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The supercloseness property of the Stoke projection for the transient Navier–Stokes equations and global superconvergence analysis

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Abstract

In this paper, we derive the supercloseness properties and global superconvergence results for the implicit Euler scheme of the transient Navier–Stokes equations. Using a prior estimate of finite element solutions, the properties of the Stokes projection and Stokes operator, the derivative transforming skill and the \(H^{-1}\)-norm estimate, we deduce the supercloseness properties of the Stokes projection for the velocity in \(L^\infty (H^1)\)-norm and pressure in \(L^\infty (L^2)\)-norm. Then the supercloseness properties of the interpolation operators are obtained for two pairs of rectangular element: the bilinear-constant element and the Bernadi–Raugel element. Finally, by the interpolation postprocessing technique, we obtain the global superconvergent results. Compared with previous results, no time step restriction is required in the analysis, and moreover, the supercloseness analysis is based on the Stokes projection, which makes the proof more concise.

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Acknowledgements

This work was sponsored by Natural Science Foundation of Henan (222300420143), the Innovative Funds Plan of Henan University of Technology (2021ZKCJ11), and Fundamental Research Funds for the Henan Provincial Colleges and Universities (22A130003).

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, China

    Minghao Li & Liuchao Xiao

  2. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

    Zhenzhen Li

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  1. Minghao Li
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  2. Liuchao Xiao
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  3. Zhenzhen Li
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Correspondence to Liuchao Xiao.

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Li, M., Xiao, L. & Li, Z. The supercloseness property of the Stoke projection for the transient Navier–Stokes equations and global superconvergence analysis. J. Appl. Math. Comput. 69, 2041–2057 (2023). https://doi.org/10.1007/s12190-022-01824-w

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