= Two-grid Poisson factor with weighted Jacobi
{title2=$\rho(E)=1/9$}
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 $C=\left(\begin{smallmatrix}s&-c\\-s&c\end{smallmatrix}\right)$, and the smoother is $S=\operatorname{diag}(1-4s/3,1-4c/3)$. 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.
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