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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ii/paper-2/39b/i/solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-14

Expand the two central second-difference operators. The given stencil identity yields

Γ_{9} u = h^{2} Δ u + \frac{h ^{4}}{12} Δ^{2} u + h^{6} [\frac{u _{xxxxxx} + u _{yyyyyy}}{360} + \frac{u _{xxxx yy} + u _{xx yyyy}}{72}] + O (h^{8}) .

(1)

Consequently its unnormalized local residual is

Γ_{9} u - h^{2} f = (h^{4} /12) Δ f + O (h^{6})

. The normalized Laplacian approximation has leading error $(h^{2} /12) Δ f$ and is second order. These Taylor statements require the indicated smooth derivatives; boundary accuracy is a separate matter.

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