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An energy- and helicity-conserving finite element scheme for the Navier-Stokes equations. (English) Zbl 1141.76039
Summary: We present a new finite element scheme for solving Navier-Stokes equations that exactly conserves both energy $\left({\int }_{{\Omega }}{u}^{2}\right)$ and helicity $\left({\int }_{{\Omega }}u·\left(\nabla ×u\right)\right)$ 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 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.
##### MSC:
 76M10 Finite element methods (fluid mechanics) 76D05 Navier-Stokes equations (fluid dynamics) 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M15 Error bounds (IVP of PDE)
##### Keywords:
stability; convergence; trapezoidal time discretization