Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 4 / b / Solution

A $C^2$ piecewise cubic supported on $[0,ph]$ has the truncated-power representation with indices $0,\dots ,p$. Vanishing for $x\ge ph$ imposesafter expanding its cubic exterior polynomial; powers of $h$ cancel. For $p\le 3$, the Vandermonde matrix columns are independent, forcing every coefficient to be zero. For $p=4$, coefficients proportional to $(1,-4,6,-4,1)$ satisfy these conditions because a fourth finite difference annihilates cubics.

Normalize to get the Cardinal cubic B-splineAt its five support knots its values are $0,1/6,2/3,1/6,0$, so their sum is one. Thus four intervals are necessary and sufficient.

For $h=1/2$, the five translates active in the interior of $[0,1]=[0,2h]$ are indexed $-3,-2,-1,0,1$. Write $s(x)={\sum }_{j=-3}^1{c}_j{N}_j^3(x)$. Nodal interpolation givesAt the left endpoint the first derivatives of the three active functions are $-1/(2h),0,1/(2h)$, and their second derivatives are $1/h^2,-2/h^2,1/h^2$. The two remaining equations areTogether these five equations form the requested system; insert $h=1/2$ if numerical coefficients are desired. The first, fourth and fifth determine the first three coefficients, and the other two then determine $c_0,c_1$, so the system is nonsingular.

The source only assumes $u\in C[0,1]$, which need not have either endpoint derivative. The Hermite conditions require those derivatives to exist, or to be supplied as separate interpolation data. The nodal conditions alone remain meaningful for every continuous $u$.