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

We consider the Stokes problem in a three-dimensional axisymmetric bounded domain with non standard conditions which involve the normal component of the velocity and tangential component of the vorticity. We reduce the three-dimensional problem into a two-dimensional one and we write a variational formulation of it with three independent unknowns: the vorticity, the velocity and the pressure. Then we propose a discretization by spectral methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.