Numerical Solution of Partial Differential Equations: Algorithms, Analysis, and Applications

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Much of modern technology is dependent on fast and accurate computer simulation of physical phenomena. For example, computer simulations enable the development of high performance vehicles, medical devices, geophysical monitoring and prediction, and countless other applications on which the health and welfare of our society depend. Once such a physical system is 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 effective simulation algorithms exist for many important physical systems, there remain many important applications for which fast and reliable algorithms do not yet exist. Moreover, it is crucial to develop not only the algorithms, but a robust and rigorous understanding of their performance, in order to be able to assess and certify their accuracy and delineate their limitations. This project will develop, improve, and validate simulation algorithms for two important and challenging classes of physical phenomena. The first concerns the propagation of waves through complex and disordered media, such as arises on tiny scales when electrical signals travel through semiconductors, on massive scales when seismic waves travel through the earth, and in many other relevant regimes. The second application area is in the burgeoning field of gravitational wave astronomy, made possible by the recent detection of gravitational waves from a black hole collision, and highly dependent on numerical simulation of Einstein's equations of general relativity. This project will advance the algorithms used to simulate complex physical phenomena modeled by partial differential equations. A key focus of effort will be on the propagation of waves through complex media, as, for example, the conduction of electrons in a semiconductor with impurities. This work will study the remarkable effect known as localization of eigenfunctions, first discovered over 50 years ago in Nobel prize winning research, and with major ramifications for many systems involving wave propagation. Until now, the understanding needed to accurately predict and control localization was lacking, but recent theoretical advances bring it within reach. A major goal of the research will be fast accurate prediction of localization from inexpensive processing of the media. This will then open the way for control of localization: the design of media with desired properties for applications. The second main direction concerns the simulation of Einstein's equations of general relativity. The goal will be to develop structure-preserving finite element methods for the equations, with the potential to greatly improve accuracy, efficiency, and robustness of simulations of general relativity. The work will focus on two types of approaches. First will be methods related to the Regge calculus introduced over 50 yeas ago as a discrete analogue of general relativity, overcoming the low accuracy of Regge calculus by developing a family of high order Regge elements and attacking the instabilities encountered, by using a 3+1 decomposition to separate time-like from space-like behavior. The second approach will be based on a system known as the Einstein-Bianchi equations, which hold the potential for leveraging advances in the numerical simulation of electromagnetic systems for the benefit of numerical relativity.