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 $(\int_{\Omega}u^{2})$ and helicity $(\int_{\Omega} u\cdot(\nabla \times u))$ in the absence of viscosity and external force. We prove 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 in a trapezoidal time discretization. Numerical results are presented that compare the scheme to the usual trapezoidal schemes.