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Least-Squares Virtual Element Method for Stokes Problems on Polygonal Meshes

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Abstract

In this paper, a least-squares virtual element method on polygonal meshes is proposed for the stress-velocity formulation of the linear Stokes problem. The \(\mathbb {H}(\textrm{div})\)-and \(\textbf{H}^1\)-conforming virtual elements are used to approximate the stress and velocity variables, respectively. Benefiting from the virtual element method and the least-squares formulation, our method allows the use of general polygonal meshes and leads to a symmetric and positive definite system. The a priori error estimates are established for the stress in \(\mathbb {H}(\textrm{div})\) norm and for the velocity in \(\textbf{H}^1\) norm. Additionally, the least-squares functional naturally offers an a posteriori error estimator without extra effort, which together with the great flexibility of mesh can guide the adaptive mesh refinement to resolve the singularity. We also extend the present method to the nonlinear Stokes problem and show the corresponding least-squares virtual element method. A series of numerical examples supporting the theoretical results are presented.

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Acknowledgements

The work of Gang Wang was supported by China Postdoctoral Science Foundation (No. 2021M692648) and National Natural Science Foundation of China (No. 12001433, 12371405). The work of Ying Wang was supported by National Natural Science Foundation of China (No. 12201485).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Gang Wang

  2. School of Science, Xi’an University of Architecture and Technology, Xi’an, 710055, People’s Republic of China

    Ying Wang

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  1. Gang Wang
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Correspondence to Gang Wang.

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Wang, G., Wang, Y. Least-Squares Virtual Element Method for Stokes Problems on Polygonal Meshes. J Sci Comput 98, 46 (2024). https://doi.org/10.1007/s10915-023-02436-7

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  • DOI: https://doi.org/10.1007/s10915-023-02436-7

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