Computer simulation of physical systems involving the deformation of solid materials, the motion of fluids, electromagnetism, and other phenomena is applied in countless ways every day in areas as varied as geophysics, medicine, and civil engineering. 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 to obtain accurate solutions of the model's equations. While such algorithms exist for many important physical systems -- and moreover have been certified by mathematical analysis so that we can have confidence in the results -- there remains substantial room for improvement, with large potential payoffs. Even more important is the need to develop accurate, fast, and certifiable algorithms for important applications for which they do not yet exist. This project focuses on a new approach to the development and analysis of computational algorithms for simulation that has in recent years achieved great success for simulations involving the deformation of solid materials ranging from auto bodies to bones. A primary goal of the project is to increase the range of systems that can be simulated accurately and confidently. One emphasis will be on complex materials that combine solid and fluid aspects together, such as the tissue in the human brain, or the saturated subsurface soil and sand in which groundwater flows. A second emphasis will be on the simulation of gravity on an astrophysical scale, which is at the heart of a new class of astronomical observatories.
The Principal Investigator will devise, improve, and validate algorithms for the computer simulation of complex physical phenomena modeled by partial differential equations. The algorithms to be developed and studied are finite element methods, which are an indispensable tool for simulation of a wide variety of phenomena in science and engineering, with the tremendous asset 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 computed solutions, and thus the possibility to develop validated methods. The present work will be based on a theory called finite element exterior calculus, initiated by the Principal Investigator and developed over the past decade, which has greatly enhanced our understanding of finite element methods and extended the range of problems for which validated finite element methods can be used. A major direction of the research will be the extension of newly discovered methods for linearly elastic materials to more complex materials such as nonlinear, viscoelastic, and poroelastic materials. A second major direction will be the development of finite element methods suited to the Einstein equations of numerical relativity. In a third direction the Principal Investigator will participate as the computational scientist in a mathematics/physics theoretical/computational team seeking to elucidate the important but poorly understood phenomenon of localization of eigenfunctions in disordered media. Although this phenomenon, known as Anderson localization, was discovered 50 years ago in Nobel prize winning research, we still lack the understanding needed to accurately predict and control it.
