Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems

Abstract. 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 nonsymmetric version of the DG-method are well-posed also without penalization of the interelement solution jumps provided boundary conditions are imposed weakly. 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.