This talk includes two parts:
Part I: Since its first observation in 1995, Bose-Einstein condensation (BEC) has been a popular topic. Recently, quantized vortices in BEC have attracted lots of attention from researchers. In this talk, the stationary states in the Thomas-Fermi regime are investigated by using an asymptotic method. The dynamical laws are derived for the quantized vortices, and other properties of the condensates are also discussed. Efficient and accurate numerical methods are developed and applied to study the properties of BEC. The structures of vortex lattices in both 2D and 3D are numerically studied, and the generation of these lattices is also presented.
Part II: Quasicontinuum methods using representative particles provide a simplified model to study huge molecular systems. The objective of this study is to develop quadrature-rule type approximations to further simplify the quasicontinuum method. For both short and long-range interatomic interactions, the complexity of this method depends on the number of representative particles but not on the total number of particles. Numerical experiments and complexity estimates illustrate that the quadrature-rule type method preserves much of accuracy of the quasicontinuum method, but at a much lower cost.