Enforcing energy, helicity and strong mass conservation in finite element computations for incompressible Navier-Stokes simulations

We study a finite element scheme for the 3D Navier-Stokes equations (NSE) that globally conserves energy and helicity and, through the use of Scott-Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor-Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme.