Uniform Error Analysis for Lagrange--Galerkin Approximations of Convection-Dominated Problems

Abstract

In this paper we present a rigorous error analysis for the Lagrange--Galerkin method applied to convection-dominated diffusion problems. We prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit. This is in contrast to other results in this field. For the time discretization, uniform convergence with respect to the diffusion parameter of order O(k//tn) is shown for initial values in L2 and O(k) for initial values in H2 . For the spatial discretization with linear finite elements, we verify uniform convergence of order O(h2 +min{h,h2 /k) for data in H2 . By interpolation of Banach spaces, suboptimal convergence rates are derived under less restrictive assumptions. The analysis is heavily based on a priori estimates, uniform in the diffusion parameter, for the solution of the continuous and the semidiscrete problem. They are derived in a Lagrangian framework by transforming the Eulerian coordinates completely into subcharacteristic coordinates. Finally, we illustrate the error estimates by some numerical results.