Why is the class of functions with a finite Dirichlet integral not complete? It is complete provided that elements $w$ are included whose first partial derivatives are only required to exist in the weak sense.
Dirichlet Integral: $$||w||^2 = \int_D \left[ \left( \frac{\partial w}{\partial x_1} \right)^2 + \cdots + \left( \frac{\partial w}{\partial x_n} \right)^2 + Pw^2 \right ] dx$$
Take $f(x)=|x|$ on the intervall $(-1,1)$. We have $f \in W^{1,2}((-1,1))$, but $f \notin C^1((-1,1))$. To check this, it is enough to proof that $f'(x)=sign(x)$ in the weak sense.
Furthermore, there exists a sequence of functions $\{ f_n \} \subset C^{\infty}((-1,1)) \cap W^{1,2}((-1,1))$ such that $ f_n \to f$ in $W^{1,2}((-1,1)$, which implies the convergence of the Dirichlet integral. But here we have a contradiction: Our sequence $f_n$ is smooth in the classical sense, but our limit is not differentiable in the classical sense (but in the weak sense).