Stability and geometric conservation laws for ALE formulations

The aim of this paper is to investigate a model ALE scheme, with respect to various possible choices of time discretizations. For each time scheme, we investigate the relationships between stability and the so-called geometric conservation laws (GCL). We shall see that GCL condition proves neither necessary nor sufficient for stability. In doing so, we review some known theoretical results and we prove some new stability results for space-time ALE discretizations. Some new error estimates are also presented. Several numerical experiments confirm the theory.