Finite Element Methods for Incompressible Flow Yielding Divergence-Free Approximations

该研究项目旨在开发实用的、保结构的计算方法来模拟不可压缩流动。不可压缩流动系统在计算流体力学(CFD)中普遍存在，并出现在一些工程和生物模型中，如用于飞机设计、天气预报和脂膜模拟的模型。该项目的主要重点是确定在离散水平上继承所研究系统的内在结构和不变量的兼容计算方法。这样的离散化产生了具有增强的稳定性和守恒性的数值格式，从而产生物理上相关的和精确的近似。预计该项目的结果将直接惠及CFD领域的研究人员和从业者，并在需要忠实近似的应用中发挥作用。除了构建这些数值格式和算法外，该项目还将分析方法的稳定性和收敛；理论分析将为未来提高效率、精度和保真度的数值格式的发展提供洞察力。该项目的中心是开发用于精确执行无散度约束的不可压缩Stokes方程和Navier-Stokes方程的有限元方法。这类格式具有一些理想的性质，包括改进的误差估计、增强的时间步进格式的长期稳定性和精度、无散子空间的显式刻画和局部基，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)发展稳定的三维Navier-Stokes方程的有限元对，加强严格的质量守恒；(Ii)将单纯的Bernstein-Bezier理论，一个强大的多项式样条分析工具，应用于混合有限元框架；(Iii)构建稳健且计算有吸引力的轴对称流体模型方法；以及(Iv)通过将流体流动模型结合到有限元外部微积分框架中，在表面上建立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.