Journal of Scientific Computing

, Volume 58, Issue 3, pp 574–591 | Cite as

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

  • Jun HuEmail author
  • Yunqing Huang
  • Quan Shen


This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated \(Q_1\) element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.


Lower bound Upper bound Eigenvalue Nonconforming finite element 


  1. 1.
    Armentano, M.G., Duran, R.G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA 17, 93–101 (2004)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 641–787. North-Holland, Amsterdam (1991)Google Scholar
  3. 3.
    Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numerica 19, 1–120 (2010)Google Scholar
  4. 4.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM Classics in Applied Mathematics, Philadelphia (2002)CrossRefGoogle Scholar
  5. 5.
    Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7, 33–76 (1973)MathSciNetGoogle Scholar
  6. 6.
    Hu, J., Huang, Y.Q., Lin, Q.: The lower bounds for eigenvalues of elliptic operators-by nonconforming finite element methods. arXiv:1112.1145v1[math.NA] (2011)Google Scholar
  7. 7.
    Hu, J., Huang, Y.Q., Shen, Q.: A high accuracy post-processing algorithm for the eigenvalues of elliptic operators. J. Sci. Comput. 52, 426–445 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Hu, J., Huang, Y.Q., Shen, Q.: Constructing both lower and upper bounds for the eigenvalues of the elliptic operators by the nonconforming element (under review)Google Scholar
  9. 9.
    Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the \(P_0\) and \(P_1\) triangular finite elements. Comput. Methods Appl. Mech. Eng. 196, 3750–3758 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Li, Y.A.: Lower approximation of eigenvalue by the nonconforming finite element method. Math. Numer. Sin. 30, 195–200 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Lin, Q., Xie, H.H., Luo, F.S., Li, Y., Yang, Y.D.: Stokes eigenvalue approximations from below wirh nonconforming mixed finite element methods. Math. Pract. Theory 40, 157–168 (2010)MathSciNetGoogle Scholar
  12. 12.
    Lin, Q., Xie, H.H., Xu, J.C.: Lower Bounds of the Discretization for Piecewise Polynomials. arXiv:1106.4395v1 [Math.NA] (22 Jun 2011)Google Scholar
  13. 13.
    Liu, H.P., Yan, N.N.: Four finite element solutions and comparison of problem for the poisson equation eigenvalue. Chin. J. Numer. Meth. Comput. Appl. 2, 81–91 (2005)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Morley, L.S.D.: The triangular equilibrium element in the solutions of plate bending problem. Aero. Quart. 19, 149–169 (1968)Google Scholar
  15. 15.
    Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33, 23–42 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods PDEs 8, 97–111 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Shen, Q.: High-Accuracy Algorithms for the Eigenvalue Problems of the Elliptic Operators and the Vibration Frequencies of the Cavity Flow (In Chinese). PhD. Dissertation in School of Mathematical Science, Peking University (June 2012)Google Scholar
  18. 18.
    Shi, Z.C., Wang, M.: The Finite Element Method (In Chinese). Science Press, Beijing (2010)Google Scholar
  19. 19.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  20. 20.
    Wang, M., Shi, Z.C., Xu, J.C.: A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer. Math. 106, 335–347 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Yang, Y.D.: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comp. Math. 18, 413–418 (2000)zbMATHGoogle Scholar
  22. 22.
    Yang, Y.D., Lin, Q., Bi, H., Li, Q.: Eigenvalue approximations from below using Morley elements. Adv. Comput. Math. 36, 443–450 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Yang, Y.D., Zhang, Z.M., Lin, F.B.: Eigenvalue approximation from below using nonforming finite elements. Sci. China: Math. 53, 137–150 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Zhang, Z., Yang, Y., Chen, Z.: Eigenvalue approximation from below by Wilson’s elements. Chin. J. Numer. Math. Appl. 29, 81–84 (2007)Google Scholar
  25. 25.
    Zienkiewicz, O.C., Cheung, Y.K.: The Finite Element Method in Structrural and Continuum Mechanics. McGraw-Hill, New York (1967)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LMAM and School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  2. 2.Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtanPeople’s Republic of China

Personalised recommendations