# Interface condition for 1D Helmholtz equation using finite element method

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?

• There is still a $c^2$ to much in the third term. Feb 26 at 8:57
• Many thanks. deleted that term. Sorry for my confusion. Feb 26 at 9:30
• A good starting point would be to state what the equations and boundary conditions would be if you posed the problem on the two subdomains individually, with solutions $p^-$ on the left subdomain and $p^+$ on the right subdomain. Feb 27 at 22:10
• I think you should use the Discontinuous Galerkin (DG) method to solve this problem. As a brief but good introduction to the method, you could consult this link.
– Masa
Feb 28 at 14:49
• I want to model this problem using CG elements Mar 2 at 21:36