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

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