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Superconvergence of the stable \(P_1-P_1\) finite element pair for Stokes problem

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Abstract

We study the superconvergence of the stable lowest equal-order finite element pair solving the Stokes problem. The superclose property is proved for the interpolation function; then a superconvergence rate of \(O(h^{\frac{3}{2}})\)-order is obtained for the velocity gradient approximation by using the post-processing technique, and an \(O(h^{\frac{3}{2}})\)-order error estimate is derived for the pressure approximation. In addition, the asymptotically exact a posteriori error estimate also is given by means of the superconvergence result. Finally, some numerical experiments are provided to illustrate the theoretical analysis.

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Acknowledgments

The authors would like to thank the anonymous referees for many helpful suggestions which improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110004, China

    Tie Zhang

  2. The State Key Laboratory of Synthetical Automation for Process Industries, Research Center of National Metallurgical Automation, Northeastern University, Shenyang, 110004, China

    Lixin Tang

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  1. Tie Zhang
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Correspondence to Tie Zhang.

Additional information

This work was supported by the National Natural Science Funds of China, No. 11371081; and the SAPI Fundamental Research Funds, No. 2013ZCX02.

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Zhang, T., Tang, L. Superconvergence of the stable \(P_1-P_1\) finite element pair for Stokes problem. Calcolo 53, 35–49 (2016). https://doi.org/10.1007/s10092-014-0134-8

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