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Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$.

If the coefficient would be constant, then the weak form (see step-74 from deal.ii library) is: $$\begin{align*} &\sum_{K\in {\mathbb T}_h} (\nabla v_h, \nu \nabla u_h)_K\\ &-\sum_{F \in F_h^i} \left\{ \left< [{v_h}], \nu \{ \nabla u_h\} \cdot \mathbf n \right>_F +\left<\{ \nabla v_h \}\cdot \mathbf n,\nu [{u_h}]\right>_F -\left< [{v_h}],\nu \sigma [{u_h}] \right>_F \right\}\\ &-\sum_{F \in F_h^b} \left\{ \left<v_h, \nu \nabla u_h\cdot \mathbf n \right>_F + \left< \nabla v_h \cdot \mathbf n , \nu u_h\right>_F - \left< v_h,\nu \sigma u_h\right>_F \right\}\\ &=(v_h, f)_\Omega - \sum_{F \in F_h^b} \left\{ \left< \nabla v_h \cdot \mathbf n, \nu g_D\right>_F - \left<v_h,\nu \sigma g_D\right>_F \right\}. \end{align*}$$

However, the authors wrote (before writing the weak form)

For simplicity, we assume that the diffusion coefficient $\nu$ is constant here. Note that if $\nu$ is discontinuous, we need to take this into account when computing jump terms on cell faces.

Does this mean that I only have to write $[\nu]$ when I have the over the interior faces $F_h^i$?

My goal is to obtain a weak formulation when $\nu$ is not only constant, but also a function of space, i.e. $\nu(x,y)$

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I don't think so, your would violate the product rule.

enter image description here

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  • $\begingroup$ Thanks @ConvexHull for your answer. So what do you think they mean when they say "you have to take this into account"? $\endgroup$ Aug 15 at 18:37
  • $\begingroup$ @bobinthebox In their derivation they put $\nu$ outside the brackets. If $\nu$ is a function of space, the weak form would miss some extra terms. $\endgroup$
    – ConvexHull
    Aug 15 at 18:40
  • $\begingroup$ The first summand they have is okay even if $\nu=\nu(x,y)$, but I have to try to write down all the steps from the beginning. Do you have some reference so that I can check if what I find is the correct weak form? $\endgroup$ Aug 15 at 18:50

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