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

The five-point Laplacian applied to $f$ has expansionThe corrected right side $h^{2}f+(h^2/12){\Gamma }_{5}f$ cancels the $h^4\Delta f/12$ term in ${\Gamma }_{9}u$. ConsequentlyMore precisely, the unscaled residual's $h^6$ coefficient is $-(u_{xxxxxx}+u_{yyyyyy})/240+(u_{xxxxyy}+u_{xxyyyy})/144$. This fourth-order correction of the nine-point Poisson stencil does not require $f$ to be harmonic.