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An Adaptive \(\varvec{P_1}\) Finite Element Method for Two-Dimensional Transverse Magnetic Time Harmonic Maxwell’s Equations with General Material Properties and General Boundary Conditions

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Abstract

We present an adaptive \(P_1\) finite element method for two-dimensional transverse magnetic time harmonic Maxwell’s equations with general material properties and general boundary conditions. It is based on reducing the boundary value problems for Maxwell’s equations to standard second order scalar elliptic problems through the Hodge decomposition. We allow inhomogeneous and anisotropic electric permittivity, sign changing magnetic permeability, and both the perfectly conducting boundary condition and the impedance boundary condition. The optimal convergence of the adaptive finite element method is demonstrated by numerical experiments. We also present results for a semiconductor simulation, a cloaking simulation and a flat lens simulation that illustrate the robustness of the method.

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References

  1. Bonnet-Ben Dhia, A.-S., Chesnel, L., Ciarlet Jr, P.: \(T\)-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM Math. Model. Numer. Anal 46, 1363–1387 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenner, S.C., Cui, J., Nan, Z., Sung, L.-Y.: Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations. Math. Comput. 81, 643–659 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S.C., Gedicke, J., Sung, L.-Y.: An adaptive \(P_1\) finite element method for two-dimensional Maxwell’s equations. J. Sci. Comput. 55, 738–754 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, S.C., Gedicke, J., Sung, L.-Y.: Hodge decomposition for two-dimensional time harmonic Maxwell’s equations: impedance boundary condition. Math. Methods Appl. Sci. (2015). doi:10.1002/mma.3398

  6. Cui, J.: Multigrid methods for two-dimensional Maxwell’s equations on graded meshes. J. Comput. Appl. Math. 255, 231–247 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J., Huang, Y., Yang, W.: An adaptive edge finite element method for electromagnetic cloaking simulation. J. Comput. Phys. 249, 216–232 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 1803–1827 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)

    Book  MATH  Google Scholar 

  12. Nader, E., Ziolkowski, R.W.: Metamaterials: Physics and Engineering Exploarations. Wiley, New York (2006)

    Google Scholar 

  13. Nicaise, S.: Polygonal interface problems. Verlag Peter D. Lang, Frankfurt am Main (1993)

    MATH  Google Scholar 

  14. Nicaise, S., Venel, J.: A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients. J. Comput. Appl. Math. 235, 4272–4282 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Senior, T.B.A., Volakis, J.L.: Approximate Boundary Condition in Electromagnetics. IEEE Press, New York (1995)

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    S. C. Brenner & L.-Y. Sung

  2. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    J. Gedicke

Authors
  1. S. C. Brenner
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  2. J. Gedicke
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  3. L.-Y. Sung
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Corresponding author

Correspondence to S. C. Brenner.

Additional information

The work of the first and third authors was supported in part by the National Science Foundation under Grant No. DMS-13-19172. The work of the second author was supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD).

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Brenner, S.C., Gedicke, J. & Sung, LY. An Adaptive \(\varvec{P_1}\) Finite Element Method for Two-Dimensional Transverse Magnetic Time Harmonic Maxwell’s Equations with General Material Properties and General Boundary Conditions. J Sci Comput 68, 848–863 (2016). https://doi.org/10.1007/s10915-015-0161-x

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  • DOI: https://doi.org/10.1007/s10915-015-0161-x

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