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

The paper presents the first rigorous error analysis of an unfitted finite element method for a linear parabolic problem posed on an evolving domain $\Omega(t)$ 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 $\Omega(t)$ 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.