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Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes

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.