Discontinuous Galerkin Methods for Friedrichs’ Systems. Part II. Second-order Elliptic PDEs

This paper is the second part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The setting under scrutiny is that of Friedrichs’ systems endowed with a particular $2 \times 2$ structure in which one unknown can be eliminated to yield a system of second-order elliptic-like PDEs for the remaining unknown. A general discontinuous Galerkin method for approximating such systems is proposed and analyzed. The key feature is that the unknown that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems. All the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated. Examples are given for advection-diffusion-reaction, linear continuum mechanics, and a simplified version of the magnetohydrodynamics equations. Comparisons with well-known discontinuous Galerkin approximations for the Poisson equation are presented.