We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formulae (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and N\"urnberg (J. Comput. Phys., 222 (2007), pp.~441--467), we successfully develop high-order BGN/BDF$k$ schemes. The proposed BGN/BDF$k$ schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired $k$th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF$k$ schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.

翻译：我们提出了一类新颖的时间高阶参数化有限元方法，用于求解曲线和曲面几何流的广泛问题。通过将向后微分公式（BDF）引入时间离散化到BGN公式（最初由Barrett、Garcke和Nürnberg提出，参见J. Comput. Phys., 222 (2007), pp. 441–467），我们成功开发了高阶BGN/BDF$k$格式。所提出的BGN/BDF$k$格式不仅保留了经典一阶BGN格式的几乎所有优势（如计算效率和良好的网格质量），还在形状度量方面展现出期望的$k$阶时间精度，从二阶到四阶精度。此外，我们通过大量数值算例验证了所提出的BGN/BDF$k$格式的性能，展示了它们在处理各类几何流时的高阶时间精度，同时在整个演化过程中保持良好的网格质量。