In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method (BSDG) for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous finite element space elementwise by quadratic non-conforming bubbles. This approach leads to optimal convergence in the space and time discretization parameters. Moreover the divergence of the diffusive fluxes converges in the $L^2$ -norm independently of the geometry of the domain.