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Nonconforming finite element Stokes complexes in three dimensions

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Abstract

Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P1-P0 element for the Stokes equation in three dimensions are studied. Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem, and the optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P1-P0 element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12171300 and 11771338), the Natural Science Foundation of Shanghai (Grant No. 21ZR1480500) and the Fundamental Research Funds for the Central Universities (Grant No. 2019110066).

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Authors and Affiliations

  1. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Xuehai Huang

  2. College of Mathematics and Physics, Wenzhou University, Wenzhou, 325035, China

    Xuehai Huang

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  1. Xuehai Huang
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Correspondence to Xuehai Huang.

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Huang, X. Nonconforming finite element Stokes complexes in three dimensions. Sci. China Math. 66, 1879–1902 (2023). https://doi.org/10.1007/s11425-021-2026-7

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  • DOI: https://doi.org/10.1007/s11425-021-2026-7

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