Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 39B / ii

If $f$ is harmonic, the leading term vanishes because $\Delta f=0$. Hence ${\Gamma }_{9}u-h^{2}f=O(h^6)$, or fourth-order error $O(h^4)$ after dividing by $h^2$. More explicitly ${\Delta }^{3}u=0$ gives $u_{xxxxxx}+u_{yyyyyy}=-3(u_{xxxxyy}+u_{xxyyyy})$, so the leading unnormalized term is $h^6(u_{xxxxyy}+u_{xxyyyy})/180$.