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

协作研究：具有非线性约束和规范自由度的离散几何偏微分方程的有限元方法

• 批准号: 0715146
• 负责人: Michael Holst
• 金额: $ 18万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0715146&HistoricalAwards=false
• 关键词: Collaborative; Research; Finite; Element; Methods

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.

该项目涉及数学物理和几何分析交叉点中出现的某些演化偏微分方程组 (PDE) 的近似解。此类方程组称为几何 PDE，具有约束和规范自由度，广泛出现在一系列物理和数学问题；例子包括麦克斯韦方程组（或者更一般地说，弯曲背景上的杨-米尔斯方程组）、爱因斯坦场方程组和其他具有无限维对称群的哈密顿系统。这种系统的柯西公式产生了一个约束演化系统，必须用规范固定条件来增强该系统，以获得独特的演化向量场。该项目将涉及构建有限元离散化来求解此类几何偏微分方程系统；将分析处理具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（其中数值解在一定数量的时间步之后被投影回约束流形），使用特殊的有限元自动求解线性化约束，从而保持约束流形的分段线性近似，最后是仅控制约束而不是强制约束的最小二乘方法。特别是，至少对于线性化方程，将导出保证数值解收敛于连续统解的稳定性结果。 将导出后验误差估计，用于研究离散化的属性，并构建自适应方法。该项目涉及新的数学和计算技术的设计、开发和实施，用于解决多尺度中的一类重要的、具有挑战性的和紧迫的数学问题以及多物理场建模和仿真。 所开发的技术将通过更可靠和准确地模拟引力塌缩、变形旋转黑洞的非线性稳定性、双黑洞碰撞和由于最近建造了引力波探测器，例如路易斯安那州和华盛顿州 NSF 资助的 LIGO 设备，这些问题中的大多数目前引起了人们的极大兴趣。 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。 这里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，所产生的技术将为探索天体物理学和相对论以及纯数学的某些领域的模型提供强大的工具，例如几何分析。