Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 40E / d

For the high-frequency indices $(n+1)/2\le k\le n$, the angle ${\theta }_k=k\pi /(n+1)$ ranges from $\pi /2$ to $\pi -\pi /(n+1)$. Sincethe large-$n$ high-frequency smoothing factor of weighted Jacobi isFor $0<\omega <1$, this is strictly less than one independently of $n$, so high-frequency error decays geometrically much faster than the low-frequency factor $1-O(n^{-2})$.

To minimize ${\mu }_{\omega }$, balance the endpoint magnitudes:Thus the optimal relaxation and attenuation factors areFor finite $n$, retaining the endpoint $\cos (\pi -\pi /(n+1))=-\cos (\pi /(n+1))$ giveswhich tend to $2/3$ and $1/3$.