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

这个项目关注的是在数学物理和几何分析的交叉点上产生的静态和演化偏微分方程（PDE）系统的近似解。 这样的方程组，被称为几何偏微分方程，既有约束又有额外的自由度，出现在广泛的物理和数学问题中;例子包括麦克斯韦方程（或更一般的弯曲背景下的杨-米尔斯方程），爱因斯坦场方程和其他哈密顿系统。这样的系统的初始值制定产生一个约束的发展系统，必须增加侧条件，以获得一个独特的发展。 非动力学几何偏微分方程（作为约束或其他）本身就有很大的兴趣;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Monge-Ampere方程等。 这类问题的最具挑战性的特点之一，数学分析和计算模拟，是潜在的空间域具有潜在的复杂拓扑结构的流形的结构。 此外，取决于特定模型，几何形状和拓扑结构都可以随时间演变。 该项目的结果有可能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。 本文所提出的方法将有助于复杂三维约束非线性动力学数值模拟方法的发展。 所产生的模拟技术将为天体物理学和相对论中的数学和计算模型的探索以及几何分析等纯数学领域提供强有力的工具。该项目为研究生提供了研究培训机会。这个项目的主要技术目标是开发新的离散化技术的一类几何偏微分方程，其中包括爱因斯坦方程。 重点是建模的情况下，目前的最先进的方法和软件目前用于爱因斯坦方程，如极端质量比的二元黑洞系统的情况下，提出了特别的挑战。 这些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PDE类的内在离散化。 迄今为止，大多数的方法，如二维问题的表面有限元方法，是基于开发的表面嵌入到三个空间，然后使用线的方法离散分离的空间和时间discetizations。 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。 该项目研究真正内在的离散化的发展，不使用外部信息来产生离散化，以允许发展数值方法，用于在具有任意拓扑结构的黎曼2-和3-流形上发展PDE，而不施加先验的离散空间切片。 该方法是开发基于地图集的离散技术和时空离散的基础上显式帐篷搭帐篷的方法或完全隐式的时空discetizations。 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表面上的数值方法，并通过使用有限元外部微积分框架。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.