A mass, energy, and helicity conserving dual-field discretization of the incompressible Navier-Stokes problem

Abstract

A structure-preserving dual-field discretization of the 3D incompressible Navier-Stokes problem is derived, which conserves mass, kinetic energy, and helicity on a periodic domain. In this approach, a conservative mixed variational formulation is introduced. It is based on two systems of equations with dual representations of velocity, vorticity, and pressure. Then, astaggered temporal discretization is constructed in order to integrate the evolution equations and to decouple the two systems. That way, the convective terms are linearzed, resulting in two discrete, algebraic systems. Furthermore, a spatial Galerkin Finite Element discretizationis introduced, that follows a mimetic approach: The finite dimensional spaces form a discrete de Rham complex, which is essential to enable the conservation properties of the scheme. Conservation of mass, kinetic energy, and helicity at the discrete level is proven in the inviscid limit.The proposed method is expanded to a much more realistic, non-periodic Dirichlet problem, by adapting the finite-dimensional function spaces. It is proven, that mass and kinetic energy arestill conserved. Numerical tests supporting the method in both the periodic and non-periodic setting are provided.