= Discrete Poisson barrier on a square
{c}
{title2=$B(x,y)=\frac{x(1-x)+y(1-y)}4$}
For the five-point discretization $L_h$ of $-\Delta$ on the unit square, the displayed quadratic <polynomial> has $L_hB=1$ exactly, nonnegative boundary values and $0\leq B\leq1/8$. If $L_he=\tau$ and $e$ has zero boundary values, the <discrete maximum principle> applied to $\pm e-\|\tau\|_\infty B$ gives $|e|\leq\|\tau\|_\infty B\leq\|\tau\|_\infty/8$. The same estimate holds for $L_h+I$, since $(L_h+I)B=1+B\geq1$. This turns a uniform <local truncation error> bound into a pointwise <global error> bound without inverting the full matrix.
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