An Energy- and Helicity-Conserving Finite Element Scheme for the Navier-Stokes Equations

We present a new finite element scheme for solving the Navier-Stokes equations that exactly conserves both energy ( ∫ Ω u) and helicity ( ∫ Ω u · (∇× u)) in the absence of viscosity and external force. We prove [email protected], http://www.math.pitt.edu/∼ler6 Partially supported by NSF Grant DMS 0508260 and 0207627 1 stability, exact conservation, and convergence for the scheme. Energy and helicity are exactly conserved by using a combination of the usual (convective) form with the rotational form of the nonlinearity, and solving for both velocity and a projected vorticity.