A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow
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.
Keywords
Cite
@article{arxiv.2208.01324,
title = {A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow},
author = {Wei Jiang and Chunmei Su and Ganghui Zhang},
journal= {arXiv preprint arXiv:2208.01324},
year = {2023}
}
Comments
22 pages, 2 figures