    Google Scholar 

  161. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994)

    Google Scholar 

  162. Jin, J.: The Finite Element Method in Electromagnetics, 2nd edn. Wiley-IEEE, Hoboken (2002)

    MATH  Google Scholar 

  163. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, New York (1988)

    Google Scholar 

  164. Kafesaki, M., Koschny, Th., Penciu, R.S., Gundogdu, T.F., Economou, E.N., Soukoulis, C.M.: Left-handed metamaterials: detailed numerical studies of the transmission properties. J. Opt. A 7, S12–S22 (2005)

    Google Scholar 

  165. Kohn, R.V., Shipman, S.P.: Magnetism and homogenization of microresonators. Multiscale Model. Simul. 7, 62–92 (2008)

    MathSciNet  MATH  Google Scholar 

  166. Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: Cloaking via change of variables in electrical impendance tomography. Inverse Probl. 24, 015016 (2008)

    MathSciNet  Google Scholar 

  167. Kohn, R.V., Onofrei, D., Vogelius, M.S., Weinstein, M.I.: Cloaking via change of variables for the Helmholtz equation. Commun. Pure Appl. Math. 63, 973–1016 (2010)

    MathSciNet  MATH  Google Scholar 

  168. Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int. J. Numer. Mech. Eng. 53, 105–122 (2002)

    MATH  Google Scholar 

  169. Kristensson, G.: Homogenization of the Maxwell equations in an anisotropic material. Technical Report LUTEDX/(TEAT-7104)/1–12/(2001), Department of Electroscience, Lund Institute of Technology, Sweden (2001)

    Google Scholar 

  170. Krizek, M., Neittaanmaki, P.: Bibliography on superconvergence. In: Krizek, M., Neittaanmaki, P., Stenberg, R. (eds.) Finite Element Methods: Superconvergence, Postprocessing and A Posteriori Estimates, pp. 315–348. Marcel Dekker, New York (1997)

    Google Scholar 

  171. Krowne, C.M., Zhang, Y. (eds.): Physics of Negative Refraction and Negative Index Materials: Optical and Electronic Aspects and Diversified Approaches. Springer, New York (2007)

    Google Scholar 

  172. Kunert, G., Nicaise, S.: Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes. ESAIM: Math. Model Numer. Anal. 37, 1013–1043 (2003)

    MathSciNet  MATH  Google Scholar 

  173. Kuzuoglu, M., Mittra, R.: Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers. IEEE Microw. Guid. Wave Lett. 6, 447–449 (1996)

    Google Scholar 

  174. Langtangen, H.P.: Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, 2nd edn. Springer, Berlin (2003)

    Google Scholar 

  175. Laroche, M., Carminati, R., Greffet, J.-J.: Near-field thermophotovoltaic energy conversion. J. Appl. Phys. 100, 063704 (2006)

    Google Scholar 

  176. Ledger, P.D., Morgan, K.: The application of the hp-finite element method to electromagnetic problems. Arch. Comput. Methods Eng. 12, 235–302 (2005)

    MathSciNet  MATH  Google Scholar 

  177. Lee, H.-J., Yook, J.-G.: Biosensing using split-ring resonators at microwave regime. Appl. Phys. Lett. 92, 254103 (2008)

    Google Scholar 

  178. Lee, J.-F., Lee, R., Cangellaris, A.C.: Time domain finite element methods. IEEE Trans. Antennas Propag. 45, 430–442 (1997)

    MathSciNet  MATH  Google Scholar 

  179. Lee, J.-H., Xiao, T., Liu, Q.H.: A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields. IEEE Trans. Microw. Theory Tech. 54, 437–444 (2006)

    Google Scholar 

  180. Leonhardt, U.: Optical conformal mapping. Science 312, 1777–1780 (2006)

    MathSciNet  MATH  Google Scholar 

  181. Leonhardt, U., Philbin, T.: Geometry and Light: The Science of Invisibility. Dover, New York (2010)

    MATH  Google Scholar 

  182. Li, J.: Posteriori error estimation for an interiori penalty discontinuous Galerkin method for Maxwell’s equations in cold plasma. Adv. Appl. Math. Mech. 1, 107–124 (2009)

    MathSciNet  Google Scholar 

  183. Li, J.: Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 198, 3161–3172 (2009)

    MATH  Google Scholar 

  184. Li, J.: Finite element study of the Lorentz model in metamaterials. Comput. Methods Appl. Mech. Eng. 200, 626–637 (2011)

    MATH  Google Scholar 

  185. Li, J.: Development of discontinuous Galerkin methods for Maxwell’s equations in metamaterials and perfectly matched layers. J. Comput. Appl. Math. 236, 950–961 (2011)

    MathSciNet  MATH  Google Scholar 

  186. Li, J.: Optimal L 2 error estimates for the interior penalty DG method for Maxwell’s equations in cold plasma. Commun. Comput. Phys. 11, 319–334 (2012)

    MathSciNet  Google Scholar 

  187. Li, J., Chen, Y.: Computational Partial Differential Equations Using MATLAB. CRC, Boca Raton (2008)

    Google Scholar 

  188. Li, J., Huang, Y.: Mathematical simulation of cloaking metamaterial structures. Adv. Appl. Math. Mech. 4, 93–101 (2012)

    MathSciNet  Google Scholar 

  189. Li, J., Wood, A.: Finite element analysis for wave propagation in double negative metamaterials. J. Sci. Comput. 32, 263–286 (2007)

    MathSciNet  MATH  Google Scholar 

  190. Li, J., Zhang, Z.: Unified analysis of time domain mixed finite element methods for Maxwell’s equations in dispersive media. J. Comput. Math. 28, 693–710 (2010)

    MathSciNet  MATH  Google Scholar 

  191. Li, J., Chen, Y., Elander, V.: Mathematical and numerical study of wave propagation in negative-index materials. Comput. Methods Appl. Mech. Eng. 197, 3976–3987 (2008)

    MathSciNet  MATH  Google Scholar 

  192. Li, J., Chen, Y., Liu, Y.: Mathematical simulation of metamaterial solar cells. Adv. Appl. Math. Mech. 3, 702–715 (2011)

    MathSciNet  Google Scholar 

  193. Li, J., Huang, Y., Lin, Y.: Developing finite element methods for Maxwell’s equations in a Cole-Cole dispersive medium. SIAM J. Sci. Comput. 33, 3153–3174 (2011)

    MathSciNet  MATH  Google Scholar 

  194. Li, J., Huang, Y., Yang, W.: Developing a time-domain finite-element method for modeling of invisible cloaks. J. Comput. Phys. 231, 2880–2891 (2012)

    MathSciNet  MATH  Google Scholar 

  195. Li, J., Huang, Y., Yang, W.: Numerical study of the Plasma-Lorentz model in metamaterials. J. Sci. Comput. doi:10.1007/s10915-012-9608-5

    Google Scholar 

  196. Li, Z., Aydin, K., Ozbay, E.: Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients. Phys. Rev. E 79, 026610 (2009)

    Google Scholar 

  197. Liang, Z., Yao, P., Sun, X., Jiang, X.: The physical picture and the essential elements of the dynamical process for dispersive cloaking structures. Appl. Phys. Lett. 92, 131118 (2008)

    Google Scholar 

  198. Lin, Q., Li, J.: Superconvergence analysis for Maxwell’s equations in dispersive media. Math. Comput. 77, 757–771 (2008)

    MathSciNet  MATH  Google Scholar 

  199. Lin, Q., Lin, J.F.: High accuracy approximation of mixed finite element for 2-D Maxwell equations (in Chinese). Acta Math. Sci. Ser. A Chin. Ed. 23, 499–503 (2003)

    MathSciNet  MATH  Google Scholar 

  200. Lin, Q., Yan, N.: Superconvergence of mixed element methods for Maxwells equations (in Chinese). Gongcheng Shuxue Xuebao 13, 1–10 (1996)

    MathSciNet  MATH  Google Scholar 

  201. Lin, Q., Yan, N.: The Construction and Analysis of High Accurate Finite Element Methods (in Chinese). Hebei University Press, Hebei (1996)

    Google Scholar 

  202. Lin, Q., Yan, N.: Global superconvergence for Maxwells equations. Math. Comput. 69, 159–176 (1999)

    MathSciNet  Google Scholar 

  203. Lin, Q., Li, J., Zhou, A.: A rectangle test for the Stokes equations. In: Prof. of Sys. Sci. and Sys. Engrg., pp. 240–241. Culture Publish Co., Great Wall (H.K.) (1991)

    Google Scholar 

  204. Lin, Q., Yan, N., Zhou, A.: A rectangle test for interpolated finite elements. In: Prof. of Sys. Sci. and Sys. Engrg., pp. 217–229. Culture Publish Co., Great Wall (H.K.) (1991)

    Google Scholar 

  205. Liu, Z., Lee, H., Xiong, Y., Sun, C., Zhang, X.: Far-field optical hyperlens magnifying sub-diffraction-limited objects. Science 315, 1686–1686 (2007)

    Google Scholar 

  206. Liu, R., Ji, C., Mock, J.J., Chin, J.Y., Cui, T.J., Smith, D.R.: Science 323, 366–369 (2009)

    Google Scholar 

  207. Lu, T., Zhang, P., Cai, W.: Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions. J. Comput. Phys. 200, 549–580 (2004)

    MathSciNet  MATH  Google Scholar 

  208. Maradudin, A.A. (eds.): Stuctured Surfaces as Optical Metamaterials. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  209. Markos, P., Soukoulis, C.M.: Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  210. Marques, R., Martin, F., Sorolla, M.: Metamaterials with Negative Parameters: Theory, Design and Microwave Applications. Wiley-IEEE, New York (2008)

    Google Scholar 

  211. Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  212. Milton, G.W., Nicorovici, N.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059 (2006)

    MathSciNet  MATH  Google Scholar 

  213. Mittra, R., Pekel, U.: A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves. IEEE Microw. Guid. Wave Lett. 53, 84–86 (1995)

    Google Scholar 

  214. Milton, G.W., Briane, M., Willis, J.R.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006)

    Google Scholar 

  215. Monk, P.: Superconvergence of finite element approximations to Maxwells equations. Numer. Methods Partial Differ. Equ. 10, 793–812 (1994)

    MathSciNet  MATH  Google Scholar 

  216. Monk, P.: A posteriori error indicators for Maxwell’s equations. J. Comput. Appl. Math. 100, 173–190 (1998)

    MathSciNet  MATH  Google Scholar 

  217. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford Science Publications, New York (2003)

    MATH  Google Scholar 

  218. Monk, P., Parrott, A.K.: A dispersion analysis of finite element methods for Maxwell’s equations. SIAM J. Sci. Comput. 15, 916–937 (1994)

    MathSciNet  MATH  Google Scholar 

  219. Montseny, E., Pernet, S., Ferriéres, X., Cohen, G.: Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell’s equations. J. Comput. Phys. 227, 6795–6820 (2008)

    MathSciNet  MATH  Google Scholar 

  220. Munk, B.A.: Metamaterials: Critique and Alternatives. Wiley-Interscience, Hoboken (2009)

    Google Scholar 

  221. Narimanov, E.E., Shalaev, V.M.: Beyond diffraction. Nature 447, 266–267 (2007)

    Google Scholar 

  222. Nédélec, J.-C.: Mixed finite elements in ℛ 3. Numer. Math. 35, 315–341 (1980)

    MathSciNet  MATH  Google Scholar 

  223. Nédélec, J.-C.: A new family of mixed finite elements in ℛ 3. Numer. Math. 50, 57–81 (1986)

    MathSciNet  MATH  Google Scholar 

  224. Nicaise, S.: On Zienkiewicz-Zhu error estimators for Maxwell’s equations. C. R. Math. Acad. Sci. Paris 340, 697–702 (2005)

    MathSciNet  MATH  Google Scholar 

  225. Nicaise, S., Creusé, E.: A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes. Calcolo 40, 249–271 (2003)

    MathSciNet  MATH  Google Scholar 

  226. Nicolaides, R.A., Wang, D.-Q.: Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions. Math. Comput. 67, 947–963 (1998)

    MathSciNet  MATH  Google Scholar 

  227. Nochetto, R.H., Veeser, A.: Primer of adaptive finite element methods. In: Naldi, G., Russo, G. (eds.) Multiscale and Adaptivity: Modeling, Numerics and Applications: C.I.M.E. Summer School, Cetraro, Italy 2009, pp. 125–226. Springer, Berlin (2012)

    Google Scholar 

  228. Noginov, M.A., Podolskiy, V. (eds.): Tutorials in Metamaterials. Series in Nano-Optics and Nanophotonics. CRC, Boca Raton (2011)

    Google Scholar 

  229. Norris, A.N.: Acoustic cloaking theory. Proc. R. Soc. A. 464, 2411–2434 (2008)

    MathSciNet  MATH  Google Scholar 

  230. O’Hara, J.F., Singh, R., Brener, I., Smirnova, E., Han, J., Taylor, A.J., Zhang, W.: Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations. Opt. Express 16, 1786–1795 (2008)

    Google Scholar 

  231. Ouchetto, O., Zouhdi, S., Bossavit, A., Griso, G., Miara, B., Razek, A.: Homogenization of structured electromagnetic materials and metamaterials. J. Mater. Process. Technol. 181, 225–229 (2007)

    Google Scholar 

  232. Padilla, W.J.: Group theoretical description of artificial electromagnetic metamaterials. Opt. Express 15, 1639–1646 (2007)

    MathSciNet  Google Scholar 

  233. Parnell, W.J.: Nonlinear pre-stress for cloaking from antiplane elastic waves. Proc. R. Soc. A 468, 563–580 (2012)

    MathSciNet  Google Scholar 

  234. Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000)

    Google Scholar 

  235. Pendry, J.B., Holden, A.J., Stewart, W.J., Youngs, I.: Extremely low frequency plasmons in metallic meso structures. Phys. Rev. Lett. 76, 4773–4776 (1996)

    Google Scholar 

  236. Pendry, J.B., Holden, A.J., Robbins, D.J., Stewart, W.J.: Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999)

    Google Scholar 

  237. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)

    MathSciNet  MATH  Google Scholar 

  238. Pernet, S., Ferrieres, X.: HP A-priori error estimates for a non-dissipative spectral discontinuous Galerkin method to solve the Maxwell equations in the time domain. Math. Comput. 76, 1801–1832 (2007)

    MathSciNet  MATH  Google Scholar 

  239. Piperno, S., Remaki, M., Fezoui, L.: A non-diffusive finite volume scheme for the 3D Maxwell equations on unstructured meshes. SIAM J. Numer. Anal. 39, 2089–2108 (2002)

    MathSciNet  MATH  Google Scholar 

  240. Pozrikidis, C.: Introduction to Finite and Spectral Element Methods Using MATLAB. Chapman & Hall/CRC, Boca Raton (2005)

    MATH  Google Scholar 

  241. Prokopidis, K.P.: On the development of efficient FDTD-PML formulations for general dispersive media. Int. J. Numer. Model. 21, 395–411 (2008)

    MATH  Google Scholar 

  242. Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell’s equations. J. Sci. Comput. 46, 1–19 (2011)

    MathSciNet  MATH  Google Scholar 

  243. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  244. Rahm, M., Schurig, D., Roberts, D.A., Cummer, S.A., Smith, D.R., Pendry, J.B.: Design of electromagnetic cloaks and concentrations using form-invariant coordinate transformations of Maxwell’s equations. Photonics Nanostructures – Fundam. Appl. 6, 87–95 (2008)

    Google Scholar 

  245. Ramakrishna, S.A., Grzegorczyk, T.M.: Physics and Applications of Negative Refractive Index Materials. CRC, Boca Raton (2008)

    Google Scholar 

  246. Rappaport, C.M.: Perfectly matched absorbing conditions based on anisotropic lossy mapping of space. IEEE Microw. Guid. Wave Lett. 53, 90–92 (1995)

    Google Scholar 

  247. Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)

    MATH  Google Scholar 

  248. Roden, J.A., Gedney, S.D.: Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media. Microw. Opt. Technol. Lett. 27, 334–339 (2000)

    Google Scholar 

  249. Sacks, Z.S., Kingsland, D.M., Lee, R., Lee, J.-F.: A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propag. 43, 1460–1463 (1995)

    Google Scholar 

  250. Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer, Berlin (1980)

    MATH  Google Scholar 

  251. Scheid, C., Lanteri, S.: Convergence of a discontinuous Galerkin scheme for the mixed time domain Maxwell’s equations in dispersive media, IMA J Numer Anal (2012). doi: 10.1093/imanum/drs008

    Google Scholar 

  252. Schmidt, A., Siebert. K.G.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Springer, Berlin (2005)

    Google Scholar 

  253. Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77, 633–649 (2008)

    MATH  Google Scholar 

  254. Schurig, D., Mock, J.J., Justice, B.J., Cummer, S.A., Pendry, J.B., Starr, A.F.S., Smith, D.R.: Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006)

    Google Scholar 

  255. Schwab, C.: p- and hp- Finite Element Methods, Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  256. Shalaev, V.M., Sarychev, A.K.: Electrodynamics of Metamaterials. World Scientific, Hackensack (2007)

    MATH  Google Scholar 

  257. Shamonina, E., Solymar, L.: Properties of magnetically coupled metamaterial elements. J. Magn. Magn. Mater. 300, 38–43 (2006)

    Google Scholar 

  258. Shaw, S.: Finite element approximation of Maxwell’s equations with Debye memory. Adv. Numer. Anal. 2010, Article ID 923832 (2010). doi:10.1155/2010/923832

    Google Scholar 

  259. Shelby, R.A., Smith, D.R., Nemat-Nasser, S.C., Schultz, S.: Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial. Appl. Phys. Lett. 78, 489–491 (2001)

    Google Scholar 

  260. Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292, 489–491 (2001)

    Google Scholar 

  261. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, New York (2011)

    MATH  Google Scholar 

  262. Shi, Y., Li, Y., Liang, C.H.: Perfectly matched layer absorbing boundary condition for truncating the boundary of the left-handed medium. Microw. Opt. Technol. Lett. 48, 57–62 (2006)

    Google Scholar 

  263. Shvets, G., Tsukerman, I. (eds.): Plasmonics and Plasmonic Metamaterials: Analysis and Applications. World Scientific, Hackensack (2011)

    Google Scholar 

  264. Sihvola, A.H.: Electromagnetic Mixing Formulas and Applications. The Institute of Electrical Engineer, London (1999)

    Google Scholar 

  265. Silveirinha, M., Belov, P., Simovski, C.: Sub-wavelength imaging at infrared frequencies using an array of metallic nanorods. Phys. Rev. B 75, 035108 (2007)

    Google Scholar 

  266. Silveirinha, M., Belov, P., Simovski, C.: Ultimate limit of resolution of subwavelength imaging devices formed by metallic rods. Opt. Lett. 33, 1726–1728 (2008)

    Google Scholar 

  267. Silvester, P.P., Ferrari, R.L.: Finite Elements for Electrical Engineers, 3rd edn. Cambridge University Press, London (1996)

    Google Scholar 

  268. Sjöberg, D., Engström, C., Kristensson, G., Wall, D.J.N., Wellander, N.: A Floquet-Bloch decomposition of Maxwell’s equations applied to homogenization. Multiscale Model. Simul. 4, 149–171 (2005)

    MathSciNet  MATH  Google Scholar 

  269. Smith, D.R., Kroll, N.: Negative refractive index in left-handed materials. Phys. Rev. Lett. 85, 2933–2936 (2000)

    Google Scholar 

  270. Smith, D., Pendry, J.: Homogenization of metamaterials by field averaging. J. Opt. Soc. Am. B 23, 391–403 (2006)

    Google Scholar 

  271. Smith, D.R., Padilla, W.J., Vier, D.C., Nemat-Nasser, S.C., Schultz, S.: Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84, 4184–4187 (2000)

    Google Scholar 

  272. Smolyaninov, I.I., Hung, Y.-J., Davis, C.C.: Magnifying superlens in the visible frequency range. Science 315, 1699–1701 (2007)

    Google Scholar 

  273. Solin, P., Dubcova, L., Cerveny, J., Dolezel, I.: Adaptive hp-FEM with arbitrary-level hanging nodes for Maxwell’s equations. Adv. Appl. Math. Mech. 2, 518–532 (2010)

    MathSciNet  Google Scholar 

  274. Solymar, L., Shamonina, E.: Waves in Metamaterials. Oxford University Press, Oxford (2009)

    Google Scholar 

  275. Syms, R.R.A., Shamonina, E., Kalinin, V., Solymar, L.: A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves. J. Appl. Phys. 97, 064909 (2005)

    Google Scholar 

  276. Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd edn. Artech House Publishers, Boston (2000)

    MATH  Google Scholar 

  277. Teixeira, F.L.: Time-domain finite-difference and finite-element methods for Maxwell equations in complex media. IEEE Trans. Antennas Propag. 56, 2150–2166 (2008)

    MathSciNet  Google Scholar 

  278. Teixeira, F.L., Chew, W.C.: PML-FDTD in cylindrical and spherical coordinates. IEEE Microw. Guid. Wave Lett. 7, 285–287 (1997)

    Google Scholar 

  279. Tobon, L., Chen, J., Liu, Q.H.: Spurious solutions in mixed finite element method for Maxwell’s equations: dispersion analysis and new basis functions. J. Comput. Phys. 230, 7300–7310 (2011)

    MathSciNet  MATH  Google Scholar 

  280. Toselli, A., Widlund, O.: Domain Decomposition Methods: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 34. Springer, New York (2004)

    Google Scholar 

  281. Toselli, A., Widlund, O., Wohlmuth, B.: A FETI preconditioner for two dimensional edge element approximations of Maxwell’s equations on nonmatching grids. SIAM J. Sci. Comput. 23, 92–108 (2001)

    MathSciNet  MATH  Google Scholar 

  282. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadephia (2001)

    Google Scholar 

  283. Tsuji, P., Engquist, B., Ying, L.: A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements. J. Comput. Phys. 231, 3770–3783 (2012)

    MathSciNet  MATH  Google Scholar 

  284. Turkel, E., Yefet, A.: Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27, 533–557 (1998)

    MathSciNet  MATH  Google Scholar 

  285. Valentine, J., Zhang, S., Zentgraf, Th., Ulin-Avila, E., Genov, D.A., Bartal, G., Zhang, X.: Three-dimensional optical metamaterial with a negative refractive index. Nature 455, 376–380 (2008)

    Google Scholar 

  286. Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    MathSciNet  MATH  Google Scholar 

  287. Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996)

    MATH  Google Scholar 

  288. Veselago, V.G.: Electrodynamics of substances with simultaneously negative values of sigma and mu. Sov. Phys. Usp. 10, 509–514 (1968)

    Google Scholar 

  289. Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods. Springer, Berlin (1995)

    MATH  Google Scholar 

  290. Wang, B., Xie, Z., Zhang, Z.: Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media. J. Comput. Phys. 229, 8552–8563 (2010)

    MathSciNet  MATH  Google Scholar 

  291. Wellander, N.: Homogenization of the Maxwell equations. Case I. Linear theory. Appl. Math. 46, 29–51 (2001)

    MathSciNet  MATH  Google Scholar 

  292. Wheeler, M.F., Whiteman, J.R.: Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems. Numer. Methods PDEs 10, 271–294 (1994)

    MathSciNet  MATH  Google Scholar 

  293. Weinan, E.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  294. Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  295. Wu, C., Avitzour, Y., Shvets, G.: Ultra-thin, wide-angle perfect absorber for infrared frequencies. In: Noginov, M.A., Zheludev, N.I., Boardman, A.D., Engheta, N. (eds.) Metamaterials: Fundamentals and Applications, Proceedings of SPIE, vol. 7029, 70290W (2008)

    Google Scholar 

  296. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2003)

    Google Scholar 

  297. Yan, N.: Superconvergence Analysis and A Posteriori Error Estimation in Finite Element Methods. Science Press, Beijing (2008)

    Google Scholar 

  298. Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Eng. 190, 4289–4299 (2001)

    MathSciNet  MATH  Google Scholar 

  299. Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14, 302–307 (1966)

    MATH  Google Scholar 

  300. Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer. 2, 285–326 (1993)

    MathSciNet  Google Scholar 

  301. Zhang, S., Fan, W., Panoiu, N.C., Malloy, K.J., Osgood, R.M., Brueck, S.R.: Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 95, 137404 (2005)

    Google Scholar 

  302. Zhang, Y., Cao, L.-Q., Wong, Y.-S.: Multiscale computations for 3D time-dependent Maxwell’s equations in composite materials. SIAM J. Sci. Comput. 32, 2560–2583 (2010)

    MathSciNet  MATH  Google Scholar 

  303. Zhao, Y., Hao, Y.: Full-wave parallel dispersive finite-difference time-domain modeling of three-dimensional electromagnetic cloaking structures. J. Comput. Phys. 228, 7300–7312 (2009)

    MATH  Google Scholar 

  304. Zhao, Y., Argyropoulos, C., Hao, Y.: Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures. Opt. Express 16, 6717–6730 (2008)

    Google Scholar 

  305. Zheng, W.Y., Chen, Z., Wang, L.: An adaptive finite element method for the H − ψ formulation of time-dependent eddy current problems. Numer. Math. 103, 667–689 (2006)

    MathSciNet  MATH  Google Scholar 

  306. Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comput. 81, 623–642 (2012)

    MathSciNet  MATH  Google Scholar 

  307. Zhong, L., Shu, S., Wang, J., Xu, J.: Two-grid methods for time-harmonic Maxwell equations. Linear Algebra Appl. 2012, Early View. doi:10.1002/nla.1827

    Google Scholar 

  308. Zhou, A., Li, J.: The full approximation accuracy for the stream function-vorticity-pressure method. Numer. Math. 68, 427–435 (1994)

    MathSciNet  MATH  Google Scholar 

  309. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Eng. 24, 337–357 (1987)

    MathSciNet  MATH  Google Scholar 

  310. Ziolkowski, R.W.: Maxwellian material based absorbing boundary conditions. Comput. Methods Appl. Mech. Eng. 169, 237–262 (1999)

    MATH  Google Scholar 

  311. Ziolkowski, R.W.: Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs. Opt. Express 11, 662–681 (2003)

    Google Scholar 

  312. Ziolkowski, R.W., Erentok, A.: Metamaterial-based efficient electrically small antennas. IEEE Trans. Antennas Propag. AP-54(7), 2113–2130 (2006)

    Google Scholar 

  313. Ziolkowski, R.W., Heyman, E.: Wave propagation in media having negative permittivity and permeability. Phys. Rev. E 64, 056625 (2001)

    Google Scholar 

  314. Zouhdi, S., Sihvola, A., Vinogradov, A.P. (eds.): Metamaterials and Plasmonics: Fundamentals, Modelling, Applications. Springer, Berlin (2009)

    Google Scholar 

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  1. Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, USA

    Jichun Li

  2. Xiangtan University, Xiangtan, Hunan, China, People’s Republic

    Yunqing Huang

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Li, J., Huang, Y. (2013). Time-Domain Finite Element Methods for Metamaterials. In: Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials. Springer Series in Computational Mathematics, vol 43. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33789-5_3

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