## Interpolation scheme for electric current / heat flux

### Functionality to add/problem to solve

An electric current is calculated as

`J = \sigma\cdot\nabla\varphi.`

The gradient of the electric potential can not be calculated using the Gauss theorem with LINEAR interpolation of the potential . A special interpolation scheme is necessary. As described below, the potential at the face needs to be weighted by the conductivity of both cells.

### Target audience

The mentioned interpolation scheme is necessary for computing the gradient of

- electric potential (i.e. electric current)
- temperature (i.e. heat flux)

### What does success look like, and how can we measure that?

I use the mentioned interpolation scheme for several years. It is already validated and published.

### Links / references

Weber, N.; Beckstein, P.; Galindo, V.; Starace, M.; Weier, T.: Electro-vortex flow simulation using coupled meshes, Computers and Fluids 168(2018) 101-109

It is Eqn. 16/17 in the Arxiv version.

### Funding

I can provide the source code.