Section 2.18 in documentation

General Discussion
1

Apologies, I’ve asked this before, but unfortunately didn’t get a reply! So trying one more time.

Hi all, section 2.18 (Tutorial to write a transient Stokes solver in matrix form) seems like it could be quite useful to me, but it does not show the original form of the PDE problem. Could anyone tell me the original strong form of the problem being solved (including the boundary conditions). I found the transient Stokes equation online, but the weak form I have derived is not the same as the weak form featured in the code provided in section 2.18 (e.g. there are no spatial derivatives in pressure).

Many thanks!

2

yes,

the strong form is basic Stokes in tlme problem

u_t - \nu \Delta u + \nabla p = 0 in \Omega
\nabla.u = \varepsilon p in \Omega
u=u_\Gamma on \partial\Omega

remark: u_\Gamma is given by (0,0) on 3 side left, right, bottom, and (1,0) on top side.

1 Like
3

Thanks! One additional question: why is there no boundary condition for the pressure, p?

4

Hello James,

I m not sure but i think it’s only a way to have a well posed problem.
To be sure of this you can run the problem with the current BC (boundary conditions) and recover the value of p on the top side.
Then you implement the same script but with the recovered p imposed on the top, while checking the value of u on the top.

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5

The pressure is the Lagrange multiplier for the continuity constraint, not an unknown of the PDE.

6

Okay, thanks. And I was wondering why there is no integral of the normal derivatives of the velocity components over the boundary present in the weak form? Is this because there are Dirichlett BCs for the two components of velocity imposed on the boundary, meaning the test functions corresponding to each of the velocity components vanish on the boundary?

7

correct me if I’m wrong,

Freefem give the possibility to implement the Dirichlet BCs with on(mesh label, u = value). This means that the information on the side is taken into account, so you don’t need another integral on the side.

8

I shall have a read about this, thanks for your reply!