I realized that this was no longer a symmetrical problem because of the term in \philin. All the other terms are the usual stokes terms and are symmetric.
Fortunately for me, the dealII doc has an answer for everything and there is an example of using a Schur Preconditioning for an asymmetric matrix : step 57 which solves navier stokes equation using a newton method. It solves the problem using an augmented lagrangian based approach.
So here are my question. Question 1, 2 and 3 is just some confirmation to things I think I already know. Real question is number 4.
1 - Is there a simpler way to solve my stokes problem than using a Schur Preconditioner and the augmented lagrangian method ?
2 - Will the augmented Lagrangien Method also works for my terms ? I don't see an issue from the theory... In the tutorial and in the paper referenced it's used to solve the antisymetric terms that comes from the advectiv terms.
3 - In my case I have eta = 1 in comparison to step 57 ?
4 - I don't understand how I should choose the gamma parameter and why its value does not change the results of the resolution. In step 57, it's value is set to 1 and does not change at all...
5 - Why is the Stokes FE is always set to :
stokes_fe(FE_Q<2>(degree+1), 2,
FE_Q<2>(degree), 1)
Can it work with :
stokes_fe(FE_Q<2>(degree), 2,
FE_Q<2>(degree), 1)
I'm asking this because my main interest is in phi and with the degree+1, I have 8 times more dofs describing the flow than the ones describing phi.
Thanks in advance,