Structure-Preserving Finite Element Methods for Incompressible Flow on Smooth Domains and Surfaces

This project will develop numerical methods for solving equations modeling incompressible flow with applications such as predicting weather patterns, designing aircraft, and simulating blood flow. The primary objective is to design and analyze finite element methods (FEMs) that maintain key physical properties at the discrete level, specifically the conservation of mass and incompressibility of the fluid. Such FEMs possess several advantages over existing methods, including superior accuracy, robustness with respect to model parameters, and exact enforcement of multiple conservation laws. However, this class of FEMs is limited in their ability to handle various equation types and geometric domains. This research will overcome these limitations by developing new robust FEMs for incompressible fluid models that can be applied to a wider range of problems. It will focus on two main areas: improving existing FEMs for fluid flow on smooth domains and developing new FEMs for fluid flow on surfaces. The project will provide training opportunities for both undergraduate and graduate students. The research consists of two integrated components. The first focuses on developing structure-preserving FEMs for the Navier-Stokes equations on smooth domains. The investigator will use non-standard applications of divergence-conforming diffeomorphisms to construct robust schemes, with a focus on high-order schemes in two and three dimensions. The second component involves applying these structure-preserving schemes towards surface partial differential equation models of incompressible flow. The investigator will extend isoparametric schemes for smooth Euclidean domains to construct divergence-free surface FEMs based on standard nodal spaces that do not require extrinsic user-defined stabilization/penalization terms. Additionally, the investigator will extend the Finite Element Exterior Calculus framework to build discrete surface Stokes complexes with respect to approximate geometries, which will provide insight into the construction of robust FEMs based on the velocity-pressure formulation and lead to primal discretizations for the surface stream function. 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.