Mathematical Sciences: Construction and Analysis of Algorithms for Degenerate Variational and Parabolic Problems

研究人员沃金顿研究涉及相变及其数值近似的物理问题。一般来说，这些问题表现出两个不同的特征之一；要么解具有自由边界问题的特征，其中未知边界将相分离，要么相被精细地散布。前者用退化抛物型方程来模拟。虽然这些方程的理论已经发展得很好，但离散理论的数值逼近却并非如此。例如，研究者与J.Rulla合作，直到最近才在对偶Sobolev空间中建立了经典Stefan问题的最优收敛速度；然而，能量在可积函数空间中的收敛速度仍然是一个悬而未决的问题。例如，当两相之间的界面具有更多结构时，对平均曲率方程的近似所知更少。研究者还研究了呈现精细相混合的问题的近似性。由于这些问题涉及的结构比任何可行的网格都要精细得多，研究人员利用最近的杨度量理论来描述精细尺度的行为。这一理论表明，细尺度行为远不是随机的，在许多情况下可以用(宏观的)Young度量来表征。研究人员研究和分析了近似杨氏度量的算法。在这个项目中，研究人员开发了有效的方法来对环境和材料科学中出现的问题进行计算建模。这些问题的一个有趣特征是，它们经常在不同材料相遇的地方展示界面。例如，在对污染物的扩散进行建模时，最重要的量是污染物与清洁环境之间的界面的位置和运动。材料科学中的问题经常涉及到涉及同一材料不同相之间的界面的非常精细的尺度现象。这种现象无处不在，例如，在大多数金属中都可以观察到。然而，在分析这类材料时，细微的天平也存在许多障碍。研究人员开发了绕过这些问题的计算工具，向现代(和传统)材料的设计和开发的自动化又迈进了一步。

Walkington The investigator studies physical problems involving phase changes and their numerical approximation. In general these problems exhibit one of two distinct traits; either the solutions take on the character of a free boundary problem where the unknown boundary separates the phases, or the phases are finely interspersed. The former are modeled by degenerate parabolic equations. While the theory for these equations is well developed, this is not the case with the discrete theory for their numerical approximation. For example, the investigator in collaboration with J. Rulla has only recently established optimal rates of convergence of the classical Stefan problem in a dual Sobolev space; however, rates of convergence of the energy in the space of integrable functions is still an open question. Even less is known about approximations of the mean curvature equation that arises, for example, when the interface between the phases has more structure. The investigator also studies the approximation of problems that exhibit fine phase mixtures. Because these problems involve structures that are much finer than any feasible mesh, the investigator utilizes the recent theory of Young measures to characterize the fine scale behavior. This theory shows that the fine scale behavior is far from random and in many cases can be characterized by a the (macroscopic) Young measure. The investigator studies and analyzes algorithms that approximate the Young measure. In this project, the investigator develops efficient ways to computationally model problems arising in the environmental and material sciences. One interesting feature of these problems is that they often exhibit interfaces where different materials meet. For example, when modeling the spread of pollutants, the single most important quantity is the location and motion of the interface between the pollutant and the clean environment. Problems in the material sciences often involve very fine scal e phenomena involving interfaces between different phases of the same material. This phenomenon is ubiquitous, being observable, for example, in most metals. However, the fine scales also present many obstacles when analyzing such materials. The investigator develops the computational tools to circumvent these problems, providing another step towards the automation of the design and development of modern (and traditional) materials.