Finite Element Methods for the Surface Stokes Equation

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The Navier-Stokes system of partial differential equations is widely used to model fluid flows in physical applications. These equations have importance for example in modeling emulsions, foams, and biological membranes. The goal of this project is to computationally solve related equations that are posed on surfaces instead of on flat domains or spaces. For example, membranes of cells can be thought of as fluids that flow and deform, and the surface of such a cell can be modeled using a surface Navier-Stokes system. Constructing accurate and efficient numerical methods for such surface fluid problems involves overcoming some challenges different from those encountered in the well-studied case of fluids on flat domains. Various ways of solving these issues have been proposed in recent years. The project will provide foundational theoretical backing for one major class of such methods and give new insight into its practical properties. The project provides training for graduate students through involvement in the research.Surface finite element methods (SFEM) have grown into an important practical tool for simulations for physical models involving partial differential equations posed on surfaces. Many finite element methods exist for solving scalar elliptic problems on surfaces, but not much work has been done for surface vector Laplace-type operators such as surface (Navier-)Stokes system for modeling fluid flow on surfaces. The main goal of this project is to develop and analyze new finite element algorithms for surface partial differential equations involving the Stokes operator. The first part of the project will focus on algorithms for the stationary (linear) Stokes problem. A new divergence-conforming trace finite element method will be developed. This method will provide a new tool for solving problems involving surface fluid models with coupled bulk effects. In addition, further theoretical analysis will be carried out for both this new algorithm and existing ones, with the focus being mostly on geometric errors which arise in SFEM due to the approximation of the actual surface on which the problem is posed by a discrete counterpart. Their behavior is well understood for scalar elliptic problems, but not for vector Laplace-type operators. Finally, algorithms will be developed and studied for time-dependent Stokes and Navier-Stokes systems on prescribed and evolving surfaces.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.