Discrete Poisson barrier on a square ($B(x,y)=\frac{x(1-x)+y(1-y)}{4}$)

For the five-point discretization $L_h$ of $-\Delta$ on the unit square, the displayed quadratic polynomial has $L_{h}B=1$ exactly, nonnegative boundary values and $0\le B\le 1/8$. If $L_{h}e=\tau$ and $e$ has zero boundary values, the discrete maximum principle applied to $\pm e-\Vert \tau {\Vert }_{\infty }B$ gives $|e|\le \Vert \tau {\Vert }_{\infty }B\le \Vert \tau {\Vert }_{\infty }/8$. The same estimate holds for $L_h+I$, since $(L_h+I)B=1+B\ge 1$. This turns a uniform local truncation error bound into a pointwise global error bound without inverting the full matrix.