Finite Element Methods for Incompressible Flow Yielding Divergence-Free Approximations

该研究项目旨在开发实用的、结构保持的计算方法来模拟不可压缩流。 具有不可压缩流的系统在计算流体动力学（CFD）中是普遍存在的，并且出现在若干工程和生物模型中，例如用于飞机设计、天气预测和脂质膜建模的那些。 该项目的主要重点是确定兼容的计算方法，继承在离散水平的内在结构和不变量的系统研究。 这种离散化产生的数值方案具有增强的稳定性和守恒性质，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的研究人员和从业人员受益，并在需要忠实近似的应用中非常有用。 除了这些数值格式和算法的构建，该项目还将分析方法的稳定性和收敛性，理论分析将为未来开发具有更高效率、精度和保真度的数值格式提供见解。该项目的核心是开发精确执行无发散约束的不可压缩Stokes和Navier-Stokes方程的有限元方法。 这种方案有几个理想的属性，包括改进的误差估计，增强的长期稳定性和精度的时间步进计划，明确的特征和本地基地的发散自由子空间，耦合方法的精度与投影方法相结合。该项目的具体目标包括：㈠为三维Navier-Stokes方程建立稳定的有限元对，以严格执行精确的质量守恒; ㈡将单纯Bernstein-Bezier理论（多项式样条的一种强有力的分析工具）应用于混合有限元框架; ㈢为轴对称流体模型建立稳健的、在计算上有吸引力的方法;以及（iv）通过将流体流动模型并入有限元外部微积分框架中来在表面上开发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.