Discontinuous Galerkin Methods for Friedrichs’ Systems

Abstract

This work presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs' systems. A general set of boundary conditions is identified to guarantee existence and uniqueness of solutions to these systems. A formulation enforcing the boundary conditions weakly is proposed. This formulation is the starting point for the construction of Discontinuous Galerkin methods formulated in terms of boundary operators and of interface operators that mildly penalize interface jumps. A general convergence analysis is presented. The setting is subsequently specialized to two-field Friedrichs' systems endowed with a particular 2{\texttimes}2 structure in which some of the unknowns can be eliminated to yield a system of second-order elliptic-like PDE's for the remaining unknowns. A general Discontinuous Galerkin method where the above elimination can be performed in each mesh cell is proposed and analyzed. Finally, details are given for four examples, namely advection-reaction equations, advection-diffusion-reaction equations, the linear elasticity equations in the mixed stress-pressure-displacement form, and the Maxwell equations in the so-called elliptic regime.