I am looking for some LBB-stable velocity-pressure combinations for incompressible Navier-Stokes where the pressure space is element-wise discontinuous, preferably with a linear variation elementwise. I can think of two options: (i) Discontinuous Galerkin and (ii) Crouziex-Raviart element, but I am not sure which one would be better. I would also like to learn about other options. Are there any shortcomings of CR elements for incompressible Navier-Stokes? I would appreciate any insights that can help me make the decision.

Note that I will be using this element for developing a solver for the fictitious domain method in which the velocity at the boundary of the immersed solid is imposed using distributed Lagrange multipliers, similar to this paper. I am interested in both quad and tria elements for 2D problems, and tetra and hexa elements for 3D problems. I have a coupled velocity-pressure-multiplier solver already implemented for Taylor-Hood elements and b-splines.

  • $\begingroup$ Stabilized method should work for any conforming finite element spaces. Also I think the MINI-element can be generalized to P2+bubble-P1d. What do you exactly mean by CR, i.e. what is the velocity/pressure pair? $\endgroup$
    – knl
    Oct 24, 2021 at 5:43
  • $\begingroup$ Thank you, @knl! If by stabilised method, you meant pressure stabilisation, then it spoils mass conservation when used with Lagrange multipliers. If not, could you please elaborate on the kind of stabilisation you meant? $\endgroup$
    – Chenna K
    Oct 24, 2021 at 9:16
  • 1
    $\begingroup$ To be honest, I do not know the exact meaning of an extended (second-order) Crouziex-Raviart (CR) element. I have seen different sets of basis functions being used with the name Crouziex-Raviart element, the element combination you mentioned being one of them. Others use one pressure DOF + two spatial derivatives of pressure (in 2D) per element. So, I am perplexed. I would like to know what exactly is a second-order CR element. $\endgroup$
    – Chenna K
    Oct 24, 2021 at 9:22
  • $\begingroup$ @Chenna K: If I remember correctly, the lesser-known second-order element proposed in the same paper by Crouzeix and Raviart is conceptually different. It is thus quite confusing to refer to it as 2nd order CR. Adding the bubble - which they do to achieve uniform LBB stability - also sacrifices pointwise incompressibility. $\endgroup$ Oct 24, 2021 at 16:53
  • $\begingroup$ Thanks, @ChristianWaluga! Your comment summarises my confusion with the higher-order CR element. I have been going through various papers but didn't find any paper that clearly explains all the basis functions for the higher-order CR elements, in a way that we have for Lagrange elements. Please let me know if you know of any such resources. $\endgroup$
    – Chenna K
    Oct 24, 2021 at 18:08

1 Answer 1


There are many variations of the idea of Taylor-Hood elements that remain stable.

  • The Taylor-Hood element on quadrilateral and hexahedral elements are generally understood (though historically incorrect) to be the space $Q_2\times Q_1$ for velocity and pressure.
  • But this is not optimal: We can make the pressure space larger and instead use $Q_2 \times P_{-1}$ where $P_{-1}$ is the set of discontinuous functions that are linear in every coordinate. This can be interpreted in two different ways, namely that $P_{-1}$ is constructed directly on the actual cell in terms of the real-space coordinates, or one can think of it as the set of linear shape functions mapped from the reference to the real cell. There are subtle differences in the convergence theory for these two. The place to read up on these elements is the book by Volker John ("Finite Element Methods for Incompressible Flow Problems", Springer 2016), as well as the article by Matthies and Tobiska: https://link.springer.com/article/10.1007%2Fs00607-002-1451-3)
  • One can also use discontinuous elements for the velocity, and one example of such approaches can be found in the paper by Cockburn, Kanschat, and Schoetzau: https://link.springer.com/article/10.1007/s10915-006-9107-7
  • $\begingroup$ Thanks, @Wolfgang Bangerth! I thought $Q_2 \times Q_1$ is with continuous approximation. Did I learn it wrong? $\endgroup$
    – Chenna K
    Oct 25, 2021 at 9:11
  • $\begingroup$ Regarding the $Q_2 \times P_{-1}$ element, I used it with a mixed formulation for solid mechanics problems. Its accuracy is poor. But, it seems to be an easier option in terms of implementation, considering that I already have a code for $Q_2 \times Q_1$ element. $\endgroup$
    – Chenna K
    Oct 25, 2021 at 9:16
  • $\begingroup$ Concerning the $RT_{k}/Q_k$ elements considered in that paper by Cockburn et al., could you provide me with some resources with the details of basis functions? $\endgroup$
    – Chenna K
    Oct 25, 2021 at 9:18
  • $\begingroup$ $Q_2\times Q_1$ uses continuous elements for both velocity and pressure, and is what is commonly referred to as the "Taylor-Hood element". If you made the pressure discontinuous, i.e., $Q_2\times Q_{-1}$, you would get an unstable element. It has one pressure degree too many per cell in 2d; but $Q_2\times P_{-1}$ is stable. $\endgroup$ Oct 25, 2021 at 17:22
  • $\begingroup$ Regarding $RT_k \times Q_k$, what is your specific question? $RT_k$ are the usual Raviart-Thomas elements whose basis functions are easy to find. $\endgroup$ Oct 25, 2021 at 17:23

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