Numerical Methods for Geometric Partial Differential Equations with Applications in Numerical Relativity

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，具有约束和额外的自由度，出现在广泛的物理和数学问题中；例子包括麦克斯韦方程（或者更普遍的在弯曲背景下的杨-米尔斯方程），爱因斯坦的场方程和其他哈密顿系统。这种系统的初值公式产生了一个受限的演化系统，为了得到唯一的演化，必须用侧条件对其进行扩充。非动态几何偏微分方程（作为约束或其他）本身就很有趣；例子包括Yamabe问题，爱因斯坦方程中的哈密顿和动量约束，以及蒙日-安培方程等。对于数学分析和计算模拟而言，这类问题最具挑战性的特征之一是其潜在的空间域具有具有潜在复杂拓扑结构的流形结构。此外，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。该项目的结果可能会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。本文提出的方法将促进复杂三维约束非线性动力学模拟数值方法的发展。所产生的模拟技术将为探索天体物理学和相对论中的数学和计算模型以及一些纯数学领域（如几何分析）提供强大的工具。本项目为研究生提供研究训练机会。本项目的主要技术目标是为一类包含爱因斯坦方程的几何偏微分方程开发新的离散化技术。重点是建模案例，这些案例对目前用于爱因斯坦方程的最先进的方法和软件提出了特殊的挑战，例如极端质量比双黑洞系统的案例。工具将是近似理论的发展，以及可靠的和可证明收敛的自适应方法，用于黎曼2-和3-流形上的非线性几何偏微分方程的固有离散化。迄今为止的大多数方法，如二维问题的曲面有限元方法，都是基于将曲面嵌入到三维空间中，然后使用线法离散化来分离空间和时间离散。对于像广义相对论这样的应用，需要一种更一般的方法，不依赖于这种嵌入的存在，也不依赖于先验的空间切片。该项目研究了真正的内在离散化的发展，不使用外部信息来产生离散化，以允许在任意拓扑的黎曼2和3流形上发展PDE的数值方法，而不强加先验的离散空间切片。该方法是开发基于地图集的离散化技术和基于显式帐篷俯仰方法或全隐式时空离散化的时空离散化。对于这些方法的设计和分析，研究人员将利用他们的团队和合作者开发的变分犯罪框架来分析在表面上提出的数值方法，并通过使用有限元外部微积分框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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, with both constraints and extra degrees of freedom, appear in a wide range of physical and mathematical problems; examples include Maxwell's equations (or more generally the Yang-Mills equations on a curved background), and Einstein's field equations and other Hamiltonian systems. The initial-value formulation for such systems yields a constrained evolution system which has to be augmented with side conditions in order to get a unique evolution. The non-dynamical geometric PDE (as constraints or otherwise) are of great interest in their own right; examples include the Yamabe problem, the Hamiltonian and momentum constraints in the Einstein equations, and the Monge-Ampere equations, among others. 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 manifold with potentially complicated topology. Moreover, both the geometry and the topology may evolve over time, depending on the particular model. The results of this project have the potential for broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed here will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology produced will provide powerful tools for the exploration of mathematical and computational models in astrophysics and relativity, as well as in some areas of pure mathematics such as geometric analysis. This project provides research training opportunities for graduate students. The primary technical aims of this project are to develop new discretization techniques for a class of geometric PDE that includes the Einstein equations. The emphasis is on modeling cases that present particular challenges for current state-of-the-art methods and software currently used for the Einstein equations, such as the case of extreme mass ration binary black hole systems. The tools will be the development of approximation theory, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of the class of nonlinear geometric PDE on Riemannian 2- and 3- manifolds. 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 three space, and then on use of method-of-lines discretization for separating the space and time discetizations. For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding, and does not on an a priori spatial slicing. This project studies the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods for evolution PDE on Riemannian 2- and 3-manifolds with arbitrary topology and without imposing an a priori discrete spatial slicing. The approach is to develop atlas-based discretization techniques and space-time discretizations based on explicit tent-pitching methods or fully implicit space-time discetizations. For the design of such methods and their analysis, researchers will exploit variational crimes frameworks developed by their team and collaborators for analyzing numerical methods posed on surfaces, and through use of the finite element exterior calculus framework.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.