Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh L2 projection

Authors

  • Jian Li,

    Corresponding author
    1. Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, People's Republic of China
    2. College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, People's Republic of China
    3. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W. Calgary, Alberta, Canada T2N 1N4
    • Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721007, People's Republic of China
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  • Liquan Mei,

    1. Faculty of Science, Xi'an Jiaotong University, Xi'an, 710049, People's Republic of China
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  • Zhangxin Chen

    1. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W. Calgary, Alberta, Canada T2N 1N4
    2. Faculty of Science, Xi'an Jiaotong University, Xi'an, 710049, People's Republic of China
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Abstract

This article first recalls the results of a stabilized finite element method based on a local Gauss integration method for the stationary Stokes equations approximated by low equal-order elements that do not satisfy the inf-sup condition. Then, we derive general superconvergence results for this stabilized method by using a local coarse mesh L2 projection. These supervergence results have three prominent features. First, they are based on a multiscale method defined for any quasi-uniform mesh. Second, they are derived on the basis of a large sparse, symmetric positive-definite system of linear equations for the solution of the stationary Stokes problem. Third, the finite elements used fail to satisfy the inf-sup condition. This article combines the merits of the new stabilized method with that of the L2 projection method. This projection method is of practical importance in scientific computation. Finally, a series of numerical experiments are presented to check the theoretical results obtained. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 115-126, 2012

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