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

TL;DR

The paper introduces ALE-UFE, a high-order unfitted finite element framework for PDEs on time-varying domains, built around forward boundary maps and backward flow (ALE) maps to construct a multi-step time discretization. It provides a rigorous stability and error analysis for the semi- and fully discrete schemes and demonstrates optimal convergence on smooth moving domains across several model problems, including a PDE-domain coupled system, a two-phase flow, and a topology-changing domain. The framework uses an explicit interface-tracking step, an approximate ALE map construction, and a robust cut-element treatment to enable high-order accuracy without full mesh regeneration. Numerical experiments confirm the method’s effectiveness for moving interfaces and coupled problems, while highlighting a degradation to second order in the presence of topological changes, indicating both strong potential and current limitations. The approach has practical implications for moving-domain simulations in multiscale and multiphase contexts, with potential extensions to fluid-structure interaction and related applications.

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.

Figures (7)

• Figure 1: Left: the domain $D$ and its partition $\mathcal{T}_h$, the approximate domain $\Omega_\eta^n$ and its $\delta$-neighborhood $\Omega_{\delta}^n$. Right: the set of red and yellow squares $\mathcal{T}^n_h$, the set of red squares $\mathcal{T}^n_{h,B}$, the set of blue edges $\mathcal{E}_{h,B}^{n}$, the union of red and yellow squares $\tilde{\Omega}^n$.
• Figure 2: Left: A cut element with markers $\mathcal{P}^{n-1}$(blue points), $3$ segments of $\Gamma_{\eta,K}^{n-1}$, (red lines), quasi-uniformly distributed points ${\boldsymbol{A}}_i$ on $\Gamma_{K,1}^{n-1}$. Right: The black line $\tilde{\Gamma}_{K,m}^n$ is obtained by Lagrange interpolation based on the points ${\boldsymbol{A}}_i^n$, $0\leq i\leq 3$ (green points), which is approximation of $\Gamma_{\eta}^n$ (blue line).
• Figure 3: Left figure: $\tilde{\Omega}_1^n$ (red and yellow squares), $\mathcal{T}_{B,1}^n$ (the set of red squares), $\mathcal{E}_{1,B}^n$ (blue edges) and $\partial \Omega_{\delta,1}^n$ (green edge). Right figure: $\tilde{\Omega}_2^n$ (red and yellow squares), $\mathcal{T}_{B,2}^n$ (the set of red squares), $\mathcal{E}_{2,B}^n$ (blue edges) and $\partial \Omega_{\delta,2}^n$ (green edge).
• Figure 4: The moving domain $\Omega_{\eta}^n$ at $t_n=0$, $0.2$, $0.4$, $0.6$, $0.8$ and $1$ ($h=1/16$).
• Figure 5: Approximate domains at $t_n =0$, $0.6$, $1.2$, $1.8$, $2.4$ and $3$ ($h=1/16$).

Theorems & Definitions (5)

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