I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is;
$$c^2\nabla^2p + w^2p=0 \qquad \mathrm{in} \quad \Omega,$$
and
$$\nabla p(0)=\nabla p(1)=0 \qquad \mathrm{at} \quad \partial \Omega,$$ $$\nabla p^-(0.8)=K[[p]]=K(p^+(0.8)-p^-(0.8)) \qquad \mathrm{at} \quad x=0.8, $$
where $c$ is the speed of sound, $\omega$ is the eigenvalue, the $p$ is the eigenfunction. $^-$ and $^+$ superscripts denote upstream and downstream directions and $K$ denotes a complex coefficient.
The weak(finite element) formulation without jump condition term is;
$$-\int_\Omega c^2\nabla u \nabla v dx + \int_{\partial\Omega} c^2\nabla u v ds + w^2\int_\Omega u v dx = 0$$
where $u$ and $v$ are test and trial functions. I just wonder how can I include the jump condition in the weak formulation? And how it will affect the implementation of stiffness and mass matrices?
