, 55:12 | Cite as

A locking-free stabilized mixed finite element method for linear elasticity: the high order case

  • Bei Zhang
  • Jikun Zhao
  • Shaochun Chen
  • Yongqin Yang


In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\)-order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\)-norm and for the displacement in \(L^2\)-norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.


Elasticity Stabilized mixed finite element method Locking-free method Inf-sup condition 

Mathematics Subject Classification

65N15 65N30 



We would like to thank the anonymous referee. His suggestions help us to better show our results in the current version.


  1. 1.
    Arnold, D.N., Douglas Jr., J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45, 1–22 (1984)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92, 401–419 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefMATHGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M., Marini, L.D.: Mixed finite element methods with continuous stresses. Math. Models Methods Appl. Sci. 3, 275–287 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burman, E., Ern, A.: Continuous interior penalty \(hp\)-finite element methods for advection and advection-diffusion equations. Math. Comput. 76, 1119–1140 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cai, Z., Starke, G.: First-order system least squares for the stress-displacement formulation: linear elasticity. SIAM J. Numer. Anal. 41, 715–730 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cai, Z., Ye, X.: A mixed nonconforming finite element for linear elasticity. Numer. Methods Partial Differ. Equ. 21, 1043–1051 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, L., Hu, J., Huang, X.: Stabilized mixed finite element methods for linear elasticity on simplicial grids in \(\mathbb{R}^n\). Comput. Methods Appl. Math. 17, 17–31 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)MATHGoogle Scholar
  10. 10.
    Girault, V., Raviart, P.: Finite Element for Navier–Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  11. 11.
    Wu, S., Gong, S., Xu, J.: Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor. Math. Models Methods Appl. Sci. 27, 2711–2743 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in \(R^n\): the higher order case. J. Comput. Math. 33, 283–296 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Huang, X., Huang, J.: The compact discontinuous Galerkin method for nearly incompressible linear elasticity. J. Sci. Comput. 56, 291–318 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Karakashian, O., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, M., Shi, D., Dai, Y.: The Brezzi–Pitkäranta stabilization scheme for the elasticity problem. J. Comput. Appl. Math. 286, 7–16 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Bei Zhang
    • 1
  • Jikun Zhao
    • 1
  • Shaochun Chen
    • 1
  • Yongqin Yang
    • 1
  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

Personalised recommendations