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Typically I have seen P2/P1 elements used for Stokes equation, but I want to use P2/(P1+P0) and P2/P0 elements because I want to ensure local mass conservation. When I say P2/(P1+P0), I simply mean the concatenation of the basis for P1 and P0 separately.

I ran simple problems in FEniCS with both element pairs and while the velocity works great in both cases (i.e., element-wise conservative), the pressure does get a little screwed up.

I have been told mixed opinions about using anything with piecewise constants for pressure. I have heard that P2/P0 is not LBB stable for 3D elements (however, I fail to see this happening because I do not get node-to-node spurious oscilliations) although this could potentially explain why my pressure looks screwed up. I have also heard that P2/P0 should be doable in 3D (hence P2/(P1+P0) should work as well).

So I guess my question is, theoretically are P2/P0 and/or P2(P1+P0) elements stable and usable for 3D Stokes and Darcy problems?

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The tables starting on page 462 of Gresho and Sani's book "Incompressible Flow and the Finite Element Method" say that $P_2 P_0$ is stable but first-order only and $P_2(P_1+P_0)$ is also stable but has two hydrostatic modes that you should account for. This is all for Stokes, so you should look for other references for Darcy flow.

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  • $\begingroup$ thank you for pointing that book out. It says that P2P0 is stable for 2D elements but there is no mention of it for 3D. P2/(P1+P0) is listed in both, so I am guessing the latter is "the one" to use? Also, what is meant by hydrostatic modes? $\endgroup$ – Justin Jun 10 '15 at 18:58
  • $\begingroup$ @Justin, keep reading. The section starting on p. 469 and the parts especially on pp. 473-474 give some insight. Basically, the pressure may exhibit some strange behavior that is non-physical and may be exacerbated by choice of boundary conditions. $\endgroup$ – Bill Barth Jun 10 '15 at 20:16

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