Two-grid Poisson factor with weighted Jacobi ($\rho (E)=1/9$)

For the one-dimensional Poisson equation, use linear interpolation, full-weighting restriction, an exact coarse solve, and one pre- and one post-sweep of weighted Jacobi with weight $2/3$. On a pair of harmonics $\theta ,\theta +\pi$, put $s={\sin }^2(\theta /2)$ and $c=1-s$. The coarse correction is $C=\left(\begin{array}{cc}s& -c\\ -s& c\end{array}\right)$, and the smoother is $S=\mathrm{diag}(1-4s/3,1-4c/3)$. The rank-one error operator $SCS$ has nonzero eigenvalue $s(1-4s/3{)}^2+c(1-4c/3{)}^2=1/9$. This exact harmonic calculation illustrates mesh-independent two-grid contraction for the constant-coefficient model; it is not an unconditional result for arbitrary operators or transfer choices.