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

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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.