 Weighted Jacobi smoothing bound for the square Poisson problem

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 $2/3$.