Quah, JohnCrystal surfaces serve a crucial function as building blocks in small electronic devices, especially for mobile communications technology and photovoltaics. In the history of computing, for example, a crucial innovation that hastened the demise of vacuum tube computers was the etching of patterns on surfaces of semiconductor materials, which led to the integrated circuit. These early procedures typically worked with the materials at very high temperatures, where the thermally rough surface could be modeled from the perspective of continuum thermodynamics. More recently, with the drive towards smaller devices and the accompanying reduction of the lifetime of surface features, manufacturing conditions for the shaping of crystal surfaces have shifted to lower temperatures. At these lower temperatures the surface is no longer rough. In order to describe an evolving surface under typical experimental conditions today, we need to consider the processes that take place at the nanoscale. Nanoscale descriptions of surface evolution start with the motion of adsorbed atoms (adatoms). Because of their large numbers, the concentration of adatoms is a meaningful object to study. Restricted to certain bounded regions of the surface, the adatom concentration satisfies a diffusion equation. At the boundaries between these regions, the hopping of adatoms is governed by kinetic laws. Real-time observation of these nanoscale processes is difficult to achieve, and experimentalists have had to devise creative methods for inferring the relevant energy barriers and kinetic rates. In contrast, the real-time observation of macroscale surface evolution can be achieved with simpler imaging techniques. Motivated by the possibility of experimental validation, we derive an equation for the macroscale surface height, which is consistent with the motion of adatoms. We hope to inspire future comparison with experiments by reporting the novel results of simulating the evolution of the macroscale surface height. Many competing models have been proposed for the diffusion and kinetics of adatoms. Due to the difficulty of observing adatom motion at the nanoscale, few of the competing models can be dismissed outright for failure to capture the observed behavior. This dissertation takes a few of the nanoscale models and systematically derives the corresponding macroscopic evolution laws, of which some are implemented numerically to provide data sets for connection with experiments. For the modeling component of this thesis, I study the effect of anisotropic adatom diffusion at the nanoscale, the inclusion of an applied electric field, the desorption of adatoms, and the extension of linear kinetics in the presence of step permeability. Analytical conjectures based on the macroscale evolution equation are presented. For the numerical component of this thesis, I select a few representative simulations using the finite element method to illustrate the most salient features of the surface evolution.en-USDissertationMathematicsPhysics, Condensed Mattercrystal surfacefinite element methodmacroscale limitnonlinear PDE