Superconvergence for rectangular serendipity finite elements

  • Chuanmiao Chen


Based on an orthogonal expansion and orthogonality correction in an element, superconvergence at symmetric points for any degree rectangular serendipity finite element approximation to second order elliptic problem is proved, and its behaviour up to the boundary is also discussed.


rectangular serendipity elements superconvergence symmetric points 


  1. 1.
    Chen, C. M., Superconvergence of finite element solution and its derivatives, Numer. Math. J. Chinese Univ., 1981, 2: 118–125.Google Scholar
  2. 2.
    Douglas, J., Dupont, T., Wheeler, M., An L∞-estimate and a superconvergence result for a Galerkin method for elliptic equation based on tensor products of piecewise polynomials, RAIRO Model. Math. Anal. Numer., 1974, 8: 61–66.MathSciNetGoogle Scholar
  3. 3.
    Lesaint, P., Zlamal, M., Superconvergence of the gradient of finite element solutions, RAIRO Model. Math. Anal. Numer., 1979, 13: 139–166.MATHMathSciNetGoogle Scholar
  4. 4.
    Zlamal, M., Some superconvergence results in the finite element method, Lecture Notes in Math. 606, Springer-Verlag, 1977, 353-362.Google Scholar
  5. 5.
    Zlamal, M., Superconvergence and reduced integration in the finite element method, Math. Comp., 1978, 32: 663–685.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Schatz, A., Sloan, I., Wahlbin, L., Superconvergence in finite element method and mesh that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 1996,33: 505–526.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Wahlbin, L., Superconvergence in Galerkin Finite Element Methods, Berlin: Springer-Verlag, 1995.MATHGoogle Scholar
  8. 8.
    Babuska, I., Strouboulis, T., Upadhyay, C. et al., Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutons of Laplace’s, Poisson’s and the elasticity equations, Numer. Methods for PDE, 1996, 12: 347–392.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Babuska, I., Strouboulis, T., The Finite Element Method and Its Reliability, London: Oxford University Press, 2001.Google Scholar
  10. 10.
    Zhang, Z. M., Derivative superconvergent points in finite element solutions of Poisson’s equation for the serendipity and intermediate families—A theoretical justification, Math. Comp., 1998, 67: 541–552.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, C. M., The element analysis method and superconvergence, in Finite Element Methods: Supercomver-gence, Post-processing and a Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics (eds. Krizek, M., Neittaanmaki, P., Stenberg, R.), Vol. 196, New York: Marcel Dekker, Inc., 1998, 71–84.Google Scholar
  12. 12.
    Chen, C. M., Superconvergence for triangular finite elements, Science in China, Ser. A, 1999, 42(6): 917–924.MATHCrossRefGoogle Scholar
  13. 13.
    Chen, C. M., Structure Theory of Superconvergence for Finite Elements, Changsha: Hunan Science and Technique Press, 2001.Google Scholar
  14. 14.
    Frehse, J., Rannacher, R., Eine L1-Fehlerabschatzung diskreter Grundlosungen in der Methods der finiten Elemente, Tagungaband “Finite Elemente”, Bonn. Math. Schrift., 1975, 89: 92–114.Google Scholar
  15. 15.
    Rannacher, R., Scoot, R., Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 1982, 38: 437–445.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chen, C. M., Optimal points of the approximation solution for Galerkin method for two-point boundary value problem, Numer. Math. J. Chinese Univ., 1979, 1: 73–79.MATHMathSciNetGoogle Scholar
  17. 17.
    Chen, C. M., Xiong, Z. G., Interior superconvergence for a rectangular finite element with 12 parameters, Acta Sci. Nat. Univ. Norm. Hunan, 1999, 22: 1–7.MathSciNetGoogle Scholar
  18. 18.
    Fufaev, V., Dirichlet problem for domain with cusps, Soviet Math. Doklady, 1960, 113: 37–39.MathSciNetGoogle Scholar
  19. 19.
    Volkov, E., Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle, Steklov Institute publications, 1965, 77: 89–112.MATHGoogle Scholar
  20. 20.
    Chen, C. M., W1,∞-interior estimates for finite element methods on regular mesh, J. Comput. Math., 1985, 3: 1–7.MATHMathSciNetGoogle Scholar
  21. 21.
    Bramble, J. H., Nitsche, J. A., Schatz, A. H., Maximum-norm interior estimates for Ritz-Galerkin methods, Math. Comp., 1975, 29: 677–688.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2003

Authors and Affiliations

  • Chuanmiao Chen
    • 1
  1. 1.Institute of ComputationHunan Normal UniversityChangshaChina

Personalised recommendations