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Journal of Scientific Computing

, Volume 70, Issue 2, pp 922–964 | Cite as

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

  • Zhongjie Lu
  • A. Cesmelioglu
  • J. J. W. Van der Vegt
  • Yan Xu
Article

Abstract

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.

Keywords

Discontinuous Galerkin methods Mixed finite element methods Maxwell equations Discrete compactness property Eigenvalue problems Photonic crystals Band structure 

References

  1. 1.
    Boffi, D.: Fortin operator and discrete compactness for edge elements. Numer. Math. 87(2), 229–246 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics. Springer, New York (2013)CrossRefMATHGoogle Scholar
  3. 3.
    Boffi, D., Conforti, M., Gastaldi, L.: Modified edge finite elements for photonic crystals. Numer. Math. 105(2), 249–266 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boffi, D., Fernandes, P., Gastaldi, L., Perugia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36(4), 1264–1290 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bossavit, A.: A rationale for edge-elements in 3-D fields computations. Magn. IEEE Trans. 24, 74–79 (1988)CrossRefGoogle Scholar
  6. 6.
    Bossavit, A.: Solving Maxwell equations in a closed cavity, and the question of ‘spurious modes’. Magn. IEEE Trans. 26(2), 702–705 (1990)CrossRefGoogle Scholar
  7. 7.
    Brenner, S.C., Li, F., Sung, L.Y.: A locally divergence-free interior penalty method for two-dimensional curl-curl problems. SIAM J. Numer. Anal. 46(3), 1190–1211 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buffa, A., Houston, P., Perugia, I.: Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes. J. Comput. Appl. Math. 204(2), 317–333 (2007). Special Issue: The Seventh International Conference on Mathematical and Numerical Aspects of Waves (WAVES05)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44(5), 2198–2226 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Busch, K., Knig, M., Niegemann, J.: Discontinuous Galerkin methods in nanophotonics. Laser Photon. Rev. 5(6), 773–809 (2011)CrossRefGoogle Scholar
  11. 11.
    Caorsi, S., Fernandes, P., Raffetto, M.: On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal. 38(2), 580–607 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cockburn, B., Li, F., Shu, C.W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194(2), 588–610 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Descloux, J., Nassif, N., Rappaz, J.: On spectral approximation. Part 1. The problem of convergence. RAIRO Anal. Numer. 12(2), 97–112 (1978)MathSciNetMATHGoogle Scholar
  14. 14.
    Dobson, D.C.: An efficient method for band structure calculations in 2D photonic crystals. J. Comput. Phys. 149(2), 363–376 (1999)CrossRefMATHGoogle Scholar
  15. 15.
    Dobson, D.C., Gopalakrishnan, J., Pasciak, J.E.: An efficient method for band structure calculations in 3D photonic crystals. J. Comput. Phys. 161(2), 668–679 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dobson, D.C., Pasciak, J.E.: Analysis of an algorithm for computing electromagnetic Bloch modes using Nédélec spaces. Comput. Methods Appl. Math. 1(2), 138–153 (2001)MathSciNetMATHGoogle Scholar
  17. 17.
    Dörfler, W., Lechleiter, A., Plum, M., Schneider, G., Wieners, C.: Photonic Crystals: Mathematical Analysis and Numerical Approximation. Springer, NewYork (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Fernandes, P., Gilardi, G.: Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7(07), 957–991 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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(2), 375–386 (2007). Special Issue: The Seventh International Conference on Mathematical and Numerical Aspects of Waves (WAVES05)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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(3), 440–468 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hesthaven, J.S., Warburton, T.: High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362(1816), 493–524 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100(3), 485–518 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: the indefinite case. ESAIM Math. Model. Numer. Anal. 39(4), 727–753 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Houston, P., Perugia, I., Schötzau, D.: hp-DGFEM for Maxwell’s equations. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, Aeditors (eds.) Numerical Mathematics and Advanced Applications, pp. 785–794. Springer, Milan (2003)CrossRefGoogle Scholar
  26. 26.
    Houston, P., Perugia, I., Schotzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42(1), 434–459 (2004)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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(25), 499–510 (2005). Selected papers from the 11th Conference on The Mathematics of Finite Elements and ApplicationsMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator: non-stabilized formulation. J. Sc. Comput. 22–23(1–3), 315–346 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2011)MATHGoogle Scholar
  30. 30.
    Kantorovich, L.: Quantum Theory of the Solid State: An Introduction, vol. 136. Springer, NewYork (2004)CrossRefMATHGoogle Scholar
  31. 31.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kikuchi, F.: Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput. Methods Appl. Mech. Eng. 64(1), 509–521 (1987)MathSciNetMATHGoogle Scholar
  33. 33.
    Kikuchi, F.: On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(3), 479–490 (1989)MATHGoogle Scholar
  34. 34.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRefMATHGoogle Scholar
  35. 35.
    Monk, P., Demkowicz, L.: Discrete compactness and the approximation of Maxwell’s equations in \(\mathbb{R}^3\). Math. Comput. 70(234), 507–523 (2001)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35(3), 315–341 (1980)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Nelson, D., Jacob, T.S.: Linear Operators. Part 1 General Theory. Interscience publishers, NewYork (1958)MATHGoogle Scholar
  38. 38.
    Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys. 230(19), 7151–7175 (2011)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Osborn, J.E.: Spectral approximation for compact operators. Math. Comput. 29(131), 712–725 (1975)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Perugia, I., Schötzau, D.: The \(hp\)-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72(243), 1179–1214 (2003)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Perugia, I., Schötzau, D., Monk, P.: Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng. 191(41), 4675–4697 (2002)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Sármány, D., Izsák, F., van der Vegt, J.J.W.: Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. J. Sci. Comput. 44(3), 219–254 (2010)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Soukoulis, C.M.: Photonic Band Gap Materials. Kluwer, Dordrecht (1996)CrossRefMATHGoogle Scholar
  44. 44.
    Sözüer, H.S., Haus, J.W.: Photonic bands: simple-cubic lattice. JOSA B 10(2), 296–302 (1993)CrossRefGoogle Scholar
  45. 45.
    Sun, D., Manges, J., Yuan, X., Cendes, Z.: Spurious modes in finite-element methods. Antennas Propag. Mag. IEEE 37(5), 12–24 (1995)CrossRefGoogle Scholar
  46. 46.
    Taflove, A., Hagness S.C.: Computational electrodynamics: the finite-difference time-domain method. Artech House, Inc., Boston, MA, second edition, 2000. With 1 CD-ROM (Windows)Google Scholar
  47. 47.
    Warburton, T., Embree, M.: The role of the penalty in the local discontinuous Galerkin method for Maxwell’s eigenvalue problem. Comput. Methods Appl. Mech. Eng. 195(2528), 3205–3223 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Zhongjie Lu
    • 1
  • A. Cesmelioglu
    • 2
  • J. J. W. Van der Vegt
    • 3
  • Yan Xu
    • 1
  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochester HillsUSA
  3. 3.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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