Mathematics of Computation

Author(s): L. Vardapetyan; L. Demkowicz.
Journal: Math. Comp. 72 (2003), 105-129.
MSC (2000): Primary 65N30, 35L15
Posted: May 1, 2002
Retrieve article in: PDF DVI PostScript

Abstract: New variational formulation to compute propagation constants is proposed. Based on it, vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property. Convergence analysis is conducted in the framework of collectively compact operators. Reported theoretical results apply to a wide class of vector finite elements including two families of Nedelec and their generalization, the

-edge elements. Numerical experiments fully support theoretical estimates for convergence rates.

1.: N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover, NY, 1993. MR 94i:47001
2.: A. Alonso and A. Valli, ``An Optimal Domain Decomposition Preconditioner for Low-Frequency Time-Harmonic Maxwell Equations'', Math. Comp., 68, 226, 607-631, April 1999. MR 99i:78002
3.: D. N. Arnold, R. S. Falk, and R. Winther, ``Multigrid in H(div) and H(curl)'', Numer. Anal., 85, 197-217, 2000. MR 2001d:65161
4.: I. Babuska, ``Error Bounds for Finite Element Method'', Numer. Math., 16, 322-333, 1971. MR 44:6166
5.: I. Babuska and A. Aziz, ``Survey Lectures on the Mathematical Foundations of the Finite Element Method'', in A. K. Aziz, ed., The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, 5-359, Academic Press, NY, 1973, pp. 1-359. MR 54:9111
6.: I. Babuska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis, vol.2, Elsevier-North Holland, Amsterdam, 1991, pp. 641-787. MR 92f:65001
7.: A. Bermúdez and D. G. Pedreira, ``Mathematical Analysis of a Finite Element Method without Spurious Solutions for Computation of Dielectric Waveguides'', Numer. Math., 61, 39-57, 1992. MR 92m:65139
8.: D. Boffi, ``A note on the de Rham complex and a discrete compactness property'', Appl. Math. Lett. 14, 33-38, 2001. MR 2001g:65145
9.: -, ``Fortin operator and discrete compactness for edge elements'', Numer. Math. 87, 229-246, 2000. CMP 2001:06
10.: F. Chatelin Spectral Approximations of Linear Operators, Academic Press, NY, 1983. MR 86d:65071
11.: L. Demkowicz, ``Asymptotic Convergence in Finite and Boundary Element Methods: Part 1: Theoretical Results", Comput. Math. Appl., 27, 12, 69-84, 1994. MR 95h:65080
12.: L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz, ``De Rham Diagram for Finite Element Spaces'', Comput. Math. Appl., 39, no. 7/8, 29-38, 2000. MR 2000m:78052
13.: L. Demkowicz, P. Monk, Ch. Schwab, and L. Vardapetyan, ``Maxwell Eigenvalues and Discrete Compactness in Two Dimensions'', Comput. Math. Appl., 40, no. 4/5, 589-605, 2000. CMP 2000:16
14.: L. Demkowicz, L. Vardapetyan , ``Modeling of Electromagnetic Absoption/Scattering Problems Using -adaptive Finite Elements'', Comput. Methods Appl. Mech. Engrg., 152, 1-2, 103-124, 1998. MR 99b:78003
15.: S. Caorsi, P. Fernandes, M. Raffetto, ``On the Convergence of Galerkin Finite Element Approximations of Electromagnetic Eigenproblems'', SIAM J. Numer. Anal., 38, 580-607, 2000. MR 2001e:65172
16.: V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. MR 88b:65129
17.: P. Joly, C. Poirier, J. E. Roberts, and P. Trouve, ``A New Nonconforming Finite Element Method for the Computation of Electromagnetic Guided Waves I: Mathematical Analysis'', SIAM J. Numer. Anal., 33 4, 1494-1525, 1996. MR 97f:78035
18.: F. Kikuchi, ``Mixed and Penalty Formulations for Finite Element Analysis of an Eigenvalue Problem in Electromagnetism'', Comput. Methods Appl. Mech. Engrg., 64, 509-521, 1987. MR 89g:78005
19.: F. Kikuchi, ``On a Discrete Compactness Property for the Nedelec Finite Elements'', J. Fac. Sci., Univ. Tokyo, Sect. IA Math, 36, 479-490, 1989. MR 91h:65173
20.: F. Kikuchi, ``Discrete Compactness of the Linear Rectangular Nedelec Element'', Abstracts of Presentations at 1999 Spring Meeting of Math. Soc. Japan, 108-111, 1999.
21.: F. Kikuchi, M. Yamamoto, H. Fujio, ``Theoretical and Computational Aspects of Nedelec's Edge Elements for Electromagnetics'', in Computational Mechanics - New Trends and Applications, Eds.:E. Oñate and S. R. Idelsohn, CIMNE, Barcelona, Spain, 1998.
22.: J.-F. Lee, ``Finite Element Analysis of Lossy Dielectric Waveguides'', IEEE Transactions on Microwave Theory and Techniques, MTT-42, 1025-1031, 1994
23.: J.F. Lee, D.K. Sun, and Z.J. Cendes, ``Full-Wave Analysis of Dielectric Waveguides Using Tangential Vector Finite Elements'', IEEE Transactions on Microwave Theory and Techniques, 39, 8, 1991.
24.: R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley and Sons, New York 1986. MR 87h:35003
25.: P. Monk and L. Demkowicz, ``Discrete Compactness and the Approximation of Maxwell's Equations in $\mathbb{R}^3$ '', Math. Comp., 70, 507-523, 2001. MR 2001g:65156
26.: J.C. Nedelec, ``Mixed Finite Elements in $\mathbb{R}^3$ '', Numer. Math., 35, 315-341, 1980. MR 81k:65125
27.: J.C. Nedelec, ``A New Family of Mixed Finite Elements in $\mathbb{R}^3$ '', Numer. Math., 50, 57-81, 1986. MR 88e:65145
28.: J.T. Oden and L.F. Demkowicz, Applied Functional Analysis for Science and Engineering, CRC Press, Boca Raton, 1996. MR 97h:00001
29.: J. E. Osborn, ``Spectral Approximation for Compact Operators'', Math. Comp., 29, 131, 712-725, 1975. MR 52:3998
30.: W. Rachowicz and L. Demkowicz, ``A Two-Dimensional -Adaptive Finite Element Package for Electromagnetics'', TICAM Report 98-15, July 1998, accepted, Comput. Methods Appl. Mech. Engrg..
31.: P. P. Silvester and G. Pelosi (eds.), Finite Elements for Wave Electromagnetics, IEEE Press, NY, 1994.
32.: L. Vardapetyan, -Adaptive Finite Element Method for Electromagnetics with Applications to Waveguiding Structures, Ph.D. thesis, Graduate School of The University of Texas at Austin, December 1999.
33.: L. Vardapetyan and L. Demkowicz, ``-Adaptive Finite Elements in Electromagnetics'', Comput. Methods Appl. Mech. Engrg., 169, 331-344, 1999. MR 99k:78004
34.: C. Weber, ``A Local Compactness Theorem for Maxwell's Equations'', Math. Meth. Appl Sci., 2, 12-25, 1980. MR 81f:78005

L. Vardapetyan
Affiliation: The Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Taylor Hall 2.400, Austin, Texas 78712
Email: leonv@research.bell-labs.com

L. Demkowicz
Affiliation: The Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Taylor Hall 2.400, Austin, Texas 78712
Email: leszek@ticam.utexas.edu

DOI: 10.1090/S0025-5718-02-01411-4
PII: S 0025-5718(02)01411-4
Keywords: Maxwell's equations, waveguide eigenmodes, full-wave analysis, $hp$ finite elements
Received by editor(s): January 11, 2000
Received by editor(s) in revised form: February 20, 2001
Posted: May 1, 2002
Copyright of article: Copyright 2002, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS