Unweighted grid error for the five-point Poisson formula ($\Vert e{\Vert }_{{\ell }^2}\le Ch$)

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 $8{\sin }^2(\pi h/2)\ge 8h^2$. 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.