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

该项目将开发数值方法来解决不可压缩流的方程建模，应用于预测天气模式、设计飞机和模拟血液流动。主要目标是设计和分析在离散水平上保持关键物理特性的有限元方法（fem），特别是质量守恒和流体的不可压缩性。与现有的方法相比，这种fem具有几个优点，包括更高的精度，相对于模型参数的鲁棒性，以及对多个守恒定律的精确执行。然而，这类fem在处理各种方程类型和几何域的能力方面受到限制。这项研究将通过为不可压缩流体模型开发新的鲁棒fem来克服这些限制，这些模型可以应用于更广泛的问题。它将集中在两个主要领域：改进现有的光滑域流体流动fem和开发新的表面流体流动fem。该项目将为本科生和研究生提供培训机会。本研究由两部分组成。第一个重点是在光滑域上建立Navier-Stokes方程的保结构有限元。研究者将使用非标准应用的发散符合的差分同态来构建鲁棒方案，重点是在二维和三维的高阶方案。第二部分涉及将这些结构保持方案应用于不可压缩流动的表面偏微分方程模型。研究者将扩展光滑欧几里得域的等参数方案，以构建基于标准节点空间的无发散曲面fem，不需要外部用户定义的稳定/惩罚项。此外，研究人员将扩展有限元外部微积分框架，以建立关于近似几何的离散表面Stokes复合体，这将为基于速度-压力公式的鲁棒fem的构建提供见解，并导致表面流函数的原始离散化。该奖项反映了美国国家科学基金会的法定使命，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。

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.