Computational Multiscale Methods for Defects: 1. Line Defects in Liquid Crystals; 2. Electron Scattering in Defected Crystals

In the first part of this thesis, we demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight as well as loop disclination solutions. The possibility of static balance of forces in the presence of a disclination and in the absence of ow and body forces is discussed. The work exploits an implicit conceptual connection between the Weingarten-Volterra characterization of possible jumps in certain potential fields and the Stokes-Helmholtz resolution of vector fields. The theoretical basis of our work is compared and contrasted with the theory of Volterra disclinations in elasticity. Physical reasoning precluding a gauge-invariant structure for the model is also presented. In part II of the thesis, the time-harmonic Schrodinger equation with periodic potential is considered. We derive the asymptotic form of the scattering wave function in the periodic space and investigate the possibility of its application as a DtN non-reflecting boundary condition. Moreover, we study the perfectly matched layer method for this problem and show that it is a reliable method, which converges rapidly to the exact solution, as the thickness of the absorbing layer increases. Moreover, we use the tight-binding method to numerically solve the Schrodinger equation for Graphene sheets, symmetry-adapted Carbon nanotubes and DNA molecules to demonstrate their electronic behavior in the presence of local defects. The results for Y-junction Carbon nanotubes depict very interesting properties and confirms the predictions for their application as new transistors.