Discretizations of incompressible fluids yielding divergence-free approximations

Friday, September 11, 2015 - 15:30 to 16:30
704 Thackeray Hall

Abstract or Additional Information

Abstract: A classical finite element method for the Navier-Stokes equations (NSE) poses the variational formulation onto finite dimensional subspaces of piecewise polynomials with respect to a partition of the domain. Such discretizations belong to the class of mixed finite element methods in which more than one independent variable is introduced in the discretization. A fundamental result of this framework is the inf-sup condition, a necessary criterion to ensure the existence and stability of the discrete problem. For NSE, the inf-sup condition implies that the finite element spaces must be compatible, namely, a surjective property of the divergence operator between piecewise polynomial spaces must be satisfied. While several finite element pairs have been developed that uniformly satisfy the inf–sup condition, they do so at the cost of violating the intrinsic structure and invariants in NSE.  As such, most methods may suffer from hidden instabilities unrelated to the discrete inf-sup condition. 

The focus of this talk is the construction of inf-sup stable and structure preserving discretization for NSE, with an emphasis on the divergence-free constraint. We develop the first conforming finite element spaces for NSE in two and three dimensions that yield divergence-free velocity approximations on general simplicial meshes. The derivation of the finite element pairs is motivated by a smooth de Rham complex which is well-suited for NSE and Stokes. The advantages of the proposed methods are discussed, and the stability and convergence properties are shown.