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Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

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Central European Journal of Mathematics

Abstract

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.

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References

  1. Brenner S.C., Scott L.R., The Mathematical Theory of Finite Element Methods, Texts Appl. Math., 15, Springer-Verlag, New York, 1994

    MATH  Google Scholar 

  2. Ciarlet P. G., The Finite Element Method for Ellptic Problem, North-Holland, Amsterdam, 1978

    Google Scholar 

  3. Guan X.F., Li M.X., Chen S.C., Some numerical quadrature schemes of a non-conforming quadrilateral finite element, Acta Math. Appl. Sin., 2012, 28, 117–126

    Article  MathSciNet  Google Scholar 

  4. He W.M., Chen W.Q., Zhu Q.D., Local superconvergence of the derivative for tensor-product block FEM, Numer. Methods Partial Differential Equations, 2012, 28, 457–475

    Article  MATH  MathSciNet  Google Scholar 

  5. He W.M., Cui J.Z., The local superconvergence of the linear finite element method for the Poisson problem, Numer. Methods Partial Differential Equations (in press), DOI: 10.1002/num.21842

  6. Krasovskiĭ J.P., Isolation of singularities of the Green’s function, Math. USSR Izv., 1967, 1, 935–966

    Article  Google Scholar 

  7. Lin Q., Lin J., Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006 (in Chinese)

    Google Scholar 

  8. Lin Q., Yan N., Construction and Analysis of Finite Element Methods, Hebei University Press, 1996 (in Chinese)

    Google Scholar 

  9. Lin Q., Zhang S., Yan N., An acceleration method for integral equations by using interpolation post-processing, Adv. Comp. Math., 1998, 9, 117–128

    Article  MATH  MathSciNet  Google Scholar 

  10. Lin Q., Zhou J., Superconvergence in high-order Galerkin finite element methods, Comput. Method Appl. Mech. Engrg., 2007, 196, 3779–3784

    Article  MATH  MathSciNet  Google Scholar 

  11. Lin Q., Zhu Q., The Preprocessing and Postprocessing for the Finite Element Methods, Scientific and Technical Publishers, Shanghai, 1994 (in Chinese)

    Google Scholar 

  12. Mao S.P., Chen S.C., Shi D.Y., Convergence and superconvergence of a nonconforming finite element on anisotropic meshes, Int. J. Numer. Anal. Model., 2007, 4, 16–38

    MATH  MathSciNet  Google Scholar 

  13. Mao S.P., Chen S.C., Sun H., A quadrilateral, anisotropic, superconvergent, nonconforming double set parameter element, Appl. Numer. Math., 2006, 56, 937–961

    Article  MATH  MathSciNet  Google Scholar 

  14. Mao S.P., Shi Z.-C., High accuracy analysis of two nonconforming plate elements, Numer. Math., 2009, 111, 407–443

    Article  MATH  MathSciNet  Google Scholar 

  15. Meng L., Zhu Q., The ultraconvergence of derivative for bicubic finite element, Comput. Method Appl. Mech. Engrg., 2007, 196, 3771–3778

    Article  MATH  MathSciNet  Google Scholar 

  16. Schatz A.H., Sloan I.H., Wahlbin L.B., Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 1996, 33, 505–521

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang Z., Ultraconvergence of the patch recovery technique, Math. Comp., 1996, 65, 1431–1437

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhang Z., Ultraconvergence of the patch recovery technique(II), Math. Comp., 2000, 69, 141–158

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhang Z., Lin R., Ultraconvergence of ZZ patch recovery at mesh symmetry points, Numer. Math., 2003, 95, 781–801

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang Z., Zhu J., Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method(I), Comput. Methods Appl. Mech. Engrg., 1995, 123, 173–187

    Article  MATH  MathSciNet  Google Scholar 

  21. Zhu Q., Superconvergence and Postprocessing Theory of Finite Element Theory, Science Press, 2008 (in Chinese)

    Google Scholar 

  22. Zhu Q., Meng L., The derivative ultraconvergence for quadratic triangular finite elements, J. Comput. Math., 2004, 22, 857–864

    MATH  MathSciNet  Google Scholar 

  23. Zhu Q., Meng L., New construction and ultraconvergence of derivative recovery operator for odd-degree rectangular elements, Sci. China Math., 2004, 47, 940–949

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhu Q., Zhao Q., SPR technique and finite element correction, Numer. Math., 2003, 96, 185–196

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, 200092, Shanghai, China

    Xiaofei Guan

  2. School of Science, China University of Geosciences (Beijing), 100083, Beijing, China

    Mingxia Li

  3. Department of Mathematics, Wenzhou University, 325035, Wenzhou, China

    Wenming He

  4. Key Laboratory of Advanced Civil Engineering Materials of Ministry of Education, Tongji University, 200092, Shanghai, China

    Zhengwu Jiang

Authors
  1. Xiaofei Guan
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  2. Mingxia Li
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  3. Wenming He
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  4. Zhengwu Jiang
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Correspondence to Xiaofei Guan.

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Guan, X., Li, M., He, W. et al. Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients. centr.eur.j.math. 12, 1733–1747 (2014). https://doi.org/10.2478/s11533-014-0440-z

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  • DOI: https://doi.org/10.2478/s11533-014-0440-z

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