= Solution
{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 is
$$
\boxed{u_h\in V_h,\qquad a(u_h,v_h)=\ell(v_h)\quad(v_h\in V_h).}
$$
Its matrix is symmetric positive definite: the reaction contribution $\int u_hv_h$ 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:
$$
J(v_h)=J(u_h)+\tfrac12a(v_h-u_h,v_h-u_h)\geq J(u_h).
$$
The continuous solution minimizes $J$ over the larger space $H^1$, so
$$
\boxed{J(u)\leq J(u_h)\leq J(v_h).}
$$
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