Uniform Convergence Rates for Eulerian and Lagrangian Nite Element Approximations of Convection-dominated Diiusion Problems

Standard error estimates for nite element approximations of nonstationary convection-diiusion problems depend reciprocally on the diffusion parameter ". Therefore, the estimates become worthless in the case of strong convection-dominance 0 < " 1. This work provides an "-uniform convergence theory for nite element discretizations of convection-dominated diiusion problems in Eulerian and Lagrangian coordinates. Here, the constants which arise in the error estimates do not tend to innnity when passing to the hyperbolic limit " ! 0. In the Eulerian formulation "-uniform convergence of order one half is proven whereas in the Lagrangian framework "-uniform convergence of optimal order is established. The estimates are heavily based on "-uniform a priori estimates for the solution of the continuous problems which are derived rst.