Numerische Mathematik

, Volume 106, Issue 2, pp 335–347 | Cite as

A new class of Zienkiewicz-type non-conforming element in any dimensions



In this paper, a new class of Zienkiewicz-type non-conforming finite element, in n spatial dimensions with n ≥ 2, is proposed. The new finite element is proved to be convergent for the biharmonic equation.

Mathematics Subject Classification



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  1. 1.
    Argyris J.H., Haase M. and Mlejnek H.P. (1980). On an unconventional but natural formation of a stiffness matrix. Comput. Methods Appl. Mech. Eng. 22: 1–22 MATHCrossRefGoogle Scholar
  2. 2.
    Bazeley, G.P., Cheung, Y.K., Irons, B.M., Zienkiewicz, O.C.: Triangular elements in plate bending—conforming and non-conforming solutions. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F. Base, Ohio, pp. 547–576 (1965)Google Scholar
  3. 3.
    Bergan P.G. (1980). Finite elements based on energy orthogonal functions. Int. J. Numer. Methods Eng. 15: 1541–1555 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergan P.G. and Nygard M.K. (1984). Finite elements with increased freedom in choosing shape functions. Int. J. Numer. Methods Eng. 20: 634–664 CrossRefGoogle Scholar
  5. 5.
    Chen W., Liu Y. and Tang L. (1980). The formulation of quasi-conforming elements. J. Dalian Inst. Technol. 19(2): 37–49 Google Scholar
  6. 6.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)Google Scholar
  7. 7.
    Irons, B.M., Razzaque, A.: Experience with the patch test. In: Aziz, A.R. (ed.) Proceedings of Symposium on Mathematical Foundations of the Finite Element Method, pp. 557–587. Academic Press, London (1972)Google Scholar
  8. 8.
    Lascaux P. and Lesaint P. (1985). Some non-conforming finite elements for the plate bending problem. RAIRO Anal. Numer. R-1: 9–53 Google Scholar
  9. 9.
    Long Y. and Xin K. (1987). Generalized conforming elements. J. Civil Eng. 1: 1–14 CrossRefGoogle Scholar
  10. 10.
    Shi Z. (1984). The generalized patch test for Zienkiewicz’s triangles. J. Comput. Math. 2:   279–286 MATHMathSciNetGoogle Scholar
  11. 11.
    Shi Z. (1987). Convergence of the TRUNC plate element. Comput. Methods Appl. Mech. Eng. 62:   71–88 MATHGoogle Scholar
  12. 12.
    Shi Z. (1987). The F-E-M-Test for non-conforming finite elements. Math. Comp. 49: 391–405 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shi Z. and Chen S. (1991). Convergence of a nine degree generalized conforming element. Numer. Math. Sin. 13(2): 193–203 MATHGoogle Scholar
  14. 14.
    Shi Z., Chen S. and Zhang F. (1994). Convergence a nalysis of Bergan’s energy-orthogonal plate element. M3AS 4(4): 489–507 MATHMathSciNetGoogle Scholar
  15. 15.
    Strang G. and Fix G.J. (1973). An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs MATHGoogle Scholar
  16. 16.
    Stummel F. (1979). The generalized patch test. SIAM J. Numer. Anal. 16: 449–471 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stummel F. (1980). Basic compactness properties of non-conforming and hybrid finite element spaces. RAIRO Anal. Numer. 4(1): 81–115 MathSciNetGoogle Scholar
  18. 18.
    Tang L., Chen W. and Liu Y. (1980). Quasi-conforming elements in finite element analysis. J. Dalian Inst. Technol. 19(2): 19–35 Google Scholar
  19. 19.
    Wang M. and Xu J. (2007). Non-conforming tetrahedral finite elements for fourth order elliptic equations. Math. Comp. 76(257): 1–18 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Zhang, H.: The generalized patch test and 9-parameter quasi-congorming element. In: Feng, K. (ed.) Proceedings of the Sino-France Symposium on Finite Element Methods, pp. 566–583. Science Press, Gordan and Breach, Newark (1983)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.Institute of Computational MathematicsCASBeijingChina
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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