Convergence of a scheme for elastic flow with tangential mesh movement

TL;DR

This paper introduces a finite element method for elastic flow of closed curves that uses Dirichlet energy for mesh redistribution, providing convergence analysis and error estimates, with numerical results supporting the approach.

Contribution

The paper proposes a novel mesh redistribution scheme based on Dirichlet energy for elastic flow, simplifying analysis and ensuring convergence.

Findings

The method effectively redistributes mesh points during elastic flow.

Convergence and error estimates are established for the proposed scheme.

Numerical experiments support the theoretical results.

Abstract

Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of K Deckelnick, and G Dziuk (Math Comp 78, 266 (2009), 645-671) we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical result that support the theory.