Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 105 / 2 / b

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- Solution b

Solution

Multiply the Poisson equation by

v \in V

and apply Green's first identity. The Dirichlet boundary condition makes the trace of

v

vanish on

Γ_{1}

, while the Neumann boundary condition makes the boundary flux vanish on

Γ_{2}

. Thus the weak formulation is: find

u \in V

such that

B (u, v) = ℓ (v) for every v \in V,

where

B (u, v) = \int_{U} \nabla u \cdot \nabla v d x, ℓ (v) = \int_{U} f v d x .

Solved by gpt-5.6-sol high.

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