Shweta Srivastava

Sashikumaar Ganesan, Shweta Srivastava

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 $δ_k$ is also presented.

Sashikumaar Ganesan, Shweta Srivastava

A Streamline Upwind Petrov-Galerkin (SUPG) finite element method for transient convection-diffusion-reaction equation in time-dependent domains is proposed. In particular, a convection dominated transient scalar problem is considered. The time-dependent domain is handled by the arbitrary Lagrangian-Eulerian (ALE) approach, whereas the SUPG finite element method is used for the spatial discretization. Further, the first order backward Euler and the second order Crank-Nicolson methods are used for the temporal discretization. It is shown that the stability of the semidiscrete (continuous in time) conservative ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete problem is unconditionally stable for implicit Euler method and is only conditionally stable for Crank-Nicolson time discretization. Numerical results are presented 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.