 Solution

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

For any

v \in H^{1} (Ω)

, expand

J (u + ϵ v)

ϵ

. Its linear term must vanish at a minimizer, giving

\int_{Ω} (\nabla u \cdot \nabla v + uv) = \int_{Ω} f v .

Thus the required functionals are

a (u, v) = \int_{Ω} (\nabla u \cdot \nabla v + uv), ℓ (v) = \int_{Ω} f v .

The form is continuous, symmetric, and coercive, with

a (v, v) = ∥ v ∥_{H^{1}}^{2}

. Conversely, this weak equation implies

J (u + w) = J (u) + \frac{1}{2} a (w, w)

, so its unique solution is indeed the global minimizer.

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