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ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/b/5/c/solution

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Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 5 / c / Solution by

Codex  0 2026-10-31

Testing against compactly supported smooth functions gives the distributional identity

u^{''} = cos x u - f

. The right side belongs to

L^{2}

, since

u \in H^{1}

and

cos x

is bounded. Therefore

u^{'} \in H^{1}

and $u \in H^{2} (0, 1)$ . This also makes the endpoint derivative traces meaningful, and integrating the weak form by parts recovers the Robin conditions.

The standard piecewise linear finite element interpolation estimate is

∥ u - I_{h} u ∥_{L^{2}} \leq C_{0} h^{2} ∥ u^{''} ∥_{L^{2}}, ∥ (u - I_{h} u)^{'} ∥_{L^{2}} \leq C_{1} h ∥ u^{''} ∥_{L^{2}},

(1)

with constants independent of

h

. Hence

∥ u - I_{h} u ∥_{H^{1}} \leq C h ∥ u^{''} ∥_{L^{2}}

for

h \leq 1

. By Céa's lemma, with the fixed continuity and coercivity constants of

a

∥ u - u_{h} ∥_{H^{1}} \leq C h ∥ u^{''} ∥_{L^{2}} = O (h) .

(2)

The constant depends on the continuous problem but not on mesh refinement.

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