Discontinuous Galerkin

Abstract

In this chapter, instead of using stabilized $H^1$-conforming finite elements, we consider the discontinuous Galerkin (dG) method. The stability and convergence properties of the method rely on choosing a numerical flux across the mesh interfaces. Choosing the centered flux yields suboptimal convergence rates for smooth solutions. The stability properties of the method are tightened by penalizing the interface jumps, which corresponds to upwinding in the case of advection-reaction equations. This method gives the same error estimates as those obtained with stabilized $H^1$-conforming finite elements. Here again, the boundary conditions are enforced by a boundary penalty technique.