Laplace's and Poisson's Equations, discretized version
The discretized version reads
$$
\begin{equation*}
u_{xx}\approx \frac{u(x+h,y)-2u(x,y)+u(x-h,y)}{h^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{yy}\approx \frac{u(x,y+h)-2u(x,y)+u(x,y-h)}{h^2},
\end{equation*}
$$
which we rewrite as
$$
\begin{equation*}
u_{xx}\approx \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{yy}\approx \frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{h^2}.
\end{equation*}
$$