On the order of convergence of the discontinuous Galerkin method for hyperbolic equations

The basic error estimate for the discontinuous Galerkin method for hyperbolic equations indicates an $O(h^{n+\frac{1}{2}})$ convergence rate for $n\textrm{th}$ degree polynomial approximation over a triangular mesh of size $h$. However, the optimal $O(h^{n+1})$ rate is frequently seen in practice. Here we extend the class of meshes for which sharpness of the $O(h^{n+\frac{1}{2}})$ estimate can be demonstrated, using as an example a problem with a ``nonaligned'' mesh in which all triangle sides are bounded away from the characteristic direction. The key to realizing $h^{n+\frac{1}{2}}$ convergence is a mesh which, to the extent possible, directs the error to lower frequency modes which are approximated, not damped, as $h\rightarrow 0$.