Weak Form Deduction of Cahn Hilliard Equation on Moose Website

[js-jixu]

Dear MOOSE experts,

Recently, I've been studying the phase field module in MOOSE and noticed a discrepancy in the weak form derivation of the Cahn-Hilliard equation for the direct solution approach. Specifically, the signs of the fourth and fifth boundary terms in the residual equation ( \mathcal{R}_{c_i} ) appear to be opposite between my derivation and what's shown in the documentation (as illustrated in the attached image).

To help verify this, I also worked through the derivation with an AI assistant, and interestingly, the AI's results aligned with my own derivation. For your convenience, I've pasted the AI's derivation process below.

Would you be so kind as to take a look and let me know if there might be an error in my approach, or if this could potentially be a typo in the documentation?

Thank you very much for your time and expertise.

Best regards,

Xu

[GiudGiud]

@laagesen if you have time to take a look at the derivation

[laagesen]

@js-jixu and @GiudGiud sorry it has taken me a long time to look at this. I agree with Xu's post, the signs are incorrect. I have submitted #32864 to document this and will submit a PR to fix it shortly.