Line-Jacobi Fourier spectrum (${\lambda }_{r,s}=\cos {\theta }_s/(2-\cos {\theta }_r)$)

For the five-point Dirichlet Laplacian on an $n\times n$ square grid, a block-Jacobi splitting that solves each horizontal line exactly has blocks $\mathrm{tridiag}(-1,4,-1)$. Its tensor-product sine modes have angles ${\theta }_r=r\pi /(n+1)$ and the displayed iteration matrix eigenvalues. With $c=\cos (\pi /(n+1))$, its spectral radius is $c/(2-c)<1$, so it converges on every fixed grid. However the mode $r=1,s=n$ has multiplier $-c/(2-c)\to -1$. It is highly oscillatory between lines and is barely damped. The undamped method is therefore not a uniformly effective smoother for the multigrid method, even though it converges. Convergence and smoothing test different properties of the error spectrum.