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 $u_{i,j}$.

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