Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T08:18:53.852Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  23 January 2015

Na Liu ,
Luis Tobón ,
Yifa Tang  and
Qing Huo Liu
Na Liu
Affiliation:
Institute of Electromagnetics and Acoustics, Xiamen University, Xiamen, 361005, P.R. China
Luis Tobón
Affiliation:
Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA Departamento de Ciencias e Ingeniería de la Computación, Pontificia Universidad Javeriana, Cali, Colombia
Yifa Tang
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China
Qing Huo Liu*
Affiliation:
Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA
*
*Email addresses: [email protected] (N. Liu), [email protected] (L. Tobón), [email protected] (Y. Tang), [email protected] (Q. H. Liu)
Get access

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.

Keywords

78M1078M22Maxwell eigenvaluesspectral element methodspurious eigenvalues
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 458 - 486
Copyright
Copyright © Global Science Press Limited 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Silvester, P., Finite element solution of homogeneous waveguide problems, Alta Frequenza., 38 (1969),313–317.Google Scholar
[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.Google Scholar
[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.Google Scholar
[4]Winkler, J.R. and Davies, J.B., Elimination of spurious modes in finite element analysis, J. Comput. Phys., 56 (1984), 1–14.Google Scholar
[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.Google Scholar
[6]Sun, D., Manges, J., Yuan, X., Z. Cendes Spurious Modes in Finite Element Methods, IEEE Trans. Microw. Theory Tech., 37 (1995), 12–24.Google Scholar
[7]Nédélec, J.C., Mixed finite elements in R3, Numer. Mathem., 35 (1980), 315–341.Google Scholar
[8]Nédélec, J.C., A New Family of Mixed Finite Elements in R3, Numer. Mathem., 50 (1986),57–81.Google Scholar
[9]Mur, G., Edge Elements, their advantages and their disadvantages, IEEE Trans. Magn., 30 (1994), 3552–3557.Google Scholar
[10]Boffi, D., Finite element approximation of eigenvalue problems, Acta Numerica., 19 (2010), 1–120.Google Scholar
[11]Boffi, D., Conforti, M., and Gastaldi, L., Modified edge finite elements for photonic crystals. Numer. Mathem., 105 (2006), 249–266.CrossRefGoogle Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[15]Brenner, S.C., Li, F., Sung, L., Nonconforming Maxwell Eigensolvers, J. Sci. Comput., 40 (2009), 51–85.CrossRefGoogle Scholar
[16]Monk, P., Finite element methods for Maxwell’s equations, Oxford University Press, 2003.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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).Google Scholar
[21]Monk, P., on the p and hp extension of Nedéléc’s curl-conforming elements. J. Comp. Appl. Math., 53 (1994),117–137.Google Scholar
[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.Google Scholar
[23]Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237–339.Google Scholar
[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.Google Scholar
[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.Google Scholar
[26]perso.univ-rennes1.fr/monique.dauge/core/index.html.Google Scholar
[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.Google Scholar
[28]Graglia, R.D., and Lombardi, G., Singular higher order complete vector bases for finite methods, IEEE Trans. Antennas Propagat., 52 (2004), 1672–1685.Google Scholar
[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.Google Scholar