Approximation of eigenvalues in mixed form, Discrete Compactness Property, and application to hp mixed finite elements

Computer Methods in Applied Mechanics and Engineering

Volumn 196, Issue 37-40 SPEC. ISS., 2007, Pages 3672-3681

  Boffi, Daniele ^a  

^a UNIVERSITY OF PAVIA   (Italy)

Author keywords

Discrete compactness; Eigenvalues approximation; Mixed finite elements

Indexed keywords

APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; NUMERICAL ANALYSIS;

DISCRETE COMPACTNESS PROPERTY; EIGENVALUES APPROXIMATION; HP FINITE ELEMENTS; MAXWELL'S EIGENVALUES; MIXED FINITE ELEMENTS;

COMPUTATIONAL MECHANICS;

APPROXIMATION THEORY; COMPUTATIONAL MECHANICS; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; NUMERICAL ANALYSIS;

EID: 34447096192     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2006.10.024     Document Type: Article

References (34)

1. Anselone P.M. Collectively Compact Operator Approximation Theory and Applications to Integral Equations (1971), Prentice-Hall Inc., Englewood Cliffs, NJ (With an appendix by Joel Davis, Prentice-Hall Series in Automatic Computation)
2. Arnold D.N. Differential complexes and numerical stability. Proceedings of International Congress on Mathematicians vol. I (2002), Higher Ed. Press, Beijing 137-157
3. Arnold D.N., Falk R.S., and Winther R. Differential complexes and stability of finite element methods. I. The de rham complex. In: Arnold D.N., Bochev P.B., Lehoucq R.B., Nicolaides R.A., and Shashkov M. (Eds). Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications vol. 142 (2006), Springer, Berlin 23-46
4. Babuška I., and Osborn J.E. Eigenvalue problems. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of Numerical Analysis vol. II (1991), Elsevier Science Publishers B.V., North Holland 641-788
5. Bermúdez A., Durán R., Muschietti M.A., Rodríguez R., and Solomin J. Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32 4 (1995) 1280-1295
6. Boffi D. Fortin operator and discrete compactness for edge elements. Numer. Math. 87 2 (2000) 229-246
7. Boffi D. A note on the de Rham complex and a discrete compactness property. Appl. Math. Lett. 14 1 (2001) 33-38
8. Boffi D. Compatible discretizations for eigenvalue problems. In: Arnold D.N., Bochev P.B., Lehoucq R.B., Nicolaides R.A., and Shashkov M. (Eds). Compatible Spatial Discretizations. The IMA Volumes in Mathematics and ist Applications vol. 142 (2006), Springer, Berlin 121-142
9. Boffi D., Brezzi F., and Gastaldi L. On the convergence of eigenvalues for mixed formulations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 1-2 (1998) 131-154 (1997, Dedicated to Ennio De Giorgi)
10. Boffi D., Brezzi F., and Gastaldi L. On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 229 (2000) 121-140
11. Boffi D., Chinosi C., and Gastaldi L. Approximation of grad-div operator in non-convex domains. Comput. Model. Eng. Sci. (CMES) 1 (2000) 27-38
12. Boffi D., Costabel M., Dauge M., and Demkowicz L. Discrete compactness for the hp version of rectangular edge finite elements. SIAM J. Numer. Anal. 44 3 (2006) 979-1004
13. Boffi D., Fernandes P., Gastaldi L., and Perugia I. Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 4 (1999) 1264-1290
14. Bramble J.H., Kolev T.V., and Pasciak J.E. A least-squares approximation method for the time-harmonic Maxwell equations. J. Numer. Math. 13 4 (2005) 237-263
15. Caorsi S., Fernandes P., and Raffetto M. Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements. Math. Model. Numer. Anal. (M2AN) 35 2 (2001) 331-354
16. Chatelin F. Spectral approximation of linear operators. Computer Science and Applied Mathematics (1983), Academic Press Inc./Harcourt Brace Jovanovich Publishers, New York (With a foreword by P. Henrici, With solutions to exercises by Mario Ahués)
17. Costabel M., and Dauge M. Maxwell and Lamé eigenvalues on polyhedra. Math. Methods Appl. Sci. 22 (1999) 243-258
18. Costabel M., and Dauge M. Wheighted regularization of Maxwell equations in polyhedral domains. Numer. Math 93 2 (2002) 239-277
19. L. Demkowicz, L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Symposium on Advances in Computational Mechanics, vol. 5 (Austin, TX, 1997), Comput. Methods Appl. Mech. Engrg. 152 (1-2) (1998) 103-124.
20. Descloux J., Nassif N., and Rappaz J. On spectral approximation, part I: the problem of convergence. RAIRO Numer. Anal. 12 (1978) 97-112
21. Douglas Jr. J., and Roberts J.E. Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44 169 (1985) 39-52
22. Fernandes P., and Gilardi G. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 7 (1997) 957-991
23. 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, TX, 1986), vol. 64 (1-3), 1987, pp. 509-521.
24. Kikuchi F. On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 3 (1989) 479-490
25. 3. Math. Comput. 70 234 (2001) 507-523
26. 3. Numer. Math. 39 1 (1982) 97-112
27. 3. Numer. Math. 50 1 (1986) 57-81
28. Osborn J.E. Spectral approximation for compact operators. Math. Comput. 29 (1975) 712-725
29. Stummel F. Diskrete Konvergenz linearer Operatoren. Math Ann. 190 (1970/1971) 45-92
30. J.M. Thomas, Sur l'analyse numerique des methodes d'elements finis hybrides et mixtes. Ph. D. Thesis, Université Pierre et Marie Curie, 1977.
31. Vardapetyan L., and Demkowicz L. hp-adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 3-4 (1999) 331-344
32. Vardapetyan L., and Demkowicz L. Full-wave analysis of dielectric waveguides at a given frequency. Math. Comput. 72 241 (2003) 105-129 electronic
33. Vaǐnikko G.M. Discretely compact sequences. Ž. Vyčisl. Mat. i Mat. Fiz. 14 (1974) 575-583 (Russian)
34. Weber C. A local compactness theorem for Maxwell's equations. Math. Methods Appl. Sci. 2 1 (1980) 12-25