A new framework of high-order unfitted finite element methods using ALE maps for moving-domain problems

TL;DR

This paper develops a flexible ALE-UFE framework for high-order unfitted finite element methods on moving domains, demonstrating optimal convergence for smooth boundaries and challenges with topological changes.

Contribution

It introduces a generic high-order ALE-UFE framework for PDEs on moving domains, extending previous work to handle complex domain evolutions.

Findings

Optimal convergence for smooth boundaries with third- and fourth-order methods

Convergence deteriorates to second order with topological changes

Framework effectively handles various moving-domain problems

Abstract

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.