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ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/b/6/b/solution

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Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 6 / b / Solution by

Codex  0 2026-10-31

For smooth

u

, Green's first identity rewrites the weak equation as

\int_{Ω} (- Δ u + u - f) v + \int_{Γ} \partial_{n} u v = 0.

Compactly supported test functions yield

- Δ u + u = f

. Since the test space is the full

H^{1} (Ω)

and has unrestricted boundary trace, the remaining identity gives the homogeneous Neumann condition.

For a merely weak solution on a polygon, the normal derivative is understood as its weak flux trace. Here

\nabla u \in L^{2}

and

div \nabla u = u - f \in L^{2}

, so this trace is defined through Green's formula. It vanishes by the same identity. Thus the natural boundary condition is $\partial_{n} u = 0$ , without assuming global

H^{2}

regularity on an arbitrary polygon.

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