Let $\Omega = [0,1]$ and $f\in L^2(\Omega)$. Prove that the weak solution $u \in H_0^1(\Omega)$ of the Laplace equation with homogeneous Dirichlet boundary conditions $$-u''=f \text{ in } \Omega, \ \ \ u(0)=u(1)=0 \text{ on } \Omega$$ fulfills $u\in H^2(\Omega)$.
I was thinking $u \in H_0^1(\Omega)$ implies $u,u' \in L^2$, and $f\in L^2$ implies $u''\in L^2$ if $u$ is a solution. But I think I need to use the definition of weak derivative somehow to show the weak second derivative exists.