This thesis addresses the numerical treatment of Navier–Stokes equations on evolving surfaces, combining advanced concepts from surface fluid modeling, trace finite element approximations, and level set equation discretization schemes. Motivated by applications in biological membranes and thin film flows, the work focuses on the numerical simulation of surface fluid dynamics. Two principal systems are considered: the tangential surface Navier–Stokes system (TSNS), where the evolving surface is prescribed, and the full surface Navier–Stokes system (SNS), where both the surface evolution and the flow are unknowns. A major challenge in the SNS arises from the nonlinear coupling between the velocity field and the surface geometry evolution. The thesis provides a comprehensive review and comparison of existing derivations of the surface Navier–Stokes equations, highlighting differences in physical principles (such as conservation laws, thin film limits, and energetic variational approaches) and coordinate representations (local curvilinear vs. global Cartesian). It is shown that, while the TSNS systems derived from various approaches are equivalent, the SNS systems may differ depending on the chosen framework. The main goal of this thesis is the development and analysis of novel numerical methods based on the Trace Finite Element Method (TraceFEM) for both TSNS and SNS on evolving surfaces. TraceFEM uses a background mesh in the embedding space, allowing for robust discretization without the need for surface triangulation or remeshing as the surface evolves. The thesis details the construction of these methods, including stabilization and extension techniques. For the TSNS, an error analysis is provided, and the method’s accuracy and stability are demonstrated. The extension to the full SNS system is presented, addressing the additional challenges posed by the nonlinear coupling between surface evolution and fluid flow. Additionally, an efficient narrow band method for the level set equation is introduced to accurately represent surfaces evolved by material flow fields. This method is crucial for the SNS system, where the surface evolution is coupled with the fluid flow. Numerical experiments validate the theoretical findings, demonstrating the accuracy and efficiency of the proposed methods. This work contributes to computational fluid dynamics by advancing methodologies for discretizing fluid flows on deformable surfaces, paving the way for future research in this vital area of applied mathematics.
Paul Schwering (Wed,) studied this question.