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

该项目将开发用于求解方程的数值方法，模拟不可压缩流，并应用于预测天气模式，设计飞机和模拟血流。 主要目标是设计和分析有限元方法（FEM），在离散水平上保持关键的物理特性，特别是流体的质量守恒和不可压缩性。与现有方法相比，这种有限元法具有几个优点，包括上级精度，相对于模型参数的鲁棒性，以及多个守恒律的精确执行。然而，这类有限元有限的能力，以处理各种方程类型和几何域。这项研究将克服这些限制，开发新的强大的有限元不可压缩流体模型，可应用于更广泛的问题。它将集中在两个主要领域：改进现有的有限元流体流动的光滑域和开发新的有限元流体流动的表面。该项目将为本科生和研究生提供培训机会。第一个重点是发展保结构有限元法的Navier-Stokes方程光滑域。研究人员将使用非标准的应用程序的发散一致的同构，以构建强大的计划，重点是在二维和三维的高阶计划。第二部分涉及应用这些结构保持计划对表面偏微分方程模型的不可压缩流。研究人员将扩展等参计划光滑欧几里德域构建无发散表面有限元模型的基础上，不需要外部用户定义的稳定/惩罚条款的标准节点空间。此外，研究人员将扩展有限元外部微积分框架，以建立离散表面斯托克斯复杂的近似几何形状，这将提供深入了解的基础上的速度，鲁棒有限元模型的建设-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。

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.