We consider high order Discontinuous Galerkin (DG) discretization for steady state problems. It will be demonstrated that using a high order DG scheme to discretize a problem may result in two types of solution overshoots. The oscillations of the first type are associated with smooth approximation of solution discontinuities. In addition, the oscillations may appear in steady state problems as a result of incorrect flux approximation near the flux extremum point.
"Two Types of Solution Overshoots in Discontinuous Galerkin Discretization Schemes." Commun. Math. Sci. 3 (2) 235 - 249, June 2005.