Collaborative Research: Finite Element Methods for Discretizing Geometric PDEs with Nonlinear Constraints and Gauge Freedom

这个项目是关于数学物理和几何分析交叉产生的某些演化偏微分方程组的近似解。这种方程组被称为几何偏微分方程组，既有约束又有规范自由度，出现在广泛的物理和数学问题中；例如麦克斯韦方程(或者更常见的是弯曲背景上的杨-米尔斯方程)，以及爱因斯坦的场方程和其他具有无限维对称群的哈密顿系统。这类系统的柯西公式产生了一个约束演化系统，它必须增加固定规范的条件以得到唯一的演化矢量场。该项目将涉及为求解这类几何偏微分方程组而构造有限元离散；我们将分析处理带有约束的演化系统的各种技术，包括使用变分技术的约束投影(其中数值解在一些时间步长后被投影回约束流形)，使用特殊的有限元自动地求解线性化的约束，从而保持对约束流形的近似线性逼近，以及最后的最小二乘方法，它只控制约束而不是强制约束。特别地，至少对于线性化的方程，将得到保证数值解收敛于连续统解的稳定性结果。后验误差估计将用于研究离散化的性质和建立自适应方法。该项目涉及设计、开发和实施新的数学和计算技术，用于解决多尺度和多物理建模和模拟中的一大类重要的、具有挑战性的和紧迫的数学问题。所开发的技术将导致在相对论天体物理学等科学领域获得新知识，使诸如引力坍塌、变形旋转黑洞的非线性稳定性、双黑洞碰撞以及引力辐射的产生和发射等现象的更可靠和更准确的模拟成为可能。由于最近在路易斯安那州和华盛顿州建造了引力波探测器，如美国国家科学基金会资助的LIGO装置，这些问题中的大多数目前都引起了极大的兴趣。该项目的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进步，所产生的技术将为探索天体物理和相对论中的模型以及几何分析等纯数学领域的模型提供强有力的工具。

This project is concerned with the approximate solution of certainsystems of evolution partial differential equations (PDE) arising atthe intersection of mathematical physics and geometric analysis.Such systems of equations, known as Geometric PDE, with both constraintsand gauge degrees of freedom, appear in a wide range of physical andmathematical problems; examples include Maxwell's equations (or moregenerally the Yang-Mills equations on a curved background), andEinstein's field equations and other Hamiltonian systems with aninfinite-dimensional symmetry group. The Cauchy formulation for such systems yields a constrained evolution system which has to be augmented with gauge-fixing conditions in order to get a unique evolution vector field.The project will involve constructing finite element discretizationsfor solving such geometric PDE systems; various techniques for dealingwith evolution systems with constraints will be analyzed, includingconstraint-projection using variational techniques (where the numericalsolution is projected back to the constraint manifold aftersome number of time steps), the use of special finite elements whichautomatically solve the linearized constraints and thereby remain on apiecewise-linear approximation to the constraint manifold, and finallyleast-squares approaches which only control the constraints rather thanenforce them. In particular, stability results guaranteeing convergenceof the numerical solution to the continuum solution will be derived,at least for the linearized equations. A posteriori error estimates will be derived for studying properties of the discretizations, and for building adaptive methods.This project involves the design, development, and implementation ofnew mathematical and computational techniques for solving a large class of important, challenging, and pressing mathematical problems inmultiscale and multiphysics modeling and simulation. The techniques developed will lead to the aquisition of new knowledge in areas of science such as relatistic astrophysics, by making possible more reliable and accurate simulations of phenomena such as gravitational collapse, nonlinear stability of deformed rotating black holes, binary black hole collision, and the production and emission of gravitational radiation.Most of these problems are currently of great interest due to the recent construction of gravitational wave detectors such as the NSF-fundedLIGO devices in Lousiana and Washington. The results from this project will have a 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, and the technology produced 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.