Scheme for solving Laplace's (Poisson's) equation
In order to illustrate how we can transform the last equations into a
linear algebra problem of the type \( \mathbf{A}\mathbf{x}=\mathbf{w} \), with
\( \mathbf{A} \) a matrix and \( \mathbf{x} \) and \( \mathbf{w} \) unknown and known
vectors respectively, let us also for the sake of simplicity assume
that the number of points \( n=3 \). We assume also that \( u(x,y) = g(x,y)
\) on the border \( \delta\Omega \).
The inner values of the function \( u \) are then
given by
$$
\begin{align}
4u_{11} -u_{21} -u_{01} - u_{12}- u_{10}=&-\tilde{\rho}_{11} \nonumber \\
4u_{12} - u_{02} - u_{22} - u_{13}- u_{11}=&-\tilde{\rho}_{12} \nonumber \\
4u_{21} - u_{11} - u_{31} - u_{22}- u_{20}=&-\tilde{\rho}_{21} \nonumber \\
4u_{22} - u_{12} - u_{32} - u_{23}- u_{21}=&-\tilde{\rho}_{22}. \nonumber
\end{align}
$$