Spectral Discretization of the Vorticity, Velocity, and Pressure Formulation of the Stokes Problem

Abstract

We consider the Stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity, and the pressure. Next, we propose a discretization by spectral methods which relies on this formulation and, since it leads to an inf-sup condition on the pressure in a natural way, we prove optimal error estimates for the three unknowns. We present numerical experiments which are in perfect coherence with the analysis.