Local error estimates for a finite element method for hyperbolic and convection-diffusion equations

Local error estimates of near optimal order are derived for a finite element method for hyperbolic and convection dominated convection-diffusion equations in a domain $\Omega \subset R^2 $. The method generates, in an explicit fashion, a continuous piecewise polynomial approximation of degree $n \geqslant 2$ over a triangulation of $\Omega $ The scheme is shown to propagate disturbances a distance $O(\sqrt h \log \frac{1} {h})$ in the crosswind direction, where h is the meshsize. The analysis uses test functions which depend only on the crosswind variable. It is also shown to be applicable, in a parallel fashion, to the discontinuous Galerkin method, thus underscoring the close interrelationship of the two methods.