We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Finite Element Methods | SpringerLink

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Skip to main content
SpringerLink
Log in

Finite Element Methods

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

  • First Online:
Maxwell’s Equations in Periodic Structures

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

This chapter is concerned with numerical solutions of the grating problems which are discussed in Chaps. 2 and 3. There are two challenges for the grating problems: the solutions may have singularity due to possible nonsmooth surfaces and discontinuous media; the problems are formulated in unbounded domains. There are already many numerical methods for modeling the light diffraction by surface relief gratings, such as the S-matrix algorithm [1, 2], the rigorous coupled-wave approach [3, 4], the Fourier modal method [5,6,7,8], the method of coordinate transformation [9,10,11], the boundary perturbation method [12,13,14,15,16], the method of transformed field expansion [17,18,19,20], the boundary integral equation method [21,22,23,24,25,26,27], and the finite element method [28,29,30,31,32,33,34]. We present the adaptive finite element methods to overcome the difficulties when solving the diffraction grating problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N.P.K. Cotter, T.W. Preist, J.R. Sambles, Scattering-matrix approach to multilayer diffraction. J. Opt. Soc. Am. A 12, 1097–1103 (1995)

    Article  Google Scholar 

  2. L. Li, Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction grating. J. Opt. Soc. Am. A 13, 1024–1035 (1996)

    Article  Google Scholar 

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

    Article  Google Scholar 

  4. M.G. Moharam, D.A. Pommet, E.B. Grann, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach. J. Opt. Soc. Am. A 12, 1077–1086 (1995)

    Article  Google Scholar 

  5. L. Li, Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A 13, 1870–1876 (1996)

    Google Scholar 

  6. L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings. J. Opt. Soc. Am. 14, 2758–2767 (1997)

    Article  Google Scholar 

  7. L. Li, Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials. J. Modern Opt. 45, 1313–1334 (1998)

    Article  Google Scholar 

  8. L. Li, Justification of matrix truncation in the modal methods of diffraction gratings. J. Optics A: Pure Appl. Opt. 1, 531–536 (1999)

    Article  Google Scholar 

  9. J. Chandezon, M.T. Dupuis, G. Cornet, D. Maystre, Multicoated gratings: a differential formalism applicable in the entire optical region. J. Opt. Soc. Am. 72, 839–846 (1982)

    Article  Google Scholar 

  10. L. Li, 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 

  11. L. Li, J. Chandezon, Improvement of the coordinate transformation method for surface-relief gratings with sharp edges. J. Opt. Soc. Am. A 13, 2247–2255 (1996)

    Article  Google Scholar 

  12. O. Bruno, F. Reitich, Numerical solution of diffraction problems: a method of variation of boundaries. J. Opt. Soc. Am. A. 10, 1168–1175 (1993)

    Article  Google Scholar 

  13. O. Bruno, F. Reitich, Numerical solution of diffractive problems: a method of variation of boundaries II. Dielectric gratings, Padé approximants and singularities. J. Opt. Soc. Am. A 10, 2307–2316 (1993)

    Google Scholar 

  14. O. Bruno, F. Reitich, Numerical solution of diffraction problems: a method of variation of boundaries III. Doubly-periodic gratings. J. Opt. Soc. Am. 10, 2551–2562 (1993)

    Google Scholar 

  15. O. Bruno, F. Reitich, Accurate calculation of diffraction-grating efficiencies, in Proceedings of SPIE, vol. 1919. Mathematics in Smart Structures (1993), pp. 236–247

    Google Scholar 

  16. O. Bruno, F. Reitich, Approximation of analytic functions: a method of enhanced convergence. Math. Comput. 63, 195–213 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Malcolm, D.P. Nicholls, A field expansions method for scattering by periodic multilayered media. J. Acoust. Soc. Am. 129, 1783–1793 (2011)

    Article  Google Scholar 

  18. D.P. Nicholls, F. Reitich, Shape deformations in rough surface scattering: cancellations, conditioning, and convergence. J. Opt. Soc. Am. 21, 590–605 (2004)

    Article  Google Scholar 

  19. D.P. Nicholls, F. Reitich, Shape deformations in rough surface scattering: improved algorithms. J. Opt. Soc. Am. 21, 606–621 (2004)

    Article  Google Scholar 

  20. Y. He, D.P. Nicholls, J. Shen, An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure. J. Comput. Phys. 231, 3007–3022 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. J.A. Cox, D. Dobson, An integral equation method for biperiodic diffraction structures, in International Conference on the Applications and Theory of Periodic Structures (Proc. SPIE), vol. 1545, ed. by J. Lerner, W. McKinney (1991), pp. 106–113

    Google Scholar 

  22. W. Lu, Y.Y. Lu, High order integral equation method for diffraction gratings. J. Opt. Soc. Am. A 29, 734–740 (2012)

    Article  Google Scholar 

  23. E. Popov, B. Bozhkov, D. Maystre, J. Hoose, Integral method for echelles covered with lossless or absorbing thin dielectric layers. Appl. Opt. 38, 47–55 (1999)

    Article  Google Scholar 

  24. D.W. Prather, M.S. Mirotznik, J.N. Mait, Boundary integral methods applied to the analysis of diffractive optical elements. J. Opt. Soc. Am. A 14, 34–43 (1997)

    Article  Google Scholar 

  25. Y. Wu, Y.Y. Lu, Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction. J. Opt. Soc. Am. A 28, 1191–1196 (2011)

    Article  Google Scholar 

  26. V. Yachin, K. Yasumoto, Method of integral functionals for electromagnetic wave scattering from a double-periodic magnetodielectric layer. J. Opt. Soc. Am. A 24, 3606–3618 (2007)

    Article  Google Scholar 

  27. A. Rathsfeld, G. Schmidt, B.H. Kleemann, On a fast integral equation method for diffraction gratings. Commun. Comput. Phys. 1, 984–1009 (2006)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Bao, Numerical analysis of diffraction by periodic structures: TM polarization. Numer. Math. 75, 1–16 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. G. Bao, Variational approximation of Maxwell’s equations in biperiodic structures. SIAM J. Appl. Math. 57, 364–381 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Bao, Y. Cao, H. Yang, Numerical solution of diffraction problems by a least-square finite element method. Math. Method Appl. Sci. 23, 1073–1092 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Bao, H. Yang, A least-squares finite element analysis of diffraction problems. SIAM J. Numer. Anal. 37, 665–682 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. N.H. Lord, A.J. Mulholland, A dual weighted residual method applied to complex periodic gratings. Proc. Roy. Soc. Edinburgh Sect. A 469, 20130176 (2013)

    Google Scholar 

  34. D.W. Prather, Analysis and synthesis of finite aperiodic diffractive optical elements using rigorous electromagnetic models. Ph.D. Thesis, Department of Electrical Engineering, University of Maryland, 1997

    Google Scholar 

  35. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994)

    Book  MATH  Google Scholar 

  36. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978)

    MATH  Google Scholar 

  37. J.-M. Jin, The Finite Element Methods in Electromagnetics (Wiley, New York, 2002)

    MATH  Google Scholar 

  38. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, Oxford, 2003)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (Teubner, Stuttgart, 1996)

    MATH  Google Scholar 

  42. Z. Chen, S. Dai, Adaptive Galerkin methods with error control for a dynamic Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38, 1961–1985 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. J.M. Cascon, C. Kreuzer, R.H. Nochetto, K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Morin, R.H. Nochetto, K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Z. Chen, S. Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. K. Mekchay, R.H. Nochetto, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. P. Monk, A posteriori error indicators for Maxwell’s equations. J. Comput. Appl. Math. 100, 173–190 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Monk, E. Süli, The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal. 36, 251–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. B. Alpert, L. Greengard, T. Hagstrom, Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 180, 270–296 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Bayliss, E. Turkel, Radiation boundary conditions for numerical simulation of waves. Commun. Pure Appl. Math. 33, 707–725 (1980)

    Article  MATH  Google Scholar 

  53. B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31, 629–651 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  54. D. Givoli, Non-reflecting boundary conditions: a review. J. Comput. Phys. 94, 1–29 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  55. M.J. Grote, J.B. Keller, Exact nonreflecting boundary conditions for the time dependent wave equation. SIAM J. Appl. Math. 55, 280–297 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  56. M.J. Grote, J.B. Keller, On nonreflecting boundary conditions. J. Comput. Phys. 122, 231–243 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8, 47–106 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. G. Bao, H. Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal. 43, 2121–2143 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. J.H. Bramble, J.E. Pasciak, 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 

  61. W. Chew, W. Weedon, A 3D perfectly matched medium for modified Maxwell’s equations with stretched coordinates. Microwave Opt. Techno. Lett. 7, 599–604 (1994)

    Article  Google Scholar 

  62. F. Collino, P. Monk, The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19, 2061–2090 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Hohage, F. Schmidt, L. Zschiedrich, Solving time-harmonic scattering problems based on the pole condition II: convergence of the PML method. SIAM J. Math. Anal. 35, 547–560 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. E. Turkel, A. Yefet, Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27, 533–557 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  66. Z. Chen, H. Wu, 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 

  67. J. Chen, Z. Chen, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  69. G. Bao, P. Li, H. Wu, An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures. Math. Comput. 79, 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  70. X. Jiang, P. Li, J. Lv, W. Zheng, An adaptive finite element method for the wave scattering with transparent boundary condition. J. Sci. Comput. 72, 936–956 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  71. X. Jiang, P. Li, W. Zheng, Numerical solution of acoustic scattering by an adaptive DtN finite element method. Commun. Comput. Phys. 13, 1227–1244 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  72. Z. Wang, G. Bao, J. Li, P. Li, H. Wu, 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 

  73. X. Jiang, P. Li, J. Lv, Z. Wang, H. Wu, W. Zheng, An adaptive finite element DtN method for Maxwell’s equations in biperiodic structures, arXiv:1811.12449

  74. G.C. Hsiao, N. Nigam, J.E. Pasiak, L. Xu, Error analysis of the DtN-FEM for the scattering problem in acoustic via Fourier analysis. J. Comput. Appl. Math. 235, 4949–4965 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  75. I. Babuška, A. Aziz, Survey Lectures on Mathematical Foundations of the Finite Element Method, in the Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations. ed. by A. Aziz (Academic Press, New York, 1973)

    Google Scholar 

  76. F. Teixeira, W. Chew, Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microwave Guided Wave Lett. 7, 371–373 (1997)

    Article  Google Scholar 

  77. A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959–962 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  79. F. Hecht, New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  81. J.-C. Nédélec, Mixed finite elements in \(\mathbb{R}^3\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  82. M. Costabel, M. Dauge, Maxwell and Lamé eigenvalues on polyhedra. Math. Meth. Appl. Sci. 22, 243–258 (1999)

    Article  MATH  Google Scholar 

  83. PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/

    Google Scholar 

  84. P.R. Amestoy, I.S. Duff, J.-Y. L’Excellent, J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23, 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  85. P.R. Amestoy, A. Guermouche, J.-Y. L’Excellent, S. Pralet, Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32, 136–156 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China

    Gang Bao

  2. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Peijun Li

Authors
  1. Gang Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peijun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Bao .

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Science Press

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bao, G., Li, P. (2022). Finite Element Methods. In: Maxwell’s Equations in Periodic Structures. Applied Mathematical Sciences, vol 208. Springer, Singapore. https://doi.org/10.1007/978-981-16-0061-6_4

Download citation

Publish with us

Policies and ethics

Search

Navigation