Development and applications of the finite element exterior calculus

本项目将发展、扩展和应用有限元外演算，这是一种设计和验证各种偏微分方程稳定有限元方法的新方法。有限元外部微积分带来了几何学、拓扑学和泛函分析的工具，提供了算法和支持理论，这些理论可以形成解决计算科学中关键问题的基础。虽然最近才发展起来，但有限元外部演算已经被许多研究人员采用，并为重要应用开发了鲁棒和精确的数值方法。一个主要的研究方向是发展时间相关问题的有限元外演算。为此，霍奇热方程和霍奇波动方程将作为基本的模型问题，麦克斯韦电磁学方程和弹性动力学方程是众多应用中的两个。另一个重要的方向是发展立方体网格上的空间和有限元微分形式的复合体，以及研究这种空间在网格单元变形时的近似性质。申请者还将研究有限元外部微积分在求解爱因斯坦方程中的应用，这是广义相对论的基本方程。来自固体力学、流体动力学、电磁学和其他领域的物理系统的计算机模拟，每天以无数种方式应用于各种各样的问题，如飞机的设计、气候的预测和心脏装置的开发。一旦一个物理系统被数学方程系统建模，成功的模拟不仅依赖于强大的计算机硬件，还依赖于数学算法，这些算法可以利用计算机的高速数字运算来获得模型方程的精确解。虽然对于许多重要的方程存在这样的算法，并且已经通过数学分析证明，因此我们可以对结果有信心，但还有许多其他问题尚未发现准确和可证明的算法。这一建议的重点是一种新的方法来开发和分析计算算法的模拟，近年来取得了巨大的成功，模拟涉及固体材料的变形，如汽车车身或建筑梁。该项目的一个主要目标是增加系统的范围，可以准确和自信地模拟。

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.