Artículo de publicación ISIIn this work we propose an augmented discontinuous Galerkin method for elliptic linear problems in the plane with mixed boundary conditions. Our approach introduce Galerkin least-squares terms, arising from constitutive and equilibrium equations, which allow us to look for the flux unknown in the local Raviart-Thomas space. The unique solvability is established avoiding the introduction of lifting operators and we derive a C ́ea estimate, which let us conclude that the rate of convergence of error, measured in an appropriate norm, is optimal respect to the h−version. We emphasize that for practical computations, this method reduces the degrees of freedom, with respect to the classical discontinuous Galerkin method.