  We introduce new Hermite-style and Bernstein-style geometric decompositions
of the cubic order serendipity finite element spaces $S_3(I^2)$ and $S_3(I^3)$,
as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11
(2011), 337--344]. The serendipity spaces are substantially smaller in
dimension than the more commonly used bicubic and tricubic Hermite tensor
product spaces - 12 instead of 16 for the square and 32 instead of 64 for the
cube - yet are still guaranteed to obtain cubic order \textit{a priori} error
estimates in $H^1$ norm when used in finite element methods. The basis
functions we define have a canonical relationship both to the finite element
degrees of freedom as well as to the geometry of their graphs; this means the
bases may be suitable for applications employing isogeometric analysis where
domain geometry and functions supported on the domain are described by the same
basis functions. Moreover, the basis functions are linear combinations of the
commonly used bicubic and tricubic polynomial Bernstein or Hermite basis
functions, allowing their rapid incorporation into existing finite element
codes.
