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

$$\begin{aligned} \Gamma_{9}\left[u_{i, j}\right]:=&-\frac{10}{3} u_{i, j}+\frac{2}{3}\left(u_{i+1, j}+u_{i-1, j}+u_{i, j+1}+u_{i, j-1}\right) \\ &+\frac{1}{6}\left(u_{i+1, j+1}+u_{i+1, j-1}+u_{i-1, j+1}+u_{i-1, j-1}\right)=h^{2} f_{i, j} \end{aligned}$$

where $1 \leqslant i, j \leqslant m, u_{i, j} \approx u(i h, j h)$, and $(i h, j h)$ are the grid points with $h=\frac{1}{m+1}$.

(i) Find the order of the local truncation error $\eta_{i, j}$ of the approximation.

(ii) Prove that the order of the truncation error is smaller if $f$ satisfies the Laplace equation $\nabla^{2} f=0$.

(iii) Show that the modified nine-point scheme

$$\begin{aligned} \Gamma_{9}\left[u_{i, j}\right] &=h^{2} f_{i, j}+\frac{1}{12} h^{2} \Gamma_{5}\left[f_{i, j}\right] \\ &:=h^{2} f_{i, j}+\frac{1}{12} h^{2}\left(f_{i+1, j}+f_{i-1, j}+f_{i, j+1}+f_{i, j-1}-4 f_{i, j}\right) \end{aligned}$$

has a truncation error of the same order as in part (ii).

(iv) Let $\left(u_{i, j}\right)_{i, j=1}^{m}$ be a solution to the $m^{2} \times m^{2}$ system of linear equations $A \mathbf{u}=\mathbf{b}$ arising from the modified nine-point scheme in part (iii). Further, let $u(x, y)$ be the exact solution and let $e_{i, j}:=u_{i, j}-u(i h, j h)$ be the error of approximation at grid points. Prove that there exists a constant $c$ such that

$$\|\mathbf{e}\|_{2}:=\left[\sum_{i, j=1}^{m}\left|e_{i, j}\right|^{2}\right]^{1 / 2}<c h^{3}, \quad h \rightarrow 0$$

[Hint: The nine-point discretization of $\nabla^{2} u$ can be written as

$$\Gamma_{9}[u]=\left(\Gamma_{5}+\frac{1}{6} \Delta_{x}^{2} \Delta_{y}^{2}\right) u=\left(\Delta_{x}^{2}+\Delta_{y}^{2}+\frac{1}{6} \Delta_{x}^{2} \Delta_{y}^{2}\right) u$$

where $\Gamma_{5}[u]=\left(\Delta_{x}^{2}+\Delta_{y}^{2}\right) u$ is the five-point discretization and

$$\left.\begin{array}{l} \Delta_{x}^{2} u(x, y):=u(x-h, y)-2 u(x, y)+u(x+h, y) \\ \Delta_{y}^{2} u(x, y):=u(x, y-h)-2 u(x, y)+u(x, y+h) \end{array}\right]$$

[Hint: The matrix A of the nine-point scheme is symmetric, with the eigenvalues

$$\lambda_{k, \ell}=-4 \sin ^{2} \frac{k \pi h}{2}-4 \sin ^{2} \frac{\ell \pi h}{2}+\frac{8}{3} \sin ^{2} \frac{k \pi h}{2} \sin ^{2} \frac{\ell \pi h}{2}, \quad 1 \leqslant k, \ell \leqslant m$$