As a sequel to our previous work [C. Ma, Q. Zhang and W. Zheng, SIAM J. Numer. Anal., 60 (2022)], [C. Ma and W. Zheng, J. Comput. Phys. 469 (2022)], this paper presents a generic framework of arbitrary Lagrangian-Eulerian unfitted finite element (ALE-UFE) methods for partial differential equations (PDEs) on time-varying domains. The ALE-UFE method has a great potential in developing high-order unfitted finite element methods. The usefulness of the method is demonstrated by a variety of moving-domain problems, including a linear problem with explicit velocity of the boundary (or interface), a PDE-domain coupled problem, and a problem whose domain has a topological change. Numerical experiments show that optimal convergence is achieved by both third- and fourth-order methods on domains with smooth boundaries, but is deteriorated to the second order when the domain has topological changes.

翻译：作为我们前期工作的延续 [C. Ma, Q. Zhang and W. Zheng, SIAM J. Numer. Anal., 60 (2022)], [C. Ma and W. Zheng, J. Comput. Phys. 469 (2022)]，本文提出了一个适用于时变域偏微分方程(PDEs)的通用任意拉格朗日-欧拉非拟合有限元(ALE-UFE)方法框架。ALE-UFE方法在发展高阶非拟合有限元方法方面具有巨大潜力。通过多种移动域问题验证了该方法的有效性，包括具有显式边界（或界面）速度的线性问题、PDE-域耦合问题以及域发生拓扑变化的问题。数值实验表明，在具有光滑边界的域上，三阶和四阶方法均能达到最优收敛阶，但当域发生拓扑变化时，收敛阶下降至二阶。