Finite Element Methods for Incompressible Flow Yielding Divergence-Free Approximations

该研究项目旨在开发实用的、保留结构的计算方法来模拟不可压缩流动。不可压缩流系统在计算流体动力学（CFD）中无处不在，并且出现在一些工程和生物模型中，例如用于飞机设计、天气预报和脂膜建模的模型。该项目的主要重点是确定兼容的计算方法，这些方法在离散水平上继承所研究系统的固有结构和不变量。这种离散化产生具有增强稳定性和守恒特性的数值格式，从而产生物理相关和准确的近似。预计该项目的结果将直接使CFD的研究人员和实践者受益，并在需要可靠近似的应用中很有用。除了构建这些数值格式和算法外，本项目还将分析这些方法的稳定性和收敛性；理论分析将为未来数值格式的发展提供见解，以提高效率，精度和保真度。该项目集中于开发不可压缩Stokes和Navier-Stokes方程的有限元方法，这些方法精确地执行无散度约束。这种方案具有几个理想的特性，包括改进的误差估计，提高时间步进方案的长期稳定性和准确性，明确的无散度子空间的特征和局部基，以及与投影方法结合时的耦合方法精度。该项目的具体目标包括：(i)为三维Navier-Stokes方程开发稳定的有限元对，这些有限元对强烈执行精确的质量守恒；（ii）将简化Bernstein-Bezier理论（一种强大的多项式样条分析工具）应用于混合有限元框架；（iii）为轴对称流体模型构建稳健且计算上有吸引力的方法；(4)在有限元外微积分框架中引入流体流动模型，建立表面上稳定的混合有限元对。

This research project aims to develop practical, structure-preserving computational methods to simulate incompressible flow. Systems with incompressible flow are ubiquitous in computational fluid dynamics (CFD) and arise in several engineering and biological models such as those used in aircraft design, weather prediction, and lipid membrane modeling. The main focus of the project is to identify compatible computational methods that inherit at the discrete level the intrinsic structure and invariants of the systems under study. Such discretizations produce numerical schemes with enhanced stability and conservation properties, resulting in physically relevant and accurate approximations. It is anticipated that the results of the project will directly benefit researchers and practitioners in CFD and be useful in applications that require faithful approximations. In addition to the construction of these numerical schemes and algorithms, the project will analyze the stability and convergence of the methods; the theoretical analysis will provide insight for future development of numerical schemes with improved efficiency, accuracy, and fidelity.The project is centered on developing finite element methods for the incompressible Stokes and Navier-Stokes equations that enforce the divergence-free constraint exactly. Such schemes have several desirable properties, including improved error estimates, enhanced long-time stability and accuracy of time-stepping schemes, explicit characterizations and local bases of divergence-free subspaces, and coupled-method accuracy when combined with projection methods. Specific objectives of this project include (i) developing stable finite element pairs for Navier-Stokes equations in three dimensions that strongly enforce exact conservation of mass; (ii) applying simplicial Bernstein-Bezier theory, a powerful analytical tool for polynomial splines, to the mixed finite element framework; (iii) constructing robust and computationally attractive methods for axisymmetric fluid models; and (iv) developing stable mixed finite element pairs on surfaces by incorporating fluid flow models in a finite element exterior calculus framework.