I am facing a problem to handel non-linearity in FreeFem++

General Discussion
1

Hello all expert,
I am trying to solve Navier-stokes equation using Backward Euler scheme. I am facing problem with the convection term.

Here, is the weak formulation

ns
ns1192×723 47 KB

Backward-Euler
Backward-Euler1132×707 29.6 KB

Here, F(u,p) has u koronecker u which will given by the following figure
IN 2D:
Kr

Here, are the details formulations:

nsone
nsone1127×282 25.9 KB

nstwo
nstwo1197×775 80.1 KB

nsthree
nsthree1201×870 92 KB

Here, in two question marks terms i am facing issue to write in FreeFem++ as it has non-linearities.

nsfour
nsfour971×387 33.9 KB

Can you pls help me to write this two terms properly??.

But there will be four terms that are product of unkowns which will be not solvable by FreeFem++. Is there an way to solve this problem easily??. I need a minimal example to solve this. I will be very grateful if i get some idea to solve this.

Thanks in advance.

2

Bonjour,

Des exemples de résolution de NS avec FF++ sont disponibles
dans la documentation et aussi, et surtout, dans les exemples que FH donnent
dans ses cours. les plus récents sont ceux des FF Days de décembre.
On peut les trouver en suivant le lien
https://github.com/FreeFem/FreeFem-days/raw/master/2024/pdf/FH-ffday24.zip

Bon courage
et merci.

4

Thank you Sir. I will see.

5

Je vous en prie,

de plus,
si vous avez le temps jeter un coup d’oeil au fichier

${HOMEFF_INSTALL}/share/FreeFEM/4.14//examples/examples/NSNewton.edp

Il s’agit d’une méthode de Newton pour la non linéarité combinée avec une méthode de continuation sur la viscosité. Un exemple inspirant.

Je vous remercie.

6

Okay sir. But your file is not opening sir.

7

This file (NS by Newton with continuation) is available in your local installation of FF++.
My previous reply do not contain any link to a file. Just the name of the file that can helps you.

8

Okay sir. Thank you.

9

Your Pdf is good sir. I have seen. But, i need more. Actually, i am trying to compute the formulation that i have shared in the images. I am facing problem to write the term u koronecker u.

I am thinking that u koronecker u becomes u^{n-1} koronecker u^{n} when i will apply fully-discrete backward-euler. But i don’t know am i correct or not. Also, u^{n-1} is known .