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.