A CONVEXITY-PRESERVING AND PERIMETER-DECREASING PARAMETRIC FINITE ELEMENT METHOD FOR THE AREA-PRESERVING CURVE SHORTENING FLOW.

We propose and analyze a semidiscrete 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 (1999), pp. 1808--1830] 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 semidiscrete scheme is established, and numerical results are provided to demonstrate the structure-preserving properties as well as the accuracy of the scheme. [ABSTRACT FROM AUTHOR]