High-frequency smoothing factor of weighted Jacobi (${\mu }_{\omega }$)

For modes with $k/(n+1)\in [1/2,1]$ and large $n$, the worst attenuation factor isIt is minimized by $\omega =2/3$, for which ${\mu }_{\omega }=1/3$.

For the one-dimensional Dirichlet second-difference grid, high sine modes have eigenvalues $1-\omega t_k$, with $t_k=1-\cos (k\pi /(n+1))$ and $k\ge \lceil (n+1)/2\rceil$. Let $a$ and $b$ be the smallest and largest of these $t_k$. Convexity gives attenuation $\max (|1-\omega a|,|1-\omega b|)$. Balancing its endpoint values with opposite signs gives the displayed exact minimax choice. For odd $n$, $a=1$; for even $n$, $a=1+\sin (\pi /[2(n+1)])$, and always $b=1+\cos (\pi /(n+1))$. Only in the large-grid or uniform-in-grid limit do these become ${\omega }_{\ast }=2/3$, ${\mu }_{\ast }=1/3$.