Let $u_j=(u_{1,j},\ldots,u_{m,j})^T$ collect the unknowns in grid column $j$, and define $b_j=h^2(f_{1,j},\ldots,f_{m,j})^T$. The terms within one column form the block
$$
\boxed{B=\left[\frac23,-\frac{10}3,\frac23\right]},
$$
while coupling to either adjacent column contributes
$$
\boxed{C=\left[\frac16,\frac23,\frac16\right]}.
$$
The zero boundary values remove all blocks outside the displayed range, so the <nine-point finite-difference stencil> becomes the <block tridiagonal matrix> system
$$
\begin{pmatrix}
B&C&&\\
C&B&\ddots&\\
&\ddots&\ddots&C\\
&&C&B
\end{pmatrix}
\begin{pmatrix}u_1\\u_2\\\vdots\\u_m\end{pmatrix}
=
\begin{pmatrix}b_1\\b_2\\\vdots\\b_m\end{pmatrix}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 39C / b / Solution

A nine-point finite-difference stencil on a square grid couples a central value to its four axial neighbours and four diagonal neighbours. Ordering the grid by columns produces a block tridiagonal matrix whose diagonal and off-diagonal blocks are <symmetric tridiagonal Toeplitz matrices>.

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