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. Copyright © 2006 John Wiley & Sons, Ltd.