A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation
Year: 2020
Author: Baiju Zhang, Yan Yang, Minfu Feng
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1806-m2017-0287
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 310–336
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Weak Galerkin method Navier-Stokes equations Stream-function formulation.