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Finite element approximation of eigenvalue problems

Published online by Cambridge University Press:  10 May 2010

Daniele Boffi
Daniele Boffi
Affiliation:
Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy, E-mail: [email protected]
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We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

Type
Research Article
Information
Acta Numerica , Volume 19 , May 2010 , pp. 1 - 120
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Ainsworth, M. and Coyle, J. (2003), Computation of Maxwell eigenvalues on curvilinear domains using hp-version Nédélec elements. In Numerical Mathematics and Advanced Applications, Springer Italia, Milan, pp. 219–231.CrossRefGoogle Scholar
Alonso, A. and Russo, A. Dello (2009), ‘Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods’, J. Comput. Appl. Math. 223, 177–197.CrossRefGoogle Scholar
Anselone, P. M. (1971), Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Anselone, P. M. and Palmer, T. W. (1968), ‘Spectral properties of collectively compact sets of linear operators.’, J. Math. Mech. 17, 853–860.Google Scholar
Antonietti, P. F., Buffa, A. and Perugia, I. (2006), ‘Discontinuous Galerkin approximation of the Laplace eigenproblem’, Comput. Methods Appl. Mech. Engrg 195, 3483–3503.CrossRefGoogle Scholar
Arbenz, P. and Geus, R. (1999), ‘A comparison of solvers for large eigenvalue problems occurring in the design of resonant cavities’, Numer. Linear Algebra Appl. 6, 3–16.3.0.CO;2-I>CrossRefGoogle Scholar
Armentano, M. G. and Durán, R. G. (2004), ‘Asymptotic lower bounds for eigenvalues by nonconforming finite element methods’, Electron. Trans. Numer. Anal. 17, 93–101 (electronic).Google Scholar
Arnold, D. N. (2002), Differential complexes and numerical stability. In Proc. International Congress of Mathematicians, Vol. I: Beijing 2002, Higher Education Press, Beijing, pp. 137–157.Google Scholar
Arnold, D. N., Boffi, D. and Falk, R. S. (2002), ‘Approximation by quadrilateral finite elements’, Math. Comp. 71, 909–922 (electronic).CrossRefGoogle Scholar
Arnold, D. N., Boffi, D. and Falk, R. S. (2005), ‘Quadrilateral H (div) finite elements’, SIAM J. Numer. Anal. 42, 2429–2451 (electronic).CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2000), ‘Multigrid in H(div) and H(curl)’, Numer. Math. 85, 197–217.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006 a), Differential complexes and stability of finite element methods I: The de Rham complex. In Compatible Spatial Discretizations (Arnold, D., Bochev, P., Lehoucq, R., Nicolaides, R. and Shaskov, M., eds), Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, Berlin, pp. 23–46.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006 b), Finite element exterior calculus, homological techniques, and applications. In Acta Numerica, Vol. 15, Cambridge University Press, pp. 1–155.Google Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2010), ‘Finite element exterior calculus: From Hodge theory to numerical stability’, Bull. Amer. Math. Soc. NS 47, 281–353.CrossRefGoogle Scholar
Assous, F., Ciarlet, P. Jr, and Sonnendrücker, E. (1998), ‘Resolution of the Maxwell equations in a domain with reentrant corners’, RAIRO Modél. Math. Anal. Numér. 32, 359–389.CrossRefGoogle Scholar
Babuška, I. (1973), ‘The finite element method with Lagrangian multipliers’, Numer. Math. 20, 179–192.CrossRefGoogle Scholar
Babuška, I. and Narasimhan, R. (1997), ‘The Babuška-Brezzi condition and the patch test: An example’, Comput. Methods Appl. Mech. Engrg 140, 183–199.CrossRefGoogle Scholar
Babuška, I. and Osborn, J. (1991), Eigenvalue problems. In Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, pp. 641–787.Google Scholar
Babuška, I. and Osborn, J. E. (1989), ‘Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems’, Math. Comp. 52, 275–297.CrossRefGoogle Scholar
Bathe, K.-J., Nitikitpaiboon, C. and Wang, X. (1995), ‘A mixed displacement-based finite element formulation for acoustic fluid-structure interaction’, Comput. & Structures 56, 225–237.CrossRefGoogle Scholar
Bermúdez, A. and Rodríguez, R. (1994), ‘Finite element computation of the vibration modes of a fluid-solid system’, Comput. Methods Appl. Mech. Engrg 119, 355–370.CrossRefGoogle Scholar
Bermúdez, A., Durán, R., Muschietti, M. A., Rodríguez, R. and Solomin, J. (1995), ‘Finite element vibration analysis of fluid-solid systems without spurious modes’, SIAM J. Numer. Anal. 32, 1280–1295.CrossRefGoogle Scholar
Bermúdez, A., Gamallo, P., Nogueiras, M. R. and Rodríguez, R. (2006), ‘Approximation of a structural acoustic vibration problem by hexahedral finite elements’, IMA J. Numer. Anal. 26, 391–421.CrossRefGoogle Scholar
Birkhoff, G., de Boor, C., Swartz, B. and Wendroff, B. (1966), ‘Rayleigh-Ritz approximation by piecewise cubic polynomials’, SIAM J. Numer. Anal. 3, 188–203.CrossRefGoogle Scholar
Boffi, D. (1997), ‘Three-dimensional finite element methods for the Stokes problem’, SIAM J. Numer. Anal. 34, 664–670.CrossRefGoogle Scholar
Boffi, D. (2000), ‘Fortin operator and discrete compactness for edge elements’, Numer. Math. 87, 229–246.CrossRefGoogle Scholar
Boffi, D. (2001), ‘A note on the de Rham complex and a discrete compactness property’, Appl. Math. Lett. 14, 33–38.CrossRefGoogle Scholar
Boffi, D. (2007), ‘Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements’, Comput. Methods Appl. Mech. Engrg 196, 3672–3681.CrossRefGoogle Scholar
Boffi, D. and Gastaldi, L. (2004), ‘Analysis of finite element approximation of evolution problems in mixed form’, SIAM J. Numer. Anal. 42, 1502–1526 (electronic).CrossRefGoogle Scholar
Boffi, D. and Lovadina, C. (1997), ‘Remarks on augmented Lagrangian formulations for mixed finite element schemes’, Boll. Un. Mat. Ital. A (7) 11, 41–55.Google Scholar
Boffi, D., Brezzi, F. and Gastaldi, L. (1997), ‘On the convergence of eigenvalues for mixed formulations’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, 131–154.Google Scholar
Boffi, D., Brezzi, F. and Gastaldi, L. (2000 a), ‘On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form’, Math. Comp. 69, 121–140.CrossRefGoogle Scholar
Boffi, D., Chinosi, C. and Gastaldi, L. (2000 b), ‘Approximation of the grad div operator in nonconvex domains’, Comput. Model. Eng. Sci. 1, 31–43.Google Scholar
Boffi, D., Conforti, M. and Gastaldi, L. (2006 a), ‘Modified edge finite elements for photonic crystals’, Numer. Math. 105, 249–266.CrossRefGoogle Scholar
Boffi, D., Costabel, M., Dauge, M. and Demkowicz, L. (2006 b), ‘Discrete compactness for the hp version of rectangular edge finite elements’, SIAM J. Numer. Anal. 44, 979–1004 (electronic).CrossRefGoogle Scholar
Boffi, D., Costabel, M., Dauge, M., Demkowicz, L. and Hiptmair, R. (2009), Discrete compactness for the p-version of discrete differential forms. HAL: hal-00420150, arXiv: 0909.5079v2.Google Scholar
Boffi, D., Demkowicz, L. and Costabel, M. (2003), ‘Discrete compactness for p and hp 2D edge finite elements’, Math. Models Methods Appl. Sci. 13, 1673–1687.CrossRefGoogle Scholar
Boffi, D., Durán, R. G. and Gastaldi, L. (1999 a), ‘A remark on spurious eigenvalues in a square’, Appl. Math. Lett. 12, 107–114.CrossRefGoogle Scholar
Boffi, D., Fernandes, P., Gastaldi, L. and Perugia, I. (1999 b), ‘Computational models of electromagnetic resonators: Analysis of edge element approximation’, SIAM J. Numer. Anal. 36, 1264–1290 (electronic).CrossRefGoogle Scholar
Boffi, D., Kikuchi, F. and Schöberl, J. (2006 c), ‘Edge element computation of Maxwell‘s eigenvalues on general quadrilateral meshes’, Math. Models Methods Appl. Sci. 16, 265–273.CrossRefGoogle Scholar
Boland, J. M. and Nicolaides, R. A. (1985), ‘Stable and semistable low order finite elements for viscous flows’, SIAM J. Numer. Anal. 22, 474–492.CrossRefGoogle Scholar
Bossavit, A. (1988), ‘Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism’, IEEE Proc. A 135, 493–500.Google Scholar
Bossavit, A. (1989), ‘Un nouveau point de vue sur les éléments mixtes’, Bull. Soc. Math. Appl. Industr. 20, 22–35.Google Scholar
Bossavit, A. (1990), ‘Solving Maxwell equations in a closed cavity and the question of spurious modes’, IEEE Trans. Magnetics 26, 702–705.CrossRefGoogle Scholar
Bossavit, A. (1998), Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements, Academic Press, San Diego, CA.Google Scholar
Bramble, J. H. and Osborn, J. E. (1973), ‘Rate of convergence estimates for non-selfadjoint eigenvalue approximations’, Math. Comp. 27, 525–549.CrossRefGoogle Scholar
Bramble, J. H., Kolev, T. V. and Pasciak, J. E. (2005), ‘The approximation of the Maxwell eigenvalue problem using a least-squares method’, Math. Comp. 74, 1575–1598 (electronic).CrossRefGoogle Scholar
Brenner, S. C., Li, F. and Sung, L. Y. (2008), ‘A locally divergence-free interior penalty method for two-dimensional curl-curl problems’, SIAM J. Numer. Anal. 46, 1190–1211.CrossRefGoogle Scholar
Brezzi, F. (1974), ‘On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers’, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151.Google Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Vol. 15 of Springer Series in Computational Mathematics, Springer, New York.CrossRefGoogle Scholar
Brezzi, F., Douglas, J., Duran, R. and Fortin, M. (1987 a), ‘Mixed finite elements for second order elliptic problems in three variables’, Numer. Math. 51, 237–250.CrossRefGoogle Scholar
Brezzi, F., Douglas, J. Jr, and Marini, L. D. (1985), ‘Two families of mixed finite elements for second order elliptic problems’, Numer. Math. 47, 217–235.CrossRefGoogle Scholar
Brezzi, F., Douglas, J. Jr, and Marini, L. D. (1986), Recent results on mixed finite element methods for second order elliptic problems. In Vistas in Applied Mathematics, Optimization Software, New York, pp. 25–43.Google Scholar
Brezzi, F., Douglas, J. Jr, Fortin, M. and Marini, L. D. (1987 b), ‘Efficient rectangular mixed finite elements in two and three space variables’, RAIRO Modél. Math. Anal. Numér. 21, 581–604.CrossRefGoogle Scholar
Buffa, A. and Perugia, I. (2006), ‘Discontinuous Galerkin approximation of the Maxwell eigenproblem’, SIAM J. Numer. Anal. 44, 2198–2226 (electronic).CrossRefGoogle Scholar
Buffa, A., Costabel, M. and Dauge, M. (2005), ‘Algebraic convergence for anisotropic edge elements in polyhedral domains’, Numer. Math. 101, 29–65.CrossRefGoogle Scholar
Buffa, A., Houston, P. and Perugia, I. (2007), ‘Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes’, J. Comput. Appl. Math. 204, 317–333.CrossRefGoogle Scholar
Buffa, A., Perugia, I. and Warburton, T. (2009), ‘The mortar-discontinuous Galerkin method for the 2D Maxwell eigenproblem’, J. Sci. Comput. 40, 86–114.CrossRefGoogle Scholar
Cangiani, A., Gardini, F. and Manzini, G. (2010), ‘Convergence of the mimetic finite difference method for eigenvalue problems in mixed form’, Comput. Methods Appl. Mech. Engrg. Submitted.Google Scholar
Canuto, C. (1978), ‘Eigenvalue approximations by mixed methods’, RAIRO Anal. Numér. 12, 27–50.CrossRefGoogle Scholar
Caorsi, S., Fernandes, P. and Raffetto, M. (2000), ‘On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems’, SIAM J. Numer. Anal. 38, 580–607 (electronic).CrossRefGoogle Scholar
Caorsi, S., Fernandes, P. and Raffetto, M. (2001), ‘Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements’, M2AN Math. Model. Numer. Anal. 35, 331–354.CrossRefGoogle Scholar
Carstensen, C. (2008), ‘Convergence of an adaptive FEM for a class of degenerate convex minimization problems’, IMA J. Numer. Anal. 28, 423–439.CrossRefGoogle Scholar
Chatelin, F. (1973), ‘Convergence of approximation methods to compute eigenelements of linear operations’, SIAM J. Numer. Anal. 10, 939–948.CrossRefGoogle Scholar
Chatelin, F. (1983), Spectral Approximation of Linear Operators, Computer Science and Applied Mathematics, Academic Press, New York.Google Scholar
Chatelin, F. and Lemordant, M. J. (1975), ‘La méthode de Rayleigh-Ritz appliquée à des opérateurs différentiels elliptiques: Ordres de convergence des éléments propres’, Numer. Math. 23, 215–222.CrossRefGoogle Scholar
Chen, H. and Taylor, R. (1990), ‘Vibration analysis of fluid-solid systems using a finite element displacement formulation’, Interna t. J., Numer. Methods Engrg 29, 683–698.Google Scholar
Christiansen, S. H. (2007), ‘Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension’, Numer. Math. 107, 87–106.CrossRefGoogle Scholar
Christiansen, S. H. (2009), personal communication.CrossRefGoogle Scholar
Ciarlet, P. Jr, and Zou, J. (1999), ‘Fully discrete finite element approaches for time-dependent Maxwell's equations’, Numer. Math. 82, 193–219.CrossRefGoogle Scholar
Ciarlet, P. G. and Raviart, P.-A. (1974), A mixed finite element method for the biharmonic equation. In Mathematical Aspects of Finite Elements in Partial Differential Equations: Proc. Sympos., Madison 1974, Academic Press, New York, pp. 125–145.CrossRefGoogle Scholar
Costabel, M. and Dauge, M. (2002), ‘Weighted regularization of Maxwell equations in polyhedral domains: A rehabilitation of nodal finite elements’, Numer. Math. 93, 239–277.CrossRefGoogle Scholar
Costabel, M. and Dauge, M. (2003), Computation of resonance frequencies for Maxwell equations in non-smooth domains. In Topics in Computational Wave Propagation, Vol. 31 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, pp. 125–161.CrossRefGoogle Scholar
Creusé, E. and Nicaise, S. (2006), ‘Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system’, M2AN Math. Model. Numer. Anal. 40, 413–430.CrossRefGoogle Scholar
Dauge, M. (2003), Benchmark computations for Maxwell equations. http://perso.univ-rennes1.fr/monique.dauge/benchmax.html.Google Scholar
Demkowicz, L. (2005), ‘Fully automatic hp-adaptivity for Maxwell‘s equations’, Comput. Methods Appl. Mech. Engrg 194, 605–624.CrossRefGoogle Scholar
Demkowicz, L., Monk, P., Schwab, C. and Vardapetyan, L. (2000 a), ‘Maxwell eigenvalues and discrete compactness in two dimensions’, Comput. Math. Appl. 40, 589–605.CrossRefGoogle Scholar
Demkowicz, L., Monk, P., Vardapetyan, L. and Rachowicz, W. (2000 b), ‘De Rham diagram for hp finite element spaces’, Comput. Math. Appl. 39, 29–38.CrossRefGoogle Scholar
Descloux, J., Nassif, N. and Rappaz, J. (1978 a), ‘On spectral approximation I: The problem of convergence’, RAIRO Anal. Numér. 12, 97–112.CrossRefGoogle Scholar
Descloux, J., Nassif, N. and Rappaz, J. (1978 b), ‘On spectral approximation II: Error estimates for the Galerkin method’, RAIRO Anal. Numér. 12, 113–119.CrossRefGoogle Scholar
Douglas, J. Jr, and Roberts, J. E. (1982), ‘Mixed finite element methods for second order elliptic problems’, Mat. Apl. Comput. 1, 91–103.Google Scholar
Durán, R. G., Gastaldi, L. and Padra, C. (1999), ‘A posteriori error estimators for mixed approximations of eigenvalue problems’, Math. Models Methods Appl. Sci. 9, 1165–1178.CrossRefGoogle Scholar
Durán, R. G., Padra, C. and Rodríguez, R. (2003), ‘A posteriori error estimates for the finite element approximation of eigenvalue problems’, Math. Models Methods Appl. Sci. 13, 1219–1229.CrossRefGoogle Scholar
Falk, R. S. and Osborn, J. E. (1980), ‘Error estimates for mixed methods’, RAIRO Anal. Numér. 14, 249–277.CrossRefGoogle Scholar
Fix, G. J., Gunzburger, M. D. and Nicolaides, R. A. (1981), ‘On mixed finite element methods for first-order elliptic systems’, Numer. Math. 37, 29–48.CrossRefGoogle Scholar
Fortin, M. (1977), ‘An analysis of the convergence of mixed finite element methods’, RAIRO Anal. Numér. 11, 341–354.CrossRefGoogle Scholar
Gamallo, P. (2002), Contribución al estudio matemático de problemas de simulación elastoacústica y control activo del ruido. PhD thesis, Universidade de Santiago de Compostela, Spain.Google Scholar
Garau, E. M., Morin, P. and Zuppa, C. (2009), ‘Convergence of adaptive finite element methods for eigenvalue problems’, Math. Models Methods Appl. Sci. 19, 721–747.CrossRefGoogle Scholar
Gardini, F. (2004), ‘A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction’, Istit. Lombardo Accad. Sci. Lett. Rend. A 138, 17–34.Google Scholar
Gardini, F. (2005), ‘Discrete compactness property for quadrilateral finite element spaces’, Numer. Methods Partial Differential Equations 21, 41–56.CrossRefGoogle Scholar
Gastaldi, L. (1996), ‘Mixed finite element methods in fluid structure systems’, Numer. Math. 74, 153–176.CrossRefGoogle Scholar
Giani, S. and Graham, I. G. (2009), ‘A convergent adaptive method for elliptic eigenvalue problems’, SIAM J. Numer. Anal. 47, 1067–1091.CrossRefGoogle Scholar
Glowinski, R. (1973), Approximations externes, par éléments finis de Lagrange d‘ordre un et deux, du problème de Dirichlet pour l‘opérateur biharmonique: Méthode itérative de résolution des problèmes approchés. In Topics in Numerical Analysis: Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972, Academic Press, London, pp. 123–171.Google Scholar
Grigorieff, R. D. (1975 a), ‘Diskrete Approximation von Eigenwertproblemen I: Qualitative Konvergenz’, Numer. Math. 24, 355–374.CrossRefGoogle Scholar
Grigorieff, R. D. (1975 b), ‘Diskrete Approximation von Eigenwertproblemen II: Konvergenzordnung’, Numer. Math. 24, 415–433.CrossRefGoogle Scholar
Grigorieff, R. D. (1975 c), ‘Diskrete Approximation von Eigenwertproblemen III: Asymptotische Entwicklungen’, Numer. Math. 25, 79–97.CrossRefGoogle Scholar
Gross, P. W. and Kotiuga, P. R. (2004), Electromagnetic Theory and Computation: A Topological Approach, Vol. 48 of Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Grubišić, L. and Ovall, J. S. (2009), ‘On estimators for eigenvalue/eigenvector approximations’, Math. Comp. 78, 739–770.CrossRefGoogle Scholar
Hackbusch, W. (1979), ‘On the computation of approximate eigenvalues and eigen-functions of elliptic operators by means of a multi-grid method’, SIAM J. Numer. Anal. 16, 201–215.CrossRefGoogle Scholar
Hesthaven, J. S. and Warburton, T. (2004), ‘High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem’, Philos. Trans. Roy. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 362, 493–524.CrossRefGoogle ScholarPubMed
Heuveline, V. and Rannacher, R. (2001), ‘A posteriori error control for finite approximations of elliptic eigenvalue problems’, Adv. Comput. Math. 15, 107–138.CrossRefGoogle Scholar
Hiptmair, R. (1999 a), ‘Canonical construction of finite elements’, Math. Comp. 68, 1325–1346.CrossRefGoogle Scholar
Hiptmair, R. (1999 b), ‘Multigrid method for Maxwell‘s equations’, SIAM J. Numer. Anal. 36, 204–225 (electronic).CrossRefGoogle Scholar
Hiptmair, R. (2002), Finite elements in computational electromagnetism. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 237–339.Google Scholar
Hiptmair, R. and Ledger, P. D. (2005), ‘Computation of resonant modes for axisym-metric Maxwell cavities using hp-version edge finite elements’, Internat. J. Numer. Methods Engrg 62, 1652–1676.CrossRefGoogle Scholar
Johnson, C. and Pitkäranta, J. (1982), ‘Analysis of some mixed finite element methods related to reduced integration’, Math. Comp. 38, 375–400.CrossRefGoogle Scholar
Kato, T. (1966), Perturbation Theory for Linear Operators, Vol. 132 of Die Grundlehren der Mathematischen Wissenschaften, Springer, New York.Google Scholar
Kato, T. (1995), Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin. Reprint of the 1980 edition.CrossRefGoogle Scholar
Kikuchi, F. (1987), ‘Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism’, Comput. Methods Appl. Mech. Engrg 64, 509–521.Google Scholar
Kikuchi, F. (1989), ‘On a discrete compactness property for the Nédélec finite elements’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 479–490.Google Scholar
Knyazev, A. V. and Osborn, J. E. (2006), ‘New a priori FEM error estimates for eigenvalues’, SIAM J. Numer. Anal. 43, 2647–2667 (electronic).CrossRefGoogle Scholar
Kolata, W. G. (1978), ‘Approximation in variationally posed eigenvalue problems’, Numer. Math. 29, 159–171.CrossRefGoogle Scholar
Larson, M. G. (2000), ‘A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems’, SIAM J. Numer. Anal. 38, 608–625 (electronic).CrossRefGoogle Scholar
Larsson, S. and Thomée, V. (2003), Partial Differential Equations with Numerical Methods, Vol. 45 of Texts in Applied Mathematics, Springer, Berlin.Google Scholar
Mercier, B. (1974), ‘Numerical solution of the biharmonic problem by mixed finite elements of class C 0’, Boll. Un. Mat. Ital. (4) 10, 133–149.Google Scholar
Mercier, B., Osborn, J., Rappaz, J. and Raviart, P.-A. (1981), ‘Eigenvalue approximation by mixed and hybrid methods’, Math. Comp. 36, 427–453.CrossRefGoogle Scholar
Monk, P. (2003), Finite Element Methods for Maxwell‘s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press.CrossRefGoogle Scholar
Monk, P. and Demkowicz, L. (2001), ‘Discrete compactness and the approximation of Maxwell‘s equations in R3’, Math. Comp. 70, 507–523.CrossRefGoogle Scholar
Morin, P., Nochetto, R. H. and Siebert, K. G. (2000), ‘Data oscillation and convergence of adaptive FEM’, SIAM J. Numer. Anal. 38, 466–488 (electronic).CrossRefGoogle Scholar
Nédélec, J.-C. (1980), ‘Mixed finite elements in R3’, Numer. Math. 35, 315–341.CrossRefGoogle Scholar
Nédélec, J.-C. (1986), ‘A new family of mixed finite elements in R3’, Numer. Math. 50, 57–81.CrossRefGoogle Scholar
Neymeyr, K. (2002), ‘A posteriori error estimation for elliptic eigenproblems’, Numer. Linear Algebra Appl. 9, 263–279.CrossRefGoogle Scholar
Nicaise, S. (2001), ‘Edge elements on anisotropic meshes and approximation of the Maxwell equations’, SIAM J. Numer. Anal. 39, 784–816 (electronic).CrossRefGoogle Scholar
Osborn, J. E. (1975), ‘Spectral approximation for compact operators’, Math. Comput. 29, 712–725.CrossRefGoogle Scholar
Picard, R. (1984), ‘An elementary proof for a compact imbedding result in generalized electromagnetic theory’, Math. Z. 187, 151–161.CrossRefGoogle Scholar
Powell, M. J. D. (1974), Piecewise quadratic surface fitting for contour plotting. In Software for Numerical Mathematics: Proc. Conf., Inst. Math. Appl., Lough-borough 1973, Academic Press, London, pp. 253–271.Google Scholar
Qin, J. (1994), On the convergence of some low order mixed finite elements for incompressible fluids. PhD thesis, The Pennsylvania State University, Department of Mathematics.Google Scholar
Rannacher, R. (1979), ‘Nonconforming finite element methods for eigenvalue problems in linear plate theory’, Numer. Math. 33, 23–42.CrossRefGoogle Scholar
Raviart, P.-A. and Thomas, J. M. (1977), A mixed finite element method for 2nd order elliptic problems. In Mathematical Aspects of Finite Element Methods: Proc. Conf., Rome 1975, Vol. 606 of Lecture Notes in Mathematics, Springer, Berlin, pp. 292–315.CrossRefGoogle Scholar
Raviart, P.-A. and Thomas, J.-M. (1983), Introduction à l‘Analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris.Google Scholar
Reitzinger, S. and Schöberl, J. (2002), ‘An algebraic multigrid method for finite element discretizations with edge elements’, Numer. Linear Algebra Appl. 9, 223–238.CrossRefGoogle Scholar
Scholz, R. (1978), ‘A mixed method for 4th order problems using linear finite elements’, RAIRO Anal. Numér. 12, 85–90.CrossRefGoogle Scholar
Simoncini, V. (2003), ‘Algebraic formulations for the solution of the nullspace-free eigenvalue problem using the inexact shift-and-invert Lanczos method’, Numer. Linear Algebra Appl. 10, 357–375.CrossRefGoogle Scholar
Strang, G. and Fix, G. J. (1973), An Analysis of the Finite Element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Stummel, F. (1970), ‘Diskrete Konvergenz linearer Operatoren I’, Math. Ann. 190, 45–92.CrossRefGoogle Scholar
Stummel, F. (1971), ‘Diskrete Konvergenz linearer Operatoren II’, Math. Z. 120, 231–264.CrossRefGoogle Scholar
Stummel, F. (1972), Diskrete Konvergenz linearer Operatoren III. In Linear Operators and Approximation: Proc. Conf., Oberwolfach 1971, Vol. 20 of Internat. Ser. Numer. Math., Birkhäuser, Basel, pp. 196–216.CrossRefGoogle Scholar
Stummel, F. (1980), ‘Basic compactness properties of nonconforming and hybrid finite element spaces’, RAIRO Anal. Numér. 14, 81–115.CrossRefGoogle Scholar
Vardapetyan, L. and Demkowicz, L. (1999), ‘hp-adaptive finite elements in electromagnetics’, Comput. Methods Appl. Mech. Engrg 169, 331–344.CrossRefGoogle Scholar
Vainikko, G. M. (1964), ‘Asymptotic error bounds for projection methods in the eigenvalue problem’, Z. Vyčisl. Mat. i Mat. Fiz. 4, 405–425.Google Scholar
Vainikko, G. M. (1966), ‘On the rate of convergence of certain approximation methods of Galerkin type in eigenvalue problems’, Izv. Vysš. Učebn. Zaved. Matematika 2, 37–45.Google Scholar
Wang, Y., Monk, P. and Szabo, B. (1996), ‘Computing cavity modes using the p version of the finite element method’, IEEE Trans. Magnetics 32, 1934–1940.CrossRefGoogle Scholar
Warburton, T. and Embree, M. (2006), ‘The role of the penalty in the local discontinuous Galerkin method for Maxwell‘s eigenvalue problem’, Comput. Methods Appl. Mech. Engrg 195, 3205–3223.CrossRefGoogle Scholar
Werner, B. (1981), Complementary variational principles and nonconforming Trefftz elements. In Numerical Treatment of Differential Equations, Vol. 3: Clausthal 1980, Vol. 56 of Internat. Schriftenreihe Numer. Math., Birkhäuser, Basel, pp. 180–192.Google Scholar
Whitney, H. (1957), Geometric Integration Theory, Princeton University Press.CrossRefGoogle Scholar
Wong, S. H. and Cendes, Z. J. (1988), ‘Combined finite element-modal solution of three-dimensional eddy current problems’, IEEE Trans. Magnetics 24, 2685–2687.CrossRefGoogle Scholar