Numerical Methods for Geometric PDE on Manifolds with Arbitrary Topology

任意拓扑流形上几何偏微分方程的数值方法

• 批准号: 1620366
• 负责人: Michael Holst
• 金额: $ 21.45万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620366&HistoricalAwards=false
• 关键词: Numerical; Methods; Geometric; PDE; Manifolds

This project is concerned with the approximate solution of systems of stationary and evolution partial differential equations (PDE) arising at the intersection of mathematical physics and geometric analysis. Such systems of equations, known as Geometric PDE, appear in a wide range of physical and mathematical problems; one of the primary motivations for this project is the Einstein, which are of central importance to gravitational wave science. One of the most challenging features of this class of problems, for both mathematical analysis and computational simulation, is the underlying spatial domain which has the structure of a potentially complicated manifold rather than a simple shape in 3-space. Moreover, both the geometry and the topology of this manifold may evolve over time, depending on the particular model. The research results will have a broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology the PIs produce will provide powerful tools for the exploration of models in astrophysics and relativity as well as in some areas of pure mathematics such as geometric analysis. The two graduate students involved in the project will be co-trained by both investigators; this will involve regular interaction between all four members of the team.The primary technical aim of this project is to develop a general approximation theory framework, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of a general class of nonlinear geometric elliptic and evolution PDE on Riemannian 2- and 3-manifolds. While the solution theory for this class of PDE has been intensively studied over the last thirty years, progress on the development of robust numerical methods with a corresponding approximation theory has been a more recent development. Most of the approaches to date, such as surface finite element methods for two-dimensional problems, are based on exploiting the embedding of the surface into 3-dimension. For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding. In this project, the PIs will study the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods on Riemannian 2- and 3-manifolds with arbitrary topology. The PIs' approach is to develop an atlas-based discretization using techniques such as the multi-cube framework and the local simplex approximation techniques developed by the project team. To develop a corresponding error analysis framework, the PIs will exploit the variational crimes framework for methods in surfaces, such as methods based on finite element exterior calculus.

该项目与在数学物理学和几何分析的交集中产生的固定和进化部分微分方程（PDE）系统的近似解决方案有关。这样的方程系统（称为几何PDE）出现在各种物理和数学问题中。该项目的主要动机之一是爱因斯坦，这对于引力波科学至关重要。对于数学分析和计算模拟而言，这类问题的最具挑战性特征之一是基本的空间域，它具有潜在的复杂流形的结构，而不是3个空间中的简单形状。此外，根据特定模型的不同，该流形的几何形状和拓扑都可能随着时间的流逝而发展。研究结果将对数学领域（例如几何分析）以及天体物理学和总体相对论产生广泛的影响。开发的方法将有助于进步数值方法，用于复杂的三维约束非线性动力学模拟。 PIS生产的模拟技术将为探索天体物理学和相对论模型以及在纯数学（例如几何分析）的某些领域提供强大的工具。参与该项目的两名研究生将由两位调查人员共同培训。这将涉及团队的所有四个成员之间的定期互动。该项目的主要技术目的是开发一个一般近似理论框架，以及可靠且可靠的自适应方法，以对Riemannian 2-和3-manifolds上的一般非线性几何椭圆形和进化PDE进行内在离散化。 尽管在过去的三十年中，对这类PDE的解决方案理论进行了深入的研究，但具有相应近似理论的强大数值方法的发展的进展一直是最新的发展。迄今为止，大多数方法（例如用于二维问题的表面有限元方法）是基于将表面嵌入到三维中的基础。对于诸如一般相对论之类的应用程序，需要一种更通用的方法，不依赖于这种嵌入的存在。在该项目中，PI将研究真正固有的离散化的开发，这些离散化不使用外部信息来产生离散化，以允许开发有关具有任意拓扑的Riemannian 2和3个manifords的数值方法。 PIS的方法是使用项目团队开发的多立方体框架和局部单纯形近似技术等技术开发基于ATLAS的离散化。为了开发相应的误差分析框架，PIS将利用表面中方法的变异犯罪框架，例如基于有限元外部计算的方法。