Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction to Metamaterials Read chapter Read first chapter

2013 | OriginalPaper | Chapter

3. Time-Domain Finite Element Methods for Metamaterials

Authors : Jichun Li, Yunqing Huang

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

Publisher: Springer Berlin Heidelberg

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Introduction to Finite Element Methods
next chapter Discontinuous Galerkin Methods for Metamaterials
Literature
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)MathSciNetMATH
2.
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)MathSciNetMATH
4.
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience, New York (2000)MATH
5.
Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comput. 68, 607–631 (1999)MathSciNetMATH
6.
Alu, A., Engheta, N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005)
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)
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)
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)
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
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)MathSciNetMATH
12.
Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85, 175–195 (2000)MathSciNet
13.
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetMATH
14.
Avitzour, Y., Urzhumov, Y.A., Shvets, G.: Wide-angle infrared absorber based on negative index plasmonic metamaterial. Phys. Rev. B 79, 045131 (2008)
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)
16.
Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)MathSciNetMATH
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)MathSciNetMATH
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)
19.
Banerjee, B.: An Introduction to Metamaterials and Waves in Composites. CRC, Boca Raton (2011)
20.
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Solving Differential Equations. Birkhäuser, Basel (2003)
21.
Bank, R.E.: PLTMG: A Software Package for Solving Elliptic Partial Differential Equations: Users’ Guide 8.0. SIAM, Philadelphia (1998)
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
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
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)MathSciNetMATH
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)MathSciNetMATH
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)MathSciNetMATH
27.
Barbatis, G., Stratis, I.G.: Homogenization of Maxwells equations in dissipative bianisotropic media. Math. Methods Appl. Sci. 26, 1241–1253 (2003)MathSciNetMATH
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
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)MathSciNetMATH
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
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)
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
33.
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, New York (1978)MATH
34.
Berenger, J.P.: A perfectly matched layer for the absorbing EM waves. J. Comput. Phys. 114, 185–200 (1994)MathSciNetMATH
35.
Berenger, J.P.: Three-dimensional perfectly matched layer for the absorbtion of electromagnetic waves. J. Comput. Phys. 127, 363–379 (1996)MathSciNetMATH
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)
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)
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)
39.
Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer, New York (2009)MATH
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)MathSciNetMATH
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)MathSciNetMATH
42.
Bondeson, A., Rylander, T., Ingelstrom, P.: Computational Electromagnetics. Springer, New York (2010)
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)MathSciNetMATH
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)
45.
Bossavit, A.: Computational Electromagnetism. Academic, San Diego (1998)MATH
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)MathSciNetMATH
47.
Bouchitté, G., Schweizer, B.: Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8, 717–750 (2010)MathSciNetMATH
48.
Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77, 651–672 (2008)MATH
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)MathSciNetMATH
50.
Brandts, J.: Superconvergence of mixed finite element semi-discretization of two time-dependent problems. Appl. Math. 44, 43–53 (1999)MathSciNetMATH
51.
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin/Heidelberg (1994)MATH
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)MathSciNetMATH
53.
Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)MATH
54.
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin/Heidelberg (1991)MATH
55.
Buffa, A., Costabel, M., Schwab, C.: Boundary element methods for maxwell’s equations on non-smooth domains. Numer. Math. 92, 679–710 (2002)MathSciNetMATH
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)MathSciNetMATH
57.
Cai, W., Shalaev, V.: Optical Metamaterials: Fundamentals and Applications. Springer, New York (2009)
58.
Caloz, C., Itoh, T.: Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Wiley, Hoboken (2005)
59.
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2010)
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)MathSciNetMATH
61.
Capolino, F. (ed.): Metamaterials Handbook – Two Volume Slipcase Set: Theory and Phenomena of Metamaterials. CRC, Boca Raton (2009)
62.
Carey, G.F., Oden, J.T.: Finite Elements: Computational Aspects. Prentice-Hall, Englewood Cliffs (1983)
63.
Carstensen, C., Hu, J.: A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, 473–502 (2007)MathSciNetMATH
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
65.
Chen, Z.: Finite Element Methods and Their Applications. Springer, Berlin (2005)MATH
66.
Chen, H., Chen, M.: Flipping photons backward: reversed Cherenkov radiation. Materialstoday 14, 34–41 (2011)
67.
Chen, C.M., Huang, Y.: High Accuracy Theory of Finite Element Methods (in Chinese). Hunan Science Press, China (1995)
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)
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)MathSciNetMATH
70.
Chen, M.-H., Cockburn, B., Reitich, F.: High-order RKDG methods for computational electromagnetics. J. Sci. Comput. 22, 205–226 (2005)MathSciNet
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)
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)MathSciNetMATH
73.
Chen, H., Chan, C.T., Sheng, P.: Transformation optics and metamaterials. Nat. Mater. 9, 387–396 (2010)
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).
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)
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)MathSciNetMATH
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)MathSciNetMATH
78.
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH
79.
Ciarlet, P. Jr., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999)MathSciNetMATH
80.
Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris 335, 99–104 (2002)MathSciNetMATH
81.
Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)MATH
82.
Cochez-Dhondt, S., Nicaise, S.: Robust a posteriori error estimation for the Maxwell equations. Comput. Methods Appl. Mech. Eng. 196, 2583–2595 (2007)MathSciNetMATH
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)
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)MathSciNetMATH
85.
Cohen, G.C.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2001)
86.
Cohen, G.C., Monk, P.: Gauss point mass lumping schemes for Maxwell’s equations. Numer. Methods Partial Diff. Equ. 14, 63–88 (1998)MathSciNetMATH
87.
Cohen, G.C., Monk, P.: Mur-Nédélec finite element schemes for Maxwell’s equations. Comput. Methods Appl. Mech. Eng. 169, 197–217 (1999)MathSciNetMATH
88.
Correia, D., Jin, J.-M.: 3D-FDTD-PML analysis of left-handed metamaterials. Microw. Opt. Technol. Lett. 40, 201–205 (2004)
89.
Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151(3), 221–276 (2000)MathSciNetMATH
90.
Costabel, M., Dauge, M., Nicaise, S.: Singularities of eddy current problems. M2AN Math. Model. Numer. Anal. 37, 807–831 (2003)
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)MathSciNetMATH
92.
Coutts, T.J.: A review of progress in thermophotovoltaic generation of electricity. Renew. Sustain. Energy Rev. 3, 77–184 (1999)
93.
Craster, R.V., Guenneau, S. (eds.): Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking. Springer, New York (2013)
94.
Cui, T.J., Smith, D., Liu, R. (eds.): Metamaterials: Theory, Design, and Applications. Springer, New York (2009)
95.
Cummer, S.A.: Perfectly matched layer behavior in negative refractive index materials. IEEE Antennas Wirel. Propag. Lett. 3, 172–175 (2004)
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)
97.
Demkowicz, L.: Computing with hp-Adaptive Finite Elements I. One and Two-Dimensional Elliptic and Maxwell Problems. CRC, Boca Raton (2006)
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)
99.
Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)MATH
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)MathSciNetMATH
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)
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)
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)MathSciNetMATH
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)
105.
Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications. Springer, New York (2009)MATH
106.
Eleftheriades, G.V., Balmain, K.G. (eds.): Negative Refraction Metamaterials: Fundamental Principles and Applications. Wiley, Hoboken (2005)
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
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
109.
Engheta, N., Ziolkowski, R.W. (eds.): Electromagnetic Metamaterials: Physics and Engineering Explorations. Wiley, Hoboken (2006)
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)
111.
Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 4, 105–158 (1995)MathSciNet
112.
Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)MATH
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)MathSciNetMATH
114.
Fairweather, G.: Finite Element Galerkin Methods for Differential Equations. Marcel Dekker, New York-Basel (1978)MATH
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)
116.
Fang, N., Lee, H., Sun, C., Zhang, X.: Sub-diffraction-limited optical imaging with a silver superlens. Science 308, 534–537 (2005)
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)MathSciNetMATH
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)MathSciNetMATH
119.
Fernandes, P., Raffetto, M.: Realistic and correct models of impressed sources for time-harmonic electromagnetic boundary value problems involving metamaterials. Preprint, Oct. 2011
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)MathSciNetMATH
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
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)MathSciNetMATH
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)
124.
Gay-Balmaz, P., Martin, O.J.F.: Efficient isotropic magnetic resonators. Appl. Phys. Lett. 81, 939–941 (2002)
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)
126.
Giles, M.B., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2002)MathSciNetMATH
127.
Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations – Theory and Algorithms. Springer, Berlin (1986)MATH
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)MathSciNetMATH
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)MathSciNetMATH
130.
Greenleaf, A., Lassas, M., Uhlmann, G.: On non-uniqueness for Calderón’s inverse problem. Math. Res. Lett. 10, 685–693 (2003)MathSciNetMATH
131.
Greenleaf, A., Lassas, M., Uhlmann, G.: Anisotropic conductivities that cannot be detected by EIT. Physiol. Meas. 24, 413–419 (2003)
132.
Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Cloaking devices, electromagnetics wormholes and transformation optics. SIAM Rev. 51, 3–33 (2009)MathSciNetMATH
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)MathSciNetMATH
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)MathSciNetMATH
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)
136.
Hackbusch, W.: Multi-Grid Methods and Applications. Springer, New York (1985)MATH
137.
Hao, Y., Mittra, R.: FDTD Modeling of Metamaterials: Theory and Applications. Artech House Publishers, Boston (2008)
138.
Harrington, R.F.: Field Computation by Moment Methods. Wiley-IEEE, Hoboken (1993)
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
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)MathSciNetMATH
141.
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)MATH
142.
Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)MATH
143.
Hetmaniuk, U., Liu, H.Y., Uhlmann, G.: On three dimensional active acoustic cloaking devices and their simulation. Preprint, University of Washington (2009)
144.
145.
146.
Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)MathSciNetMATH
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
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)MathSciNetMATH
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)MathSciNetMATH
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)MathSciNetMATH
151.
Huang, Y., Li, J.: Interior penalty discontinuous Galerkin method for Maxwell’s equation in cold plasma. J. Sci. Comput. 41, 321–340 (2009)MathSciNetMATH
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)MathSciNetMATH
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
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
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)MathSciNetMATH
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)MathSciNetMATH
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)
158.
Hughes, T.J.R.: Finite Element Method – Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)MATH
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
160.
Jiao, D., Jin, J.-M.: Time-domain finite-element modeling of dispersive media. IEEE Microw. Wirel. Compon. Lett. 11, 220–222 (2001)
161.
Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994)
162.
Jin, J.: The Finite Element Method in Electromagnetics, 2nd edn. Wiley-IEEE, Hoboken (2002)MATH
163.
Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, New York (1988)
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)
165.
Kohn, R.V., Shipman, S.P.: Magnetism and homogenization of microresonators. Multiscale Model. Simul. 7, 62–92 (2008)MathSciNetMATH
166.
Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: Cloaking via change of variables in electrical impendance tomography. Inverse Probl. 24, 015016 (2008)MathSciNet
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)MathSciNetMATH
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
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)
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)
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)
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)MathSciNetMATH
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)
174.
Langtangen, H.P.: Computational Partial Differential Equations: Numerical Methods and Diffpack Programming, 2nd edn. Springer, Berlin (2003)
175.
Laroche, M., Carminati, R., Greffet, J.-J.: Near-field thermophotovoltaic energy conversion. J. Appl. Phys. 100, 063704 (2006)
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)MathSciNetMATH
177.
Lee, H.-J., Yook, J.-G.: Biosensing using split-ring resonators at microwave regime. Appl. Phys. Lett. 92, 254103 (2008)
178.
Lee, J.-F., Lee, R., Cangellaris, A.C.: Time domain finite element methods. IEEE Trans. Antennas Propag. 45, 430–442 (1997)MathSciNetMATH
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)
180.
181.
Leonhardt, U., Philbin, T.: Geometry and Light: The Science of Invisibility. Dover, New York (2010)MATH
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
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
184.
Li, J.: Finite element study of the Lorentz model in metamaterials. Comput. Methods Appl. Mech. Eng. 200, 626–637 (2011)MATH
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)MathSciNetMATH
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
187.
Li, J., Chen, Y.: Computational Partial Differential Equations Using MATLAB. CRC, Boca Raton (2008)
188.
Li, J., Huang, Y.: Mathematical simulation of cloaking metamaterial structures. Adv. Appl. Math. Mech. 4, 93–101 (2012)MathSciNet
189.
Li, J., Wood, A.: Finite element analysis for wave propagation in double negative metamaterials. J. Sci. Comput. 32, 263–286 (2007)MathSciNetMATH
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)MathSciNetMATH
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)MathSciNetMATH
192.
Li, J., Chen, Y., Liu, Y.: Mathematical simulation of metamaterial solar cells. Adv. Appl. Math. Mech. 3, 702–715 (2011)MathSciNet
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)MathSciNetMATH
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)MathSciNetMATH
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
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)
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)
198.
Lin, Q., Li, J.: Superconvergence analysis for Maxwell’s equations in dispersive media. Math. Comput. 77, 757–771 (2008)MathSciNetMATH
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)MathSciNetMATH
200.
Lin, Q., Yan, N.: Superconvergence of mixed element methods for Maxwells equations (in Chinese). Gongcheng Shuxue Xuebao 13, 1–10 (1996)MathSciNetMATH
201.
Lin, Q., Yan, N.: The Construction and Analysis of High Accurate Finite Element Methods (in Chinese). Hebei University Press, Hebei (1996)
202.
Lin, Q., Yan, N.: Global superconvergence for Maxwells equations. Math. Comput. 69, 159–176 (1999)MathSciNet
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)
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)
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)
206.
Liu, R., Ji, C., Mock, J.J., Chin, J.Y., Cui, T.J., Smith, D.R.: Science 323, 366–369 (2009)
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)MathSciNetMATH
208.
Maradudin, A.A. (eds.): Stuctured Surfaces as Optical Metamaterials. Cambridge University Press, Cambridge (2011)
209.
Markos, P., Soukoulis, C.M.: Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials. Princeton University Press, Princeton (2008)MATH
210.
Marques, R., Martin, F., Sorolla, M.: Metamaterials with Negative Parameters: Theory, Design and Microwave Applications. Wiley-IEEE, New York (2008)
211.
Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)MATH
212.
Milton, G.W., Nicorovici, N.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059 (2006)MathSciNetMATH
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)
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)
215.
Monk, P.: Superconvergence of finite element approximations to Maxwells equations. Numer. Methods Partial Differ. Equ. 10, 793–812 (1994)MathSciNetMATH
216.
Monk, P.: A posteriori error indicators for Maxwell’s equations. J. Comput. Appl. Math. 100, 173–190 (1998)MathSciNetMATH
217.
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford Science Publications, New York (2003)MATH
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)MathSciNetMATH
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)MathSciNetMATH
220.
Munk, B.A.: Metamaterials: Critique and Alternatives. Wiley-Interscience, Hoboken (2009)
221.
Narimanov, E.E., Shalaev, V.M.: Beyond diffraction. Nature 447, 266–267 (2007)
222.
223.
224.
Nicaise, S.: On Zienkiewicz-Zhu error estimators for Maxwell’s equations. C. R. Math. Acad. Sci. Paris 340, 697–702 (2005)MathSciNetMATH
225.
Nicaise, S., Creusé, E.: A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes. Calcolo 40, 249–271 (2003)MathSciNetMATH
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)MathSciNetMATH
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)
228.
Noginov, M.A., Podolskiy, V. (eds.): Tutorials in Metamaterials. Series in Nano-Optics and Nanophotonics. CRC, Boca Raton (2011)
229.
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)
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)
232.
Padilla, W.J.: Group theoretical description of artificial electromagnetic metamaterials. Opt. Express 15, 1639–1646 (2007)MathSciNet
233.
Parnell, W.J.: Nonlinear pre-stress for cloaking from antiplane elastic waves. Proc. R. Soc. A 468, 563–580 (2012)MathSciNet
234.
Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000)
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)
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)
237.
Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)MathSciNetMATH
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)MathSciNetMATH
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)MathSciNetMATH
240.
Pozrikidis, C.: Introduction to Finite and Spectral Element Methods Using MATLAB. Chapman & Hall/CRC, Boca Raton (2005)MATH
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
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)MathSciNetMATH
243.
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)MATH
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)
245.
Ramakrishna, S.A., Grzegorczyk, T.M.: Physics and Applications of Negative Refractive Index Materials. CRC, Boca Raton (2008)
246.
Rappaport, C.M.: Perfectly matched absorbing conditions based on anisotropic lossy mapping of space. IEEE Microw. Guid. Wave Lett. 53, 90–92 (1995)
247.
Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)MATH
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)
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)
250.
Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer, Berlin (1980)MATH
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
252.
Schmidt, A., Siebert. K.G.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Springer, Berlin (2005)
253.
Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comput. 77, 633–649 (2008)MATH
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)
255.
Schwab, C.: p- and hp- Finite Element Methods, Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, New York (1998)MATH
256.
Shalaev, V.M., Sarychev, A.K.: Electrodynamics of Metamaterials. World Scientific, Hackensack (2007)MATH
257.
Shamonina, E., Solymar, L.: Properties of magnetically coupled metamaterial elements. J. Magn. Magn. Mater. 300, 38–43 (2006)
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
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)
260.
Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292, 489–491 (2001)
261.
Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, New York (2011)MATH
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)
263.
Shvets, G., Tsukerman, I. (eds.): Plasmonics and Plasmonic Metamaterials: Analysis and Applications. World Scientific, Hackensack (2011)
264.
Sihvola, A.H.: Electromagnetic Mixing Formulas and Applications. The Institute of Electrical Engineer, London (1999)
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)
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)
267.
Silvester, P.P., Ferrari, R.L.: Finite Elements for Electrical Engineers, 3rd edn. Cambridge University Press, London (1996)
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)MathSciNetMATH
269.
Smith, D.R., Kroll, N.: Negative refractive index in left-handed materials. Phys. Rev. Lett. 85, 2933–2936 (2000)
270.
Smith, D., Pendry, J.: Homogenization of metamaterials by field averaging. J. Opt. Soc. Am. B 23, 391–403 (2006)
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)
272.
Smolyaninov, I.I., Hung, Y.-J., Davis, C.C.: Magnifying superlens in the visible frequency range. Science 315, 1699–1701 (2007)
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
274.
Solymar, L., Shamonina, E.: Waves in Metamaterials. Oxford University Press, Oxford (2009)
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)
276.
Taflove, A., Hagness, S.C.: Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd edn. Artech House Publishers, Boston (2000)MATH
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
278.
Teixeira, F.L., Chew, W.C.: PML-FDTD in cylindrical and spherical coordinates. IEEE Microw. Guid. Wave Lett. 7, 285–287 (1997)
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)MathSciNetMATH
280.
Toselli, A., Widlund, O.: Domain Decomposition Methods: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 34. Springer, New York (2004)
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)MathSciNetMATH
282.
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadephia (2001)
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)MathSciNetMATH
284.
Turkel, E., Yefet, A.: Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27, 533–557 (1998)MathSciNetMATH
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)
286.
Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)MathSciNetMATH
287.
Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996)MATH
288.
Veselago, V.G.: Electrodynamics of substances with simultaneously negative values of sigma and mu. Sov. Phys. Usp. 10, 509–514 (1968)
289.
Wahlbin, L.B.: Superconvergence in Galerkin Finite Element Methods. Springer, Berlin (1995)MATH
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)MathSciNetMATH
291.
Wellander, N.: Homogenization of the Maxwell equations. Case I. Linear theory. Appl. Math. 46, 29–51 (2001)MathSciNetMATH
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)MathSciNetMATH
293.
Weinan, E.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011)MATH
294.
Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)MATH
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)
296.
Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2003)
297.
Yan, N.: Superconvergence Analysis and A Posteriori Error Estimation in Finite Element Methods. Science Press, Beijing (2008)
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)MathSciNetMATH
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
300.
Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer. 2, 285–326 (1993)MathSciNet
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)
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)MathSciNetMATH
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
304.
Zhao, Y., Argyropoulos, C., Hao, Y.: Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures. Opt. Express 16, 6717–6730 (2008)
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)MathSciNetMATH
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)MathSciNetMATH
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
308.
Zhou, A., Li, J.: The full approximation accuracy for the stream function-vorticity-pressure method. Numer. Math. 68, 427–435 (1994)MathSciNetMATH
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)MathSciNetMATH
310.
Ziolkowski, R.W.: Maxwellian material based absorbing boundary conditions. Comput. Methods Appl. Mech. Eng. 169, 237–262 (1999)MATH
311.
Ziolkowski, R.W.: Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs. Opt. Express 11, 662–681 (2003)
312.
Ziolkowski, R.W., Erentok, A.: Metamaterial-based efficient electrically small antennas. IEEE Trans. Antennas Propag. AP-54(7), 2113–2130 (2006)
313.
Ziolkowski, R.W., Heyman, E.: Wave propagation in media having negative permittivity and permeability. Phys. Rev. E 64, 056625 (2001)
314.
Zouhdi, S., Sihvola, A., Vinogradov, A.P. (eds.): Metamaterials and Plasmonics: Fundamentals, Modelling, Applications. Springer, Berlin (2009)
Metadata
Title
Time-Domain Finite Element Methods for Metamaterials
Authors
Jichun Li
Yunqing Huang
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials

Print ISBN: 978-3-642-33788-8

Electronic ISBN: 978-3-642-33789-5

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-33789-5_3

Premium Partner

    Image Credits