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The unique solvability of the three-field continuous formulation, as well as the well-definiteness and stability of the proposed Galerkin and Petrov-Galerkin methods, is established thanks to the BabuSka-Brezzi theory. Optimal a priori error estimates are derived using norms robust with respect to the Lame constants, turning these numerical methods particularly appealing for nearly incompressible materials. We exemplify the accuracy (in a suitably weighted norm), as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2D and 3D, also including cases not covered by our theoretical analysis. (C) 2018 The Author(s). 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