ARBITRARY-ORDER STRUCTURE-PRESERVING DISCRETIZATIONS FOR GEOMETRIC CURVATURE FLOWS

Ganghui Zhang `|` Boris Andrews `|` Patrick Farrell

19.MAY.2026 (arXiv) `|` In review (SISC)

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[…] we present the first discretization of geometric curvature flows […] that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov–Galerkin in time. […] The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. […]

FULL ABSTRACT

Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve these properties on discretization. This has motivated a substantial body of research on structure-preserving discretizations for these flows, albeit at low order in time.

In this work, we present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov–Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. We demonstrate its structure-preserving properties and high-order convergence on several benchmark examples.

Two of the most prominent examples of geometric flows (equations governing the evolution of surfaces driven solely by their own geometry) are:

mean curvature flow (MCF), in which a surface shrinks in the direction of its curvature, and
surface diffusion (SD), in which a surface evolves to minimise its area while preserving the enclosed volume (like a soap bubble relaxing to a sphere).

Both flows satisfy fundamental structural properties:

MCF and SD dissipate area.
SD additionally conserves enclosed volume.

For reliable long-time simulation, it is desirable that numerical discretisations inherit these properties exactly, i.e. the discretisation be structure-preserving. Structure preservation for geometric flows has attracted a rich body of work. This paper presents the first structure-preserving discretisation of MCF and SD that achieves arbitrary order in both space and time.

The mesh quality problem

A crucial obstacle in geometric flow simulation is preventing mesh distortion.

On the continuous level, the evolution is entirely normal; that is, normal in the geometric sense: tangential velocity is redundant and unspecified. In a spatial discretisation, however, tangential motion redistributes mesh nodes and must be specified; left unchecked, a bad choice of tangential motion can degrade mesh quality significantly.

The minimal deformation rate (MDR) strategy (Hu & Li, 2022) addresses this by choosing the tangential motion to minimise a certain deformation energy ((\int_{\mathcal{M}} |\nabla_{\mathcal{M}} \dot{\mathbf{X}}|^2)) modelling the rate at which the mesh quality degrades. This is known to give vastly improved mesh quality comparable to classical method, but at the cost of the stability in area and volume.

Our approach: auxiliary variables

A key insight of this work is that area and volume stability can be re-introduced to MDR discretisations of MCF and SD via:

the systematic introducion of auxiliary variables, and
a special kind of time discretisation called continuous Petrov–Galerkin (CPG).

This yields fully discrete, structure-preserving discretisations for MCF and SD at arbitrary polynomial orders in space and time; in particular, we impose practically no restrictions on the finite element spaces used. This strategy (using auxiliary variables and CPG to replicate preserve structures after discretisation) is taken directly from my general framework with Patrick Farrell, described in an earlier companion paper.

THE SCHEMES IN BRIEF

Over a given timestep \(T_n\), the \(s\)-stage MCF discretisation is defined as follows: Find \(\mathbf{X} \in \mathbb{P}_s(T_n; \mathbb{V}^d)\) satisfying initial data, and \((p, \mathbf{R}, \kappa) \in \mathbb{P}_{s-1}(T_n; \mathbb{V} \times \mathbb{V}^d \times \mathbb{V})\), satisfying:

\[\begin{aligned} \int_{T_n} (\dot{\mathbf{X}} \cdot \mathbf{n},\, y)_{\mathcal{M}} &= -\int_{T_n} (\kappa,\, y)_{\mathcal{M}}, \\ \int_{T_n} \big[(\nabla_{\mathcal{M}} \dot{\mathbf{X}},\, \nabla_{\mathcal{M}} \mathbf{Q})_{\mathcal{M}} &+ (p\mathbf{n},\, \mathbf{Q})_{\mathcal{M}}\big] \\ &= 0, \\ \int_{T_n} (\mathbf{R} \cdot \mathbf{n},\, s)_{\mathcal{M}} &= 0, \\ \int_{T_n} \big[(\nabla_{\mathcal{M}} \mathbf{R},\, \nabla_{\mathcal{M}} \mathbf{\Lambda})_{\mathcal{M}} &+ (\kappa\,\mathbf{n},\, \mathbf{\Lambda})_{\mathcal{M}}\big] \\ &= -\int_{T_n} (\nabla_{\mathcal{M}} \mathbf{X},\, \nabla_{\mathcal{M}} \mathbf{\Lambda})_{\mathcal{M}}, \end{aligned}\] for all \((y, \mathbf{Q}, s, \mathbf{\Lambda}) \in \mathbb{P}_{s-1}(T_n; \mathbb{V} \times \mathbb{V}^d \times \mathbb{V} \times \mathbb{V}^d)\). (Sorry I had to make the equations a little ugly to fit them into this box!)

The MDR tangential motion is encoded in the \(p\) variable. The auxiliary variables \((\kappa, \mathbf{R})\) are introduced to preserve area dissipation; for further discussion about why these variables are necessary, see the manuscript.

The SD scheme is practically identical, the difference being a modified first equation in which \((\nabla_{\mathcal{M}} \kappa,\, \nabla_{\mathcal{M}} y)_{\mathcal{M}}\) replaces \((\kappa, y)_{\mathcal{M}}\). Volume conservation then follows immediately from testing with \(y = 1\).

At the semi-discrete level, our formulation coincides with the very recently proposed dual-MDR scheme of Gao, Li & Tang (2026). The key innovations in our work lie in:

the CPG time discretisation, which carries the structure-preserving properties through to arbitrary order, and
the construction through the general framework for structure-preserving discretisations, which we believe will extend beyond MCF and SD in the future.

Numerical results

Several benchmark problems in the manuscript confirm both the structure-preserving properties and the expected convergence rates, including SD on an 8x1x1 cuboid. Here’s something that’s not in the manuscript though: a video!

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We would gladly discuss it further!

VIDEOS

Patrick Farrell’s opening lecture for the Programme on Differential Complexes at the University of Vienna (20.APR.2026) is primarily about the general auxiliary variable framework, but briefly touches on this work during the motivation:

This paper represents an application of the auxiliary variable framework for structure-preserving time discretisations introduced in my earlier paper with Patrick Farrell.