Scheme for solving Laplace's (Poisson's) equation, with Jacobi's method

We assume now that we have an estimate for the unknown functions $u_{11}$, $u_{12}$, $u_{21}$ and $u_{22}$. We will call this the zeroth value and label it as $u^{(0)}_{11}$, $u^{(0)}_{12}$, $u^{(0)}_{21}$ and $u^{(0)}_{22}$. We can then set up an iterative scheme where the next solution is defined in terms of the previous one as $$\begin{align} u^{(1)}_{11} =&\frac{1}{4}(b_1-u^{(0)}_{12} -u^{(0)}_{21}) \nonumber \\ u^{(1)}_{12} =&\frac{1}{4}(b_2-u^{(0)}_{11}-u^{(0)}_{22}) \nonumber \\ u^{(1)}_{21} =&\frac{1}{4}(b_3-u^{(0)}_{11}-u^{(0)}_{22}) \nonumber \\ u^{(1)}_{22}=&\frac{1}{4}(b_4-u^{(0)}_{12}-u^{(0)}_{21}), \nonumber \end{align}$$ where we have defined the vector $$\begin{equation*} \mathbf{b}= \begin{bmatrix} u_{01}+u_{10}-\tilde{\rho}_{11}\\ u_{13}+u_{02}-\tilde{\rho}_{12}\\ u_{31}+u_{20}-\tilde{\rho}_{21} \\ u_{32}+u_{23}-\tilde{\rho}_{22}\\ \end{bmatrix}. \end{equation*}$$

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