Journal of Scientific Computing

, Volume 68, Issue 2, pp 848–863 | Cite as

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

Article

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.

Keywords

Adaptivity Error estimators Finite element method Hodge decomposition Maxwell’s equations Impedance boundary condition Metamaterials Cloaking Flat lens 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 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. 6.
    Cui, J.: Multigrid methods for two-dimensional Maxwell’s equations on graded meshes. J. Comput. Appl. Math. 255, 231–247 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li, J., Huang, Y., Yang, W.: An adaptive edge finite element method for electromagnetic cloaking simulation. J. Comput. Phys. 249, 216–232 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Nader, E., Ziolkowski, R.W.: Metamaterials: Physics and Engineering Exploarations. Wiley, New York (2006)Google Scholar
  13. 13.
    Nicaise, S.: Polygonal interface problems. Verlag Peter D. Lang, Frankfurt am Main (1993)MATHGoogle Scholar
  14. 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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000)CrossRefGoogle Scholar
  16. 16.
    Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Senior, T.B.A., Volakis, J.L.: Approximate Boundary Condition in Electromagnetics. IEEE Press, New York (1995)CrossRefMATHGoogle Scholar
  19. 19.
    Solymar, L., Shamonina, E.: Waves in Metamaterials. Oxford University Press, Oxford (2009)Google Scholar
  20. 20.
    Ziolkowski, R.W.: Pulsed and cw gaussian beam interactions with double negative metamaterial slabs. Opt. Express 11, 662–681 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)Universität HeidelbergHeidelbergGermany

Personalised recommendations