Adaptive moving mesh methods for the planar Willmore flow

In this paper, we propose adaptive moving mesh methods for the planar Willmore flow by incorporating a tangential velocity into the original geometric evolution. The tangential velocity is designed based on a monitor function constructed from the curvature and its variation, enabling dynamic mesh redistribution along the evolving interface. This adaptive redistribution enhances spatial resolution in regions of high geometric complexity while preserving mesh regularity. The resulting moving mesh formulation is discretized using the kth-order backward differentiation formula (BDFk) in time and finite difference methods in space. Moreover, a class of new relaxed Lagrange multiplier approaches is further incorporated into the adaptive moving mesh framework to construct energy-stable adaptive moving mesh methods. Furthermore, to enhance the adaptivity and flexibility of the proposed framework, we develop an adaptive strategy for selecting the monitor function and introduce an alternative redistribution approach. Finally, extensive numerical experiments demonstrate that the proposed BDFk-based adaptive scheme accurately captures the geometric evolution of the planar Willmore flow and exhibits excellent robustness and computational efficiency for problems involving complex interface geometries.