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ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/b/8/a/solution

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Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 8 / a / Solution by

Codex  0 2026-10-31

Use the full continuous piecewise linear space

V_{h} \subset H^{1} (0, 1)

, including endpoint basis functions, because the boundary conditions are Neumann. Define

a (w, v) = \int_{0}^{1} (w^{'} v^{'} + e^{x} w v) .

Choose

u_{h}^{0}

as the

L^{2}

projection of

u_{0}

onto

V_{h}

. For each time step seek

u_{h}^{m + 1} \in V_{h}

with

(\frac{u _{h}^{m + 1} - u _{h}^{m}}{Δ t}, v_{h}) + a (\frac{u _{h}^{m + 1} + u _{h}^{m}}{2}, v_{h}) = 0 (v_{h} \in V_{h}) .

This is the conforming Crank-Nicolson method. The reaction coefficient and the diffusion are both treated at the midpoint; integration by parts has removed the boundary flux through the homogeneous Neumann condition.

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