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

本项目研究数学物理与几何分析交叉产生的某些发展偏微分方程组的近似解，这类方程组称为几何偏微分方程，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，为了得到一个唯一的演化向量场，必须用规范固定条件对其进行增广。将分析用于处理具有约束的进化系统的各种技术，包含约束投影变分法（其中数值解在一定数量的时间步长之后被投影回约束流形），使用特殊的有限元自动解决线性化的约束，从而保持对约束流形的逐点线性近似，最后使用最小二乘方法，只控制约束，而不是强制约束。特别是，稳定性的结果，保证收敛的数值解的连续性解决方案将推导出，至少对于线性化方程。 后验误差估计将用于研究离散化的性质，并用于建立自适应方法。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场建模和模拟中的一大类重要，具有挑战性和紧迫的数学问题。 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的知识，使引力坍缩、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。由于最近的建设，引力波探测器，如路易斯安那州和华盛顿的NSF资助的LIGO设备。 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的影响。 这里开发的方法将有助于复杂的三维约束非线性动力学模拟的数值方法的进步，所产生的技术将为天体物理学和相对论以及一些纯数学领域（如几何分析）的模型探索提供强大的工具。

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.