Analysis of a finite element method for PDEs in evolving domains with topological changes

TL;DR

The paper addresses linear parabolic PDEs posed on evolving domains that may undergo instantaneous topological changes. It develops a rigorous error analysis for an unfitted Eulerian FEM, leveraging a structured domain-evolution framework with a single critical time $t_c$, a level-set representation, and stability tools anchored by a narrow-band extension. The main contributions include a uniform stability estimate across singularities, optimal-order energy-norm error bounds (Theorem Th2), and a key narrow-band estimate, all substantiated by a Morse-theory–informed analysis of level-set domain changes and numerical validation on a 2D splitting example. This work provides the first rigorous numerical analysis for moving domains with topology changes and informs future development of space-time or alternative formulations for broader topological scenarios.

Abstract

The paper presents the first rigorous error analysis of an unfitted finite element method for a linear parabolic problem posed on an evolving domain $Ω(t)$ that may undergo a topological change, such as, for example, a domain splitting. The domain evolution is assumed to be $C^2$-smooth away from a critical time $t_c$, at which the topology may change instantaneously. To accommodate such topological transitions in the error analysis, we introduce several structural assumptions on the evolution of $Ω(t)$ in the vicinity of the critical time. These assumptions allow a specific stability estimate even across singularities. Based on this stability result we derive optimal-order discretization error bounds, provided the continuous solution is sufficiently smooth. We demonstrate the applicability of our assumptions with examples of level-set domains undergoing topological transitions and discuss cases where the analysis fails. The theoretical error estimate is confirmed by the results of a numerical experiment.

Figures (7)

• Figure 1: $d=2$. A splitting scenario with $\lambda_1<0<\lambda_2$, $\frac{\partial \phi}{\partial t}(x_c,t_c) > 0$. The left plot shows $\Gamma_{\!\mathcal{Q}}$ near $(x_c,t_c)$, while the right shows snapshots of $\Gamma(t)$ near $(x_c,t_c)$.
• Figure 2: A splitting scenario with $\phi(x,t) =-|x_1|^{6} +x_2^2+t$ in $\mathcal{O}(x_c,t_c)$, corresponding to a degenerate critical point. The left plot visualizes $\Gamma_{\!\mathcal{Q}}$, while the right plot shows snapshots of $\Gamma(t)$. Both in a neighborhood of $(x_c,t_c)$.
• Figure 3: Domain merging with a 'non-smooth' transition. The left plot visualizes $\Gamma_{\!\mathcal{Q}}$, while the right plot shows snapshots of $\Gamma(t)$, both in a neighborhood of $(x_c, t_c)$.
• Figure 4: One-dimensional domains merging/splitting with Assumption \ref{['Ass4']} true (left plot) and false (right plot).
• Figure 5: BDF1 results for $m=1$ (left panel) and $m=2$ (right panel).

Theorems & Definitions (28)

• definition 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• Remark 4.1
• lemma 4
• proof
• Remark 4.2