$H^2-$ conforming finite element on cube

Question

I read in a book about an $H^2$-conforming element on a rectangle, the Bogner-Fox-Schmit Rectangle element, and I was wondering if it has a three-dimensional extension to a cube. The degree of freedom of the BFS element is defined as $\frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial^2 \varphi}{\partial x^2}, \frac{\partial^2 \varphi}{\partial y^2}, \frac{\partial^2 \varphi}{\partial x \partial y}$.

Answer 1

In this paper "MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle", they define it as products of Hermite polynomials in 2D, so you can try to generalize this to 3D. It should yield $4^3=64$ degreees of freedom per element. I suppose natural ones would be pointwise values ($8$), gradients ($3\times 8 = 24$), second mixed derivatives $\partial_{xy}, \partial_{xz}, \partial_{yz}$ ($3\times 8 = 24$), and triple mixed derivative $\partial_{xyz}$ ($8$).

You can generalize this to $n$ dimensions by multiplying $n$ Hermite polynomials. You have $2^n$ vertices of the $n$-dimensional hypercube. There are $4^n$ basis functions. Pointwise values $2^n$, first derivatives $n\times 2^n$, second mixed derivatives ${n \choose 2}\times 2^n$, third mixed derivatives ${n \choose 3} \times 2^n$, ..., $(n-1)$-st mixed derivatives ${n \choose n-1} \times 2^n$, and $n$-th mixed derivative $2^n$. Note that the mixed derivatives above are without coordinate repetition (e.g. $\partial_{xy}$ but not $\partial_{xx}$), that's why I consider combinations). It may be possible to prescribe $\partial_{xx}$ and friends, but it's not the natural boundary condition for a Hermite polynomial.