 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007/b/5/iii/solution

The weak equation implies

u^{''} = x^{2} u - f \in L^{2} (a, b)

u \in H^{2} (a, b)

. Its nodal interpolant satisfies

∥ (u - I_{h} u)^{'} ∥_{2} \leq \frac{h}{π} ∥ u^{''} ∥_{2}, ∥ u - I_{h} u ∥_{2} \leq \frac{h ^{2}}{π ^{2}} ∥ u^{''} ∥_{2} .

These follow interval by interval: the interpolation error vanishes at both endpoints, and its derivative has mean zero, so two Poincaré inequalities apply. If

M

and

γ

are the continuity and coercivity constants, Céa's lemma gives

∥ u - u_{h} ∥_{H^{1}} \leq \frac{M}{γ} ∥ u - I_{h} u ∥_{H^{1}} \leq \frac{M}{γ} \frac{h ^{2}}{π ^{2}} + \frac{h ^{4}}{π ^{4}} ∥ u^{''} ∥_{2} = O (h) .

All constants are independent of the mesh size.

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