Discrete compactness and the approximation of Maxwell's equations in ℝ3

Mathematics of Computation

Volumn 70, Issue 234, 2001, Pages 507-523

  Monk, P ^a   Demkowicz, L ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Discrete compactness; Eigenvalues; Error estimates; Finite element methods

Indexed keywords

EID: 0035318036     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-00-01229-1     Document Type: Article

References (29)

1. A. ALONSO AND A. VALLI, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp, 68 (1999), 607-631. MR 99i:78002
2. P. ANSELONE, Collectively compact operator approximation theory, Prentice-Hall, 1971. MR 56:1753
3. 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), 131-154. MR 99i:65121
4. _, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. to appear. CMP 98:17
5. J. BRAMBLE, J. PASCIAK, AND J. XU, The analysis of multigrid algorithms with nonnested or noninherited quadratic forms, Math. Comp. 56 (1991), pp. 1-34. MR 91h:65159
6. M. COSTABEL, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci., 12 (1990), pp. 365-368. MR 91c:35028
7. M. COSTABEL AND M. DAUGE, Singularities of electromagnetic fields in polyhedral domains, tech. report, IRMAR, Université de Rennes 1, France, 1997. Report available at http://www.maths.univ-rennes1.fr/~costabel/.
8. L. DEMKOWICZ, Asymptotic convergence in finite and boundary element methods. Part 1: Theoretical results, Comput. Math. Appl. 27 (1994), pp. 69-84. MR 95h:70653
9. L. DEMKOWICZ AND L. VARDAPETYAN, Modelling electromagnetic absorbtion/scattering problems using hp-adaptive finite elements, Comput. Methods Appl. Mech. Engrg., 152 (1998), pp. 103-124. MR 99b:78003
10. 3, Math. Comp., 51 (1988), pp. 55-74. MR 90e:65155
11. 3, in The Navier-Stokes equations, Theory and Numerical Methods, Lecture Notes 1431, Springer, 1990, pp. 201-218. MR 91k:65143
12. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer Verlag, New York, 1986. MR 83b:65122
13. H. IHLENBURG AND I. BABUŠKA, Finite element solution of the Helmholtz equation with high wavenumber Part I: The h-version of the FEM, Comput. Math. Appl., 30 (1995), pp. 9-37. MR 96j:65123
14. P. JOLY, C. POIRIER, J.-E. ROBERTS, AND P. TROUVE, A new non-conforming finite element method for computation of electromagnetic guided waves, SIAM J. Numer. Anal., 33 (1996), 1494-1525. MR 97f:78035
15. 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. MR 88k:46028
16. _, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comput. Methods Appl. Mech. Eng., 64 (1987), pp. 509-521. MR 89g:78005
17. _, On a discrete compactness property for the Nedelec finite elements, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math, 36 (1989), pp. 479-490. MR 91h:65173
18. R. KRESS, Linear Integral Equations, Springer-Verlag, second ed., 1999. MR 90j:45001
19. M. KŘÍŽEK AND P. NEITTAANMÄKI, On the validity of Friedrichs' inequalities, Math. Scand., 54 (1984), pp. 17-26. MR 86b:35020
20. R. LEIS, Initial Boundary Value Problems in Mathematical Physics, John Wiley, New York, 1986. MR 87h:35003
21. V. LEVILLAIN, Eigenvalue appoximation by mixed methods for resonant inhomogenous cavities with metallic boundaries, Math. Comp., 58 (1992), pp. 11-20. MR 92f:65130
22. P. MONK, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math., 63 (1992), pp. 243-261. MR, 94b:65134
23. 3, Numer. Math., 35 (1980), pp. 315-341. MR 81k:65125
24. 3, Numer. Math., 39 (1982), pp. 97-112. MR 83g:65111
25. 3, Numer. Math., 50 (1986), pp. 57-81. MR 88e:65145
26. J. OSBORN, Spectral approximation for compact operators, Mathematics of Computation, 29 (1975), pp. 712-725. MR 52:3998
27. W. RACHOWICZ AND L. DEMKOWICZ, A two-dimensional hp-adaptive finite element package for electromagnetics, Tech. Report 98-15, TICAM, University of Texas, 1998. submitted to Computer Methods in Applied Mechanics and Engineering.
28. A. SCHATZ, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), pp. 959-962. MR 51:9526
29. L. VARDAPETYAN AND L. DEMKOWICZ, hp-Adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg., 169 (1999), pp. 331-344. MR 99k:78004