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 Ω ⊂ R². The method generates, in an explicit fashion, a continuous piecewise polynomial approximation of degree n 2 over a triangulation of Ω The scheme is shown to propagate disturbances a distance O(√ h log 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.