= Solution
A <multigrid method> combines a fine-grid smoother with a <coarse-grid correction>. Starting with $u_h$, apply a few smoothing iterations, compute $r_h=b_h-A_hu_h$, restrict this residual to a coarser grid, and approximately solve $A_He_H=Rr_h$. Prolong the coarse error and update $u_h\leftarrow u_h+Pe_H$, then post-smooth. The coarse solve is itself performed recursively, with a direct solve at the coarsest level; visiting each coarser level once gives a V-cycle. The smoother damps rapidly varying error, while error smooth on the fine grid is efficiently represented and corrected on the coarse grid.
For ordinary Jacobi, $\omega=1$, the highest-frequency sine-product error has <eigenvalue> approaching $-1$, not zero. In particular for the discrete Poisson equation it is $-\cos(\pi h)$, whose magnitude approaches one. This nearly alternating error changes sign between sweeps but is barely damped, and ordinary coarse-grid restriction does not represent it well. Thus \b[with the standard point-Jacobi smoother and coarse-grid transfers one chooses $\omega\ne1$, usually $0<\omega<1$, for fast multigrid convergence]. For example $\omega=2/3$ makes the extreme high-frequency <eigenvalue> approach $-1/3$. Convergence of Jacobi as a standalone iteration is therefore insufficient to guarantee that it is an effective smoother. This conclusion concerns that standard multigrid choice; other smoothers or specially designed transfers can change it.
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