Advancements in Divergence-Free Approximations for Incompressible Flow

流体流动模型的精确计算直接影响到各种应用的模拟和预测，如天气和气候、飞机设计等。通常，这样的计算是基于偏微分方程的离散化，这些偏微分方程模拟了底层系统的物理特性。这些近似将误差引入到模拟中，为了进行可靠的模拟和预测，这些误差需要严格量化。该项目将构建精确的、保留结构的计算方案，在流体流动模型的离散水平上精确地执行潜在的物理定律。这个属性导致了高保真度方案，相对于几个模型参数是鲁棒的。该项目的一个特别重点是开发和分析计算算法，以减少几何误差，从而产生可证明的准确答案。此外，该项目旨在提供用户友好的方法；特别是，该项目将把模块整合到当前的计算和开源软件中，以提高可用性、可移植性和扩展能力。本项目将构建Stokes方程和Navier-Stokes方程的有限元方法，这些方法在离散水平上精确地执行无散度约束。这样的离散化有几个理想的特性，例如，关于模型参数的改进的稳定性和误差估计，任何离散化参数的精确守恒特性，以及离散无散度子空间的表征。然而，目前的无散度有限元方法是不切实际的高阶且难以实现。该项目将识别和构建简单的无发散有限元方法，重点是三维设置和可纳入现有有限元软件的方法。在具有曲面边界的域上，对模型参数和几何形状都具有鲁棒性的无发散有限元方法也将被开发和分析。这将包括研究等参无发散有限元方法和虚拟域方法。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。

Accurate computations of fluid flow models have a direct impact on simulations and predictions in various applications, e.g., weather and climate, aircraft design, etc. Typically, such computations are based on discretizations of partial differential equations that model the physics of the underlying system. These approximations introduce errors into the simulations that need to be rigorously quantified in order to make reliable simulations and predictions. This project will construct accurate, structure-preserving computational schemes that exactly enforce the underlying physical laws at the discrete level in fluid flow models. This attribute leads to high fidelity schemes that are robust with respect to several model parameters. A particular focus of the project is to develop and analyze computational algorithms that reduce geometric error and thus yield provably accurate answers. In addition, the project aims to provide user-friendly methods; in particular, the project will incorporate modules into current computational and open source software to improve usability, portability, and outreach.This project will construct finite element methods for the Stokes and Navier-Stokes equations that exactly enforce the divergence-free constraint at the discrete level. Such discretizations have several desirable properties, for example, improved stability and error estimates with respect to model parameters, exact conservation properties for any discretization parameter, and characterizations of discrete divergence-free subspaces. However, current divergence-free finite element methods are impractically high-order and arduous to implement. This project will identify and construct simple divergence-free finite element methods with an emphasis on the three-dimensional setting and on methods that can be incorporated into current finite element software. Divergence-free finite element methods on domains with curved boundaries that are robust with respect to both model parameters and the geometry will also be developed and analyzed. This will consist of studying both isoparametric divergence-free finite element methods and fictitious domain approaches.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.