A $C^0$-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation (Journal of Computational Mathematics, 2020)
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 $ψ$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $ ablaψ$ 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.