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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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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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