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Numerische Mathematik

, Volume 136, Issue 1, pp 215–248 | Cite as

Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

  • Wen-ming HeEmail author
  • Zhimin Zhang
  • Qingsong Zou
Article
  • 343 Downloads

Abstract

In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth \((k\ge 2)\) order finite element solution converges to the gradient of the exact solution with order \(\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)\). The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green’s function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings.

Mathematics Subject Classification

65N30 65N25 65N15 

Notes

Acknowledgments

The authors would like to thank both the anonymous referees for their careful reading of the paper and their valuable comments which leads to a significant improvement of the paper. The first author is supported in part by the National Natural Science Foundation of China (11671304, 11171257, 11301396), the Zhejiang Provincial Natural Science Foundation, China (No. LY15A010015). The second author is supported in part by the National Natural Science Foundation of China (11471031,91430216) and the US National Science Foundation through grant U1530401. The third author is supported in part by the National Natural Science Foundation of China through grants 11571384 and 11428103, by the Fundamental Research Funds for the Central Universities through grant 16lgjc80, and by Guangdong Provincial Natural Science Foundation of China through grant 2014A030313179.

References

  1. 1.
    Asadzadeh, M., Schatz, A., Wendland, W.: Asymptotic error expansions for the finite element method for second order elliptic problems in \(R^{N}(N\ge 2), I\): Local interior expansions. SIAM J. Numer. Anal. 48, 2000–2017 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asadzadeh, M., Schatz, A., Wendland, W.: A non-standard approach to Richardson extrapolation in the finite element method for second order elliptic problems. Math. Comp. 78, 1951–1973 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bacuta, C., Nistor, V., Zikatanov, L.T.: Improving the rate of convergence of ‘high order finite elements on polyhedra I: a priori estimates. Numer. Funct. Anal. Optim. 26, 613–639 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, H., Lin, Q., Rannacher, R.: Asymptotic error expansions and Richardson extrapolation for linear finite elements. Numer. Math. 49, 11–37 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bank, R.E., Xu, J.: Asymptotic exact a posteriori error estimates, Part I: grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, C., Huang, Y.: High accuracy theory of finite element methods. Hunan Science and Technology Press, People’s Republic of China (1995). (in Chinese)Google Scholar
  7. 7.
    Chen, C., Lin, Q.: Extrapolation of finite element approximations in a rectangular domain. J. Comput. Math. 7, 235–255 (1989)MathSciNetGoogle Scholar
  8. 8.
    Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78, 35–53 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in \(L^{p}\)-norm. Math. Comput. 76, 179–204 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grisvard, P.: Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, numerical solution of partial differential equations III (Hubbard, B., ed.), Academic Press, New York (1976)Google Scholar
  11. 11.
    John F.: General properties of solutions of linear elliptic partial differential equations. Proc. Sympos. on Spectral Theory and Differential Problems, Oklahoma A & M College, Stillwater, Okla., pp. 113–175. MR 13, 349 (1951)Google Scholar
  12. 12.
    Miranda, C.: Partial differential equations of elliptic type, 2nd edn. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  13. 13.
    He, W., Guan, X., Cui, J.: The Local Superconvergence of the trilinear element for the three-dimensional Poisson problem. J. Math. Anal. Appl. 388, 863–872 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, C., Hu, S.: The highest order superconvergence for bi-\(k\) degree rectangular elements at nodes- a proof of \(2k\)-conjecture. Math. Comp. 82, 1337–1355 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huang, Y., Xu, J.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comp. 77(263), 1253–1268 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krasovskii, J.P.: Isolation of singularities of the Green’s function. Math. USSR-IZV 1, 935–966 (1967)CrossRefGoogle Scholar
  17. 17.
    Lin, Q.: Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners. Numer. Math. 58, 631–640 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lin, Q., Yan, N.: Construction and analysis for finite element methods, Hebei University (1996)Google Scholar
  19. 19.
    Lin, Q., Zhou, J.: Superconvergence in high-order Galerkin finite element methods. Comput. Method Appl. Mech. Eng. 196, 3779–3784 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lin, Q., Zhu, Q.: The preprocessing and postprocessing for finite element methods. Shanghai Scientific and Technical Publishers, Shanghai (1994) (in Chinese)Google Scholar
  21. 21.
    Nitsche, J., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comp. 28, 937–955 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods, part II. Math. Comp. 64, 907–928 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Schatz, A.H., Sloan, L.H., Wahlbin, L.B.: Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point. SIAM J. Numer. Anal. 33, 505–521 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comp. 31, 414–442 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schatz, A.H., Wahlbin, L.B.: Asymptotically exact a posterior estimators for the pointwise gradient error on each element in irregular meshes. part II: The piecewise linear case. Math. Comp. 73, 517–523 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wahlbin, L.B.: Superconvergence in Galerkin finite element methods. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wahlbin, L.B.: General principles of superconvergence in Galerkin finite element methods. Lect. Notes Pure Appl. Math. 198, 269–285 (1998)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimates for mildly structured grids. Math. Comp. 73, 1139–1152 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhang, T.: The derivative patch interpolating recovery technique and superconvergence. Numer. Math. Appl. 2, 1–10 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zhang, Z.: Recovery technique in finite element methods. In: Adaptive computations: theory and algorithm, Edited by Tao Tang, Jinchao Xu, Mathematics Monograph Series 6. Science Publisher, Beijing, People’s Republic of China, pp. 333–412 (2007)Google Scholar
  31. 31.
    Zhang, Z.: Polynomial preserving recovery for anisotropic and irregular grids. J. Comp. Math. 22, 331–340 (2004)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zhang, Z.: Ultraconvergence of the patch recovery technique II. Math. Comp. 69, 141–158 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence Property. SIAM J. on Sci. Comput. 26, 1192–1213 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zienkiewicz, O.C., Zhu, J.: The superconvergence patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Eng. 101, 207–224 (1992)CrossRefzbMATHGoogle Scholar
  35. 35.
    Zhu, Q., Lin, Q.: Theory of superconvergence of finite elements. Hunan Science and Technology Press, Hunan (1989). (in Chinese)Google Scholar
  36. 36.
    Zhu, Q.: High precision and postprocessing theory of finite element method. Science Publisher, Beijing (2008). (in Chinese)Google Scholar
  37. 37.
    Zienkiewicz, O.C., Zhu, J.: The superconvergence patch recovery and a posteriori estimates Part I: the recovery technique. Int. J. Num. Methods Eng. 33, 1331–1364 (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsWenzhou UniversityWenzhouChina
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.Department of MathematicsWayne state UniversityDetroitUSA
  4. 4.School of Mathematics and Computational ScienceSun Yat-sen UniversityGuangzhouChina

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