Finite element exterior calculus and applications

要模拟、理解、预测并最终控制一个复杂的自然或人为的物理系统，甚至社会系统，第一步通常是对系统进行数学建模。对于各种各样的系统，例如那些基于固体力学、流体力学、量子力学、电磁学、万有引力、声学、热力学、某些随机过程和许多其他的系统，最好的数学模型采用一组偏微分方程的形式。对于结构工程、气候学、生物学或许多其他领域中出现的复杂现代应用，所产生的偏微分方程只能使用快速计算机近似求解。当我们处理新的和更复杂的应用时，在计算机上近似求解这些方程的精确算法的发展仍然是一个巨大的挑战。随着有限元法的发展，偏微分方程的计算机解的最大进步之一是在过去的一个世纪里，有限元法已经成为模拟科学和工程中出现的各种现象的不可或缺的工具。有限元的一个巨大优势在于，它不仅提供了一种开发模拟数值算法的方法，而且还提供了一个评估计算解的准确性的理论框架。该项目旨在开发扩展有限元方法适用性的新算法，并开发新的工具，以便更好地理解有限元算法的性能，特别是允许对其准确性进行精确认证。我们将特别强调弹性的偏微分方程，它描述了固体在重力、载荷和风等力作用下的变形和可能的断裂，以及电磁学的偏微分方程，它被广泛地用于各种建模情况，包括电力传输、光传输、无线电波和磁力。但是所考虑的框架将是非常通用的，并且这些技术将扩展到许多其他应用领域。在许多工业和工程应用中，特别是在最具挑战性的设计应用中，例如飞机、先进建筑和桥梁以及海上石油平台，都需要求解弹性方程的稳健可靠的方法。最近的设计失败，其中一些是灾难性的，已经被追溯到不充分的弹性数值算法。求解电磁学方程的同样可靠的方法是许多现代技术的基础。因此，该项目有可能对国家竞争力和公共安全作出贡献。

The first step in simulating, understanding, predicting, and eventually controlling a complex natural or man-made physical system, or even social system, is often to model the system mathematically. For a wide variety of systems--for example those based on solid mechanics, fluid mechanics, quantum mechanics, electromagnetism, gravitation, acoustics, thermodynamics, certain stochastic processes, and many others--the best mathematical models take the form of a set of partial differential equations. For the complex modern applications that arise in structural engineering, climatology, biology, or many other fields, the resulting partial differential equations can only be solved approximately using fast computers. The development of accurate algorithms to approximately solve these equations on computers remains a tremendous challenge as we tackle new and more complex applications. One of the greatest advances for the computer solution of partial differential equations came in the past century with the development of the finite element method, which has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering. A tremendous asset of finite elements is that they not only provide a methodology to develop numerical algorithms for simulation, but also a theoretical framework in which to assess the accuracy of the computed solutions. This project aims to develop new algorithms which extend the applicability of the finite element method, and to develop new tools which allow for better understanding of the performance of finite element algorithms, and, in particular, allow precise certification of their accuracy. A particular emphasis will be on the partial differential equations of elasticity, which describe the deformation and possible fracture of a solid body subject to forces like gravity, loading, and wind, and on the partial differential equations of electromagnetism, which are used in a wide variety modeling situations involving electric power transmission, transmission of light, radio waves, and magnetism. But the framework considered will be quite general and the techniques will extend to numerous other application areas.Robust and reliable methods for solving the equations of elasticity are needed in many industrial and engineering applications, especially in the most challenging design applications, for example for aircraft, advanced buildings and bridges, and offshore oil platforms. Recent design failures, some of them catastrophic, have been traced to inadequate numerical algorithms for elasticity. Similarly reliable methods for solving the equations of electromagnetism are at the basis of much of modern technology. Thus this project has the potential to contribute to national competitiveness and public safety.