We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation originally proposed by Barrett, Garcke and N\"urnberg (BGN) in \cite{BGN07}. Under discretization in space with piecewise linear elements this leads to a stable continuous-in-time semidiscrete scheme, which retains the equidistribution property from the BGN methods. Furthermore, two fully discrete schemes can be shown to satisfy unconditional energy stability estimates. Numerical examples are presented to showcase the good properties of the introduced schemes, including an asymptotic equidistribution of vertices.

翻译：本文提出并分析了平面曲线Willmore流的稳定有限元逼近格式。所提格式基于一种新颖的弱形式，该形式将曲率演化方程与Barrett、Garcke和Nürnberg（BGN）在文献\cite{BGN07}中最初提出的曲率表述相结合。在空间上采用分段线性元离散后，该框架导出了一个稳定的连续时间半离散格式，该格式保留了BGN方法的等分布特性。此外，可以证明两种全离散格式均满足无条件能量稳定性估计。数值算例展示了所引入格式的良好性质，包括顶点渐近等分布特性。