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Volume 38, Issue 2
A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation

Baiju Zhang, Yan Yang & Minfu Feng

DOI: 10.4208/jcm.1806-m2017-0287

J. Comp. Math., 38 (2020), pp. 310-336.

Published online: 2020-02

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  • Abstract

We propose and analyze a $C^0$-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 piecewise-polynomial approximations of degree $k+2$ for the stream-function $\psi$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $\nabla\psi$ 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 $L^2$-norm, $H^1$-norm and broken $H^2$-norm. Numerical tests are presented to demonstrate the theoretical results.

  • Keywords

Weak Galerkin method, Navier-Stokes equations, Stream-function formulation.

  • AMS Subject Headings

65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhangbaiju1990@163.com (Baiju Zhang)

yyan2011@163.com (Yan Yang)

fmf@scu.edu.cn (Minfu Feng)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-310, author = {Zhang , BaijuYang , Yan and Feng , Minfu}, title = {A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {2}, pages = {310--336}, abstract = {

We propose and analyze a $C^0$-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 piecewise-polynomial approximations of degree $k+2$ for the stream-function $\psi$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $\nabla\psi$ 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 $L^2$-norm, $H^1$-norm and broken $H^2$-norm. Numerical tests are presented to demonstrate the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1806-m2017-0287}, url = {http://global-sci.org/intro/article_detail/jcm/14519.html} }
TY - JOUR T1 - A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation AU - Zhang , Baiju AU - Yang , Yan AU - Feng , Minfu JO - Journal of Computational Mathematics VL - 2 SP - 310 EP - 336 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1806-m2017-0287 UR - https://global-sci.org/intro/article_detail/jcm/14519.html KW - Weak Galerkin method, Navier-Stokes equations, Stream-function formulation. AB -

We propose and analyze a $C^0$-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 piecewise-polynomial approximations of degree $k+2$ for the stream-function $\psi$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $\nabla\psi$ 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 $L^2$-norm, $H^1$-norm and broken $H^2$-norm. Numerical tests are presented to demonstrate the theoretical results.

Zhang , BaijuYang , Yan and Feng , Minfu. (2020). A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation. Journal of Computational Mathematics. 38 (2). 310-336. doi:10.4208/jcm.1806-m2017-0287
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