  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.
