# Mixed form Poisson with in-homogeneous conductivity

**URL:** <https://fenicsproject.discourse.group/t/mixed-form-poisson-with-in-homogeneous-conductivity/17232>\
**Category:** mathematics\
**Tags:** variational-form\
**Created:** [March 18, 2025, 11:12am UTC](https://fenicsproject.discourse.group/t/mixed-form-poisson-with-in-homogeneous-conductivity/17232 "2025-03-18T11:12:21Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Kiri](https://avatars.discourse-cdn.com/v4/letter/k/a183cd/32.png) [@Kiri](https://fenicsproject.discourse.group/u/Kiri)\
**Post date:** [March 18, 2025, 11:12am UTC](https://fenicsproject.discourse.group/t/mixed-form-poisson-with-in-homogeneous-conductivity/17232/1 "2025-03-18T11:12:21Z")

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Hi,

What is the correct method for including spatially varying conductivity (i.e. multiple materials), \kappa(\boldsymbol{x}), in the mixed form of the Poisson equation?

In particular, in my derivation below I am unsure about the \int (\nabla\cdot\kappa(\boldsymbol{x})\tau)u\,d\Omega term in Equation (6). Is it correct to include \kappa within the divergence operator? This is currently solving but I want to make sure I am not missing something.

**Strong form**

\begin{aligned} \nabla\cdot (\kappa(\boldsymbol{x})\nabla u) &= -f \;\;\;\;\;\;\;\;\;\;\;\;\;(1) \\ u &= u\_D\; \textrm{on}\; \Gamma\_D\;\;\;(2) \\ \kappa(\boldsymbol{x})\nabla u\cdot n &= g \; \textrm{on}\; \Gamma\_N \;\;\;\;\;\;(3) \end{aligned}

Converting to mixed form:

\begin{aligned} q - \kappa(\boldsymbol{x})\nabla u &= 0\;\;\;\;\;\;(4)\\ \nabla\cdot q &= -f\;\;\;(5) \end{aligned}

**Weak form** (after integration by parts)  
Find q, u \in W = Q \times U , with test functions: \tau, v \in W such that:

\begin{aligned} a &= \int q\cdot \tau\,d\Omega + \int (\nabla\cdot q)v \,d\Omega + \int (\nabla\cdot\kappa(\boldsymbol{x})\tau)u\,d\Omega \;\;\;(6) \\ L &= \int u\_D \kappa(\boldsymbol{x})\tau \cdot n\,ds - \int fv\,d\Omega \;\;\;(7) \end{aligned}

A similar [question](https://fenicsproject.discourse.group/t/heat-equation-in-non-dimensional-form-behaving-differently-than-in-usual-format/4355/2) was posted a few years ago, but an explicit solution was not provided.

Thanks!

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<div class="post-metadata">

**Author:** ![Stein](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/stein/32/8751_2.png) [@Stein](https://fenicsproject.discourse.group/u/Stein)\
**Post date:** [March 18, 2025, 12:44pm UTC](https://fenicsproject.discourse.group/t/mixed-form-poisson-with-in-homogeneous-conductivity/17232/2 "2025-03-18T12:44:09Z")

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It is more common to write

\begin{aligned} \kappa(\mathbf{x})^{-1} q -\nabla u &= 0\;\;\;\;\;\;(4)\\ \nabla\cdot q &= -f\;\;\;(5) \end{aligned}

so that you get the nicely symmetric:

\begin{aligned} a &= \int \kappa(\mathbf{x})^{-1} q\cdot \tau\,d\Omega + \int (\nabla\cdot q)v \,d\Omega + \int (\nabla\cdot \tau)u\,d\Omega \\ L &= \int u\_D\, \tau \cdot n\,ds - \int fv\,d\Omega\end{aligned}

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<div class="post-metadata">

**Author:** ![Kiri](https://avatars.discourse-cdn.com/v4/letter/k/a183cd/32.png) [@Kiri](https://fenicsproject.discourse.group/u/Kiri)\
**Post date:** [March 18, 2025, 12:55pm UTC](https://fenicsproject.discourse.group/t/mixed-form-poisson-with-in-homogeneous-conductivity/17232/3 "2025-03-18T12:55:35Z")

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Thanks - this looks much cleaner!