An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures

Bao, Gang; Li, Peijun; Wu, Haijun

doi:10.1090/S0025-5718-09-02257-1

An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures
HTML articles powered by AMS MathViewer

by Gang Bao, Peijun Li and Haijun Wu PDF

Math. Comp. 79 (2010), 1-34 Request permission

Abstract:

An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.

References

Toufic Abboud, Formulation variationnelle des équations de Maxwell dans un réseau bipériodique de $\textbf {R}^3$, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 3, 245–248 (French, with English and French summaries). MR 1233420
Habib Ammari and Gang Bao, Maxwell’s equations in periodic chiral structures, Math. Nachr. 251 (2003), 3–18. MR 1960800, DOI 10.1002/mana.200310026
H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal. 31 (2000), no. 4, 836–861. MR 1752419, DOI 10.1137/S0036141098343604
Douglas N. Arnold, Arup Mukherjee, and Luc Pouly, Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput. 22 (2000), no. 2, 431–448. MR 1780608, DOI 10.1137/S1064827597323373
Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
Gang Bao, Variational approximation of Maxwell’s equations in biperiodic structures, SIAM J. Appl. Math. 57 (1997), no. 2, 364–381. MR 1438758, DOI 10.1137/S0036139995279408
Gang Bao, Zhiming Chen, and Haijun Wu, Adaptive finite-element method for diffraction gratings, J. Opt. Soc. Amer. A 22 (2005), no. 6, 1106–1114. MR 2188104, DOI 10.1364/JOSAA.22.001106
Gang Bao, Lawrence Cowsar, and Wen Masters (eds.), Mathematical modeling in optical science, Frontiers in Applied Mathematics, vol. 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. MR 1831328, DOI 10.1137/1.9780898717594
Gang Bao and David C. Dobson, On the scattering by a biperiodic structure, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2715–2723. MR 1694448, DOI 10.1090/S0002-9939-00-05509-X
Gang Bao, David C. Dobson, and J. Allen Cox, Mathematical studies in rigorous grating theory, J. Opt. Soc. Amer. A 12 (1995), no. 5, 1029–1042. MR 1335399, DOI 10.1364/JOSAA.12.001029
Gang Bao and Haijun Wu, Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal. 43 (2005), no. 5, 2121–2143. MR 2192334, DOI 10.1137/040604315
Gang Bao and Aihui Zhou, Analysis of finite dimensional approximations to a class of partial differential equations, Math. Methods Appl. Sci. 27 (2004), no. 17, 2055–2066. MR 2099817, DOI 10.1002/mma.539
Rudi Beck, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal. 34 (2000), no. 1, 159–182 (English, with English and French summaries). MR 1735971, DOI 10.1051/m2an:2000136
Jean-Pierre Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185–200. MR 1294924, DOI 10.1006/jcph.1994.1159
Jean-Pierre Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 127 (1996), no. 2, 363–379. MR 1412240, DOI 10.1006/jcph.1996.0181
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
James H. Bramble and Joseph E. Pasciak, Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems, Math. Comp. 76 (2007), no. 258, 597–614. MR 2291829, DOI 10.1090/S0025-5718-06-01930-2
James H. Bramble and Joseph E. Pasciak, Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem, Math. Comp. 77 (2008), no. 261, 1–10. MR 2353940, DOI 10.1090/S0025-5718-07-02037-6
Yia-Chung Chang, Guangwei Li, Hanyou Chu, and Jon Opsal, Efficient finite-element, Green’s function approach for critical-dimension metrology of three-dimensional gratings on multilayer films, J. Opt. Soc. Amer. A 23 (2006), no. 3, 638–645. MR 2232900, DOI 10.1364/JOSAA.23.000638
Junqing Chen and Zhiming Chen, An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp. 77 (2008), no. 262, 673–698. MR 2373175, DOI 10.1090/S0025-5718-07-02055-8
Zhiming Chen and Shibin Dai, On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients, SIAM J. Sci. Comput. 24 (2002), no. 2, 443–462. MR 1951050, DOI 10.1137/S1064827501383713
Zhiming Chen and Xuezhe Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal. 43 (2005), no. 2, 645–671. MR 2177884, DOI 10.1137/040610337
Zhiming Chen and Haijun Wu, An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal. 41 (2003), no. 3, 799–826. MR 2005183, DOI 10.1137/S0036142902400901
F. Collino and P. B. Monk, Optimizing the perfectly matched layer, Comput. Methods Appl. Mech. Engrg. 164 (1998), no. 1-2, 157–171. Exterior problems of wave propagation (Boulder, CO, 1997; San Francisco, CA, 1997). MR 1662777, DOI 10.1016/S0045-7825(98)00052-8
Martin Costabel and Monique Dauge, Maxwell and Lamé eigenvalues on polyhedra, Math. Methods Appl. Sci. 22 (1999), no. 3, 243–258. MR 1672271, DOI 10.1002/(SICI)1099-1476(199902)22:3
David Dobson and Avner Friedman, The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl. 166 (1992), no. 2, 507–528. MR 1160941, DOI 10.1016/0022-247X(92)90312-2
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
R. E. Edwards, Fourier series. A modern introduction. Vol. 1, 2nd ed., Graduate Texts in Mathematics, vol. 64, Springer-Verlag, New York-Berlin, 1979. MR 545506
S.D. Gedney, An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices, IEEE Trans. Antennas Propagat. 44 (1996), 1630–1639.
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
M. Lassas and E. Somersalo, On the existence and convergence of the solution of PML equations, Computing 60 (1998), no. 3, 229–241. MR 1621305, DOI 10.1007/BF02684334
L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings, J. Opt. Soc. Amer. A 14 (1997), 2758–2767.
Khamron Mekchay and Ricardo H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), no. 5, 1803–1827. MR 2192319, DOI 10.1137/04060929X
Peter Monk, A posteriori error indicators for Maxwell’s equations, J. Comput. Appl. Math. 100 (1998), no. 2, 173–190. MR 1659117, DOI 10.1016/S0377-0427(98)00187-3
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
Peter Monk and Endre Süli, The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals, SIAM J. Numer. Anal. 36 (1999), no. 1, 251–274. MR 1654563, DOI 10.1137/S0036142997315172
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR 1980447, DOI 10.1137/S0036144502409093
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
Roger Petit (ed.), Electromagnetic theory of gratings, Topics in Current Physics, vol. 22, Springer-Verlag, Berlin-New York, 1980. MR 609533
Hans-Jürgen Schmeisser and Hans Triebel, Topics in Fourier analysis and function spaces, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. MR 891189
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
F. Teixera and W. Chew, Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates, IEEE microwave and guided wave letters 7 (1997), 371–373.
E. Turkel and A. Yefet, Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math. 27 (1998), no. 4, 533–557. Absorbing boundary conditions. MR 1644675, DOI 10.1016/S0168-9274(98)00026-9
R. Verfürth, A review of a posteriori error estimation and adaptive mesh refinement techniques, Teubner, Stuttgart, 1996.
Haijun Wu and Zhimin Zhang, Can we have superconvergent gradient recovery under adaptive meshes?, SIAM J. Numer. Anal. 45 (2007), no. 4, 1701–1722. MR 2338406, DOI 10.1137/060661430
V. Yachin and K. Yasumoto, Method of integral functionals for electromagnetic wave scattering from a double-periodic magnetodielectric layer, J. Opt. Soc. Amer. A 24 (2007), 3606–3618.