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\(H^2\)-Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier–Stokes Equations

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Abstract

This paper considers the \(H^2\)-stability results for the first order fully discrete schemes based on the mixed finite element method for the time-dependent Navier–Stokes equations with the initial data \(u_0\in H^\alpha \) with \(\alpha =0,~1\) and 2. A mixed finite element method is used to the spatial discretization of the Navier–Stokes equations, and the temporal treatments of the spatial discrete Navier–Stokes equations are the first order implicit, semi-implicit, implicit/explicit(the semi-implicit/explicit in the case of \({\alpha }=0\)) and explicit schemes. The \(H^2\)-stability results of the schemes are provided, where the first order implicit and semi-implicit schemes are the \(H^2\)-unconditional stable, the first order explicit scheme is the \(H^2\)-conditional stable, and the implicit/explicit scheme (the semi-implicit/explicit scheme in the case of \({\alpha }=0\)) is the \(H^2\)-almost unconditional stable. Moreover, this paper makes some numerical investigations of the \(H^2\)-stability results for the first order fully discrete schemes for the time-dependent Navier–Stokes equations. Through a series of numerical experiments, it is verified that the numerical results are shown to support the developed \(H^2\)-stability theory.

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Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions which helped to improve the quality of our present paper.

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Authors and Affiliations

  1. Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an , 710049, People’s Republic of China

    Yinnian He

  2. College of Mathematics and System Sciences, Xinjiang University, Ürümqi , 830046, People’s Republic of China

    Pengzhan Huang & Xinlong Feng

Authors
  1. Yinnian He
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  2. Pengzhan Huang
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  3. Xinlong Feng
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Corresponding author

Correspondence to Pengzhan Huang.

Additional information

The first author is partially support by the NSF of China (Grant Nos. 11271298, 11362021). The second author is partially support by the China Postdoctoral Science Foundation (Grant No. 2013M530438) and the NSF of Xinjiang Province (Grant No. 2013211B01). The third author is partially support by the Distinguished Young Scholars Fund of Xinjiang Province (Grant No. 2013711010) and the NSF of China (Grant No. 11271313).

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He, Y., Huang, P. & Feng, X. \(H^2\)-Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier–Stokes Equations. J Sci Comput 62, 230–264 (2015). https://doi.org/10.1007/s10915-014-9854-9

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  • DOI: https://doi.org/10.1007/s10915-014-9854-9

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