Finite Element Methods for the Surface Stokes Equation

偏微分方程的Navier-Stokes系统在物理应用中被广泛用于模拟流体流动。 这些方程例如在模拟乳液、泡沫和生物膜中具有重要性。这个项目的目标是计算解决相关的方程，而不是在平面域或空间的表面。 例如，细胞膜可以被认为是流动和变形的流体，并且这种细胞的表面可以使用表面Navier-Stokes系统来建模。 构建准确和有效的数值方法，这样的表面流体问题涉及克服一些挑战，不同于那些在平坦域上的流体的情况下，所遇到的。 近年来，已经提出了解决这些问题的各种方法。 该项目将为此类方法的一个主要类别提供基础理论支持，并对其实际特性提供新的见解。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）已经发展成为模拟涉及表面上的偏微分方程的物理模型的重要实用工具。 已有许多有限元方法用于求解曲面上的标量椭圆问题，但对于曲面向量Laplace型算子如曲面（Navier-）Stokes系统用于模拟曲面上的流体流动的研究还不多。 这个项目的主要目标是开发和分析新的有限元算法的表面偏微分方程涉及斯托克斯算子。 该项目的第一部分将集中在稳态（线性）斯托克斯问题的算法。 发展了一种新的发散协调迹有限元方法。 该方法将为解决具有耦合体效应的表面流体模型问题提供一种新的工具。 此外，将进行进一步的理论分析，这两个新的算法和现有的，重点主要是在SFEM中出现的几何误差，由于近似的实际表面上的问题是由一个离散的对应。他们的行为是很好理解的标量椭圆问题，但不是向量拉普拉斯型算子。 最后，将在规定的和不断变化的表面上开发和研究与时间相关的斯托克斯和纳维尔-斯托克斯系统的算法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.