 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007/b/6/ii/solution

Let

V = {v \in H^{1} (Ω) : v = 0 on Γ_{D}}

, with boundary values interpreted as traces. The outward unit normal on

Γ_{N}

(- 1, - 1) / 2

, so the stated derivative condition is precisely

\partial_{n} u = 0

u \in V, \int_{Ω} \nabla u \cdot \nabla v d x d y = \int_{Ω} v d x d y (v \in V) .

The Dirichlet boundary has positive length, so the Poincaré inequality makes the form coercive. Let

V_{h} \subset V

be the continuous piecewise affine space on the four triangles. The finite element approximation satisfies the same identity for every

v_{h} \in V_{h}

Nodes

A, C, D, E, F

have zero values, leaving

V_{h} = span {ϕ_{B}}

. Although isolated endpoints have zero boundary measure, continuity along the Dirichlet edges forces the nodal values at

A

and

C

to vanish.

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