Let $f \in C^0([0,1])$ and $$u''(x) = f(x)$$ in $ \Omega = (0,1)$ $$u'(0) = u'(1) = 0$$
If we set $f(x) = (x- \frac{1}{2})$ and add the boundary condition $u(0) = 0$. What basis (for a finite dimensional) subspace $S \subset S^1_G := \{ \phi \in C^0([0,1]) | \forall r \in G : \phi |_r \in \mathbb P_1 \}$ would you choose to approximate $u$ numerically?