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We consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure, and the pseudostress are the main unknowns. Alternatively, the pressure can be eliminated from the original equations, thus yielding an equivalent formulation with only two unknowns. In this paper we develop a priori and a posteriori error analyses of both approaches. We first apply the Babuška-Brezzi theory to prove the well-posedness of the continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k P 0 for the pseudostresses, and piecewise polynomials of degree k for the velocities and the pressures, ensure unique solvability and stability of the associated Galerkin schemes. Then, we derive reliable and efficient residual-based a posteriori error estimators for both schemes, without and with pressure unknown. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimators, and showing the behaviour of the associated adaptive algorithms.