Advancements in Divergence-Free Approximations for Incompressible Flow

流体流动模型的精确计算对各种应用中的模拟和预测具有直接影响，例如，天气和气候、飞行器设计等。典型地，这样的计算基于对基础系统的物理特性建模的偏微分方程的离散化。这些近似值将误差引入到需要严格量化的模拟中，以便进行可靠的模拟和预测。这个项目将构建精确的，结构保持的计算方案，在流体流动模型的离散水平上精确地执行基本的物理定律。这个属性导致高保真度的计划，是强大的几个模型参数。该项目的一个特别重点是开发和分析计算算法，减少几何误差，从而产生可证明准确的答案。此外，该项目旨在提供用户友好的方法;特别是，该项目将把模块纳入当前的计算和开放源码软件，以提高可用性、可移植性和推广范围。该项目将为Stokes和Navier-Stokes方程构建有限元方法，在离散水平上严格执行无发散约束。这样的离散化有几个理想的属性，例如，改进的稳定性和误差估计相对于模型参数，精确的保守性的任何离散化参数，和离散无发散子空间的特征。 然而，目前的无发散有限元方法是不切实际的高阶和艰巨的实现。本项目将确定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当前有限元软件的方法。还将开发和分析具有弯曲边界的域上的无发散有限元方法，这些方法在模型参数和几何形状方面都是鲁棒的。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.