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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.
Lu et al. (Tue,) studied this question.