Novel Surface and Trace Finite Element Methods for Incompressible Fluids

Incompressible fluid flows on curved surfaces govern a wide range of phenomena central to biotechnology, biomedical engineering, and advanced materials. Examples include biological membranes, drug-delivery vesicles, the fluid coating the air sacs in the lungs, biofilms, and fluid transport along complex interfaces. As computational modeling becomes increasingly indispensable for biotechnology innovation, biomanufacturing, and health technologies, there is a growing need for accurate and reliable simulation tools for these systems. However, existing computational methods for these surface-flow models often rely on artificial stabilization parameters that can compromise robustness and predictive capability. This project develops next-generation mathematical and algorithmic tools for simulating incompressible flows on surfaces using structure-preserving computational methods that exactly preserve key physical laws such as mass conservation. The resulting algorithms will improve the reliability and accuracy of simulations central to biotechnology research and advanced engineering applications, while reducing dependence on ad hoc numerical tuning. The research develops structure-preserving finite element methods for incompressible surface PDEs through two complementary directions. First, it constructs divergence-free and tangential surface finite element spaces based on classical Euclidean pairs such as Scott-Vogelius and adapted to surfaces using novel Piola-type mappings. These spaces exactly enforce tangentiality and incompressibility, maintain weak interelement continuity, and achieve optimal-order convergence without requiring superparametric geometry approximations. Second, the project develops a divergence-free unfitted TraceFEM for incompressible viscous surface flows using bulk divergence-free finite element pairs and a modified continuity equation to enforce surface incompressibility exactly, thereby ensuring pressure robustness and avoiding pressure stabilization while retaining the geometric flexibility of TraceFEM. These methods are designed to be accurate, parameter-free, robust, and applicable to high-fidelity simulations in biophysics, materials science, and fluid mechanics. The algorithms will be implemented in open-source software. The project will support graduate and undergraduate training, contribute to PhD dissertation work, and motivate a new graduate course on numerical methods for surface PDEs at the University of Pittsburgh. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2607685 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Michael Neilan | Institution: University of Pittsburgh, PITTSBURGH, PA | Award Amount: $400,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2607685 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2607685.html