We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first one involves the coupling of the outward unit normal vector $\boldsymbol{n}$ and the normal velocity $V$, and the second one concerns the time derivative of the mean curvature $\kappa$. Based on them, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully-discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate its efficiency and validate its unconditional energy stability.

翻译：我们提出了一种用于平面Willmore流的能量稳定参数有限元方法（PFEM），并建立了其全离散格式的无条件能量稳定性。关键在于引入了两个新的几何恒等式来描述平面Willmore流：第一个涉及单位外法向量$\boldsymbol{n}$与法向速度$V$的耦合，第二个涉及平均曲率$\kappa$的时间导数。基于这些恒等式，我们推导出了一组用于平面Willmore流的新几何偏微分方程，从而发展出新的全离散且无条件能量稳定的PFEM。我们的稳定性分析同样基于这两个新的几何恒等式。大量数值实验验证了该方法的效率及其无条件能量稳定性。