Second order in time finite element schemes for curve shortening flow and curve diffusion

We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.

💡 Research Summary

The paper presents and rigorously analyses second‑order‑in‑time finite element schemes for two fundamental geometric evolution equations: curve shortening flow (CSF) and curve diffusion (CD). Both flows are described in a parametric form x(ρ,t) on the periodic interval I = ℝ/ℤ. By applying the DeTurck trick, the authors rewrite the purely normal velocity laws Vₙₒᵣ = κ (CSF) and Vₙₒᵣ = –∇ₛ²κ (CD) into strictly parabolic systems that contain a carefully chosen tangential velocity component. This tangential motion improves mesh quality by promoting asymptotic equidistribution of the nodes.

For CSF the underlying first‑order scheme (mass‑lumped backward Euler) is known from earlier work (Deckelnick & Nürnberg, 2009). The authors embed this scheme into a predictor‑corrector framework inspired by a Crank‑Nicolson–type approach (Mikula & Sevcovic, 2011). In each time step the algorithm proceeds as follows: (i) a half‑step predictor x_{m+½}^h is computed using the first‑order method with the current mesh length |x_{m,ρ}^h|; (ii) a full‑step corrector computes x_{m+1}^h by solving a linear system that uses the averaged mesh length |x_{m+½,ρ}^h|². Both substeps involve only linear systems with symmetric positive definite matrices, so exactly two linear solves are required per time step.

The authors prove unconditional stability: the discrete Dirichlet energy ½‖x_{m+1}^h‖² plus a term proportional to the inner product of the node‑wise length and the time increment is non‑increasing. Under the regularity assumption that the exact solution satisfies c₀ ≤ |x_ρ| ≤ C₀, they establish optimal error bounds provided the time step obeys Δt ≤ γ h^{1/4}. Specifically,

• L²‑error: ‖x(·,t_m) – x_m^h‖ = O(h² + Δt²),
• H¹‑error: |x(·,t_m) – x_m^h|₁ = O(h + Δt²),
• Time‑derivative error: ‖x_t(·,t_{m+½}) – (x_{m+1}^h – x_m^h)/Δt‖ = O(h⁴ + Δt⁴).

The proof introduces an energy‑like error measure E_m that combines the L² error, a term involving the interpolated exact solution, and a discrete gradient contribution. By induction and careful handling of the mass‑lumped quadrature errors, the authors bound E_m by C(h⁴+Δt⁴), which yields the super‑convergence of the H¹‑norm.

For curve diffusion, the fourth‑order PDE is split into two coupled second‑order equations by introducing an auxiliary variable y = x_{ρρ}/|x_ρ|². The same predictor‑corrector idea is applied: a half‑step predictor for (x,y) is obtained, then a full‑step corrector solves a linear system that respects the coupling. Although the authors cannot provide a full theoretical error analysis for CD due to the more intricate coupling, they present extensive numerical experiments that demonstrate second‑order convergence in time and asymptotic mesh equidistribution, mirroring the CSF results.

Numerical tests include shrinking circles (where the exact solution is known) and more irregular initial curves. The experiments confirm the predicted convergence rates and show that the node distribution becomes nearly uniform as the simulation proceeds, validating the benefit of the DeTurck‑induced tangential velocity.

In conclusion, the paper delivers a practical, linear‑solve‑only algorithm that attains second‑order accuracy in time for both CSF and CD while preserving high mesh quality. The rigorous error analysis for CSF is, to the authors’ knowledge, the first of its kind for a second‑order‑in‑time scheme in this context. Future work may extend the analysis to higher‑order spatial elements, to surface diffusion in higher dimensions, and to fully nonlinear error bounds for the curve diffusion scheme.