ALE-SUPG finite element method for convection-diffusion problems in time-dependent domains: Non-conservative form

Stability estimates for Streamline Upwind Petrov-Galerkin (SUPG) finite element method with different time integration schemes for the solution of a scalar transient convection-diffusion-reaction equation in a time-dependent domain are derived. The deformation of the domain is handled with the arbitrary Lagrangian-Eulerian (ALE) approach. In particular, the non-conservative form of the ALE scheme is considered. The implicit Euler, the Crank-Nicolson, and the backward-difference~(BDF-2) methods are used for temporal discretization. It is shown that the stability of the semi-discrete (continuous in time) ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete problem is only conditionally stable. The theoretical considerations are illustrated by a numerical example. Further, the dependence of numerical solution on the choice of stabilization parameter $\delta_k$ is also presented.