Nonconforming finite element methods of order two and order three for the Stokes flow in three dimensions

In this study, the nonconforming finite elements of order two and order three are constructed and exploited for the Stokes problem. The moments of order up to $k-1$ ($k=2,3$) on all the facets of the tetrahedron are used for DoFs (degrees of freedom) to construct the unisolvent $k$-order nonconforming finite element with the bubble function space of $P_{k+1}$ explicitly represented. The pair of the $k$-order element and the discontinuous piecewise $P_{k}$ is proved to be stable for solving the Stokes problem with the element-wise divergence-free condition preserved. The main difficulty in establishing the discrete inf-sup condition comes from the fact that the usual Fortin operator can not be constructed. Thanks to the explicit representation of the bubble functions, its divergence space is proved to be identical to the orthogonal complement space of constants with respect to $P_k$ on the tetrahedron, which plays an important role to overcome the aforementioned difficulty and leads to the desirable well-posedness of the discrete problem. Furthermore, a reduced $k$-order nonconforming finite element with a discontinuous piecewise $P_{k-1}$ is designed and proved to be stable for solving the Stokes problem. The lack of the Fortin operator causes difficulty in analyzing the discrete inf-sup condition for the reduced third-order element pair. To deal with this problem, the so-called macro-element technique is adopted with a crucial algebraic result concerning the property of functions in the orthogonal complement space of the divergence of the discrete velocity space with respect to the discrete pressure space on the macro-element. Numerical experiments are provided to validate the theoretical results.