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

该项目将开发数值方法，用于求解模拟不可压缩流动的方程，并应用于预测天气模式、设计飞机和模拟血液流动。其主要目标是设计和分析在离散水平上保持关键物理性质的有限元方法(FEMS)，特别是流体的质量守恒和不可压缩。这种FEMS与现有方法相比具有几个优点，包括较高的精度、对模型参数的稳健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上是有限的。这项研究将通过开发适用于更广泛问题的不可压缩流体模型的新的健壮的FEMS来克服这些限制。它将集中在两个主要领域：改进现有的光滑区域流体流动的FEMS和开发新的表面流体流动的FEMS。该项目将为本科生和研究生提供培训机会。这项研究包括两个综合部分。第一个重点是发展光滑区域上的N-S方程的保结构有限元方法。研究人员将使用散度协调差同胚的非标准应用来构造健壮的格式，重点是二维和三维的高阶格式。第二部分涉及将这些结构保持格式应用于不可压缩流动的表面偏微分方程组模型。研究人员将扩展光滑欧氏区域的等参格式，以构造基于不需要外部用户定义的稳定化/惩罚项的标准节点空间的无散度曲面有限元模型。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斯托克斯复合体，这将深入了解基于速度-压力公式的健壮FEMS的构建，并导致表面流函数的原始离散化。该奖项反映了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。

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.