Adaptive Lagrange-Galerkin methods for unsteady convection-dominated diffusion problems

Abstract

In this paper we derive an a posteriori error estimate in the L²(L²) norm for the Lagrange-Galerkin discretisation of the unsteady two-dimensional convection-diffusion problem. The proof of the error estimate is based on so-called strong stability estimates of an associated backward dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori error estimate, we design an adaptive algorithm for determining both the space and time discretisation parameters in such a way as to yield global error control in the L²(L²) norm with respect to a pre-determined tolerance. Moreover, the reliability and efficiency of this adaptive strategy are numerically verified on test problems with known analytic solutions.