A Finite Element Formulation Satisfying the Discrete Geometric Conservation Law Based on Averaged Jacobians

Abstract

In this article a new methodology for developing DGCL (for Discrete Geometric Conservation Law) compliant formulations is presented. It is carried out in the context of the Finite Element Method (FEM) for general advective-diffusive systems on moving domains using an Arbitrary Lagrangian Eulerian (ALE) scheme.
There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed-grid counterpart. However, only a few works propose a methodology for obtaining a compliant scheme.
In this work, a DGCL compliant scheme based on an Averaged ALE Jacobians Formulation (AJF) is obtained. This new formulation is applied to the theta-family of time integration methods. In addition, an extension to the three-point Backward Difference Formula (BDF) is given. With the aim to validate the AJF formulation a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements, and the 2D Euler flow over a pitching NACA0012 airfoil.