Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 6 / c

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 isIts 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 $v_h\in V_h$, expand the quadratic functional about $u_h$ and use the discrete weak equation to remove its linear term:The continuous solution minimizes $J$ over the larger space $H^1$, so