On uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems

Abstract

Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter e or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < e ≪ 1. In this work we develop a rigorous e-uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit e → 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation e-uniform convergence of order one half is proven whereas in the Lagrangian framework e-uniform convergence of optimal order is established. The estimates are based on e-uniform a priori estimates for the solution of the continuous problems which are derived first.