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.