The variational formulation of the Stokes problem with three independent unknowns, the vorticity, the velocity and the pressure, was born 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. Résumé:La formulation variationnelle du problème de Stokes avec trois inconnues indépendantes, le tourbillon, la vitesse et la pression, est née pour traiter des conditions aux limites non usuelles portant sur la composante normale de la vitesse et les composantes tangentielles du tourbillon. Nous proposons une extension de cette formulation au cas de conditions aux limites mixtes dans un domaine tri-dimensionnel. Nous considérons ensuite une discrétisation obtenue par méthode spectrale. Une analyse numérique détaillée permet d'établir des majorations d'erreur pour les trois inconnues et des expériences numériques confirment l'intérêt de la discrétisation.
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