A C^0-Weak Galerkin Finite Element Method for the Two-Dimensional Navier-Stokes Equations in Stream-Function Formulation

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 del 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.