Four-neighbour mean expansion
= Four-neighbour mean expansion
{title2=$\operatorname{mean}_hu=u+\tfrac14h^2\Delta u+O(h^4)$}
Averaging a smooth <function> at the four points $x\pm he_1,x\pm he_2$ cancels odd <Taylor expansion> terms and leaves $h^2\Delta u/4$. The fourth-order correction is $h^4(u_{1111}+u_{2222})/48$. Equality to the center value at every sufficiently small radius therefore forces $\Delta u=0$ at that point, but does not establish harmonicity throughout a neighbourhood.