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Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blum, H., Lin, Q., Rannacher, R.: Asymptotic error expansions and Richardson extrapolation for linear finite elements. Numer. Math. 49, 11–37 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chen, C., Lin, Q.: Extrapolation of finite element approximations in a rectangular domain. J. Comput. Math. 7, 235–255 (1989)

    MathSciNet  Google Scholar 

  8. Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78, 35–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in \(L^{p}\)-norm. Math. Comput. 76, 179–204 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  12. Miranda, C.: Partial differential equations of elliptic type, 2nd edn. Springer, Berlin (1970)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Y., Xu, J.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comp. 77(263), 1253–1268 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krasovskii, J.P.: Isolation of singularities of the Green’s function. Math. USSR-IZV 1, 935–966 (1967)

    Article  Google Scholar 

  17. Lin, Q.: Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners. Numer. Math. 58, 631–640 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Lin, Q., Yan, N.: Construction and analysis for finite element methods, Hebei University (1996)

  19. Lin, Q., Zhou, J.: Superconvergence in high-order Galerkin finite element methods. Comput. Method Appl. Mech. Eng. 196, 3779–3784 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lin, Q., Zhu, Q.: The preprocessing and postprocessing for finite element methods. Shanghai Scientific and Technical Publishers, Shanghai (1994) (in Chinese)

  21. Nitsche, J., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comp. 28, 937–955 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods, part II. Math. Comp. 64, 907–928 (1995)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comp. 31, 414–442 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wahlbin, L.B.: Superconvergence in Galerkin finite element methods. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  27. Wahlbin, L.B.: General principles of superconvergence in Galerkin finite element methods. Lect. Notes Pure Appl. Math. 198, 269–285 (1998)

    MathSciNet  MATH  Google Scholar 

  28. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimates for mildly structured grids. Math. Comp. 73, 1139–1152 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, T.: The derivative patch interpolating recovery technique and superconvergence. Numer. Math. Appl. 2, 1–10 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

  31. Zhang, Z.: Polynomial preserving recovery for anisotropic and irregular grids. J. Comp. Math. 22, 331–340 (2004)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence Property. SIAM J. on Sci. Comput. 26, 1192–1213 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  35. Zhu, Q., Lin, Q.: Theory of superconvergence of finite elements. Hunan Science and Technology Press, Hunan (1989). (in Chinese)

    Google Scholar 

  36. Zhu, Q.: High precision and postprocessing theory of finite element method. Science Publisher, Beijing (2008). (in Chinese)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, Wenzhou University, Wenzhou, 320035, Zhejiang, China

    Wen-ming He

  2. Beijing Computational Science Research Center, Beijing, 100084, China

    Zhimin Zhang

  3. Department of Mathematics, Wayne state University, Detroit, MI, 48202, USA

    Zhimin Zhang

  4. School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Qingsong Zou

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  1. Wen-ming He
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Correspondence to Wen-ming He.

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He, Wm., Zhang, Z. & Zou, Q. Ultraconvergence of high order FEMs for elliptic problems with variable coefficients. Numer. Math. 136, 215–248 (2017). https://doi.org/10.1007/s00211-016-0838-6

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  • DOI: https://doi.org/10.1007/s00211-016-0838-6

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