I am trying to use finite elements method to solve a PDE containing two domains, i.e., $\Omega_1$ and $\Omega_2$. These two domains have different PDE equations, with the same variables, i.e., $$L_1(u_1)=f_1,\\ L_2(u_2)=f_2. $$ with $u_1$ and $u_2$ have same physical meaning, e.g., electric potential. And $u_1\in H^1(\Omega_1)$, $u_2\in H^1(\Omega_2)$.
Let $\Gamma=\overline{\Omega}_1\cap\overline{\Omega}_2$ be the interface. $u_1$ and $u_2$ have relation on this interface, e.g., $$ u_1(x)=u_2(x)+g(x). $$ On other parts of boundaries, i.e., $\partial\Omega_1-\Gamma$ and $\partial\Omega_2-\Gamma$, we have well-defined boundary conditions.
My questions are as follows:
How should we view the problem when using FEM? Shall we view the problem as solving PDEs in the whole domain $\Omega=\Omega_1\cup\Omega_2$ having interface $\Gamma$? Or shall we view the problem as solving two PDEs in domain $\Omega_1$ and $\Omega_2$, respectively, but have relations?
In the first view, how shall we choose the trial function space? Note that $u$ does not belong to $H^1(\Omega)$.
In the second view, we need to set unknown nodes at $\Gamma$ for both $u_1$ and $u_2$. The relation $$ u_1(x)=u_2(x)+g(x). $$ can be used to establish one equation for each interface node. But there are now two unknowns each interface node. If I select test functions confined at these nodes, then for each domain, it can establish one equation, which finally contributes 3 equations to one interface node, larger than the number of unknowns.
Is there any mature method for such kind of problems? Thanks very much!