Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment

In this Note we prove that in two and three space dimensions, the symmetric and non-symmetric discontinuous Galerkin method for second order elliptic problems is stable when using piecewise linear elements enriched with quadratic bubbles without any penalization of the interelement jumps. The method yields optimal convergence rates in both the broken energy norm and, in the symmetric case, the $L^2$-norm. Moreover the method can be written in conservative form with fluxes independent of any stabilization parameter.