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 in 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.

中文翻译：

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曲线表面扩散流的减周面积守恒算法

针对二维平面内闭合曲线的表面扩散流，提出了一种基于新的弱公式和新的时间步长方案的全离散有限元方法。证明该方法可以在离散水平上同时保留两个几何结构，即曲线的周长随时间减小而曲线所包围的面积保持不变。数值例子证明了所提出方法的收敛性以及该方法在保留两个几何结构方面的有效性。