Computation of elastic structures in contact is performed by means of a dual analysis combining displacement-based and equilibrium-based finite elements. Contact conditions are formulated in the framework of second-order cone programmaing (SOCP) and an efficient interior point method (IPM) algorithm is presented for the resolution of the associated optimization problems. The dual approach allows the user to assess the quality of convergence and to efficiently calculate a discretization error estimator which includes a contact error term. An efficient remeshing scheme, based on the local contributions of the elements to the global error, can then be used to efficiently improve the solution accuracy. The whole process is illustrated on some examples and applied to a typical steel assembly. Its efficiency, in particular concerning the IPM solver, is demonstrated in comparison with the industrial finite element code Abaqus.