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

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Abstract

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.

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References

  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

    Article  MATH  Google Scholar 

  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)

  3. Bergan P.G. (1980). Finite elements based on energy orthogonal functions. Int. J. Numer. Methods Eng. 15: 1541–1555

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  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. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

  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)

  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. Long Y. and Xin K. (1987). Generalized conforming elements. J. Civil Eng. 1: 1–14

    Article  Google Scholar 

  10. Shi Z. (1984). The generalized patch test for Zienkiewicz’s triangles. J. Comput. Math. 2:   279–286

    MATH  MathSciNet  Google Scholar 

  11. Shi Z. (1987). Convergence of the TRUNC plate element. Comput. Methods Appl. Mech. Eng. 62:   71–88

    MATH  Google Scholar 

  12. Shi Z. (1987). The F-E-M-Test for non-conforming finite elements. Math. Comp. 49: 391–405

    Article  MATH  MathSciNet  Google Scholar 

  13. Shi Z. and Chen S. (1991). Convergence of a nine degree generalized conforming element. Numer. Math. Sin. 13(2): 193–203

    MATH  Google Scholar 

  14. Shi Z., Chen S. and Zhang F. (1994). Convergence a nalysis of Bergan’s energy-orthogonal plate element. M3AS 4(4): 489–507

    MATH  MathSciNet  Google Scholar 

  15. Strang G. and Fix G.J. (1973). An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  16. Stummel F. (1979). The generalized patch test. SIAM J. Numer. Anal. 16: 449–471

    Article  MATH  MathSciNet  Google Scholar 

  17. Stummel F. (1980). Basic compactness properties of non-conforming and hybrid finite element spaces. RAIRO Anal. Numer. 4(1): 81–115

    MathSciNet  Google Scholar 

  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. Wang M. and Xu J. (2007). Non-conforming tetrahedral finite elements for fourth order elliptic equations. Math. Comp. 76(257): 1–18

    Article  MATH  MathSciNet  Google Scholar 

  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)

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, China

    Ming Wang & Jinchao Xu

  2. Institute of Computational Mathematics, CAS, Beijing, China

    Zhong-ci Shi

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Jinchao Xu

Authors
  1. Ming Wang
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  2. Zhong-ci Shi
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  3. Jinchao Xu
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Corresponding author

Correspondence to Ming Wang.

Additional information

The work was supported by the National Natural Science Foundation of China (10571006).

This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.

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Wang, M., Shi, Zc. & Xu, J. A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer. Math. 106, 335–347 (2007). https://doi.org/10.1007/s00211-007-0063-4

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  • DOI: https://doi.org/10.1007/s00211-007-0063-4

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