Advances in Computational Mathematics

, Volume 19, Issue 1–3, pp 277–291 | Cite as

A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity

  • Chang-Ock Lee
  • Jongwoo Lee
  • Dongwoo Sheen


We propose a locking-free nonconforming finite element method based on quadrilaterals to solve for the displacement variable in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is optimal and robust in the sense that the convergence estimates in the energy and L2-norms are independent of the Lamé parameter λ.

nonconforming finite element method planar linear elasticity locking effects 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D.N. Arnold, Discretization by finite elements of a model parameter dependent problem, Numer. Math. 37 (1981) 405–421.Google Scholar
  2. [2]
    D.N. Arnold, D. Boffi, R.S. Falk and L. Gastaldi, Finite element approximation on quadrilateral meshes, Comm. Numer. Methods Engrg. 17 (2001) 805–812.Google Scholar
  3. [3]
    D.N. Arnold, F. Brezzi and J. Douglas, Jr., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984) 347–367.Google Scholar
  4. [4]
    I. Babuška and M. Suri, Locking effect in the finite element approximation of elasticity problem, Numer. Math. 62 (1992) 439–463.Google Scholar
  5. [5]
    I. Babuška and M. Suri, On locking and robustness in the finie element method, SIAM J. Numer. Anal. 29 (1992) 1261–1293.Google Scholar
  6. [6]
    D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics (Cambridge Univ. Press, Cambridge, 1997).Google Scholar
  7. [7]
    S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin/New York, 1994).Google Scholar
  8. [8]
    S. Brenner and L. Sung, Linear finite element methods for planar elasticity, Math. Comp. 59 (1992) 321–338.Google Scholar
  9. [9]
    Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: A correction, Calcolo 37 (2000) 253–254.Google Scholar
  10. [10]
    Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations, Calcolo 36 (1999) 215–232.Google Scholar
  11. [11]
    X. Cheng, W. Han and H. Huang, Finite element methods for Timoshenko beam, circular arch and Reissner–Mindlin plate problems, J. Comput. Appl. Math. 79(2) (1997) 215–234.Google Scholar
  12. [12]
    P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).Google Scholar
  13. [13]
    M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Modél. Math. Anal. Numér. 3 (1973) 33–75.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Chang-Ock Lee
    • 1
  • Jongwoo Lee
    • 2
  • Dongwoo Sheen
    • 3
  1. 1.Division of Applied MathematicsKAISTDaejeonKorea
  2. 2.Department of MathematicsKwangwoon UniversitySeoulKorea
  3. 3.Department of MathematicsSeoul National UniversitySeoulKorea

Personalised recommendations