Discontinuous Galerkin methods for friedrichs' systems. I. General theory

This paper presents a unified analysis of discontinuous Galerkin methods to approximate Friedrichs' systems. An abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator. Then a general discontinuous Galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed. All the design constraints of the method are fully stated, and an abstract error analysis in the spirit of Strang's Second Lemma is presented. Finally, the method is formulated locally using element fluxes, and links with other formulations are discussed. Details are given for three examples, namely, advection-reaction equations, advection-diffusion-reaction equations, and the Maxwell equations in the so-called elliptic regime.