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

沃金顿 研究员研究涉及相变及其数值近似的物理问题。 在一般情况下，这些问题表现出两个不同的特点之一;要么解决方案采取的自由边界问题的特点，未知的边界分离的阶段，或阶段是细穿插。 前者用退化抛物方程来模拟。 虽然这些方程的理论是发达国家，这是不是与离散理论的情况下，其数值近似。 例如，与J. Rulla合作的研究者最近才建立了对偶Sobolev空间中经典Stefan问题的最优收敛速度;然而，可积函数空间中能量的收敛速度仍然是一个悬而未决的问题。 甚至更少的是已知的近似的平均曲率方程，例如，当界面之间的相位有更多的结构。 调查员还研究了近似的问题，表现出良好的相混合物。 由于这些问题涉及的结构比任何可行的网格精细得多，研究人员利用最近的杨氏测度理论来表征精细尺度行为。 这一理论表明，精细尺度行为远非随机的，在许多情况下可以用（宏观）杨氏测度来表征。 研究人员研究和分析近似Young测度的算法。 在这个项目中，研究人员开发了有效的方法来计算模拟环境和材料科学中出现的问题。 这些问题的一个有趣的特征是，它们通常表现出不同材料相遇的界面。 例如，在对污染物的扩散进行建模时，最重要的一个量是污染物与清洁环境之间的界面的位置和运动。 材料科学中的问题通常涉及到非常细微的现象，这些现象涉及到同一材料不同相之间的界面。 这种现象普遍存在，例如在大多数金属中都可以观察到。 然而，当分析这样的材料时，精细的尺度也存在许多障碍。 研究人员开发了计算工具来规避这些问题，为现代（和传统）材料的设计和开发自动化迈出了又一步。

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.