Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 38A / i / Solution

Assume sufficient smoothness up to the boundary for uniform Taylor remainders, for example $u\in C^8(\overline{\Omega })$ and $f=\Delta u\in C^6(\overline{\Omega })$. Direct Taylor expansion of the nine-point finite-difference stencil givesThus the normalized local truncation error $h^{-2}{\Gamma }_{9}u-f$ is $\boxed{O(h^2)}$. The unscaled residual ${\Gamma }_{9}u-h^{2}f$ is $O(h^4)$. Stating the normalization avoids ambiguity in the meaning of truncation order.