Low order discontinuous Galerkin methods for second order elliptic problems

We consider DG-methods for 2nd order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the non-symmetric version of the DG-method have regular system matrices also without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a discontinuous Galerkin method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and non- symmetric DG-method without stabilization. All these proposed methods share the feature that they conserve mass locally independent of the penalty parameter.