A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements

Abstract

The Arbitrary Lagrangian Eulerian formulation is widely used when problems in moving domains have to be approximated. Here we consider a time dependent linear advection–diffusion problem in a moving two-dimensional domain and discretize it by linear finite elements in space and a modification of the implicit Euler scheme, based on the mid point rule, in time. The resulting space–time discretization satisfies the so called Geometric Conservation Laws which prevent from numerical instabilities due to inaccurate calculations of geometric quantities. We derive a priori error estimates which are optimal both in space and time using slightly more regularity than that necessary for the case of non moving domains.