Federal Grant Award Summary

The University of Pittsburgh received a $400,000 Project Grant from the National Science Foundation's Division of Mathematical Sciences (CFDA 47.049) effective July 1, 2026, through June 30, 2029, to develop novel surface and trace finite element methods for simulating incompressible fluid flows on curved surfaces. The research delivers next-generation mathematical and algorithmic computational tools designed to accurately model fluid dynamics in biotechnology, biomedical engineering, and advanced materials applications—including biological membranes, drug-delivery vesicles, lung fluid coatings, biofilms, and complex fluid interfaces. The primary deliverables include structure-preserving finite element methods that exactly preserve key physical laws such as mass conservation, eliminating reliance on artificial stabilization parameters that compromise robustness and predictive capability in existing simulation models.

The project advances two complementary computational methodologies: (1) construction of divergence-free and tangential surface finite element spaces based on Scott-Vogelius pairs adapted to curved surfaces using novel Piola-type mappings that maintain optimal convergence without superparametric geometry approximations, and (2) development of a divergence-free unfitted TraceFEM (Trace Finite Element Method) for incompressible viscous surface flows that enforces surface incompressibility exactly. These improved simulation tools will enhance the reliability and accuracy of computational modeling critical to biotechnology research and innovation while reducing dependence on ad hoc numerical tuning, thereby supporting federal objectives in advancing mathematical sciences and strengthening the nation's scientific enterprise.