This project is devoted to finding solutions to the two-dimensional eigenvalue problems with the Schrodinger operator as it appears in the Kohn-Sham equation. Algorithms allowing an efficient solution for the two-dimensional Schrodinger operator with variable coefficients will be developed using multigrid methods for solving partial differential equations and multiscale eigenfunction representation. The Kohn-Sham equations are used to calculate electronic structures of atoms, the number and distribution of electrons about a central nucleus. And, while a complete description of the electronic structure of an atom would include more than just the number and spatial distribution of electrons within the atom, most of the rest of information about the electronic structure is dictated by these two properties. Once the spatial distribution of electrons is known, other important properties follow. For instance, the energies, ionization enthalpies, sizes, and magnetic properties of atoms all depend on the number and arrangement of the electrons within the atom.
The first goal of this project is to find efficient solvers for model problems, i.e., the development of mathematical and computational tools for solving Schrodinger equations in general. The next step for the investigator will be to extend these methods for finding the electronic structure of atoms in real materials, allowing their better understanding and control. In addition, the methodology developed for the Schrodinger equations will be extended for solving a much wider range of problems: for instance, the Helmholtz equations that describe processes arising in seismology, acoustics, and others.
