A perimeter-decreasing and area-conserving algorithm for surface diffusion flow of curves

TL;DR

This paper introduces a novel finite element method for surface diffusion flow of closed curves in 2D, which simultaneously decreases perimeter and conserves enclosed area, with proven geometric structure preservation and demonstrated effectiveness.

Contribution

It presents a new weak formulation and time-stepping scheme that ensure simultaneous perimeter decrease and area conservation at the discrete level.

Findings

Method preserves geometric structures numerically.

Convergence of the proposed scheme is demonstrated.

Effective in maintaining perimeter and area properties.

Abstract

A fully discrete finite element method, based on a new weak formulation and a new time-stepping scheme, is proposed for the surface diffusion flow of closed curves in the two-dimensional plane. It is proved that the proposed method can preserve two geometric structures simultaneously at the discrete level, i.e., the perimeter of the curve decreases in time while the area enclosed by the curve is conserved. Numerical examples are provided to demonstrate the convergence of the proposed method and the effectiveness of the method in preserving the two geometric structures.