 Solution

For homogeneous error boundary values, the tensor-product sine vectors $U_{j,k}=\sin (r\pi jh)\sin (s\pi kh)$ form a basis. The sine addition formula gives $U_{j+1,k}+U_{j-1,k}=2\cos (r\pi h)U_{j,k}$, and similarly in the other coordinate. Hence the five-point matrix hasThere is a printing error in the displayed iteration. Taken literally, its error multiplier is $1/20-(2/5)(\cos (r\pi h)+\cos (s\pi h))$, and its fixed-point equation is $(19/4)U_{j,k}+{\sum }_{\mathrm{neighbors}}U=h^2{f}_{j,k}$, rather than the stated Poisson equation. Thus the printed affine update cannot be used as a consistent relaxation for this system.

The intended weighted Jacobi iteration with $\omega =4/5$ isIts iteration matrix is $I-(\omega /4)A$, whose mode multiplier isIf at least one of the two cosines is negative, their sum lies strictly between $-2$ and $1$, so $-3/5<{\mu }_{r,s}<3/5$. Thus every such high-frequency mode contracts by at most $\boxed{3/5}$ per sweep, uniformly in grid spacing. This weighted Jacobi smoothing bound for the square Poisson problem justifies the corrected method as a smoother. The literal printed update also damps modes, but toward the wrong fixed-point equation.