(a) The boundary value problem $-\Delta u+c u=f$ on the unit square $[0,1]^{2}$ with zero boundary conditions and scalar constant $c>0$ is discretised using finite differences as

$$\begin{array}{r} -u_{i-1, j}-u_{i+1, j}-u_{i, j-1}-u_{i, j+1}+4 u_{i, j}+c h^{2} u_{i, j}=h^{2} f(i h, j h), \\ i, j=1, \ldots, m \end{array}$$

with $h=1 /(m+1)$. Show that for the resulting system $A u=b$, for a suitable matrix $A$ and vectors $u$ and $b$, both the Jacobi and Gauss-Seidel methods converge. [You may cite and use known results on the discretised Laplace operator and on the convergence of iterative methods.]

Define the Jacobi method with relaxation parameter $\omega$. Find the eigenvalues $\lambda_{k, l}$ of the iteration matrix $H_{\omega}$ for the above problem and show that, in order to ensure convergence for all $h$, the condition $0<\omega \leqslant 1$ is necessary.

[Hint: The eigenvalues of the discretised Laplace operator in two dimensions are $4\left(\sin ^{2} \frac{\pi k h}{2}+\sin ^{2} \frac{\pi l h}{2}\right)$ for integers $k, l$.]

(b) Explain the components and steps in a multigrid method for solving the Poisson equation, discretised as $A_{h} u_{h}=b_{h}$. If we use the relaxed Jacobi method within the multigrid method, is it necessary to choose $\omega \neq 1$ to get fast convergence? Explain why or why not.