| Description |
Interface problems and models with evolving geometry arise in the mathematical descriptions of various engineering and scientific questions, for example, in materials science, fluid dynamics, biology, and geoscience. Prototypical physical examples include heat conduction in layered media (with jump discontinuities in the diffusion coefficient) and models of phase change in materials. Typically, these models cannot be solved analytically, and solutions may have low regularity (or discontinuities) at interfaces. Consequently, the development of methods for the numerical approximation of the given model is required. However, conventional methods - which typically involve an underlying regularity assumption - may perform poorly or completely fail to provide an accurate approximation. This dissertation is two-fold - first, we design high-order accurate and efficient methods based on Difference Potentials for the numerical approximation of interface problems and models with evolving geometry. We focus for now on parabolic interface models in 1D or 2D with explicitly- or implicitly-defined geometry and on elliptic models in evolving domains. However, the proposed methods are by no means limited to these models. Indeed, there are exciting avenues for future research. Secondly, we mathematically model and analyze macro- and microscale fluid properties of sea ice, or frozen seawater. As a material, sea ice is a porous, multiscale composite of pure ice with brine inclusions. The polar ice packs play key roles in Earth's climate and ecological systems. On the macroscale, we study the geometry of Arctic melt ponds - pools of melt water on the surface of sea ice floes - and characterize a striking transition in the fractal geometry of the ponds as they grow and coalesce. On the microscale, we turn our attention to the interplay between biology and physics. Significant differences between sea ice with and without entrained algae have been observed - including decreased fluid permeability of the ice - due to extracellular substances secreted by the algae that modify pore structure. In order to investigate this phenomenon, we propose a random network model for fluid transport through the ice, involving a new bimodal-lognormal distribution for the inclusion sizes, and derive rigorous upper bounds. |
