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In this study, the nonconforming finite elements of order two and order threeare constructed and exploited for the Stokes problem. The moments of order upto $k-1$ ($k=2,3$) on all the facets of the tetrahedron are used for DoFs(degrees of freedom) to construct the unisolvent $k$-order nonconforming finiteelement with the bubble function space of $P_{k+1}$ explicitly represented. Thepair of the $k$-order element and the discontinuous piecewise $P_{k}$ is provedto be stable for solving the Stokes problem with the element-wisedivergence-free condition preserved. The main difficulty in establishing thediscrete inf-sup condition comes from the fact that the usual Fortin operatorcan not be constructed. Thanks to the explicit representation of the bubblefunctions, its divergence space is proved to be identical to the orthogonalcomplement space of constants with respect to $P_k$ on the tetrahedron, whichplays an important role to overcome the aforementioned difficulty and leads tothe 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 ofthe Fortin operator causes difficulty in analyzing the discrete inf-supcondition for the reduced third-order element pair. To deal with this problem,the so-called macro-element technique is adopted with a crucial algebraicresult concerning the property of functions in the orthogonal complement spaceof the divergence of the discrete velocity space with respect to the discretepressure space on the macro-element. Numerical experiments are provided tovalidate the theoretical results.

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