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Journal of Computational Mathematics ›› 2020, Vol. 38 ›› Issue (2): 310-336.DOI: 10.4208/jcm.1806-m2017-0287

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A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION

Baiju Zhang1, Yan Yang2, Minfu Feng3   

  1. 1. School of Mathematics, Sichuan University, Chengdu 610064, China;
    2. School of Sciences, Southwest Petroleum University, Chengdu 610500, China;
    3. School of Mathematics, Sichuan University, Chengdu 610064, China
  • Received:2017-12-06 Revised:2018-05-11 Online:2020-03-15 Published:2020-03-15
  • Supported by:

    This research is supported by the National Natural Science Foundation of China (No.11271273). The authors would like to thank the editors and referees for careful reading of the research and their criticism, valuable suggestions and constructive comments which lead to great improvement of this paper.

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Baiju Zhang, Yan Yang, Minfu Feng. A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION[J]. Journal of Computational Mathematics, 2020, 38(2): 310-336.

We propose and analyze a C0-weak Galerkin (WG) finite element method for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, the system of Navier-Stokes equations is reduced to a single fourth-order nonlinear partial differential equation and the incompressibility constraint is automatically satisfied. The proposed method uses continuous piecewisepolynomial approximations of degree k+2 for the stream-function ψ and discontinuous piecewise-polynomial approximations of degree k+1 for the trace of ∇ψ on the interelement boundaries. The existence of a discrete solution is proved by means of a topological degree argument, while the uniqueness is obtained under a data smallness condition. An optimal error estimate is obtained in L2-norm, H1-norm and broken H2-norm. Numerical tests are presented to demonstrate the theoretical results.
Key words: Weak Galerkin method, Navier-Stokes equations, Stream-function formulation

CLC Number: 

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