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+1/2)) convergence rate for nth 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+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+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 -> 0.