Domain decomposition

Question

In order to solve the $1$-D problem \begin{align*} u^{''}=f \in (0,1),\\ u(0)=u(1)=0, \end{align*} $f\in L^{2}(0,1)$, if we choose our solution space to be $H^{1}(0,1/2)\times H^{1}(1/2,1)$ and then get a variational formulation. Does it coincide with the weak formulation on $H^{1}_{0}(0,1)$? Or some interface condition to be imposed to get a solution? Here I assume our original solution is enough regular i.e. in $H^{2}$. I have some confusion. Please some body help