Spectral element discretization of the vorticity, velocity and pressure formulation of the Navier-Stokes problem

The two-dimensional Navier-Stokes equations, when subject to non-standard boundary conditions which involve the normal component of the velocity and the vorticity, admit a variational formulation with three independent unknowns, the vorticity, velocity and pressure. We propose a discretization of this problem by spectral element methods. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.