Speaker
Description
The talk concerns linear parabolic problems posed on time-dependent domains that may undergo topological changes. The domains are represented by smooth level-set functions and may split, merge, or develop or lose components and holes. Using the classification of generic transitions provided by Morse theory, we introduce anisotropic space-time function spaces adapted to the resulting singular geometries. This framework yields existence, uniqueness, and a priori estimates for weak solutions.
The talk then focuses on an unfitted finite element method for such and on its error analysis. Several structural assumptions on the domain evolution near the critical time provide control over the variation of solution norms, even across the singularity, and form the foundation of the stability and error analysis. Their applicability is illustrated by level-set domains undergoing different topological transitions. Optimal-order error estimates are supported by numerical experiments, and limitations of the current analysis and open questions will be discussed.