Skip to main content
SpringerLink
Log in

EQ rot1 nonconforming finite element approximation to Signorini problem

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we apply EQ rot1 nonconforming finite element to approximate Signorini problem. If the exact solution \(u \in H^{\tfrac{5} {2}} \left( \Omega \right)\), the error estimate of order O(h) about the broken energy norm is obtained for quadrilateral meshes satisfying regularity assumption and bi-section condition. Furthermore, the superconvergence results of order \(O\left( {h^{\tfrac{3} {2}} } \right)\) are derived for rectangular meshes. Numerical results are presented to confirm the considered theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Belhachmi Z, Ben Belgacem F. Quadratic finite element approximation of the Signorini problem. Math Comp, 2001, 72: 83–104

    Article  MathSciNet  Google Scholar 

  2. Ben Belgacem F. Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods. SIAM J Numer Math, 2000, 37: 1198–1216

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben Belgacem F, Renard Y. Hybrid finite element methods for the Signorini problem. Math Comp, 2003, 72: 1117–1145

    Article  MathSciNet  MATH  Google Scholar 

  4. Brezzi F, Hager W W, Raviart P A. Error estimates for the finite element solution of variational inequalities, Part I, Primal theory. Numer Math, 1977, 28: 431–443

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciarlet P G. The finite element method for elliptic problem. Amsterdam: North-Holland, 1987

    Google Scholar 

  6. Haslinger J, Hlaváček I, Nečas J. Numerical methods for unilateral problems in solid mechanics. In: Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, vol. 4. Amsterdam: North-Holland, 1996, 313–485

    Google Scholar 

  7. Hua D Y, Wang L H. The nonconforming finite element method for Signorini problem. J Comput Math, 2007, 25: 67–80

    MathSciNet  MATH  Google Scholar 

  8. Kikuchi N, Oden J T. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Philadelphia, PA: SIAM, 1988

    Book  MATH  Google Scholar 

  9. Li M X, Chen H T, Mao S P. Accuracy enhancement for the Signorini problem with finite element method. Acta Math Sci Ser B, 2011, 31: 897–908

    MathSciNet  MATH  Google Scholar 

  10. Li M X, Lin Q, Zhang S H. Superconvergence of finite element method for Signorini problem. J Comput Appl Math, 2008, 222: 284–292

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin Q, Lin J F. Finite Element Methods: Accuracy and Improvement. Beijing: Science Press, 2006

    Google Scholar 

  12. Lin Q, Tobiska L, Zhou A H. Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J Numer Anal, 2005, 25: 160–181

    Article  MathSciNet  MATH  Google Scholar 

  13. Luo F S, Lin Q, Xie H H. Computing the lower and upper bounds of Laplace eigenvalue problem: By combining conforming and nonconforming finite element methods. Sci China Math, 2012, 55: 1069–1082

    Article  MathSciNet  MATH  Google Scholar 

  14. Mao S P, Shi Z C. On the error bounds of nonconforming finite elements. Sci China Math, 2010, 53: 2917–2926

    Article  MathSciNet  MATH  Google Scholar 

  15. Ming P B, Shi Z C. Quadrilateral mesh. Chin Ann Math Ser B, 2002, 23: 235–252

    Article  MathSciNet  MATH  Google Scholar 

  16. Risch Uwe. Superconvergence of a nonconforming low order finite element. Appl Numer Math, 2005, 54: 324–338

    Article  MathSciNet  Google Scholar 

  17. Scarpini F, Vivaldi M A. Error estimates for the approximation of some unilateral problems. RAIRO Model Math Anal Numer, 1977, 11: 197–208

    MathSciNet  MATH  Google Scholar 

  18. Shi D Y, Mao S P, Chen S C. A class of anisotropic Crouzeix-Raviart type finite element approximations to Signorini variational inequality problem. Math Numer Sin, 2005, 27: 45–54

    MathSciNet  Google Scholar 

  19. Shi D Y, Mao S P, Chen S C. Anisotropic nonconforming finite element with some superconvergence results. J Comput Math, 2005, 23: 261–274

    MathSciNet  MATH  Google Scholar 

  20. Shi D Y, Mao S P, Chen S C. A locking-free anisotropic nonconforming rectangular finite element for planar linear elasticity problems. Acta Math Sci Ser B, 2007, 27: 193–202

    MathSciNet  MATH  Google Scholar 

  21. Shi D Y, Pei L F. Low order Crouzeix-Raviart type nonconforming finite element methods for approximating Maxwell’s equations. Int J Numer Anal Model, 2008, 5: 373–385

    MathSciNet  MATH  Google Scholar 

  22. Shi D Y, Peng Y C, Chen S C. Superconvergence of a nonconforming finite element approximation to Viscoelasticity type equations on anisotropic meshes. Numer Math J Chin Univ, 2006, 15: 375–384

    MathSciNet  MATH  Google Scholar 

  23. Shi D Y, Ren J C. Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes. Nonlinear Anal, 2009, 71: 3842–3852

    Article  MathSciNet  MATH  Google Scholar 

  24. Shi D Y, Ren J C, Gong W. Convergence and superconvergence analysis of a nonconforming finite element method for solving the Signorini problem. Nonlinear Anal, 2012, 75: 3493–3502

    Article  MathSciNet  MATH  Google Scholar 

  25. Shi D Y, Wang H H, Du Y P. An anisotropic nonconforming finite element method for approximating a class of nonlinear Sobolev equations. J Comput Math, 2009, 27: 299–314

    MathSciNet  MATH  Google Scholar 

  26. Shi D Y, Wang X L. Two low order characteristic finite element methods for a convection-dominated transport problem. Comput Math Appl, 2010, 59: 3630–3639

    Article  MathSciNet  MATH  Google Scholar 

  27. Shi Z C. A convergence condition for the quadrilateral Wilson element. Numer Math, 1984, 44: 349–361

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu X J. On the accuracy of nonconforming quadrilateral Q 1 element approximation for the Navier-Stokes problem. SIAM J Numer Anal, 2000, 38: 17–39

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang Y D, Zhang Z M, Lin F B. Eigenvalue approximation from below using non-conforming finite elements. Sci China Math, 2010, 53: 137–150

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, China

    DongYang Shi

  2. Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang, 471023, China

    Chao Xu

Authors
  1. DongYang Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to DongYang Shi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shi, D., Xu, C. EQ rot1 nonconforming finite element approximation to Signorini problem. Sci. China Math. 56, 1301–1311 (2013). https://doi.org/10.1007/s11425-013-4615-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4615-z

Keywords

MSC(2010)

Search

Navigation