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A divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes

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Abstract

In this paper, we present a divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes. The velocity and the pressure are discretized by the H(div) virtual element and discontinuous piecewise polynomials, respectively. An additional polynomial space that lives on the element edges is introduced to approximate the tangential trace of the velocity. The velocity at the discrete level is point-wise divergence-free and thus the exact mass conservation is preserved in the discretization. Given suitable data conditions, the well-posedness of the discrete problem is proved and a rigorous error analysis of the method is derived. The error with respect to a mesh dependent norm for the velocity depends on the smoothness of the velocity and the approximation of the load term. A series of numerical experiments are reported to validate the performance o f the method.

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Acknowledgements

The authors would like to thank the two anonymous reviewers for giving them many valuable suggestions which improved the quality of this paper.

Funding

The work of Gang Wang was supported by the Fundamental Research Funds for the Central Universities (No. G2019KY05104) and National Natural Science Foundation of China (No. 12001433). The work of Feng Wang was supported by National Natural Science Foundation of China (Nos. 12071227, 11871281) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 20KJA110001). The work of Yinnian He was supported by the Major Research and Development Program of China (No. 2016YFB0200901) and National Natural Science Foundation of China (No. 11771348).

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Authors and Affiliations

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

    Gang Wang

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Feng Wang

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China

    Yinnian He

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  1. Gang Wang
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  2. Feng Wang
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Wang, G., Wang, F. & He, Y. A divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes. Adv Comput Math 47, 83 (2021). https://doi.org/10.1007/s10444-021-09909-z

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