Numerical Solution of Partial Differential Equations and Applications
These researchers are continuing to study and explore Anderson localization, the localization of eigenfunctions of the Schrodinger equation with a sufficiently random potential. A new theory, which is being developed with a strong interdisciplinary group of collaborators, proposes a way to predict the location of such standing waves without having to solve the expensive eigenvalue problem that determines them. In so doing, the theory clarifies the relationship between the localized wave geometry and the structure of the disordered media. The development of the new theory is being strongly guided by computations.
Another project is pursuing two different new approaches to the numerical solution of the Einstein equations of general relativity and related problems. One approach uses the Einstein-Bianchi formulation in which the Einstein equations take the form of a higher order tensorial variant of Maxwell's equations. This approach requires the researchers to discover and implement a new type of finite elements. The second approach is based on Regge calculus, which approximates curved space time by piecewise flat space with curvature concentrated on lower dimensional structures.
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