Geometric variational finite element discretizations for fluids

We present an overview of finite element variational integrators for compressible and incompressible fluids with variable density. The numerical schemes are derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. We illustrate the conservation properties of the scheme with the Rayleigh-Taylor instability test.