Finite element exterior calculus and applications

模拟、理解、预测并最终控制复杂的自然或人造物理系统，甚至社会系统的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，引力，声学，热力学，某些随机过程，以及许多其他-最好的数学模型采取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的现代应用，所产生的偏微分方程只能使用快速计算机近似求解。 在计算机上近似求解这些方程的精确算法的开发仍然是一个巨大的挑战，因为我们要处理新的和更复杂的应用。 在过去的世纪，随着有限元法的发展，偏微分方程的计算机解法取得了最大的进步，有限元法已成为模拟科学和工程中出现的各种现象的不可缺少的工具。 有限元的一个巨大的资产是，他们不仅提供了一种方法来开发模拟的数值算法，而且还提供了一个理论框架，以评估计算的解决方案的准确性。 该项目旨在开发新的算法，扩展有限元法的适用性，并开发新的工具，使更好地了解有限元算法的性能，特别是，允许精确的认证其准确性。 一个特别的重点将是弹性偏微分方程，它描述了变形和可能的断裂的固体受到像重力，负荷，和风的力量，并在偏微分方程的电磁，这是用于各种建模的情况下，涉及电力传输，光，无线电波的传输，和磁性。 但是，所考虑的框架将是相当普遍的，技术将扩展到许多其他的应用领域，鲁棒性和可靠的方法来解决弹性方程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，先进的建筑物和桥梁，海上石油平台。 最近的设计失败，其中一些是灾难性的，已被追溯到弹性数值算法不足。 同样，求解电磁方程的可靠方法是许多现代技术的基础。 因此，该项目有可能促进国家竞争力和公共安全。

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.