Local discontinuous Galerkin method for diffusion equations with reduced stabilization

We extend the results on minimal stabilization of Burman and Stamm [J. Sci. Comp., 33 (2007), pp.~183-208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the form of a discrete inf-sup condition is proved and optimal convergence follows. Some numerical examples using high order approximation spaces illustrate the theory.