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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?

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  • $\begingroup$ There is still a $c^2$ to much in the third term. $\endgroup$
    – ConvexHull
    Commented Feb 26, 2023 at 8:57
  • $\begingroup$ Many thanks. deleted that term. Sorry for my confusion. $\endgroup$ Commented Feb 26, 2023 at 9:30
  • $\begingroup$ 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. $\endgroup$ Commented Feb 27, 2023 at 22:10
  • $\begingroup$ 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. $\endgroup$
    – Masa
    Commented Feb 28, 2023 at 14:49
  • $\begingroup$ I want to model this problem using CG elements $\endgroup$ Commented Mar 2, 2023 at 21:36

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