Laplace's and Poisson's Equations, boundary conditions
The boundary condtions read
\begin{equation*}
u_{i,0} = g_{i,0} \hspace{0.5cm} 0\le i \le n+1,
\end{equation*}
\begin{equation*}
u_{i,L} = g_{i,0} \hspace{0.5cm} 0\le i \le n+1,
\end{equation*}
\begin{equation*}
u_{0,j} = g_{0,j} \hspace{0.5cm} 0\le j \le n+1,
\end{equation*}
and
\begin{equation*}
u_{L,j} = g_{L,j} \hspace{0.5cm} 0\le j \le n+1.
\end{equation*}
With n+1 mesh points the equations for u result in a system of (n+1)^2 linear equations in the (n+1)^2 unknown u_{i,j} .