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.

该项目涉及数学物理和几何分析交叉领域中出现的平稳和演化偏微分方程（PDE）系统的近似解。这种方程组，称为几何偏微分方程，出现在广泛的物理和数学问题中。该项目的主要动机之一是爱因斯坦，它对引力波科学至关重要。对于数学分析和计算模拟来说，此类问题最具挑战性的特征之一是底层空间域，它具有潜在复杂流形的结构，而不是 3 空间中的简单形状。此外，该流形的几何形状和拓扑结构都可能随着时间的推移而演变，具体取决于特定的模型。研究成果将对几何分析等数学领域以及天体物理学和广义相对论产生广泛影响。所开发的方法将有助于复杂三维约束非线性动力学模拟数值方法的进步。 PI 开发的模拟技术将为探索天体物理学和相对论以及几何分析等纯数学领域的模型提供强大的工具。参与该项目的两名研究生将由两位研究者共同培养；这将涉及团队所有四名成员之间的定期互动。该项目的主要技术目标是开发一个通用逼近理论框架，以及可靠且可证明收敛的自适应方法，用于一般类非线性几何椭圆的内在离散化黎曼 2 流形和 3 流形上的演化偏微分方程。 虽然此类偏微分方程的解理论在过去三十年中得到了深入研究，但鲁棒数值方法及其相应的近似理论的发展却是最近才取得的进展。迄今为止的大多数方法，例如用于二维问题的表面有限元方法，都是基于利用表面嵌入到 3 维中。对于广义相对论等应用，需要一种不依赖于这种嵌入的存在的更通用的方法。在该项目中，PI 将研究真正内在离散化的发展，这种离散化不使用外在信息来产生离散化，从而允许开发具有任意拓扑的黎曼 2 流形和 3 流形的数值方法。 PI 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来开发基于图集的离散化。为了开发相应的误差分析框架，PI 将利用表面方法的变分犯罪框架，例如基于有限元外微积分的方法。