Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 40E / b / Solution

After multiplying the five-point equations by $-1$ if necessary, their matrix $A$ is symmetric positive definite. For any ordering write $A=D+L+L^T$, where $D=4I$. Gauss-Seidel method uses $M=D+L$ and has iteration matrix $-M^{-1}{L}^T$. In part (a) take $B=L^T$; then $A-B=M$ andwhich is positive definite. Therefore the iteration matrix has spectral radius below one for every ordering, proving convergence.