# How to enforce the boundary conditions

I want to solve a poisson problem on two domains,(interface problem: Let $\Omega$ be a square, $\Omega=\Omega1 \cup \Omega2$ and let $\Omega1$ be a circle inside the square. $\Gamma$ is the boundary of circle. $$\Delta u_1=f_1 ~~in ~~\Omega1$$ $$\Delta u_2=f_2 ~~in ~~\Omega2$$ $$u_2=g~~on ~~\partial \Omega$$ $$u_2-u_1=w ~~on~~\Gamma$$ $$\frac{\partial u_2}{\partial n_2}-\frac{\partial u_1}{\partial n_1}=v~~on~~\Gamma$$

What should I do by the conditions on $\Gamma$, they are boundary conditions? If yes, How to enforce these conditions, when the test space is $H^1(\Omega)$.

The conditions on $\Gamma$ are called interface conditions. You could incorporate the jump conditions into your weak form. Typically, the trial and test spaces are $H^{1}(\Omega_{1}) \times H^{1}(\Omega_{2})$. Numerically, the interfacial condition is easier to treat if $\Gamma$ is aligned with a mesh of $\Omega$. One example of how you might treat the embedded interface condition can be found in Chapter 4 of the PhD thesis by Chandrasekhar Annavarapu, out of John Dolbow's group at Duke.