 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/b/6/c/solution

Take

V_{h}

to be all continuous functions affine on each triangle of the stated negative-diagonal mesh. No boundary value is fixed, because the Neumann condition is natural. The discrete problem is

u_{h} \in V_{h}, a (u_{h}, v_{h}) = ℓ (v_{h}) (v_{h} \in V_{h}) .

Its matrix is symmetric positive definite: the reaction contribution

\int u_{h} v_{h}

removes the constant null mode that would occur for pure Neumann Poisson equations. Thus the solution exists uniquely.

For every

v_{h} \in V_{h}

u_{h}

and use the discrete weak equation to remove its linear term:

J (v_{h}) = J (u_{h}) + \frac{1}{2} a (v_{h} - u_{h}, v_{h} - u_{h}) \geq J (u_{h}) .

The continuous solution minimizes

J

over the larger space

H^{1}

, so

J (u) \leq J (u_{h}) \leq J (v_{h}) .

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