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You define a function $u$ on $\Omega=\Omega_1 \cup \Omega_2$ so that on $\Omega_1$ you have $u=u_1$ and similarly on the other part of the domain. You'd do the same with a function $v$. Then the term you have trouble with is simply $$ (\nabla u, \nabla v)_\Omega. $$


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