STRONGLY ENSTROPHY-STABLE INTEGRATORS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

Boris Andrews `|` Matin Shams `|` Patrick Farrell

SEP.2026 (arXiv) `|` In review (FoCM)

CHECK OUT ON ARXIV!

We propose a mixed finite element discretisation for the incompressible Navier–Stokes equations that preserves the evolution laws of both energy and enstrophy […]. In two dimensions, [this leads] to a Reynolds-number-independent bound on the velocity gradient that naturally stabilises the scheme, even on severely under-resolved meshes. In three dimensions, the scheme preserves both dissipation and the generation of enstrophy through vortex stretching. […]

FULL ABSTRACT

We propose a mixed finite element discretisation for the incompressible Navier–Stokes equations that preserves the evolution laws of both energy and enstrophy, in a stronger sense than previous discretisations. In two dimensions, the evolution law for enstrophy only permits dissipation for thermodynamically isolated systems, leading to a Reynolds-number-independent bound on the velocity gradient that naturally stabilises the scheme, even on severely under-resolved meshes. In three dimensions, the scheme preserves both dissipation and the generation of enstrophy through vortex stretching.

We enforce these evolution laws by systematically introducing auxiliary variables into the discretisation. While conforming implementations of these schemes require discrete Stokes complexes with enhanced regularity, we introduce both (i) equivalent reparametrisations and (ii) penalty formulations that require only the typical curl- and div-conforming spaces from the standard discrete de Rham complex. The scheme handles different types of boundary conditions and curved domains. The robust stabilisation properties of the proposed scheme are demonstrated through numerical simulations of a shear flow, a spherical vortex, and flow past an obstacle. We observe numerically that preserving the discrete evolution of enstrophy in this way has a strong stabilising effect on the numerical solution, especially in two dimensions.

(Further details available soon!)

This represents a particularly exciting application of my earlier work with Patrick Farrell, on general constructions for conservative and dissipative finite element integrators, using the ideas to help in stabilisation efforts at high Reynolds numbers.

Problem	Reward
Stable reduced order models from the auxiliary variable framework	★☆☆☆☆

CO-AUTHORS

Matin Shams

Affiliation | University of Oxford
Joint publications | Strongly enstrophy-stable integrators for the incompressible Navier–Stokes equations

Patrick Farrell

Affiliation | University of Oxford / Charles University
Joint publications | Helicity-preserving finite element discretization for magnetic relaxation / Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables / Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems / Arbitrary-order structure-preserving discretizations for geometric curvature flows / Automated Galerkin time stepping in Irksome / Strongly enstrophy-stable integrators for the incompressible Navier–Stokes equations / Energy-stable discretisations of viscoelastic flows* / Energy- and entropy-stable discretisations for the quasi-incompressible Maxwell–Stefan equations*

VIDEOS

Check out my talk at the Programme on Differential Complexes at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) within the University of Vienna (MAY.2026) below:

TALKS

2026

SciCADE, University of Edinburgh
Programme on Differential Complexes, Erwin Schrödinger International Institute (ESI), Vienna
Workshop on Finite Element Tensor Calculus, Tsinghua University

2025

Numerical Analysis Group Internal Seminar, University of Oxford
ACOMEN, Ghent University
ECCOMAS MFET, Aachen, Germany