I want to solve the ODE \begin{array}{ll} -u''(x)=f(x) & x\in (0,1) \\ u(0)=g(0) \\ u(1)=g(1)\, \\ \end{array}
with finite differences using $u''(x)\approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2}$. To approximate $[0,1]$ I use a grid $T=\{0=x_0<\dots , x_n=1\}$.
This leads to the equations $\frac{1}{h^2}(-u_{i-1}+2u_i-u_{i+1})=f(x_i)$ for $1\leq i\leq n-1$, $u_0=g(x_0), u_n=g(x_n)$ and the linear system \begin{align} A=\frac{1}{h^2}\begin{pmatrix} 1 & 0 & 0 & &\cdots & 0\\ -1 & 2 & -1 & 0 & \cdots & 0 \\ 0 & -1 & 2 & -1 & \cdots & 0 \\ 0 & 0 & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & -1 & 2 & -1 \\ 0 & \cdots & 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} u_0\\ \vdots \\ u_n \end{pmatrix}= \begin{pmatrix} \frac{g(0)}{h^2} \\ f(x_1)\\ \vdots\\ f(x_{n-1})\\ \frac{g(1)}{h^2} \end{pmatrix} \end{align} I already showed that $(Av)_i<0 \Rightarrow v_i<0$ for all $i=0,\dots ,n$. How can I show that there exists $C>0$ so that $\|u\|_\infty\leq C\big (\|f\|_\infty+\|g\|_\infty\big )$ ?($u$ is a solution of the linear system)