An energy- and helicity-conserving enriched Galerkin method for the incompressible Navier-Stokes equations

Speaker: ZHANG Qian

Venue: Building 2, room 312, Haina yuan, Zijingang Campus

Abstract: In the ideal limit and in the absence of external forces, the incompressible Navier–Stokes equations preserve fundamental invariants, including kinetic energy and helicity. From a numerical perspective, failure to respect these invariants may reduce the physical reliability of long-time simulations of three-dimensional flows. However, numerical methods that simultaneously preserve kinetic energy and helicity remain relatively limited; many existing approaches are computationally expensive, require nonlinear solvers, or are applicable only under restrictive boundary conditions.

In this talk, I will present an efficient enriched Galerkin method that preserves both invariants. The method enriches a first-order continuous Galerkin velocity space with the lowest-order Raviart–Thomas space and uses a piecewise-constant pressure approximation. The enrichment corrects the normal component of the continuous Galerkin velocity field, yields an inf-sup stable velocity–pressure discretization, and retains the same number of global degrees of freedom as the classical Bernardi–Raugel element. Together with a carefully designed rotational-form discretization of the convective term, this formulation leads to a fully nonlinear Crank–Nicolson scheme and two linearized variants. All three schemes exactly preserve discrete kinetic energy and helicity in the inviscid limit, and each Picard iteration step of the nonlinear scheme preserves the invariants. I will also present numerical examples demonstrating the accuracy and structure-preserving performance of the proposed method.