Let $\Omega_1$ and $\Omega_2$ be a non-overlapping partition of a domain $\Omega$. Suppose that a function $u \in L^2(\Omega)$ has $H^2$ regularity on the two subdomains (i.e. $u|_{\Omega_1} \in H^2(\Omega_1)$ and $u|_{\Omega_1} H^2(\Omega_2)$). I would like to prove that, provided that $$ \begin{split} [\![ u ]\!] &= u|_{\Omega_1} - u|_{\Omega_2} = 0 \\ [\![ \nabla u ]\!] &= \nabla u|_{\Omega_1} \cdot \mathbf{n}_1 + \nabla u|_{\Omega_2} \cdot \mathbf{n}_2 = 0 \end{split} $$ then $u$ is globally $H^2$ (i.e. $u \in H^2(\Omega)$).
Do you have any literature reference where this result is proven?