Abstract:We formulate a new discrete-exterior calculus based discretisation of the incompressible Euler and Navier-Stokes equations that preserves the geometric structure of the continuum, and establish a rigorous convergence and structure theory for a new discretisation. The discretisation operates on prismatic Delaunay-Voronoi meshes over closed Riemannian manifolds. The geometry of Euler and Navier-Stokes equations is maintained via a discrete Lie derivative that is built from an extrusion-based contraction for the nonlinear term in vector-invariant form. Conservation of energy and Kelvin circulation links the discrete scheme to the continuum: at the discrete level, energy conservation is a stability property, and in the vanishing-resolution limit it becomes both a constructive route into the conservative weak-solution theory of the continuum equations and a selection principle on the limits the scheme can reach. This correspondence appears in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|)$ in dimensions $d=2,3$, uniformly in viscosity $\nu \ge 0$; first order on general meshes, second order under centroid proximity and reconstruction symmetry. \emph{Leray-Hopf weak regime}: subsequential $L^2$ limits of the discrete Navier-Stokes system are weak solutions of the viscous equations. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions, with concentration defect vanishing above the Onsager threshold $\alpha > 1/3$ provided the discrete solutions admit a uniform $C^{0,\alpha}$ bound; the scheme reaches the energy-conserving side of the Onsager landscape but not the dissipative side. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any Hölder regularity, an exclusion that follows from discrete energy conservation.
From: Peter Korn [view email]
[v1]
Wed, 13 May 2026 06:12:26 UTC (111 KB)
[v2]
Fri, 15 May 2026 05:18:58 UTC (111 KB)
[v3]
Thu, 21 May 2026 05:19:25 UTC (112 KB)
[v4]
Wed, 24 Jun 2026 05:25:13 UTC (87 KB)
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