  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.
