Analysis of an algorithm for computing electromagnetic bloch modes using nedelec spaces

Computational Methods in Applied Mathematics

Volumn 1, Issue 2, 2001, Pages 138-153

  Dobson, David C ^a   Pasciak, Joseph E ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Convergence analysis; Mixed finite elements; Periodic structure; Photonic band gap

Indexed keywords

EID: 85010571824     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2001-0010     Document Type: Article

References (26)

1. D. N. Arnold, R. S. Falk, and R. Winther, Multigrid in h(div) and h(curl), Numer. Math. (to appear).
2. W. Axmann and P. Kuchment, An efficient finite element method for computing spectra of photonic and acoustic band-gap materials, J. Comp. Phys., 150 (1999), pp. 468-481.
3. D. Boffi, A note on the De Rham complex and a discrete compactness property, (preprint).
4. D. Boffi, F. Brezzi, and L. Gastaldi, On the convergence of eigenvalues for mixed formulations, Ann. Scuola Norm. Sup. Pisa C. Sci., 25 (1997), No. 4, pp. 131-154.
5. D. Boffi, F. Brezzi, and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp., 69 (2000), pp. 121-140.
6. D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal., 36 (1999), pp. 1264-1290.
7. C. M. Bowden, J. P. Dowling, and H. O. Everitt, eds., Development and applications of materials exhibiting photonic band gaps, J. Opt. Soc. Amer. B, 10 (1993), No. 2.
8. J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp., 56 (1991), pp. 1-34.
9. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
10. M. Costabel, M. Dauge, and S. Nicaise, Singularities of Maxwell's interface problems, Model. Math. Anal. Numér., 33 (1999), pp. 627-649.
11. D. Dobson, J. Gopalakrishnan, and J. Pasciak, An efficient method for band structure calculations in 3D photonic crystals, J. Comp. Phys. (to appear).
12. A. Figotin and Y. A. Godin, The computation of spectra of some 2D photonic crystals, J. Comp. Phys., 136 (1997), pp. 585-598.
13. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, Berlin, 1986.
14. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Princeton University Press, Princeton NJ, 1995.
15. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
16. F. Kikuchi, An isomorphic property of two Hilbert spaces appearing in electromagnetism: Analysis by the mixed formulation, Japan J. Appl. Math., 3 (1986), pp. 53-58.
17. F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comp. Meth. Appl. Mech. Eng., 64 (1987), pp. 509-521.
18. F. Kikuchi, On a discrete compactness property for the Nedelec finite elements, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math.,, 36 (1989), pp. 479-490.
19. R. D. Lazarov, J. E. Pasciak, and P. Vassilevski, Iterative solution of a combined mixed and standard Galerkin discretization method for elliptic problems, (preprint).
20. J. C. Nedelec, Mixed finite elements in ℝ3, Numer. Math., 35 (1980), pp. 315-341.
21. J. C. Nedelec, A new family of mixed finite elements in ℝ3, Numer. Math., 50 (1986), pp. 57-81.
22. J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, 1967.
23. M. Reed and B. Simon, Methods of Modern Mathematics Physics, Vol IV: Analysis of Operators, Academic Press, 1978.
24. C. M. Soukoulis, ed., Photonic Band Gap Materials, Kluwer, Dordrecht, 1996.
25. H. S. Sözüer, J. W. Haus, and R. Inguva, Photonic bands: Convergence problems with the plane-wave method, Phys. Rev. B, 45 (1992), No. 24, pp. 13962-13972.
26. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 1997.