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We propose and analyze a semi-discrete parametric finite element scheme forsolving the area-preserving curve shortening flow. The scheme is based onDziuk's approach (SIAM J. Numer. Anal. 36(6): 1808-1830, 1999) for theanisotropic curve shortening flow. We prove that the scheme preserves twofundamental geometric structures of the flow with an initially convex curve:(i) the convexity-preserving property, and (ii) the perimeter-decreasingproperty. To the best of our knowledge, the convexity-preserving property ofnumerical schemes which approximate the flow is rigorously proved for the firsttime. Furthermore, the error estimate of the semi-discrete scheme isestablished, and numerical results are provided to demonstrate thestructure-preserving properties as well as the accuracy of the scheme.

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