Project abstract for group arnoldd

Numerical Solution of Partial Differential Equations and Applications

These researchers are continuing to study a newly developed approach to predicting and explaining Anderson localization, the localization of eigenfunctions of the Schrodinger equation with a sufficiently random potential. The 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 researchers recently tested the new approach against direct eigenvalue calculations on Itasca. They are pursuing two different new approaches to the numerical solution of the Einstein equations of general relativity. They are also developing new methods for a posteriori error estimation and for the numerical approximation of problems with Robin and impedance boundary conditions. All the computations are being performed with the FEniCS software environment, which allows highly efficient use of Itasca resources while providing a high-level Python programming interface.