Weighted Jacobi method for the one-dimensional Poisson equation

For the centered second-difference matrix with diagonal $-2$ and off-diagonals $1$, the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when $0<\omega \le 1$.

Multiplying both $A$ and $b$ by $-1$ leaves $D^{-1}A$ and the weighted Jacobi method unchanged. Thus the same formula applies to the positive second-difference matrix $T_n$ with diagonal $2$ and neighboring entries $-1$, the finite difference form of the one-dimensional Poisson equation with Dirichlet boundary conditions. Its eigenvectors have components $\sin (jk\pi /(n+1))$, and $D^{-1}{T}_n$ has eigenvalues $1-\cos (k\pi /(n+1))$. At $\omega =2/3$, upper-half modes have factors between $-1/3$ and $1/3$, while low-frequency factors remain close to one on fine grids. This explains why the weighted Jacobi method helps a multigrid method by reducing oscillatory errors, even when it is a slow solver on its own.^[ref]

For modes with $k/(n+1)\in [1/2,1]$ and large $n$, the worst attenuation factor isIt is minimized by $\omega =2/3$, for which ${\mu }_{\omega }=1/3$.

For the one-dimensional Dirichlet second-difference grid, high sine modes have eigenvalues $1-\omega t_k$, with $t_k=1-\cos (k\pi /(n+1))$ and $k\ge \lceil (n+1)/2\rceil$. Let $a$ and $b$ be the smallest and largest of these $t_k$. Convexity gives attenuation $\max (|1-\omega a|,|1-\omega b|)$. Balancing its endpoint values with opposite signs gives the displayed exact minimax choice. For odd $n$, $a=1$; for even $n$, $a=1+\sin (\pi /[2(n+1)])$, and always $b=1+\cos (\pi /(n+1))$. Only in the large-grid or uniform-in-grid limit do these become ${\omega }_{\ast }=2/3$, ${\mu }_{\ast }=1/3$.