Can the dolfinx-external-operator package (GitHub - a-latyshev/dolfinx-external-operator: Extension of DOLFINx implementing the concept of external operator) be used in combination with mixed_element in a variational formulation – to solve for a coupled system of equations? If so, would it be possible to share a simple example?
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Bumping this! Having the same issue right now 
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See Using Python functions in FEniCS variational formulation - #24 by Thomas_Petersen for my attempted implementation of dolfinx-external-operator within a mixed_element construction.
What I can confirm is we (Expressing general constitutive models in FEniCSx using external operators and algorithmic automatic differentiation - Archive ouverte HAL) haven’t yet tried mixed forms with the DOLFINx implementation; the underlying UFL external operator has unit tests covering the mixed case, so there we are already OK.
In terms of providing a didactic example of using external operators and mixed forms, there is a simple example here, currently in primal form that (I think) could be duplicated into a new notebook in mixed form:
https://a-latyshev.github.io/dolfinx-external-operator/demo/demo_nonlinear_heat_equation_part2.html
Users could then use this example as a basis for their own work when using a mixed form.
Please let me know if you’d be interested in doing this and I would be happy to assist you in getting a pull request defined, reviewed, and merged into the main branch.