Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / B / 6 / ii / Solution

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 $(-1,-1)/\sqrt{2}$, so the stated derivative condition is precisely ${\partial }_{n}u=0$. integration by parts givesThe 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 $V_h=\mathrm{span}\{{\varphi }_B\}$. Although isolated endpoints have zero boundary measure, continuity along the Dirichlet edges forces the nodal values at $A$ and $C$ to vanish.