Abstract
The choice of discrete spaces for a variationally posed symmetric and compact eigenvalue problem corresponding a source problem is discussed. Any standard Galerkin discretization space that is convergent for the source problem automatically performs well for the eigenvalue problem. On the other hand, mixed discretizations that are convergent (satisfying the classical Brezzi conditions) exhibit spurious low frequency eigenmodes. Examples of discretizations with spurious modes are presented. Moreover, necessary and sufficient conditions on mixed discretization are established for the (ordered) discrete eigenvalues to converge to the corresponding continuous eigenvalues. The theory is applied to the determination of band gaps for photonic crystals and evolution problems.
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References
C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), pp. 823–864.
D.N. Arnold, Differential complexes and numerical stability, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Beijing, 2002, Higher Ed. Press, pp. 137–157.
I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, P. Ciarlet and J. Lions, eds., vol. II, Elsevier Science Publishers B.V., North Holland, 1991, pp. 641–788.
D. Boffi, Fortin operator and discrete compactness for edge elements, Numer. Math., 87 (2000), pp. 229–246.
—, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett., 14 (2001), pp. 33–38.
D. Boffi, F. Brezzi, and L. Gastaldi, On the convergence of eigenvalues for mixed formulations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997), pp. 131–154 (1998). Dedicated to Ennio De Giorgi.
—, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp., 69 (2000), pp. 121–140.
D. Boffi, A. Buffa, and L. Gastaldi, Convergence analysis for hyperbolic evolution problems in mixed form. submitted.
D. Boffi, M. Conforti, and L. Gastaldi, Modified edge finite elements for photonic crystals. submitted.
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.
D. Boffi and L. Gastaldi, Interpolation estimates for edge finite elements and application to band gap computation. Applied Numerical Mathematics, to appear.
D. Boffi and L. Gastaldi, Analysis of finite element approximation of evolution problems in mixed form, SIAM J. Numer. Anal., 42 (2004), pp. 1502–1526 (electronic).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991.
S. Caorsi, P. Fernandes, and M. Raffetto, Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 331–354.
M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal., 151 (2000), pp. 221–276.
L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Comput. Methods Appl. Mech. Engrg., 152 (1998), pp. 103–124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997).
D.C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, An efficient method for band structure calculations in 3D photonic crystals, J. Comput. Phys., 161 (2000), pp. 668–679.
G.J. Fix, M.D. Gunzburger, and R.A. Nicolaides, On mixed finite element methods for first-order elliptic systems, Numer. Math., 37 (1981), pp. 29–48.
M. Fortin, An analysis of the convergence of mixed finite element methods, R.A.I.R.O., Anal. Numer., 11 (1977), pp. 341–354.
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), pp. 237–339.
T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, in Proceedings of the first world congress on computational mechanics (Austin, Tex., 1986), vol. 64, 1987, pp. 509–521.
P. Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.
H.L. Royden, Real analysis, Macmillan Publishing Company, New York, second ed., 1988.
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Boffi, D. (2006). Compatible Discretizations for Eigenvalue Problems. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_6
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DOI: https://doi.org/10.1007/0-387-38034-5_6
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