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

研究者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明显的特征之一；解要么具有自由边界问题的特征，其中未知边界将相分开，要么相精细地穿插。前者用退化抛物方程建模。虽然这些方程的理论发展得很好，但离散理论的数值近似却不是这样。例如，与J. Rulla合作的研究者直到最近才建立了对偶Sobolev空间中经典Stefan问题的最优收敛率；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。对于出现的平均曲率方程的近似，例如，当相之间的界面具有更多的结构时，我们所知的就更少了。研究者还研究了表现出细相混合的问题的近似。由于这些问题涉及的结构比任何可行的网格都要细得多，因此研究者利用最新的杨氏测量理论来表征精细尺度的行为。该理论表明，精细尺度的行为远非随机的，在许多情况下可以用（宏观的）杨氏测度来表征。研究者研究和分析近似杨测度的算法。在这个项目中，研究者开发了有效的方法来计算环境和材料科学中出现的问题。这些问题的一个有趣的特点是，它们经常表现出不同材料相遇的界面。例如，当模拟污染物的扩散时，最重要的一个量是污染物和清洁环境之间界面的位置和运动。材料科学中的问题常常涉及非常精细的尺度现象，涉及同一材料的不同相之间的界面。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，细尺度也存在许多障碍。研究者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提供了又一步。

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.