Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / B / 8 / i

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- Solution i

Solution

Use

V = {v \in H^{1} (0, 1) : v (0) = 0}

and its continuous piecewise linear subspace

V_{h} = span {ϕ_{1}, \dots, ϕ_{N}}

. The condition at

x = 1

is natural, so the last hat function remains free. Set

u_{h}^{0}

to the

L^{2}

orthogonal projection of

u_{0}

onto

V_{h}

: the stated

L^{2}

initial data need not have nodal values.

For each

m

, find

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

such that

(\frac{u _{h}^{m + 1} - u _{h}^{m}}{Δ t}, v_{h}) + ((u_{h}^{m + 1})^{'}, v_{h}^{'}) + (u_{h}^{m + 1}, v_{h}) = 0 (v_{h} \in V_{h}) .

This applies Backward Euler method to diffusion and reaction together. The step bilinear form is coercive, so each step has a unique solution.

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