# Heat equation in non-dimensional form behaving differently than in usual format

**URL:** <https://fenicsproject.discourse.group/t/heat-equation-in-non-dimensional-form-behaving-differently-than-in-usual-format/4355>\
**Category:** Uncategorized\
**Created:** [October 18, 2020, 10:10pm UTC](https://fenicsproject.discourse.group/t/heat-equation-in-non-dimensional-form-behaving-differently-than-in-usual-format/4355 "2020-10-18T22:10:33Z")\
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**Author:** ![kamensky](https://avatars.discourse-cdn.com/v4/letter/k/e95f7d/32.png) [@kamensky](https://fenicsproject.discourse.group/u/kamensky)\
**Post date:** [October 20, 2020, 12:20am UTC](https://fenicsproject.discourse.group/t/heat-equation-in-non-dimensional-form-behaving-differently-than-in-usual-format/4355/2 "2020-10-20T00:20:09Z")

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Note that the weak form \int\_\Omega k\nabla u\cdot\nabla v\,d\mathbf{x} corresponds to the strong form -\nabla\cdot(k\nabla u), which is not the same as -k\nabla^2 u when k varies in space. Likewise, when c\_p^{-1} varies in space, it cannot be pulled inside the divergence operator, so

-c\_p^{-1}\nabla\cdot(k\nabla u) \neq -\nabla\cdot(c\_p^{-1}k\nabla u)\text{ ,}

where the left side is what you get by dividing the original conservation law through by c, but the right side is what the weak form in the code corresponds to when `normal == False`.

While k and c\_p^{-1} are piecewise-constant, there are Dirac deltas in their spatial derivatives at the discontinuity. (You could pull c\_p^{-1} inside the divergence operator on either side of the internal interface where c\_p and k are discontinuous, but then when you integrate by parts on either side to get a weak form, there would be non-cancelling boundary terms at the interface.)

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