Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard h-weighted graphnorm and obtain optimal order error estimates with respect to mesh-size. Discontinuous Galerkin method, advection-reaction equation, local mass conservation, interior penalty.