Deriving the weak form of my problem, is it correct?

[raboynics]

@Stein

I don’t see how that follows from your text.

\vec J_U \cdot n = -\kappa \nabla T \cdot n +T(S\vec J) \cdot n +V\vec J \cdot n = 0
Given J \cdot n = 0, you would get \kappa \nabla T \cdot n = T(S\vec J) \cdot n .

You’re absolutely right, this time an overlook from my part.

@Stein

Is this problem formulation what you’re trying to achieve?
\begin{cases} \nabla\cdot \vec J_U = 0 \text{ in }\Omega\\ \nabla\cdot \vec J = 0 \text{ in }\Omega\\ J\cdot n = 0 \text{ on }\partial\Omega \\ T = T_{D} \text{ on }\partial\Omega_D \\ -\kappa\nabla T \cdot n= 0 \text{ on }\partial\Omega \setminus \partial\Omega_D \end{cases}
with \vec J_U = -\kappa \nabla T +T(S\vec J) +V\vec J and \vec J =-\sigma \nabla V - \sigma S \nabla T

Hmmm, I’ve puzzled for a bit, but I cannot come up with a clean weak form (without variational crimes) that corresponds to that PDE. To me, that hints at the incompatibility of those BCs, but I could be wrong.

I think you are right about your conclusion.
Ok, thinking more about it, the 4 first conditions are an absolute must. The 5th condition is probably wrong (sorry about that!), I would rather have \vec J_U \cdot \hat n=0 on the boundaries where no Dirichlet conditions are enforced.

I checked up quickly last night (between midnight and 1 am… I was tired to say the least) and I think my current problem is indeed like this, and I was wrong in claiming that \vec J_U \cdot \hat n \neq 0 on the boundaries without any specified boundary conditions (FEniCSx returned numbers like 1e-4 or 1e-5 so I wasn’t sure if this was supposed to vanish or not, but since I used the usual non accurate way to compute fluxes, it may indeed be 0).

Therefore, to sum up, I think I want:
\begin{cases} \nabla\cdot \vec J_U = 0 \text{ in }\Omega\\ \nabla\cdot \vec J = 0 \text{ in }\Omega\\ J\cdot n = 0 \text{ on }\partial\Omega \\ T = T_{D} \text{ on }\partial\Omega_D \\ \vec J_U \cdot n= 0 \text{ on }\partial\Omega \setminus \partial\Omega_D \end{cases} .

This would indeed imply that \kappa \nabla T \cdot \hat n = T(S\vec J) \cdot \hat n on the boundaries where I do not prescribe the Dirichlet b.c.s.

I am getting convinced this is what happens physically. It means that there are 2 boundaries from which energy enters and leave the system, which makes sense since the system is in thermal contact with reservoirs there, and the other 2 boundaries are thermally and electrically insulated. On those insulated boundaries, strangely \nabla T \neq \vec 0, but this is required for no energy exchange with the exterior, because TS\vec J \cdot \hat n \neq 0 there, too, which is a bit weird but apparently possible.

So nice, I think you basically either figured out yourself or at least helped me to understand what is going on in this uncommon system I am dealing with. Things that we take for granted are not necessarily valid anymore, like the thermal insulation implying \nabla T =\vec 0 does not hold in this thermoelectric system.

Thanks a lot Stein, for all the information and your patience.

My question would be, then, whether I am correctly modelling the 5 conditions with my current weak form (with the sigma term corrected)? I think so but I prefer to be 100% sure.