Applied Mathematics and Mechanics

, Volume 7, Issue 5, pp 443–459 | Cite as

On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

  • Zhang Hong-qing
  • Wang Ming
Article

Abstract

In this paper, the compactness of quasi-conforming element spaces and the—convergence of quasi-conforming element method are discussed. The well-known Rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties, and the generalized Poincare inequality. The generalized Friedrichs inequality and the generalized inequality of Poincare-Friedrichs are proved true for them. The error estimates are also given. It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity, and satisfy the rank condition of element and pass the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L2′2(Ω)-errors are O(hτ), O(hτ), O(hτ2), O(hτ2), O(hτ), and O(hτ4) respectively.

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References

  1. [1]
    Zhang Hong-qing, and Wang Ming, Finite element approximations with multiple sets of functions and quasi-conforming element for plate bending problems,Applied Mathematics and Mechanics,6, 1 (1985), 41–52.Google Scholar
  2. [2]
    Zhang Hong-qing, and Wang Ming, Finite element approximations with multiple sets of functions and quasi-conforming elements,Fifth international symposium on differential geometry and differential equations-computation of partial differential equations, Beijing, China (1984).Google Scholar
  3. [3]
    Tang Li-min, Chen Wan-ji and Liu Ying-xi, Quasi-conforming elements for finite element analysis,Journal of Dalian Institute of Technology,19, 2 (1980), 19–35. (in Chinese)Google Scholar
  4. [4]
    Chen Wan-ji, Liu Ying-xi and Tang Li-min, The formulation of quasi-conforming elements.Journal of Dalian Institute of Technology,19, 2(1980), 37–49. (in Chinese)Google Scholar
  5. [5]
    Jiang He-yong, Derivation of higher precision triangular plate element by quasi-conforming element method,Journal of Dalian Institute of Technology,20, Suppl. 2 (1981), 21–28. (in Chinese)Google Scholar
  6. [6]
    Stummel, F., The generalized patch test,SIAM J. Num. Anal. 16 (1979), 449–471.Google Scholar
  7. [7]
    Feng Kang, On the theory of discontinuous finite elements,Mathematicae Numericae Sinica.1, 4 (1979), 378–385. (in Chinese)Google Scholar
  8. [8]
    Stummel, F., Basic compactness properties of nonconforming and hybrid finite element spaces,RAIRO, Analyse, Numeriqe, Numerical Analysis,4, 1 (1980), 81–115.Google Scholar
  9. [9]
    Ciarlet, P.C.,The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford (1978).Google Scholar
  10. [10]
    Yosida Kosaku,Functional Analysis, Fifth edition, Springer-Verlag (1978).Google Scholar

Copyright information

© Shanghai University of Technology (SUT) 1986

Authors and Affiliations

  • Zhang Hong-qing
    • 1
  • Wang Ming
    • 1
  1. 1.Department of Applied MathematicsDalian Institute of TechnologyDalian

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