A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow

TL;DR

This paper introduces a semi-discrete finite element scheme for the area-preserving curve shortening flow that rigorously preserves convexity and decreases perimeter, with proven error estimates and numerical validation.

Contribution

It is the first to rigorously prove convexity preservation in a numerical scheme for this flow, extending Dziuk's approach to a new geometric setting.

Findings

The scheme preserves convexity of initially convex curves.

The scheme guarantees perimeter decreases over time.

Numerical results confirm the theoretical properties and accuracy.

Abstract

We propose and analyze a semi-discrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk's approach (SIAM J. Numer. Anal. 36(6): 1808-1830, 1999) for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental geometric structures of the flow with an initially convex curve: (i) the convexity-preserving property, and (ii) the perimeter-decreasing property. To the best of our knowledge, the convexity-preserving property of numerical schemes which approximate the flow is rigorously proved for the first time. Furthermore, the error estimate of the semi-discrete scheme is established, and numerical results are provided to demonstrate the structure-preserving properties as well as the accuracy of the scheme.