The nine-point method for the Poisson equation $\nabla^{2} u=f$ (with zero Dirichlet boundary conditions) in a square, reads

$$\begin{array}{r} \frac{2}{3}\left(u_{i-1, j}+u_{i+1, j}+u_{i, j-1}+u_{i, j+1}\right)+\frac{1}{6}\left(u_{i-1, j-1}+u_{i-1, j+1}+u_{i+1, j-1}+u_{i+1, j+1}\right) \\ -\frac{10}{3} u_{i, j}=h^{2} f_{i, j}, \quad i, j=1, \ldots, m, \end{array}$$

where $u_{0, j}=u_{m+1, j}=u_{i, 0}=u_{i, m+1}=0$, for all $i, j=0, \ldots, m+1$.

(i) By arranging the two-dimensional arrays $\left\{u_{i, j}\right\}_{i, j=1, \ldots, m}$ and $\left\{f_{i, j}\right\}_{i, j=1, \ldots, m}$ into column vectors $u \in \mathbb{R}^{m^{2}}$ and $b \in \mathbb{R}^{m^{2}}$ respectively, the linear system above takes the matrix form $A u=b$. Prove that, regardless of the ordering of the points on the grid, the matrix $A$ is symmetric and negative definite.

(ii) Formulate the Jacobi method with relaxation for solving the above linear system.

(iii) Prove that the iteration converges if the relaxation parameter $\omega$ is equal to $1 .$

[You may quote without proof any relevant result about convergence of iterative methods.]