About Scott–Vogelius element

General Discussion
21

Running your new plugin I observe that the Powell-Sabin mesh is correct, but there is some difficulty:

When solving the Stokes linear system it takes 6 times more in cpu to get the solution. This indicates a stiff problem.
I think that making the computations of the intersection between the edge and the line joining the two triangle centers in C++ is not accurate enough compared to the same computation in FreeFem++ (that I did).
The consequence is that the points are not accurately enough aligned.
I had such problems before when saving my mesh into mesh1.msh:
I had to indicate a large number of digits when writing into the file.
Is it possible to use higher accuracy in your splitmesh6.cpp script ?
Francois.

22

Ok, so I will put this new version in the develop version to day.

23

Con you send me you script, because I am suprise about the Stokes probleme.
The code is trivial, I think the problem is “ailleurs”.

mon script test …
Scott–Vogelius.edp (753 Bytes)

24

Dear teacher, the command “splitmesh6” does produce the powell-sabin mesh (in most situations), but it only suits two dimensions. Is there a similar command for producing the powell-sabin mesh in a three-dimensional area?

25

splitmesh12 is what you are looking for.

26

In 3d this is called Worsey-Farin, see
Low-order divergence-free approximations for the Stokes problem on
Worsey–Farin and Powell–Sabin splits, Comput. Methods Appl. Mech. Engrg. 390 (2022) 114444

The Powell-Sabin split for arbitrary 2d mesh splitmesh6PowellSabin,
and the Worsey-Farin split for arbitrary 3d mesh splitmesh12WorseyFarin are available on the developer branch of FreeFem, but you can get them as plugins
https://perso.math.u-pem.fr/bouchut.francois/splitmesh6.cpp
https://perso.math.u-pem.fr/bouchut.francois/splitmesh12.cpp
(thanks Frederic Hecht for these update versions).
You can call them by (for example)

load “splitmesh6”
mesh Th=splitmesh6PowellSabin(Th0);

load "splitmesh12"
mesh3 Th=splitmesh12WorseyFarin(Th0);

the previous commands (still available) splitmesh6 and splitmesh12 are simplified versions that work only for square() or cube() meshes.

1 Like
28

Dear teacher, I am very grateful for your reply. And I get your idea. But when I use the Scott-V element (e.g., P2-P1disc), I need some restrictions on mesh for the inf_sup condition. In 2D, we need the barycenter refinement and the powell-sabin refinement for 3D. So, using W-F mesh (splitmesh12) can also satisfy the LBB condition for the SV element ? Thank you again honestly.

29

For short, the adapted split to satisfy the inf-sup condition is

– For P1/P0,
splitmesh6PowellSabin in 2d
splitmesh12WorseyFarin in 3d
– For P2/P1dc
splitmesh3 in 2d
splitmesh4 in 3d

Indeed the inf-sup condition is satisfied on V_h\times M_h with V_h either P1 or P2 according to the above, and M_h a certain subspace of P0 or P1dc, containing the image of V_h by the div operator. The precise definition of M_h is not important (it is not the full space), what is important is that it contains \mathop{\rm div}V_h.

30

I used splitmesh4 and P2/P1dc for the 3D-Cavity_Driven test, and it’s worked! Appreciation from my heart!