Skip to main content Accesibility Help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Cited by 6
  • Cited by
    Crossref Citations
    Crossref logo
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Liu, Jie Jiang, Wei Lin, Fubiao Liu, Na and Liu, Qing Huo 2017. A Two-Grid Vector Discretization Scheme for the Resonant Cavity Problem With Anisotropic Media. IEEE Transactions on Microwave Theory and Techniques, Vol. 65, Issue. 8, p. 2719.
    Shi, Linlin Liu, Na Zhou, Jianyang Zhou, Yuanguo Wang, Jiamin and Liu, Qing Huo 2016. Spectral element method for band-structure calculations of 3D phononic crystals. Journal of Physics D: Applied Physics, Vol. 49, Issue. 45, p. 455102.
    Shi, Linlin Zhou, Yuanguo Wang, Jia-Min Zhuang, Mingwei Liu, Na and Liu, Qing Huo 2016. Spectral element method for elastic and acoustic waves in frequency domain. Journal of Computational Physics, Vol. 327, Issue. , p. 19.
    Zhou, Yuanguo Shi, Linlin Liu, Na Zhu, Chunhui Liu, Hai and Liu, Qing Huo 2016. Spectral Element Method and Domain Decomposition for Low-Frequency Subsurface EM Simulation. IEEE Geoscience and Remote Sensing Letters, Vol. 13, Issue. 4, p. 550.
    Liu, Na Cai, Guoxiong Zhu, Chunhui Huang, Yueqin and Liu, Qing Huo 2016. The Mixed Finite-Element Method With Mass Lumping for Computing Optical Waveguide Modes. IEEE Journal of Selected Topics in Quantum Electronics, Vol. 22, Issue. 2, p. 187.
    Liu, Na Cai, Guoxiong Zhu, Chunhui Tang, Yifa and Liu, Qing Huo 2015. The Mixed Spectral-Element Method for Anisotropic, Lossy, and Open Waveguides. IEEE Transactions on Microwave Theory and Techniques, Vol. 63, Issue. 10, p. 3094.
    ×
  • Get access
    Add to cart USD35.00
    Check if you have access via personal or institutional login
    Log in Register

Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem

  • Na Liu (a1), Luis Tobón (a2) (a3), Yifa Tang (a4) and Qing Huo Liu (a2)
Abstract

It is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

Copyright
COPYRIGHT: © Global Science Press Limited 2015 
Corresponding author
*Email addresses: liuna@xmu.edu.cn (N. Liu), let12@duke.edu (L. Tobón), tyf@lsec.cc.ac.cn (Y. Tang), qhliu@duke.edu (Q. H. Liu)
References
Hide All
[1]Silvester, P., Finite element solution of homogeneous waveguide problems, Alta Frequenza., 38 (1969),313–317.
[2]Cendes, Z.J., Hudak, D., Lee, J.F., and Sun, D.K, Development of New Methods for Predicting the Bistatic Electromagnetic Scattering from Absorbing Shapes, RADC Final Report, Hansom Air Force Base, MA, 1986.
[3]Rahman, B.M.A., Davies, J.B., Penalty Function Improvement of Waveguide Solution by Finite Elements, IEEE Trans. Microw. Theory Tech., 32 (1984), 922–928.
[4]Winkler, J.R. and Davies, J.B., Elimination of spurious modes in finite element analysis, J. Comput. Phys., 56 (1984), 1–14.
[5]Kobelansky, A.J. and Webb, J.P., Eliminating spurious modes in finite-element waveguide problems by using divergence-free fields, Electron. Lett., 22 (1986), 569–570.
[6]Sun, D., Manges, J., Yuan, X., Z. Cendes Spurious Modes in Finite Element Methods, IEEE Trans. Microw. Theory Tech., 37 (1995), 12–24.
[7]Nédélec, J.C., Mixed finite elements in R3, Numer. Mathem., 35 (1980), 315–341.
[8]Nédélec, J.C., A New Family of Mixed Finite Elements in R3, Numer. Mathem., 50 (1986),57–81.
[9]Mur, G., Edge Elements, their advantages and their disadvantages, IEEE Trans. Magn., 30 (1994), 3552–3557.
[10]Boffi, D., Finite element approximation of eigenvalue problems, Acta Numerica., 19 (2010), 1–120.
[11]Boffi, D., Conforti, M., and Gastaldi, L., Modified edge finite elements for photonic crystals. Numer. Mathem., 105 (2006), 249–266.
[12]D., Boffi, Brezzi, F., Gastaldi, L., On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa Cl. Sci., 25 (1997), 131C154.
[13]Hiptmair, R. and Ledger, P.D., Computation of resonant modes for axisymmetric Maxwell cavities using hp-version edge finite elements. Int. J. Numer. Meth. Engng, 62 (2005), 1652–1676.
[14]Kikuchi, F., Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comput. Methods Appl. Mech. Engrg., 64 (1987), 509–521.
[15]Brenner, S.C., Li, F., Sung, L., Nonconforming Maxwell Eigensolvers, J. Sci. Comput., 40 (2009), 51–85.
[16]Monk, P., Finite element methods for Maxwell’s equations, Oxford University Press, 2003.
[17]Adam, S., Arbenz, P., Geus, R., Arbenz, P., Eigenvalue solvers for electromagnetic fields in cavities, Technical Report 275, Institute of Scientific Computing, ETH zürich, 1997.
[18]Lee, J.H., Xiao, T., and Liu, Q.H., A 3-D Spectral-Element Method Using Mixed-Order Curl Conforming Vector Basis Functions for Electromagnetic Fields, IEEE Trans. Microw. Theory Tech., 54 (2006), 437–444.
[19]Luo, M., Liu, Q.H., and Li, Z., Spectral element method for band structures of two-dimensional anisotropic photonic crystals, Phys. Rev. E 79, 026705 (2009), 1–8.
[20]Liu, N., Tang, Y., Zhu, X., Tobón, L. and Liu, Q.H., Higher-order Mixed Spectral Element Method for Maxwell Eigenvalue Problem, IEEE International Symposium on Antennas and Propagation (APS 2013).
[21]Monk, P., on the p and hp extension of Nedéléc’s curl-conforming elements. J. Comp. Appl. Math., 53 (1994),117–137.
[22]Suri, M., on the stability and convergence of higher-order mixed finite element methods for second-order elliptic problems, Math. Comp., 54 (1990), 1–19.
[23]Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237–339.
[24]Bespalov, A.N., Finite element method for the eigenmode problem of a RF cavity resonator, Sov. J. Numer. Anal. Appl. Math. Model., 3 (1988), 163–178.
[25]Brenner, S.C., Li, F. and Sung, L., A Locally Divergence-free Interior Penalty Method for Two-dimensional Curl-curl Problems, SIAM J. Numer. Anal., 46 (2008), 1190–1211.
[26]perso.univ-rennes1.fr/monique.dauge/core/index.html.
[27]Tanner, Z.P., Savage, S., Tanner, D.R., and Peterson, A.F., Two-dimensional singular vector elements for finite-element analysis, IEEE Trans. Microw. Theory Tech., 46 (1998), 178–184.
[28]Graglia, R.D., and Lombardi, G., Singular higher order complete vector bases for finite methods, IEEE Trans. Antennas Propagat., 52 (2004), 1672–1685.
[29]Peverini, O.A., Addamo, G., Virone, G., Tascone, R., and Orta, R., A Spectral-Element Method for the Analysis of 2-D Waveguide Devices With Sharp Edges and Irregular Shapes, IEEE Trans. Microw. Theory Tech., 59 (2011), 1685–1695.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Cancel
Confirm
×