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 - <p style="text-align: justify;">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.</p>