Some recent developments in high order finite element methods for incompressible flow

Over the past 30-40 years, high order finite element methods (FEMs) have gained significant traction. While much of the theory and implementation of high order continuous FEMs are well-understood, FEMs with increased smoothness are less popular in the literature and in practice. Nevertheless, engineering problems involving plates and shells give rise to fourth order elliptic equations, whose conforming approximations typically entail the Argyris elements, which are globally C1 and C2 at element vertices. The Argyris elements and their high order counterparts can then be used to construct the mass-conserving Falk-Neilan elements for incompressible flow problems. In particular, the Falk-Neilan elements inherit a level of extra smoothness at element vertices. In this talk, we will give a brief overview of some recent developments concerning the uniform hp-stability and preconditioning of the Falk-Neilan elements.