# SIPG method for $-\nabla \cdot (\nu \nabla u)=f$

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_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_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. $$\nu(x,y)$$

• @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. Aug 15 at 18:40
• 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? Aug 15 at 18:50