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.