Convergent adaptive finite element methods for photonic crystal applications. (English) Zbl 1256.65094
Photonic cristals (PC) exhibit interesting properties in the propagation of electromagnetic waves, such as spectral band gaps. In 2D PCs theory, Maxwell’s equations reduce to a two-dimensional one-component wave equation which determines either the electric field (TH mode) or the magnetic field (TE mode). Because of periodicity, one splits each mode into a family of eigenvalue problems which involve non-coercive elliptic operators with discontinuous coefficients. The author extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to PC problems dealing with complications arising essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. The assumptions regarding the conformity and shape regularity of the initial mesh suffice to prove the convergence of the adaptive method in an oscillatory-free way.
The author also presents and proves the convergence of an adaptive method to compute efficiently an entire band in the spectrum. Using this method, the maximum and the minimum of the computed approximation of the band converge to the true global maximum and minimum of the band. The numerical results cover both the cases of periodic structures with and without compact defects. According to the author’s opinion, the paper is the first contribution to the topic of oscillation-free convergence of adaptive finite element methods for PC applications.
The author also presents and proves the convergence of an adaptive method to compute efficiently an entire band in the spectrum. Using this method, the maximum and the minimum of the computed approximation of the band converge to the true global maximum and minimum of the band. The numerical results cover both the cases of periodic structures with and without compact defects. According to the author’s opinion, the paper is the first contribution to the topic of oscillation-free convergence of adaptive finite element methods for PC applications.
Reviewer: Adrian Carabineanu (Bucureşti)
MSC:
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
35P15 | Estimates of eigenvalues in context of PDEs |
35Q61 | Maxwell equations |
35R05 | PDEs with low regular coefficients and/or low regular data |
82D25 | Statistical mechanics of crystals |
78A25 | Electromagnetic theory (general) |
78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
second-order problems; finite element methods; adaptive algorithm; convergence; photonic crystals; oscillation-free; electromagnetic waves; spectral band gaps; Maxwell’s equations; electric field; magnetic field; non-coercive elliptic operators; discontinuous coefficients; elliptic eigenvalue problems; numerical resultsReferences:
[1] | Ammari, H.Santosa, F., SIAM J. Appl. Math.64, 2018 (2004). · Zbl 1060.35140 |
[2] | Ashcroft, N. W.; Mermin, N. D., Solid State Physics, 1976, Brooks/Cole |
[3] | Axmann, W.Kuchment, P., J. Comput. Phys.150, 468 (1999). · Zbl 0963.78028 |
[4] | Bernardi, C.Verfürth, R., Numer. Math.85, 579 (2000). · Zbl 0962.65096 |
[5] | Boffi, D.Conforti, M.Gastaldi, L., Numer. Math.105, 249 (2006). · Zbl 1129.65082 |
[6] | Cao, Y.Hou, Z.Liu, Y., Phys. Lett. A327, 247 (2004). · Zbl 1138.82362 |
[7] | C. Carstensen and J. Gedicke, An oscillation-free adaptive FEM for symmetric eigenvalue problems, preprint 489, DFG Research Center Matheon, Berlin (2008). |
[8] | Cox, S. J.Dobson, D. C., SIAM J. Appl. Math.59, 2108 (1999). · Zbl 1027.78521 |
[9] | Dai, X.Xu, J.Zhou, A., Numer. Math.110, 313 (2008). · Zbl 1159.65090 |
[10] | Dobson, D. C., J. Comput. Phys.149, 363 (1999). · Zbl 0924.65133 |
[11] | Dobson, D. C.Gopalakrishnan, J.Pasciak, J. E., J. Comput. Phys.161, 668 (2000). · Zbl 1034.78013 |
[12] | Engström, C.Hafner, C.Schmidt, K., J. Comput. Theor. Nanosci.6, 775 (2009). |
[13] | Figotin, A.Goren, V., Phys. Rev. E64, 056623 (2001). |
[14] | Figotin, A.Gorentsveig, V., Phys. Rev. B58, 180 (1998). |
[15] | Figotin, A.Klein, A., J. Stat. Phys.86, 165 (1997). · Zbl 0952.76543 |
[16] | Figotin, A.Klein, A., SIAM J. Appl. Math.58, 1748 (1998). · Zbl 0963.78005 |
[17] | Garau, E. M.Morin, P., IMA J. Numer. Anal. (2010), DOI: 10.1093/imanum/drp055. |
[18] | Garau, E. M.Morin, P.Zuppa, C., Math. Models Methods Appl. Sci.19, 721 (2009). · Zbl 1184.65100 |
[19] | S. Giani, Convergence of adaptive finite element methods for elliptic eigenvalue problems with application to photonic crystals, Ph.D. Thesis, University of Bath (2008). |
[20] | S. Giani and I. G. Graham, A convergent adaptive method for elliptic eigenvalue problems and numerical experiments, Bath Institute for Complex Systems, preprint number 14/08, University of Bath (2008). |
[21] | Giani, S.Graham, I. G., SIAM J. Numer. Anal.47, 1067 (2009). · Zbl 1191.65147 |
[22] | Giani, S.; Graham, I. G., Numer. Math. |
[23] | B. Hiett, Photonic Crystal modelling using finite element analysis, Ph.D. Thesis, University of Southampton (2000). |
[24] | Joannopoulos, J. D.Johnson, S. G., Opt. Express8, 173 (2001). |
[25] | Joannopoulos, J., Photonic Crystals: Molding the Flow of Light, 2008, Princeton Univ. Press · Zbl 1144.78303 |
[26] | A. Klöckner, On the computation of maximally localized Wannier functions, Ph.D. Thesis, Karlsruhe University (2004). |
[27] | Kuchment, P., Floquet Theory for Partial Differential Equations, 1993, Birkhäuser · Zbl 0789.35002 |
[28] | Kuchment, P., Frontiers Appl. Math.22, 207 (2001). |
[29] | Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, 1998, SIAM · Zbl 0901.65021 |
[30] | Morin, P.Siebert, K.Veeser, A., Math. Models Methods Appl. Sci.18, 707 (2008). · Zbl 1153.65111 |
[31] | R. A. Norton, Numerical computation of band gaps in photonic crystal fibres, preprint, University of Bath (2008). |
[32] | Norton, R.Scheichl, R., SIAM J. Numer. Anal.47, 4356 (2010). · Zbl 1218.65124 |
[33] | Pearce, G. J.Hedley, T. D.Bird, D. M., Phys. Rev. B71, 195108 (2005). |
[34] | Petzoldt, M., Adv. Comput. Math.16, 47 (2002). · Zbl 0997.65123 |
[35] | Sakoda, K., Optical Properties of Photonic Crystals, 2001, Springer |
[36] | Schmidt, K.Kappeler, R., Opt. Express18, 7307 (2010). |
[37] | Schmidt, K.Kauf, P., Comput. Methods Appl. Mech. Engrg.198, 1249 (2009). · Zbl 1157.78355 |
[38] | Scott, J. A., Sparse Direct Methods: An Introduction, 535, 2000, Springer |
[39] | Scott, R. L.Zhang, S., Math. Comput.54, 483 (1990). · Zbl 0696.65007 |
[40] | Soussi, S., SIAM J. Numer. Anal.43, 1175 (2005). |
[41] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method, 1973, Prentice-Hall · Zbl 0356.65096 |
[42] | Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, 1996, Wiley-Teubner · Zbl 0853.65108 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.