Exercise 4 (2+2+2=6 points)
Let $\Omega_h = \{(ih, jh) \mid i, j = 1, \dots, n - 1\}$ with $h = 1/n$, $\Omega_h = \{(ih, jh) \mid i, j = 0, \dots, n\}$,
$\partial \Omega_h = \Omega_h \setminus \Omega_h$ and $H_0 = \{U: \Omega_h \to \mathbb{R} \mid U(x) = 0 \, \forall x \in \partial \Omega_h\}$. Then $H_0$ is isomorphic to
$\mathbb{R}^N$ with $N = (n - 1)^2$ and it holds
$H_0 \ni U \iff U = (U(x_i))_{i=1}^N \in \mathbb{R}^N$
with some numbering
$\Omega_h = \{x_i \mid i = 1, \dots, N\}$.
(1)
a) Find a numbering (1) such that
$-\Delta_h U = A_h U \quad \forall U \in H_0$
holds with a symmetric matrix $A_h \in \mathbb{R}^{N \times N}$.
b) Show that
$4 \|U\|_\infty \le \|A_h U\|_\infty \le \frac{8}{h^2} \|U\|_\infty \quad \forall U \in \mathbb{R}^N$.
c) Show that the condition number $\kappa(A_h) = \|A_h\|_2 \|A_h^{-1}\|_2$ grows with decreasing mesh
size according to
$\kappa(A_h) \le \frac{2}{h^2}$.
