= Solution
{c}
Let $V=\{v\in H^1(\Omega):v=0\text{ on }\Gamma_D\}$, with boundary values interpreted as traces. The outward <unit normal> on $\Gamma_N$ is $(-1,-1)/\sqrt2$, so the stated <derivative> condition is precisely $\partial_nu=0$. <integration by parts> gives
$$
u\in V,\qquad \int_\Omega\nabla u\cdot\nabla v\,dx\,dy=\int_\Omega v\,dx\,dy\quad(v\in V).
$$
The Dirichlet boundary has positive length, so the <Poincaré inequality> makes the form coercive. Let $V_h\subset V$ be the continuous piecewise affine space on the four triangles. The <finite element> approximation satisfies the same identity for every $v_h\in V_h$.
Nodes $A,C,D,E,F$ have zero values, leaving $V_h=\operatorname{span}\{\phi_B\}$. Although isolated endpoints have zero boundary measure, continuity along the Dirichlet edges forces the nodal values at $A$ and $C$ to vanish.
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