ALE-type FEM formulation for PDEs on time-dependent domains with vanishing discrete SCL

Abstract

The aim of this paper is to introduce a finite element formulation within Arbitrary Lagrangian Eulerian framework with vanishing discrete {\it Space Conservation Law} (SCL) for differential equations on time dependent domains. The novelty of the formulation is the method for temporal integration which results in preserving the SCL property and retaining the higher order accuracy at the same time. Once the time derivative is discretized (based on integration or differentiation formula), the common approach for terms in differential equation which do not involve temporal derivative is classified to be a kind of "time averaging" between time steps. In the spirit of classical approaches, this involves evaluating these terms in several points in time between the current and the previous time step ($[t_n,t_{n+1}]$), and then averaging them in order to provide the satisfaction of discrete SCL. Here, we fully use the polynomial in time form of mapping through which evolution of domain is realized -- the so called ALE map -- in order to avoid the problematics arising due to the moving grids. We give a general recipe on temporal schemes that have to be employed once the discretization for the temporal derivative is chosen. Numerical investigations on stability, accuracy and convergence are performed and the simulated results are compared with benchmark problems set up by other authors.