Journal of Scientific Computing

, Volume 76, Issue 3, pp 1813–1838 | Cite as

An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations

  • Weiqi Zhou
  • Haijun WuEmail author


The diffraction grating problem is modeled by a Helmholtz equation with PML boundary conditions. The PML is truncated by some few-mode Dirichlet to Neumann boundary conditions so that those Fourier modes that cannot be well absorbed by the PML pass through without reflections. Convergence of the truncated PML solution is proved, whose rate is exponential with respect to the PML parameters and uniform with respect to all modes. An a posteriori error estimate is derived for the finite element discretization. The a posteriori error estimate consists of two parts, the finite element discretization error and the PML truncation error which decays exponentially with respect to the PML parameters and uniformly with respect to all modes. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the PML parameters are determined through the PML truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm.


Adaptivity Perfectly matched layer Few-mode DtN operator A posteriori error estimates Diffraction gratings 

Mathematics Subject Classification

65N12 65N15 65N30 78A40 



The authors would like to thank Professor Zhiming Chen for suggesting this topic of research and thank the anonymous referees for their detailed comments and suggestions that improved the paper.


  1. 1.
    Abboud, T.: Electromagnetic waves in periodic media. In: Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, pp. 1–9. Newark, DE (1993)Google Scholar
  2. 2.
    Ammari, H., Bao, G.: Maxwell’s equations in periodic chiral structures. Mathematische Nachrichten 251, 3–18 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ammari, H., Nédélec, J.: Low-frequency electromagnetic scattering. SIAM J. Math. Anal. 31, 836–861 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babuška, I., Aziz, A.: Survey Lectures on Mathematical Foundations of the Finite Element Method. In: Aziz, A. (ed.) The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, pp. 5–359. Academic Press, New York (1973)Google Scholar
  5. 5.
    Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bao, G.: Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32, 1155–1169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bao, G.: Numerical analysis of diffraction by periodic structures: TM polarization. Numerische Mathematik 75, 1–16 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bao, G., Cao, Y., Yang, H.: Numerical solution of diffraction problems by a least-square finite element method. Math. Methods Appl. Sci. 23, 1073–1092 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bao, G., Chen, Z., Wu, H.: Adaptive finite-element method for diffraction gratings. J. Opt. Soc. Am. A 22, 1106–1114 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bao, G., Cowsar, L., Masters, W.: Mathematical Modeling in Optical Science. Frontiers Appl. Math. 22. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bao, G., Dobson, D.C., Cox, J.A.: Mathematical studies in rigorous grating theory. J. Opt. Soc. Am. A 12, 1029–1042 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bao, G., Li, P., Wu, H.: An adaptive edge element method with perfectly matched absorbing layers for the wave scattering by periodic structures. Math. Comp. 79, 1–34 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bao, G., Wu, H.: Convergence analysis of the PML problems for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal. 43, 2121–2143 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numerische Mathematik 97, 219–268 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bramble, J.H., Pasciak, J.E.: Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comput. 76, 597–614 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chandezon, J., Dupuis, M.T., Cornet, G., Maystre, D.: Multicoated gratings: a differential formalism applicable in the entire optical region. J. Opt. Soc. Am. 72, 839–846 (1982)CrossRefGoogle Scholar
  19. 19.
    Chen, Z., Chen, J.: An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems. Math. Comput. 77, 673–698 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. (USA) 24, 443–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, Z., Liu, X.: An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43, 645–671 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41, 799–826 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chen, Z., Zheng, Z.: Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in layered media. SIAM J. Numer. Anal. 48, 2158–2185 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chew, W., Jin, J., Michielssen, E.: Complex coordinate stretching as a generalized absorbing boundary condition. Microw. Opt. Technol. Lett. 15, 363–369 (1997)CrossRefGoogle Scholar
  25. 25.
    Dobson, D.C.: Optimal design of periodic antireflective structures for the Helmholtz equation. Eur. J. Appl. Math. 4, 321–340 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Dobson, D., Friedman, A.: The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166, 507–528 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Dörfler, W.: A convergent adaptive algorithm for Possion’s equations. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ebbesen, T.W., Lezec, H.J., Ghaemi, H.F., Thio, T., Wolff, P.A.: Extraordinary optical transmission through subwavelength hole arrays. Nature (London) 391, 667–669 (1998)CrossRefGoogle Scholar
  29. 29.
    Gaylord, T.K., Moharam, M.G.: Analysis and applications of optical diffraction by gratings. Proc. IEEE 73, 894–937 (1985)CrossRefGoogle Scholar
  30. 30.
    Ji, R.: A posteriori analysis for the finite element method with PML truncated by Neumann boundary condition for diffraction gratings. MA.Sc Thesis, Nanjing University, Nanjing, China (2011)Google Scholar
  31. 31.
    Lassas, M., Somersalo, E.: On the existence and convergence of the solution of PML equations. Computing 60, 229–241 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, L.: Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings. J. Opt. Soc. Am. A 13, 1024–1035 (1996)CrossRefGoogle Scholar
  33. 33.
    Li, L.: Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings. J. Opt. Soc. Am. A 16, 2521–2531 (1999)CrossRefGoogle Scholar
  34. 34.
    Lord, N.H., Mulholland, A.J.: A dual weighted residual method applied to complex periodic gratings. Proc. R. Soc. A 469, 20130176 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Moharam, M.G., Gaylord, T.K.: Diffraction analysis of dielectric surface-relief gratings. J. Opt. Soc. Am. 72, 1385–1392 (1982)CrossRefGoogle Scholar
  37. 37.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nevire, M., Cerutti-Maori, G., Cadilhac, M.: Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur. Opt. Commun. 3, 48–52 (1971)CrossRefGoogle Scholar
  39. 39.
    Petit, R. (ed.): Electromagnetic Theory of Gratings. Topics in Current Physics 22. Springer, Heidelberg (1980)Google Scholar
  40. 40.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wang, Z., Bao, G., Li, J., Li, P., Wu, H.: An adaptive finite element method for the diffraction grating problem with transparent boundary condition. SIAM J. Numer. Anal. 53, 1585–1607 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang, S.S., Magnusson, R.: Multilayer waveguidegrating filters. Appl. Opt. 34, 2414–2420 (1995)CrossRefGoogle Scholar
  44. 44.
    Zschiedrich, L.: Transparent boundary conditions for Maxwell’s equations: numerical concepts beyond the PML method. Dissertion thesis, vorgelegt am Fachbereich Mathmatik und Informatik der Freien Universitat Berlin, Februar (2009)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina

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