= Cubic Hermite finite element
{c}
A one-dimensional <Cubic Hermite finite element> represents a cubic <polynomial> on $[a,b]$ by its value and first <derivative> at each endpoint. Put $h=b-a>0$ and $\xi=(x-a)/h$. Its four reference <shape functions> are
$$
H_{00}=1-3\xi^2+2\xi^3,\quad H_{10}=\xi-2\xi^2+\xi^3,\quad
H_{01}=3\xi^2-2\xi^3,\quad H_{11}=-\xi^2+\xi^3.
$$
The interpolating <polynomial> is
$$
p(x)=y_aH_{00}(\xi)+h s_aH_{10}(\xi)+y_bH_{01}(\xi)+h s_bH_{11}(\xi).
$$
Thus $p(a)=y_a$, $p'(a)=s_a$, $p(b)=y_b$ and $p'(b)=s_b$. The factors $h$ convert reference-coordinate derivatives to physical derivatives. These conditions determine the cubic uniquely: a difference with zero endpoint values and slopes would have two double roots, impossible for a nonzero cubic.
Sharing both value and slope between adjacent intervals produces globally $C^1$ piecewise cubics in <Sobolev space> $H^2$. With the required endpoint constraints, they form a <conforming finite element space> for fourth-order variational problems, such as a bending problem whose <energy> contains $\int|u''|^2$. Ordinary continuous piecewise-linear elements have slope jumps and do not have this $H^2$ conformity.
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