Understanding my problem (ME/accurate solution?)

[dokken]

@raboynics

I am willing to introduce new variables such as J, J_U, J_Q (some heat flux) and use RT elements using a ME scheme if it makes things better. I tested rapidly and I got even worse results by implementing J with RT elements, where the div J and div J_U values where a million times greater than in my case, for the regime where my code works fine.

Have you looked into stable mixed poisson formulations?
See for instance
Mixed formulation for the Poisson equation — DOLFINx 0.9.0 documentation or dolfinx-tutorial/chapter4/mixed_poisson.ipynb at main · jorgensd/dolfinx-tutorial · GitHub

As a general comment, you should build your model step by step. In the example provided, you have not provided enough information about the issues that you are phasing for most people to help you.

@raboynics

LLM IA tells me I can do a better post-process and project J into a RT space. Would this give me more accurate values of the volume integral of \vec J^2 and of \frac{\partial J_x}{\partial x}?

If your solution is not satisfactory, a post processing projection usually doesn’t solve the problem (it would just hide the issues).

My advise is to work on the mathematical formulation and discretisation of your problem.
I guess the strong form is written at: Deriving the weak form of my problem, is it correct? - #7 by raboynics
and you should sit down and carefully write up a mixed formulation.

To me it kind of looks like a PNP ( https://pmc.ncbi.nlm.nih.gov/articles/PMC3122111/ ) equation (steady state), where you might be able to draw inspiration from https://arxiv.org/pdf/2409.08746 or other papers solving such problems either with a primal or mixed discretisation.