Finite Element Methods for the Surface Stokes Equation

在物理应用中，Navier-Stokes偏微分方程组被广泛用于模拟流体流动。例如，这些方程在乳状液、泡沫和生物膜的建模中具有重要意义。这个项目的目标是通过计算来求解曲面上的相关方程，而不是平面区域或空间上的方程。例如，细胞膜可以被认为是流动和变形的流体，这样的细胞的表面可以使用表面Navier-Stokes系统来建模。对于这类表面流体问题，建立准确而有效的数值方法需要克服一些挑战，这些挑战不同于在平坦区域上流体的充分研究情况中所遇到的那些挑战。近年来，人们提出了解决这些问题的各种方法。该项目将为此类方法中的一大类提供基础理论支持，并对其实用性质提供新的见解。该项目通过参与研究为研究生提供培训。表面有限元方法(SFEM)已发展成为模拟涉及表面偏微分方程组的物理模型的重要实用工具。求解曲面上的标量椭圆型问题的有限元方法有很多，但对于曲面向量拉普拉斯算子，如用于模拟曲面上流体流动的曲面(Navier-Stokes)系统，所做的工作还很少。这个项目的主要目标是开发和分析涉及Stokes算子的表面偏微分方程组的新的有限元算法。项目的第一部分将集中在静态(线性)Stokes问题的算法上。提出了一种新的散度协调迹线有限元方法。该方法为求解具有耦合整体效应的地表流体模型问题提供了新的工具。此外，还将对新算法和现有算法进行进一步的理论分析，重点讨论由于离散对应面对实际曲面的逼近而产生的几何误差。它们的行为对于标量椭圆型问题是很好理解的，但对于向量拉普拉斯型算子却不是。最后，将在规定的和不断变化的表面上开发和研究依赖时间的Stokes和Navier-Stokes系统的算法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.