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A Modified Rotated-\(Q_1\) Finite Element for the Stokes Equations on Quadrilateral and Hexahedral Meshes

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Abstract

We construct a modified rotated-\(Q_1\) finite element, where we replace the \(P_2\) bubbles of the Rannacher–Turek rotated-\(Q_1\) element by multi-piece linear polynomials. In 2D, we use \(\{1,x,y,|\lambda _1|\}\) as the basis on a general quadrilateral, where \(\lambda _1\) is a linear polynomial which vanishes at the middle edge \(\textbf{x}_3\textbf{x}_4\) of two opposite mid-edge nodes \(\textbf{x}_1\) and \(\textbf{x}_2\). In 3D, we use \(\{1,x,y,z, |\lambda _1|,|\lambda _3|\}\) as the basis on a general hexahedron, where \(\lambda _1\) is a linear polynomial which vanishes at the mid plane \(\textbf{x}_3\textbf{x}_4\textbf{x}_5\textbf{x}_6\) between the two opposite mid-face nodes \(\textbf{x}_1\) and \(\textbf{x}_2\), and \(\lambda _3\) is a linear polynomial which vanishes at the mid plane \(\textbf{x}_1\textbf{x}_2\textbf{x}_5\textbf{x}_6\) between the two opposite mid-face nodes \(\textbf{x}_3\) and \(\textbf{x}_4\). The new rotated-\(Q_1\) finite element is shown inf-sup stable and quasi-optimal in solving the Stokes equations, on general quadrilateral and hexahedral meshes. Numerical tests in 2D and 3D show the method does converge quasi-optimally on non-asymptotic parallelogram or parallelepiped meshes, while the Rannacher–Turek rotated-\(Q_1\) element fails to converge on such meshes.

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Acknowledgements

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Funding

The work of Liwei Xu is supported in part by NSFC Grant 12071060. Xuejun Xu was supported by National Natural Science Foundation of China (Grant No. 12071350), Shanghai Municipal Science and Technology Major Project No. 2021SHZDZX0100, and Science and Technology Commission of Shanghai Municipality.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Liwei Xu

  2. School of Mathematical Science, Tongji University, Shanghai, 200092, China

    Xuejun Xu

  3. Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Xuejun Xu

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Liwei Xu
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  2. Xuejun Xu
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Correspondence to Shangyou Zhang.

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Xu, L., Xu, X. & Zhang, S. A Modified Rotated-\(Q_1\) Finite Element for the Stokes Equations on Quadrilateral and Hexahedral Meshes. J Sci Comput 99, 11 (2024). https://doi.org/10.1007/s10915-024-02477-6

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  • DOI: https://doi.org/10.1007/s10915-024-02477-6

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