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

A Streamline Upwind Petrov-Galerkin (SUPG) finite element method for a convection dominated transient convection-diffusion-reaction equation in time-dependent domains is proposed. The time-dependent domain is handled by the arbitrary Lagrangian-Eulerian (ALE) approach, whereas the SUPG method is used for the spatial discretization. Further, the first order modified backward Euler and the second order modified Crank-Nicolson methods are used for the temporal discretization. It is shown that the stability of the semi-discrete (continuous in time) conservative ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete scheme with the implicit Euler time discretization is unconditionally stable and is only conditionally stable (time step depends on mesh velocity) with the Crank-Nicolson method. Numerical results are presented to support the stability estimates and to show the influence of the SUPG stabilization parameter in a time-dependent domain. Further, the proposed numerical scheme is applied to a boundary/layer problem in a time-dependent domain.