Quasi-optimal error estimate for the approximation of the elastic flow of inextensible curves

TL;DR

The paper addresses the numerical approximation of elastic flow for inextensible curves by developing a space-time discretization that preserves the arc-length constraint through a higher-order discrete constraint. It introduces and analyzes both semi-discrete and fully discrete gradient-flow schemes, leveraging correction terms to enable robust error testing under the inextensibility constraint. The main result is a quasi-optimal error estimate showing $O(h^4)$ convergence in combined $z_t$ and $z$-errors under strong regularity, validated by numerical experiments that demonstrate superior convergence when using the $\mathcal{P}_2$ constraint. The work combines rigorous variational analysis with practical time-stepping schemes and provides extensive appendix material on interpolation stability and auxiliary estimates to support the theory.

Abstract

A space-discretization for the elastic flow of inextensible curves is devised and quasi-optimal convergence of the corresponding semi-discrete problem is proved for a suitable discretization of the nonlinear inextensibility constraint. Further a fully discrete time-stepping scheme that incorporates this constraint is proposed and unconditional stability and convergence of the discrete scheme are proved. Finally some numerical simulations are used to verify the obtained results experimentally.

Figures (2)

• Figure 1: Initial values $z_{h,0}$ for Example \ref{['experiment:circle']} (left) and Example \ref{['experiment:helix']} (right)
• Figure 2: Constructed continuous solution $\widetilde{z}(\cdot,t)$ from Example \ref{['experiment:circle to helix']} for $t = 0,\ 0.4,\ 0.8$.

Theorems & Definitions (16)

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