CAREER: Multilevel Discontinuous Least-Squares Finite Element Methods

对于偏微分方程（PDEs）的数值逼近，需要在逼近特性（精度、一致性、稳定性等）、求解时间（求解器速度、实现效率）和鲁棒性（可扩展性和适用性）之间寻求平衡。为此，本提案的主要目标是开发一种高阶离散化框架，适用于多层次设置中的快速解决技术。该提案的应用重点是由两个核心问题驱动的：血流中的中性粒细胞趋化性和微循环中的细胞力学。这些模型主要由耦合的各向异性扩散-对流-反应方程和耦合的Stokes方程控制。这些复杂方程的高效和有效的数值解是本文工作的核心。这包括一种有效的不连续最小二乘谱元方法的发展，与诸如不连续伽辽金等流行策略的比较，以及在这种情况下使用高阶谱元的集成代数多网格预调节器的发展。最后，为不连续最小二乘法和高阶预处理的进一步研究奠定了理论和计算基础。此外，该项目的一个内在要素是将数值偏微分方程的新方法和迭代方法整合到现有的科学计算课程中，以帮助培养未来的计算科学家。随着物理模型的复杂性和高性能计算环境的速度增长，对底层数学算法的需求也在增长。这个项目的目标是朝着包含整个仿真工具链的更多层的更一般化的数学框架取得进展。大规模计算分析在物理科学的许多领域是一个关键的实验组成部分，然而，计算科学家的工具集是有限的。Olson建议为核心应用开发一种多层近似方法，例如血液中的细胞行为，并通过外展和教育扩大新方法在该领域的更广泛适应。所提出的研究方法促进了与问题物理的一致性，允许自然和可扩展的计算实现，并产生准确和有效的解决方案。该项目将开发多层次方法的新步骤，向更广泛的科学和计算社区传播计算工具，并培训学生和科学家使用这些新兴的计算技术。

For the numerical approximation of partial differential equations (PDEs), a balance is sought between the approximation properties (accuracy, consistency, stability, etc.), the solution time (solver speed, implementation efficiency), and robustness (scalability and applicability). To this end, the principle goal in this proposal is to develop a high-order discretization framework amenable to fast solution techniques in a multilevel setting. The application focus of the proposal is motivated by two core problems: neutrophil chemotaxis in the blood stream and cellular mechanics in microcirculation. Principally, these models are governed by coupled anisotropic diffusion-convection-reaction equations and coupled Stokes equations. The efficient and effective numerical solutions of these complex equations is central to the proposed work. This includes the development of an effective discontinuous least-squares spectral element method, a comparison with popular strategies such as discontinuous Galerkin, and the development of an integrated algebraic multigrid preconditioner for use with high-order spectral elements in this situation. Ultimately, this work establishes a theoretical and computation base for further research in discontinuous least-squares methods and high-order preconditioning. Moreover, an intrinsic element of this project is the integration of new methods in numerical PDEs and iterative methods into the existing scientific computing curriculum to help train future computational scientists.As physical models grow in complexity and high-performance computing environments grow in speed, so do the demands on the underlying mathematical algorithms. The goal of this project is to make progress toward a more generalized mathematical framework that encompasses more layers of the entire simulation tool chain. Large-scale computational analysis is a critical experimental component in many areas of the physical sciences and yet, computational scientists are limited in their tool set. Olson proposes to develop a multilevel approximation method for core applications, such as cellular behavior in the blood stream, and to expand wider adaptation of new methods in the field through outreach and education. The proposed research methodology promotes conformity with the physics of the problem, allows for a natural and extensible computational implementation, and yields an accurate and efficient solution. This project will develop new steps for the multilevel methodology, disseminate the computational tools to the broader scientific and computing community, and train students and scientists on using these emerging computational technologies.