A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION |
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Authors: | Baiju Zhang Yan Yang & Minfu Feng |
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Institution: | School of Mathematics, Sichuan University, Chengdu 610064, China;School of Sciences, Southwest Petroleum University, Chengdu 610500, China |
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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. |
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Keywords: | Weak Galerkin method Navier-Stokes equations Stream-function formulation |
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