About Scott–Vogelius element

Dear all,

Hi, everyone! I have a question regarding the Scott–Vogelius lowest order element for Stokes flows, namely P2 element for velocity and P1dc element (piecewise linear discontinuous Galerkin) for pressure. This element does not satisfy the LBB or inf-sup condition, which can be circumvented using the macroelement technique by splitting a triangular element into three sub triangles. If I understand correctly, the splitting process is available in Freefem as I have seen the examples ‘splitmesh*.edp’ in the /plugin dir. I would like to know whether my above understanding is correct and whether this Scott–Vogelius P2-P1dc element can indeed be adopted in freefem while satisfying the inf-sup condition. Thanks in advance.

best,
lailai

Your understanding is correct, it’s perfectly fine to use Scott–Vogelius finite elements and Powel–Sabin mesh refinement. This works in 2D and 3D.

Nice, thank you very much for your prompt response. May I know whether we have a relevant freefem example to learn from? Maybe related to the Powel-Sabin mesh refinement you referred to? Thank you very much.
best,
lailai

I’m not sure there are any example, but you can take any Navier–Stokes example, replace a Taylor–Hood pair with a Scott–Vogelius one, and apply Powel–Sabin mesh refinement before discretizing your system.