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Unconditional stability and error estimates of modified characteristics FEMs for the Navier–Stokes equations

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Abstract

The paper is concerned with the unconditional stability and convergence of characteristics type methods for the time-dependent Navier–Stokes equations. We present optimal error estimates in \(L^2\) and \(H^1\) norms for a typical modified characteristics finite element method unconditionally, while all previous works require certain time-step restrictions. The analysis is based on an iterated characteristic time-discrete system, with which the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the difference between the numerical solution and the solution of the time-discrete system is \(\tau \)-independent, where \(\tau \) denotes the time stepsize. Thus numerical solution in \(W^{1,\infty }\) is bounded and optimal error estimates can be obtained in a traditional way. Numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time-dependent Navier–Stokes equations. The approach used in this paper can be easily extended to many other characteristics-based methods.

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Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions and comments, which helped to improve the quality of the paper.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003, People’s Republic of China

    Zhiyong Si

  2. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, People’s Republic of China

    Jilu Wang

  3. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Weiwei Sun

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  1. Zhiyong Si
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  2. Jilu Wang
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Correspondence to Zhiyong Si.

Additional information

Z. Si’s work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302915), Chinese NSF (Grant No. 11226306, 11301156 and 11401177) and the Doctoral Foundation of Henan Polytechnic University (No. B2012-56). J. Wang’s work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302915), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 15XNLF15). W. Sun’s work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302915).

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Si, Z., Wang, J. & Sun, W. Unconditional stability and error estimates of modified characteristics FEMs for the Navier–Stokes equations. Numer. Math. 134, 139–161 (2016). https://doi.org/10.1007/s00211-015-0767-9

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  • DOI: https://doi.org/10.1007/s00211-015-0767-9

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