Scheme for solving Laplace's (Poisson's) equation using Jacobi's iterative method
In setting up for example Jacobi's method, it is useful to rewrite the matrix \( \mathbf{A} \) as
$$
\begin{equation*}
\mathbf{A}=\mathbf{D}+\mathbf{U}+\mathbf{L},
\end{equation*}
$$
with \( \mathbf{D} \) being a diagonal matrix with \( 4 \) as the only value, \( \mathbf{U} \) is an upper triangular matrix and \( \mathbf{L} \)
a lower triangular matrix. In our case we have
$$
\begin{equation*}
\mathbf{D}=\begin{bmatrix}4&0 &0 &0 \\
0& 4 &0 &0 \\
0& 0 &4 &0 \\
0& 0 &0 &4 \\
\end{bmatrix},
\end{equation*}
$$
and
$$
\begin{equation*}
\mathbf{L}=\begin{bmatrix} 0&0 &0 &0 \\
-1& 0 &0 &0 \\
-1& 0 &0 &0 \\
0& -1 &-1 &0 \\
\end{bmatrix} \hspace{1cm} \mathbf{U}= \begin{bmatrix}
0&-1 &-1 &0 \\
0& 0 &0 &-1 \\
0& 0 &0 &-1 \\
0& 0 &0 &0 \\
\end{bmatrix}.
\end{equation*}
$$