Spectral Discretization of the Stokes Problem with Mixed Boundary Conditions

The variational formulation of the Stokes problem with three independent unknowns, the vorticity, the velocity and the pressure, was used to handle non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We propose an extension of this formulation to the case of mixed boundary conditions in a three-dimensional domain. Next we consider a spectral discretization of this problem. A detailed numerical analysis leads to error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.

Keywords: Darcy’s equations, divergence-free discrete velocity, error estimate, Galerkin method, Lipschitz-continuous boundary, mixed boundary conditions, nonstandard boundary conditions, spectral discretization, Stokes problem, vorticity