= Unweighted grid error for the five-point Poisson formula
{title2=$\|e\|_{\ell^2}\le Ch$}
For a smooth solution with bounded fourth derivatives on the unit square, the five-point local residual is $O(h^2)$. The smallest eigenvalue of the unscaled Dirichlet matrix is $8\sin^2(\pi h/2)\ge8h^2$. With $M^2$ interior points and $M\asymp h^{-1}$, the residual has unweighted Euclidean norm $O(h)$. The equation $Ae=h^2\tau$ therefore gives unweighted error $O(h)$, equivalent to second-order error in the area-weighted grid norm.
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