Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 4 / c

Let $\beta (q)$ be the centered nondimensional cubic B-spline, even and supported on $[-2,2]$. The outer cubic must be $b(2-|q|{)}^3$ to make the function and its first two derivatives vanish at the support edge. Write its inner part as $A+B{q}^2+C|q{|}^3$. Matching the value and first two derivatives at $|q|=1$ gives $A=4b$, $B=-6b$, $C=3b$. Its nonzero knot values are then $4b,b,b$, so requiring their sum to be one fixes $b=1/6$. ThusIn particular $\beta (0)=2/3$, $\beta (\pm 1)=1/6$.

With $h=1/m$, represent the spline on $[0,1]$ as $s(x)={\sum }_{j=-1}^{m+1}{c}_j\beta (x/h-j)$. There are $m+3$ unknown coefficients. interpolation at $x_i=ih$, $0\le i\le m$, givesThe natural endpoint conditions, using ${\beta }^{\prime \prime }(0)=-2$ and ${\beta }^{\prime \prime }(\pm 1)=1$, giveThese $m+3$ equations are the required matrix system: the $m+1$ interpolation rows have coefficients $(1,4,1)$ at consecutive positions and right sides $6u(x_i)$; the two endpoint rows have $(1,-2,1)$ and zero right sides. No solution of that system is required. The scaling uses $\beta (x/h-j)$, without an extra $1/h$, because normalization is by knot values rather than by integral.