The Poisson equation $\nabla^{2} u=f$ in the unit square $\Omega=[0,1] \times[0,1]$, with zero boundary conditions on $\partial \Omega$, is discretized with the nine-point formula

$$\begin{aligned} \frac{10}{3} u_{m, n} &-\frac{2}{3}\left(u_{m+1, n}+u_{m-1, n}+u_{m, n+1}+u_{m, n-1}\right) \\ &-\frac{1}{6}\left(u_{m+1, n+1}+u_{m+1, n-1}+u_{m-1, n+1}+u_{m-1, n-1}\right)=-h^{2} f_{m, n}, \end{aligned}$$

where $1 \leqslant m, n \leqslant M, u_{m, n} \approx u(m h, n h)$, and $(m h, n h)$ are grid points.

(a) Prove that, for any ordering of the grid points, the method can be written as $A \mathbf{u}=\mathbf{b}$ with a symmetric positive-definite matrix $A$.

(b) Describe the Jacobi method for solving a linear system of equations, and prove that it converges for the above system.

[You may quote without proof the corollary of the Householder-John theorem regarding convergence of the Jacobi method.]