Full-wave analysis of dielectric waveguides at a given frequency
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- by L. Vardapetyan and L. Demkowicz PDF
- Math. Comp. 72 (2003), 105-129 Request permission
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 $hp$-edge elements. Numerical experiments fully support theoretical estimates for convergence rates.References
- N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications, Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell; Reprint of the 1961 and 1963 translations; Two volumes bound as one. MR 1255973
- Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607, DOI 10.1090/S0025-5718-99-01013-3
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
- P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235
- Alfredo Bermúdez and Dolores G. Pedreira, Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides, Numer. Math. 61 (1992), no. 1, 39–57. MR 1145906, DOI 10.1007/BF01385496
- D. Boffi, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett. 14 (2001), no. 1, 33–38. MR 1793699, DOI 10.1016/S0893-9659(00)00108-7
- —, “Fortin operator and discrete compactness for edge elements”, Numer. Math. 87, 229-246, 2000.
- Françoise Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. MR 716134
- L. Demkowicz, Asymptotic convergence in finite and boundary element methods. I. Theoretical results, Comput. Math. Appl. 27 (1994), no. 12, 69–84. MR 1284131, DOI 10.1016/0898-1221(94)90087-6
- L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz, de Rham diagram for $hp$ finite element spaces, Comput. Math. Appl. 39 (2000), no. 7-8, 29–38. MR 1746160, DOI 10.1016/S0898-1221(00)00062-6
- 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.
- L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using $hp$-adaptive finite elements, Comput. Methods Appl. Mech. Engrg. 152 (1998), no. 1-2, 103–124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). MR 1602771, DOI 10.1016/S0045-7825(97)00184-9
- Salvatore Caorsi, Paolo Fernandes, and Mirco Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000), no. 2, 580–607. MR 1770063, DOI 10.1137/S0036142999357506
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- 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 (1996), no. 4, 1494–1525. MR 1403556, DOI 10.1137/S0036142993256817
- Fumio Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Proceedings of the first world congress on computational mechanics (Austin, Tex., 1986), 1987, pp. 509–521. MR 912525, DOI 10.1016/0045-7825(87)90053-3
- Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
- F. Kikuchi, “Discrete Compactness of the Linear Rectangular Nedelec Element”, Abstracts of Presentations at 1999 Spring Meeting of Math. Soc. Japan, 108-111, 1999.
- 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.
- J.-F. Lee, “Finite Element Analysis of Lossy Dielectric Waveguides”, IEEE Transactions on Microwave Theory and Techniques, MTT-42, 1025-1031, 1994
- 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.
- Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971, DOI 10.1007/978-3-663-10649-4
- P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in ${\Bbb R}^3$, Math. Comp. 70 (2001), no. 234, 507–523. MR 1709155, DOI 10.1090/S0025-5718-00-01229-1
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
- J. Tinsley Oden and Leszek F. Demkowicz, Applied functional analysis, CRC Series in Computational Mechanics and Applied Analysis, CRC Press, Boca Raton, FL, 1996. MR 1384069
- John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI 10.1090/S0025-5718-1975-0383117-3
- W. Rachowicz and L. Demkowicz, “A Two-Dimensional $hp$-Adaptive Finite Element Package for Electromagnetics”, TICAM Report 98-15, July 1998, accepted, Comput. Methods Appl. Mech. Engrg..
- P. P. Silvester and G. Pelosi (eds.), Finite Elements for Wave Electromagnetics, IEEE Press, NY, 1994.
- L. Vardapetyan, $hp$-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.
- L. Vardapetyan and L. Demkowicz, $hp$-adaptive finite elements in electromagnetics, Comput. Methods Appl. Mech. Engrg. 169 (1999), no. 3-4, 331–344. MR 1675687, DOI 10.1016/S0045-7825(98)00161-3
- Ch. Weber, A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci. 2 (1980), no. 1, 12–25. MR 561375, DOI 10.1002/mma.1670020103
Additional Information
- 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
- Received by editor(s): January 11, 2000
- Received by editor(s) in revised form: February 20, 2001
- Published electronically: May 1, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 105-129
- MSC (2000): Primary 65N30, 35L15
- DOI: https://doi.org/10.1090/S0025-5718-02-01411-4
- MathSciNet review: 1933815