Numerical Methods for Geometric PDE on Manifolds with Arbitrary Topology

这个项目是关于在数学物理和几何分析交叉产生的平稳和演化偏微分方程（PDE）系统的近似解。这样的方程组，被称为几何偏微分方程，出现在广泛的物理和数学问题中；这个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。这类问题最具挑战性的特征之一，无论是数学分析还是计算模拟，都是潜在的空间域，它具有潜在复杂流形的结构，而不是三维空间中的简单形状。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变化，这取决于特定的模型。研究结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的数值方法的发展。pi产生的模拟技术将为探索天体物理学和相对论中的模型以及一些纯数学领域（如几何分析）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训；这将涉及团队所有四名成员之间的定期互动。本项目的主要技术目标是开发一个通用的近似理论框架，以及可靠且可证明收敛的自适应方法，用于黎曼2-和3-流形上一般非线性几何椭圆和演化PDE的固有离散化。虽然这类PDE的解理论在过去的三十年中得到了深入的研究，但在与相应的近似理论相结合的鲁棒数值方法的发展方面取得的进展是最近的发展。迄今为止的大多数方法，如二维问题的表面有限元方法，都是基于将表面嵌入到三维中。对于像广义相对论这样的应用，需要一种不依赖于这种嵌入存在的更一般的方法。在这个项目中，pi将研究真正的内在离散化的发展，这种离散化不使用外在信息来产生离散化，从而允许在任意拓扑的黎曼2和3流形上发展数值方法。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开发基于地图集的离散化。为了开发相应的误差分析框架，pi将利用曲面方法的变分犯罪框架，例如基于有限元外部演算的方法。

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.