Discrete compactness for the hp version of rectangular edge finite elements

SIAM Journal on Numerical Analysis

Volumn 44, Issue 3, 2006, Pages 979-1004

  Boffi, Daniele ^a   Costabel, Martin ^b   Dauge, Monique ^b   Demkowicz, Leszek ^c  

^a UNIVERSITY OF PAVIA   (Italy)

^b UNIVERSITÉ DE RENNES 1   (France)

^c UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Discrete compactness; Edge elements; Eigenvalue approximation; hp finite elements; Maxwell equations

Indexed keywords

DISCRETE COMPACTNESS; EDGE ELEMENTS; EIGENVALUE APPROXIMATION; HP FINITE ELEMENTS;

ADAPTIVE SYSTEMS; APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; FINITE ELEMENT METHOD; MAXWELL EQUATIONS;

FINITE DIFFERENCE METHOD;

EID: 34447094722     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/04061550X     Document Type: Article

References (43)

1. M. AINSWORTH AND K. PINCHEDEZ, hp-approximation theory for BDFM and RT finite elements on quadrilaterals, SIAM J. Numer. Anal., 40 (2002), pp. 2047-2068.
2. D. N. ARNOLD, Differential Complexes and Numerical Stability, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Education Press, Beijing, China, 2002, pp. 137-157.
3. D. N. ARNOLD, D. BOFFI, AND R. S. FALK, Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 2004, 42 (2005), pp. 2429-245.
4. I. BABUŠKA AND B. Q. GUO, Regularity of the solution of elliptic problems with piecewise analytic data. Part I. Boundary value problems for linear elliptic equation of second order, SIAM J. Math. Anal., 19 (1988), pp. 172-203.
5. I. BABUSKA AND J. OSBORN, Eigenvalue Problems, in Handbook of Numerical Analysis, Vol. II, Handb. Numer. Anal. II, North-Holland, Amsterdam, 1991, pp. 641-787.
6. D. BOFFI, Fortin operator and discrete compactness for edge elements, Numer. Math., 87 (2000), pp. 229-246.
7. D. BOFFI, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett., 14 (2001), pp. 33-38.
8. D. BOFFI, L. DEMKOWICZ, AND M. COSTABEL, Discrete compactness for p and hp 2D edge finite elements, Math. Models Methods Appl. Sci., 13 (2003), pp. 1673-1687.
9. 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.
10. D. BOFFI AND L. GASTALDI, Edge finite elements for the approximation of Maxwell resolvent operator, M2AN Math. Model. Numer. Anal-, 36 (2002), pp. 293-305.
11. D. BOFFI, F. KIKUCHI, AND J. SCHÖBERL, Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes, Math. Models Methods Appl. Sci., 16 (2006), pp. 265-273.
12. A. BOSSAVIT, Mixed finite elements and the complex of Whitney forms, in The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 137-144.
13. A. BOSSAVIT, Un nouveau point de vue sur les éléments finis mixtes, Matapli (Bulletin de la Société de Mathématiques Appliquëes et Industrielles), 20 (1989), pp. 23-35.
14. A. BUFFA, M. COSTABEL, AND M. DAUGE, Algebraic convergence for anisotropic edge elements in polyhedral domains, Numer. Math., 101 (2005), pp. 29-65.
15. A. BUFFA, R. HIPTMAIR, T. VON PETERSDORFF, AND C. SCHWAB, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math., 95 (2003), pp. 459-485.
16. S. CAORSI, P. FERNANDES, AND M. RAFFETTO, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal., 38 (2000), pp. 580-607.
17. F. CHATELIN, Spectral Approximation of Linear Operators, Academic Press, New York, 1983.
18. M. COSTABEL AND M. DAUGE, Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal., 151 (2000), pp. 221-276.
19. M. COSTABEL AND M. DAUGE, Computation of resonance frequencies for Maxwell equations in non-smooth domains, in Topics in Computational Wave Propagation, Lect. Notes Comput. Sci. Eng. 31, Springer, Berlin, 2003, pp. 125-161.
20. M. COSTABEL, M. DAUGE, AND S. NICAISE, Singularities of Maxwell interface problems, M2AN Math. Model. Numer. Anal., 33 (1999), pp. 627-649.
21. M. COSTABEL, M. DAUGE, AND C. SCHWAB, Exponential convergence of hp-F EM for Maxwell's equations with weighted regularization in polygonal domains, Math. Models Methods Appl. Sci., 15 (2005), pp. 575-622.
22. M. COSTABEL, M. DAUGE, AND C. SCHWAB, Exponential convergence of the weighted regularization method for Maxwell eigenvalue problems, in Proceedings of the International Conference on Electromagnetics in Advanced Applications (ICEAA), Torino, Italy, 2005.
23. L. DEMKOWICZ, Fully automatic hp-adaptivity for Maxwell's equations, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 605-624.
24. L. DEMKOWICZ AND I. BABUŠKA, p interpolation error estimates for edge finite elements of variable order in two dimensions, SIAM J. Numer. Anal., 41 (2003), pp. 1195-1208.
25. 1, H(curl) and H (div]-conforming projection-based interpolation in three dimensions, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 267-296.
26. L. DEMKOWICZ, P. MONK, L. VARDAPETYAN, AND W. RACHOWICZ, de Rham diagram for hp finite element spaces, Comput. Math. Appl., 39 (2000), pp. 29-38.
27. 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.
28. J. DESCLOUX, N. NASSIF, AND J. RAPPAZ, On spectral approximation. I. The problem of convergence, RAIRO Anal. Numér., 12 (1978), pp. 97-112, iii.
29. B. M. DILLON, P. T. S. LIU, AND J. P. WEBB, Spurious modes in quadrilateral and triangular edge elements, COMPEL, 13, Suppl. A (1994), pp. 311-316.
30. P. FERNANDES AND M. RAFFETTO, Counterexamples to the currently accepted explanation for spurious modes and necessary and sufficient conditions to avoid them, IEEE Trans, on Magnetics, 38 (2002), pp. 653-656.
31. F. GARDINI, Discrete compactness property for quadrilateral finite element spaces, Numer. Methods Partial Differential Equations, 21 (2005), pp. 41-56.
32. P. GRISVARD, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992.
33. R. HIPTMAIR, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), pp. 237-339.
34. J. DOUGLAS, JR., AND J.E. ROBERTS, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput., 1 (1982), pp. 91-103.
35. F. KIKUCHI, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), pp. 479-490.
36. W. MCLEAN, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.
37. P. MONK, On the p- and hp-extension of Nédé lec's curl-conforming elements, J. Comput. Appl. Math., 53 (1994), pp. 117-137.
38. P. MONK, Finite element methods for Maxwell's equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.
39. 3, Math. Comp., 70 (2000), pp. 507-523.
40. 3, Numer. Math., 35 (1980), pp. 315-341.
41. 3, Numer. Math., 50 (1986), pp. 57-81.
42. W. RACHOWICZ AND L. DEMKOWICZ, An hp-adaptive finite element method for electromagnetics. Part I. Data structure and constrained approximation, Comput. Methods Appl. Mech. Engrg., 187 (2000), pp. 307-337.
43. CH. SCHWAB, p- and hp-finite element methods. Theory and applications in solid and fluid mechanics, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.