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Optimal \(L^2, H^1\) and \(L^\infty \) analysis of finite volume methods for the stationary Navier–Stokes equations with large data

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Abstract

Previous work on the stability and convergence analysis of numerical methods for the stationary Navier–Stokes equations was carried out under the uniqueness condition of the solution, which required that the data be small enough in certain norms. In this paper an optimal analysis for the finite volume methods is performed for the stationary Navier–Stokes equations, which relaxes the solution uniqueness condition and thus the data requirement. In particular, optimal order error estimates in the \(H^1\)-norm for velocity and the \(L^2\)-norm for pressure are obtained with large data, and a new residual technique for the stationary Navier–Stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the \(L^2\)-norm for the velocity. In addition, after proving a number of additional technical lemmas including weighted \(L^2\)-norm estimates for regularized Green’s functions associated with the Stokes problem, optimal error estimates in the \(L^\infty \)-norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor \(O(|\log h|)\) for the stationary Naiver–Stokes equations.

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Authors and Affiliations

  1. Department of Mathematics, Baoji University of Arts and Sciences, Baoji, 721013, P. R. China

    Jian Li

  2. 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

  3. Center for Computational Geoscience, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, P. R. China

    Zhangxin Chen

Authors
  1. Jian Li
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  2. Zhangxin Chen
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Correspondence to Jian Li.

Additional information

Supported in part by NSF of China (No. 11071193), the NCET-11-1041, Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (No. 2011kjxx12), the Project-Sponsored by SRF for ROCS, SEM, China Postdoctoral Science Foundation funded project 2012M511973, the Key Project of Baoji university of Arts and Sciences (No. ZK12041), and NSERC/AERI/Foundation CMG Chair and iCORE Chair Funds in Reservoir Simulation.

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Li, J., Chen, Z. Optimal \(L^2, H^1\) and \(L^\infty \) analysis of finite volume methods for the stationary Navier–Stokes equations with large data. Numer. Math. 126, 75–101 (2014). https://doi.org/10.1007/s00211-013-0556-2

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  • DOI: https://doi.org/10.1007/s00211-013-0556-2

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