(i) Consider the numerical approximation of the boundary-value problem

$$\begin{gathered} u^{\prime \prime}=f, \quad u:[0,1] \rightarrow \mathbb{R}, \\ u(0)=\varphi_{0}, \quad u(1)=\varphi_{1}, \end{gathered}$$

where $\varphi_{0}, \varphi_{1}$ are given constants and $f$ is a given smooth function on $[0,1]$. A grid $\left\{x_{1}, x_{2}, \ldots, x_{N}\right\}, N \geqslant 3$, on $[0,1]$ is given by

$$x_{1}=\alpha_{1} h, \quad x_{i}=x_{i-1}+h \text { for } i=2, \ldots, N-1, \quad x_{N}=1-\alpha_{2} h$$

where $0<\alpha_{1}, \alpha_{2}<1, \alpha_{1}+\alpha_{2}=1$ and $h=1 / N$. Derive finite-difference approximations for $u^{\prime \prime}\left(x_{i}\right)$, for $i=1, \ldots, N$, using at most one neighbouring grid point of $x_{i}$ on each side. Hence write down a numerical scheme to solve the problem, displaying explicitly the entries of the system matrix $A$ in the resulting system of linear equations $A u=b$, $A \in \mathbb{R}^{N \times N}, u, b \in \mathbb{R}^{N}$. What is the overall order of this numerical scheme? Explain briefly one strategy by which the order could be improved with the same grid.

(ii) Consider the numerical approximation of the boundary-value problem

$$\begin{aligned} &\nabla^{2} u=f, \quad u: \Omega \rightarrow \mathbb{R}, \\ &u(x)=0 \text { for all } x \in \partial \Omega \end{aligned}$$

where $\Omega \subset \mathbb{R}^{2}$ is an arbitrary, simply connected bounded domain with smooth boundary $\partial \Omega$, and $f$ is a given smooth function. Define the 9-point formula used to approximate the Laplacian. Using this formula and an equidistant grid inside $\Omega$, define a numerical scheme for which the system matrix is symmetric and negative definite. Prove that the system matrix of your scheme has these properties for all choices of ordering of the grid points.

Part II, $2014 \quad$ List of Questions

[TURN OVER