# Crouzeix-Raviart for Stokes on elasticity form

I have tried to solve the driven cavity problem with incompressible Stokes flow using the standard Crouzeix-Raviart non-conforming P1/P0 element (linear velocity, constant pressure, velocity-nodes at the midpoint of the edges). The velocity $$u = (u_1 \; u_2) = (1 \; 0)$$ on the top side of the domain and zero on the other three sides. As seen in the picture below, the problem is solved fine for Stokes on standard fluid-form, i.e. when the "velocity-block" comes from the functional

$$a(u,v) = \int_{\Omega}\mu\nabla u\cdot \nabla v$$ dx.

However, switching to the (equivalent in the continuous case) elasticity-form, with

$$a(u,v) = \int_{\Omega}\varepsilon(u)^TD\varepsilon(v)$$ dx,

where $$\varepsilon(u) = (\partial u_1/\partial x \;\; \partial u_2/\partial y \;\; (\partial u_1/\partial y+\partial u_2/\partial x)/2)^T$$ and $$D=\mbox{diag}(2\mu,\;2\mu,\;\mu)$$, the velocity and (and pressure field) looks very strange (despite the system matrix being non-singular):

I know that the elasticity form will not work for Neumann boundary conditions because we can get "rigid body motions", but in this case I only have Dirichlet boundary conditions, hence I'm a bit surprised that the element doesn't work. I have successfully solved the elasticity form with a stabilized version of the conforming P1/P0 element as well as the Q1/P0 (with unstable pressure). Hence, there should be nothing wrong with the elasticity form itself. So, do you think I've simply made some mistake in my implementation, or is there some fundamental problem arising when using Crouzeix-Raviart for the elasticity form?

I think this element is not stable for the elasticity operator; cf. the introduction in the following paper:

Kouhia, Reijo; Stenberg, Rolf, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow., Comput. Methods Appl. Mech. Eng. 124, No. 3, 195-212 (1995). ZBL1067.74578.

Here is the relevant part:

... Crouzeix and Raviart showed that a linear/constant combination could be used for the Stokes problem with Dirichlet boundary conditions provided that the linear elements for the velocity are nonconforming ... The approach will not, however, work for the equations of linear elasticity ...

Thus, the authors develop a finite element where only the $$x$$-component uses Crouzeix-Raviart basis and $$y$$-component uses the standard conforming $$P_1$$ basis.

• Thank you, that is an interesting paper! Yes, the C-R element is definitely unstable for the elasticity operator in general since it can lead to mechanisms in the mesh (see e.g. Hughes, The finite element method .., 2000, p.251 for an example), but I didn't think that that should be a problem here since I only have Dirichlet boundary conditions, so I don't see how I could get a mechanism? (The C-R is applied for elasticity with Dirichlet cond. in Sec. 11.4 of The mathematical Theory ... by Brenner and Scott, but I see now that they actually switched from the elasticity form to the fluid form) – Aage Apr 27 at 18:26
• I think on page 199 of my reference they explain that $a(u,v)$ is not coercive in the discrete CR space (math.aalto.fi/~rstenber/Publications/Kouhia-Stenberg.pdf). The only thing this issue has to do with the boundary condition is that you can go back and forth between the two weak formulations in the Dirichlet case. However, the lack of coercivity is a serious issue. – knl Apr 30 at 11:21
• What I'm trying to say that most of the time when you consider the stability of elements for Stokes you simply take it for granted that $a(u,v)$ is coercive (which is always the case if you use $H^1$ conforming discretizations) and then look only at the LBB condition. – knl Apr 30 at 11:32
• I've looked at "Equivalence of Finite Element Methods for Problems in Elasticity" referenced in your reference, and I think that what I'm experiencing could be that the constant in the Korn's inequality (as formulated in (5.1)) is very large as discussed on p.1500. If I recall it correctly, that constant must enter into any error estimates, thus allowing for a very large error. (It was interesting to read about the mechanism in Fig 1. as well, but I don't think that's my issue because the velocity-block of my system matrix is clearly positive definite) – Aage Apr 30 at 17:45
• I could mention that I've now also tried to solve the problem using an unstructured mesh, and the velocity field for the C-R looks much better (though still not "good", despite using rather fine meshes). This is consistent with the example on p. 1500 in Equivalence of Finite Element Methods for Problems in Elasticity which considered a uniform mesh. – Aage Apr 30 at 17:50