A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements

Abstract

In this paper we present some theoretical results on the Arbitrary Lagrangian Eulerian (ALE) formulation. This formulation may be used when dealing with moving domains and consists in recasting the governing differential equation and the related weak formulation in a frame of reference moving with the domain. The ALE technique is first presented in the whole generality for conservative equations and a result on the regularity of the underlying mapping is proven. In a second part of the work, the stability property of two types of finite element ALE schemes for parabolic evolution problems are analyzed and its relation with the so-called Geometric Conservation Laws is addressed.