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.