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

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Abstract

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.

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References

  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)

  2. Ammari, H., Bao, G.: Maxwell’s equations in periodic chiral structures. Mathematische Nachrichten 251, 3–18 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ammari, H., Nédélec, J.: Low-frequency electromagnetic scattering. SIAM J. Math. Anal. 31, 836–861 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, G.: Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32, 1155–1169 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bao, G.: Numerical analysis of diffraction by periodic structures: TM polarization. Numerische Mathematik 75, 1–16 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bao, G., Chen, Z., Wu, H.: Adaptive finite-element method for diffraction gratings. J. Opt. Soc. Am. A 22, 1106–1114 (2005)

    Article  MathSciNet  Google Scholar 

  10. Bao, G., Cowsar, L., Masters, W.: Mathematical Modeling in Optical Science. Frontiers Appl. Math. 22. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  11. Bao, G., Dobson, D.C., Cox, J.A.: Mathematical studies in rigorous grating theory. J. Opt. Soc. Am. A 12, 1029–1042 (1995)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numerische Mathematik 97, 219–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, Z., Liu, X.: An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43, 645–671 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chew, W., Jin, J., Michielssen, E.: Complex coordinate stretching as a generalized absorbing boundary condition. Microw. Opt. Technol. Lett. 15, 363–369 (1997)

    Article  Google Scholar 

  25. Dobson, D.C.: Optimal design of periodic antireflective structures for the Helmholtz equation. Eur. J. Appl. Math. 4, 321–340 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dobson, D., Friedman, A.: The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166, 507–528 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dörfler, W.: A convergent adaptive algorithm for Possion’s equations. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  29. Gaylord, T.K., Moharam, M.G.: Analysis and applications of optical diffraction by gratings. Proc. IEEE 73, 894–937 (1985)

    Article  Google Scholar 

  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)

  31. Lassas, M., Somersalo, E.: On the existence and convergence of the solution of PML equations. Computing 60, 229–241 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  34. Lord, N.H., Mulholland, A.J.: A dual weighted residual method applied to complex periodic gratings. Proc. R. Soc. A 469, 20130176 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Moharam, M.G., Gaylord, T.K.: Diffraction analysis of dielectric surface-relief gratings. J. Opt. Soc. Am. 72, 1385–1392 (1982)

    Article  Google Scholar 

  37. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  39. Petit, R. (ed.): Electromagnetic Theory of Gratings. Topics in Current Physics 22. Springer, Heidelberg (1980)

    Google Scholar 

  40. 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 

  41. Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, S.S., Magnusson, R.: Multilayer waveguidegrating filters. Appl. Opt. 34, 2414–2420 (1995)

    Article  Google Scholar 

  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)

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Acknowledgements

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.

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Correspondence to Haijun Wu.

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This work is supported by the National Natural Science Foundation of China Grants 11525103, 91630309, and 11621101.

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Zhou, W., Wu, H. An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations. J Sci Comput 76, 1813–1838 (2018). https://doi.org/10.1007/s10915-018-0683-0

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  • DOI: https://doi.org/10.1007/s10915-018-0683-0

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