Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ALE framework

Abstract

In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semidiscrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.