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Discontinuous Galerkin Methods for a Stationary Navier–Stokes Problem with a Nonlinear Slip Boundary Condition of Friction Type

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Abstract

In this work, several discontinuous Galerkin (DG) methods are introduced and analyzed to solve a variational inequality from the stationary Navier–Stokes equations with a nonlinear slip boundary condition of friction type. Existence, uniqueness and stability of numerical solutions are shown for the DG methods. Error estimates are derived for the velocity in a broken \(H^1\)-norm and for the pressure in an \(L^2\)-norm, with the optimal convergence order when linear elements for the velocity and piecewise constants for the pressure are used. Numerical results are reported to demonstrate the theoretically predicted convergence orders, as well as the capability in capturing the discontinuity, the ability in handling the shear layers, the capacity in dealing with the advection-dominated problem, and the application to the general polygonal mesh of the DG methods.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Feifei Jing

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

    Weimin Han, Wenjing Yan & Fei Wang

  3. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1410, USA

    Weimin Han

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  1. Feifei Jing
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  2. Weimin Han
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Correspondence to Wenjing Yan.

Additional information

This work is supported by the NSF under grant DMS-1521684 and by the NSF of China (Nos. 11371288, 11771350). The authors thank the referees for their valuable comments on the first version of the paper.

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Jing, F., Han, W., Yan, W. et al. Discontinuous Galerkin Methods for a Stationary Navier–Stokes Problem with a Nonlinear Slip Boundary Condition of Friction Type. J Sci Comput 76, 888–912 (2018). https://doi.org/10.1007/s10915-018-0644-7

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