Discontinuous Galerkin Methods for Friedrichs ’ Symmetric Systems

This paper presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs' symmetric 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 diffusive regime. 1. Introduction. Discontinuous Galerkin (DG) methods have been introduced in the 1970s, and their development has since followed two somewhat parallel routes depending on whether the PDE is hyperbolic or elliptic. For hyperbolic PDEs, the first DG method was introduced by Reed and Hill in 1973 [25] to simulate neutron transport and the first analysis of DG methods for hy-perbolic equations in an already rather general and abstract form was done by Lesaint and Raviart in 1974 [22, 21]. The analysis was subsequently improved by Johnson et al. who established that the optimal order of convergence in the L 2-norm is p +