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.