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The nonconforming virtual element method for the Navier-Stokes equations

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Abstract

In this paper a unified nonconforming virtual element scheme for the Navier-Stokes equations with different dimensions and different polynomial degrees is described. Its key feature is the treatment of general elements including non-convex and degenerate elements. According to the properties of an enhanced nonconforming virtual element space, the stability of this scheme is proved based on the choice of a proper velocity and pressure pair. Furthermore, we establish optimal error estimates in the discrete energy norm for velocity and the L2 norm for both velocity and pressure. Finally, we test some numerical examples to validate the theoretical results.

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Acknowledgments

The authors thank Rui Li for his valuable discussions during the stages of this research.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Xin Liu & Zhangxin Chen

  2. Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R, via Ferrata 5a, 27100, Pavia, Italy

    Xin Liu

  3. College of Petroleum Engineering, China University of Petroleum, Beijing, China

    Zhangxin Chen

  4. Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, T2N 1N4, Canada

    Zhangxin Chen

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  1. Xin Liu
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  2. Zhangxin Chen
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Correspondence to Zhangxin Chen.

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Communicated by: Ilaria Perugia

Research is supported in part by Foundation CMG in Xi’an Jiaotong University and the scholarship from China Scholarship Council (CSC) under the Grant CSC No.201706280335

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Liu, X., Chen, Z. The nonconforming virtual element method for the Navier-Stokes equations. Adv Comput Math 45, 51–74 (2019). https://doi.org/10.1007/s10444-018-9602-z

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  • DOI: https://doi.org/10.1007/s10444-018-9602-z

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