Scheme for solving Laplace's (Poisson's) equation

If we isolate on the left-hand side the unknown quantities $u_{11}$, $u_{12}$, $u_{21}$ and $u_{22}$, that is the inner points not constrained by the boundary conditions, we can rewrite the above equations as a matrix $\mathbf{A}$ times an unknown vector $\mathbf{x}$, that is $$\begin{equation*} Ax = b, \end{equation*}$$ or in more detail $$\begin{equation*} \begin{bmatrix} 4&-1 &-1 &0 \\ -1& 4 &0 &-1 \\ -1& 0 &4 &-1 \\ 0& -1 &-1 &4 \\ \end{bmatrix}\begin{bmatrix} u_{11}\\ u_{12}\\ u_{21} \\ u_{22} \\ \end{bmatrix}=\begin{bmatrix} u_{01}+u_{10}-\tilde{\rho}_{11}\\ u_{13}+u_{02}-\tilde{\rho}_{12}\\ u_{31}+u_{20}-\tilde{\rho}_{21} \\ u_{32}+u_{23}-\tilde{\rho}_{22}\\ \end{bmatrix}. \end{equation*}$$

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