Crouzeix-Raviart for Stokes on elasticity form

Question

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?

Answer 1

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.