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I am new in Domain Decomposition method. I am started to solve $-\Delta u = f$ in $\Omega$ and $u = 0$ on $\partial\Omega$.enter image description here

From that I get in $\Omega _1$

$$\begin{bmatrix}4&-1\\-1&4\end{bmatrix} \begin{bmatrix}u_1 ^{k +\frac{1}{2}}\\u_2 ^{k +\frac{1}{2}}\end{bmatrix} = \begin{bmatrix}f_1-u_3 ^k\\f_2-u_4 ^k\end{bmatrix}$$

In $\Omega 2$ $$\begin{bmatrix}4&-1\\-1&4\end{bmatrix} \begin{bmatrix}u_3 ^{k + 1}\\u_4 ^{k + 1}\end{bmatrix} = \begin{bmatrix}f_3-u_1 ^{k +\frac{1}{2}}\\f_4-u_2 ^{k +\frac{1}{2}}\end{bmatrix}$$

But in book consider matrix representation as $A_i = (A_{\Omega_i}, A_{\partial\Omega_i-\Gamma_i}, A_{\Gamma_i})$ and for $A_1$ it is given as

$$\begin{bmatrix}4&-1&0&-1&0&0&-1&0&0&-1&0&0\\-1&4&0&0&-1&0&0&-1&0&0&-1&0\end{bmatrix}$$

I did not understand this. Can anyone please explain explicitly?

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