Let $\Omega = (-1,1)$ be partitioned into two subdomains $\Omega_1 = (-1,0)$ and $\Omega_2 = (0,1)$.
I'd like to find the exact solution in one dimension: Find $u\in H_0^1(\Omega),\ $ such that :$\displaystyle \int_{\Omega} \kappa\nabla u\cdot \nabla v = \int_\Omega v$, $\forall v\in H_0^1(\Omega)$, where $\kappa|_{\Omega_1} = \alpha$ and $\kappa|_{\Omega_2} = 1$ .
Actually I know this equation exists a unique solution... but I don't know how to solve it , because I don't know how to describe the shape of $H_0^1(\Omega)$, and I just know its dense subset is $C_0^{\infty}(\Omega)$.
The strong form is $-\kappa \Delta u(x) =1$. Then the solution is quadratic on each subinterval. To determine the four integration constants you take the boundary condition $u(\pm 1)=0$ and the continuity condition $u(0^+) = u(0^-)$. Then one condition is missing, which can be obtained as follows: $$ \int_\Omega v = \int \kappa u'v'= \int_{-1}^0 \alpha u'v' + \int_0^1 u'v' = -\int_{-1}^0 \alpha u'' v - \int_0^1 u''v + (\alpha u'(0^-) - u'(0^+))v(0). $$ In order that this expression is zero for all $v$, the condition $\alpha u'(0^-) - u'(0^+)= 0$ is required, which fixes the last integration constant.