Discontinuous Galerkin finite element approximation of the two-dimensional Navier-Stokes equations in stream-function formulation

In this paper, we present the construction and computational assessment of an hp-version discontinuous Galerkin finite element method (DGFEM) for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, which ensures that the incompressibility constraint is automatically satisfied, we reduce the system of Navier-Stokes equations to a single fourth-order nonlinear partial differential equation. We introduce a discretization of this fourth-order nonlinear partial differential equation based on a combination of the symmetric DGFEM for the biharmonic part of the equation and a DGFEM with jump-penalty terms for the hyperbolic part of the problem, and then we solve the resulting nonlinear problem using Newton's method. Numerical examples, including the solution of the 2D lid-driven cavity flow problem, are presented to demonstrate the convergence and accuracy of the method.