 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2007/b/7/i/solution

Taking

w = z

in the dual problem, and using

e^{x} \geq 1

, gives

∥ z ∥_{H^{1}}^{2} \leq a (z, z) = \int_{0}^{1} z ψ \leq ∥ z ∥_{2} ∥ ψ ∥_{2} \leq ∥ z ∥_{H^{1}} ∥ ψ ∥_{2} .

Hence

∥ z ∥_{H^{1}} \leq ∥ ψ ∥_{2}

. Testing with smooth compactly supported functions identifies

- z^{''} + e^{x} z = ψ

in distributions. Therefore

z^{''} \in L^{2}

and

z \in H^{2} (0, 1)

, with

∥ z^{''} ∥_{2} \leq e ∥ z ∥_{2} + ∥ ψ ∥_{2} \leq (e + 1) ∥ ψ ∥_{2} .

integration by parts with arbitrary endpoint test values also gives the natural Neumann boundary conditions

z^{'} (0) = z^{'} (1) = 0

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