Development and applications of the finite element exterior calculus

这个项目将开发、推广和应用有限元外积分，这是一种设计和验证各种偏微分方程组的稳定有限元方法的新方法。有限元外微积分提供了几何、拓扑和泛函分析等工具，为解决计算科学中的关键问题提供了算法和理论支持。有限元外积分学虽然是最近才发展起来的，但已经被许多研究人员采用，并为重要应用提供了稳健和精确的数值方法的发展。发展含时问题的有限元外积分是今后研究的一个主要方向。为此，霍奇热方程和霍奇波动方程将作为基本的模型问题，麦克斯韦电磁方程和弹性动力学方程是众多应用中的两个。另一个重要方向是发展立方体网格上有限元微分形式的空间和复形，并研究这种空间在网格单元变形时的逼近性质。作者还将研究有限元外微积分在求解爱因斯坦方程中的应用，爱因斯坦方程是广义相对论的基本方程。对物理系统的计算机模拟来自固体力学、流体力学、电磁学等领域，每天都在飞机设计、气候预测和心脏设备的开发等问题上以无数种方式应用。一旦一个物理系统被一个数学方程系统建模，成功的模拟不仅取决于强大的计算机硬件，还取决于数学算法，这些算法可以利用计算机的高速数值运算来获得模型方程的准确解。虽然许多重要的方程都有这样的算法，而且已经通过数学分析得到了验证，因此我们可以对结果有信心，但还有许多其他问题还没有找到准确和可证明的算法。这项建议侧重于开发和分析模拟计算算法的新方法，近年来，该方法在涉及固体材料变形的模拟方面取得了巨大成功，如汽车车身或建筑横梁。这个项目的一个主要目标是增加能够准确和自信地模拟的系统的范围。

This project will develop, extend, and apply the finite element exterior calculus, a new approach to the design and validation of stable finite element methods for a variety of partial differential equations. Finite element exterior calculus brings tools from geometry, topology, and functional analysis to provides both algorithms and supporting theory which can form the foundation on which progress can be made on solving crucial problems in computational science. Although developed only recently, finite element exterior calculus has already been adopted by many researchers and has enabled the development of robust and accurate numerical methods for important applications. A major direction of research which will be pursued is the development of the finite element exterior calculus of time-dependent problems. For this, the Hodge heat equation and the Hodge wave equation will serve as the basic model problems, and Maxwell's equations of electromagnetism and the equations of elastodynamics are two of the numerous applications. Another important direction is the development of spaces and complexes of finite element differential forms on cubical meshes and the study of the approximation properties of such spaces when the elements of the mesh are distorted. The proposer will also study the application of the finite element exterior calculus to solving the Einstein equations, which are the fundamental equations of general relativity.Computer simulation of physical systems coming from solid mechanics, fluid dynamics, electromagnetism, and other areas, are applied in countless ways every day in problems as varied as the design of aircraft, the prediction of climate, and the development of cardiac devices. Once a physical system has been modeled by a system of mathematical equations, successful simulation depends not only on powerful computer hardware but also on mathematical algorithms that can harness the computer's high speed number-crunching to obtain accurate solutions of model's equations. While such algorithms exist for many important equations, and moreover have been certified by mathematical analysis so we can have confidence in the results, there are many other problems for which accurate and certifiable algorithms are yet to be discovered. This proposal focuses on a new approach to the development and analysis of computational algorithsm for simulation that has in recent years achieved great success for simulations involving the deformation of solid materials, like auto bodies or construction beams. A primary goal of this project is to increase the range of systems which can be simulated accurately and confidently.