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In this paper we consider the three dimensional exterior Stokes problem andstudy the solvability of the corresponding continuous and discrete formulationsthat arise from the coupling of a dual-mixed variational formulation with theboundary integral equation method. More precisely, after employing theincompressibility condition to eliminate the pressure, we consider theresulting velocity-stress-vorticity approach with different kind of boundaryconditions on an annular bounded domain, and couple the underlying equationswith either one or two boundary integral equations arising from the applicationof the usual and normal traces to the Green representation formula in theexterior unbounded region. As a result, we obtain saddle point operatorequations, which are then analyzed by the well-known Babuska-Brezzi theory. Weprove the well-posedness of the continuous formulations, identifying previouslythe space of solutions of the associated homogeneous problem, and specifyexplicit hypotheses to be satisfied by the finite element and boundary elementsubspaces in order to guarantee the stability of the respective Galerkinschemes. In particular, following a similar analysis given recently for theLaplacian, we are able to extend the classical Johnson & N\'ed\'elec procedureto the present case, without assuming any restrictive smoothness requirement onthe coupling boundary, but only Lipschitz-continuity. In addition, anddifferently from known approaches for the elasticity problem, we are also ableto extend the Costabel & Han coupling procedure to the 3D Stokes problem byproviding a direct proof of the required coerciveness property, that is withoutargueing by contradiction, and by using the natural norm of each space insteadof mesh-dependent norms. Finally, we briefly describe concrete examples ofdiscrete spaces satisfying the aforementioned hypotheses.

New developments on the coupling of mixed-FEM and BEM for the three-dimensional Stokes problem