= Weighted Jacobi smoothing bound for the square Poisson problem
{title2=$\omega=4/5\ \Longrightarrow\ |\mu_{r,s}|\leq3/5\text{ on high modes}$}
For the five-point Dirichlet Poisson matrix, <weighted Jacobi iteration> has sine-mode multiplier $1-\omega+(\omega/2)(\cos\theta_r+\cos\theta_s)$. Call a mode high-frequency if at least one cosine is nonpositive. Then the sum lies in $[-2,1]$, so for $\omega=4/5$ the multiplier lies in $[-3/5,3/5]$. This uniform damping makes the iteration a useful smoother for a <multigrid method>, even though low-frequency multipliers approach one on fine grids. Balancing $|1-2\omega|$ and $|1-\omega/2|$ shows why $4/5$ is the minimax parameter for this high-frequency interval. The two-dimensional high-frequency set gives a different optimum from the familiar one-dimensional value $2/3$.
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