| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
103 |
II |
2011 |
Numerical Analysis |
|
Paper 1, Section II, A |
2011 |
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.]