Advertisement

Journal of Scientific Computing

, Volume 56, Issue 3, pp 637–653 | Cite as

Anisotropic Nonconforming \({ EQ}_1^{rot}\) Quadrilateral Finite Element Approximation to Second Order Elliptic Problems

  • Dong-yang Shi
  • Chao Xu
  • Jin-huan Chen
Article

Abstract

The main aim of this paper is to study the nonconforming \(EQ_1^{rot}\) quadrilateral finite element approximation to second order elliptic problems on anisotropic meshes. The optimal order error estimates in broken energy norm and \(L^2\)-norm are obtained, and three numerical experiments are carried out to confirm the theoretical results.

Keywords

Anisotropy Nonconforming \(EQ_1^{rot}\) quadrilateral element  Optimal order error estimates 

Notes

Acknowledgments

The research is supported by the NSF of China (No. 10971203; No. 11271340), Research Fund for the Doctoral Program of Higher Education of China (No. 20094101110006).

References

  1. 1.
    Shi, Z.: A convergence condition for the quadrilateral Wilson element. Numer. Math. 44(3), 349–361 (1984)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Wachspress, E.: Incompatible quadrilateral basis functions. Int. J. Numer. Methods Eng. 12(4), 589–595 (1978)MATHCrossRefGoogle Scholar
  3. 3.
    Long, Y., Xu, Y.: Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom. Comput. Struct. 52(4), 749–755 (1994)MATHCrossRefGoogle Scholar
  4. 4.
    Long, Y., Huang, M.: A generalized conforming isoparametric element. Appl. Math. Mech. 9(10), 929–936 (1988)MATHCrossRefGoogle Scholar
  5. 5.
    Jiang, J., Cheng, X.: A nonconforming element like Wilson’s for second order problems. Numer. Math. 14(3), 274–278 (1992)MATHGoogle Scholar
  6. 6.
    Shi, D., Chen, S.: A kind of improved Wilson arbitrary quadrilateral elements. Numer. Math. J. Chin. Univ. (in Chinese) 16(2), 161–167 (1994)MathSciNetMATHGoogle Scholar
  7. 7.
    Shi, D., Chen, S.: A class of six-parameter nonconforming arbitrary quadrilateral elements. Appl. Math. J. Chin. Univ. (in Chinese) 11(2), 231–238 (1996)MathSciNetMATHGoogle Scholar
  8. 8.
    Chen, S., Shi, D.: Accuracy analysis for quasi-Wilson element. Acta Math. Sci. 20(1), 44–48 (2000)MathSciNetGoogle Scholar
  9. 9.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differ. Equ. 8(2), 97–111 (1992)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Xu, X.: On the accuracy of nonconforming quadrilateral \(Q_1\) element approximation for the Navier-Stokes problem. SIAM J. Numer. Anal. 38(1), 17–39 (2000)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Li, Q., Lin, J.: Finite element methods: accuracy and improvement. Science Press, Beiing (2006)Google Scholar
  12. 12.
    Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25(1), 160–181 (2005)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Uwe, Risch: Superconvergence of a nonconforming low order finite element. Appl. Numer. Math. 54(3–4), 324–338 (2005)MathSciNetMATHGoogle Scholar
  14. 14.
    Park, C., Sheen, D.: P1-nonconforming quadrilateral finite element methods for second order elliptic problems. SIAM J. Numer. Anal. 41(2), 624–640 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Douglas, J., Santos, J., Sheen, D., Ye, X.: Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problem. RAIRO Math. Modél. Numer. Anal. 33(4), 747–770 (1999)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ming, P., Shi, Z., Xu, Y.: Superconvergence studies of quadrilateral nonconforming rotated \(Q_1\) elements. Int. J. Numer. Anal. Model. 3(3), 322–332 (2006)MathSciNetMATHGoogle Scholar
  17. 17.
    Ciarlet, P.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  18. 18.
    Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47(2–3), 277–293 (1991)MathSciNetGoogle Scholar
  19. 19.
    Apel, T.: Anisotropic finite elements: local estimates and applications. B.G Teubner, Stuttgart (1999)Google Scholar
  20. 20.
    Chen, S., Shi, D., Zhao, Y.: Anisotropic interpolations and quasi-Wilson element for narrow quadrilateral meshes. IMA J. Numer. Anal. 24(1), 77–95 (2004)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Chen, S., Zhao, Y., Shi, D.: Anisotropic interpolations with application to nonconforming elements. Appl. Numer. Math. 49(2), 135–152 (2004)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Apel, T., Nicaise, S., Schöberl, J.: Crouzeix–Raviart type finite elements on anisotropic meshes. Numer. Math. 89(2), 193–223 (2001)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Shi, D., Mao, S., Chen, S.: Anisotropic nonconforming finite element with some superconvergence results. J. Comput. Math. 23(3), 261–274 (2005)MathSciNetMATHGoogle Scholar
  24. 24.
    Mao, S., Chen, S., Shi, D.: Convergence and superconvergence of a nonconforming finite element on anisotropic meshes. Int. J. Numer. Anal. Model. 4(1), 16–38 (2007)MathSciNetMATHGoogle Scholar
  25. 25.
    Shi, D., Mao, S., Chen, S.: A locking-free anisotropic nonconforming rectangular finite element for planar linear elasticity problems. Acta Math. Sci. 27B(1), 193–202 (2007)MathSciNetGoogle Scholar
  26. 26.
    Shi, D., Guan, H.: A class of Crouzeix–Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes. Hokkaido Math. J. 36(4), 687–709 (2007)MathSciNetMATHGoogle Scholar
  27. 27.
    Shi, D., Pei, L.: Low order Crouzeix–Raviart type nonconforming finite element methods for approximation Maxwell’s equations. Int. J. Numer. Anal. Model. 5(3), 373–385 (2008)MathSciNetMATHGoogle Scholar
  28. 28.
    Shi, D., Wang, H., Du, Y.: An anisotropic nonconforming finite element method for approximation a class of nonlinear Sobolev equations. J. Comput. Math. 27(2–3), 299–314 (2009)MathSciNetMATHGoogle Scholar
  29. 29.
    Shi, D., Ren, J.: Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes. Nonlinear Anal. TMA 71(9), 3842–3852 (2009)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Apel, T.: Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing 60(2), 157–174 (1998)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Ming, P., Shi, Z.: Quadrilateral mesh. Chin. Ann. Math. 23(B2), 235–252 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsZhengzhou UniversityZhengzhouChina
  2. 2.Department of Mathematics and PhysicsLuoyang Institute of Science and TechnologyLuoyangChina

Personalised recommendations