Two low-order nonconforming finite element methods for the Stokes flow in 3D

In this paper, we propose two low order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the non-conforming FEM of Kouhia and Stenberg (1995, Comput. Methods Appl. Mech. Engrg.). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous in the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf-sup condition hold uniformly in the meshsize and also for a non-empty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia-Stenberg FEM to three space dimensions: The finite element space with one non-conforming and two conforming piecewise affine components does not satisfy a discrete inf-sup condition with piecewise constant pressure approximations, while finite element functions with two non-conforming and one conforming component do not satisfy a discrete Korn inequality.