Collaborative Research: Numerical Methods for Nonlinear Diffusion Problems

这一建议涉及复杂区域上的非线性椭圆型和抛物型偏微分方程组的近似解。这类问题被用来模拟工程、数学和科学的各个领域中的各种物理现象。本项目中考虑的特殊应用出现在化学静电学、几何分析中一般表面的拓扑分类以及半导体物理中。这些给计算科学带来障碍的非线性问题的共同特征包括：多域和依赖于时间的域，多尺度和多物理，以及潜在的规则性损失和爆炸。在这个项目中，研究人员计划开发一些基本的数学工具来处理这些困难，这些工具将具有广泛的适用性。这个项目的结果将在数学以及科学和工程领域产生广泛的影响。在这个项目中，研究人员专注于几何分析中用于解决几何中的基本问题的应用，以及在现代生物、化学和物理中出现的高度复杂的数学模型的应用。然而，与复杂的、多维的区域和需要新的计算方法的多尺度、多物理的微分方程相关的计算困难在许多情况下出现。本文提出的方法将有助于复杂三维非线性动力学数值模拟方法的发展。我们生产的模拟技术将为探索物理、化学和生物学的模型以及几何分析等纯数学领域的模型提供令人兴奋和强大的工具。

This proposal is concerned with the approximate solution of systemsof nonlinear elliptic and parabolic partial differential equations posedon complicated domains. Such problems are used to model a wide varietyof physical phenomena in various fields of engineering, mathematics, andscience. Particular applications considered in this project arise inchemical electrostatics, in the topological classification of generalsurfaces in geometric analysis, and in semiconductor physics. Commoncharacteristics of such nonlinear problems presenting hurdles tocomputational science include: multi-domain and time-dependent domains,multi-scale and multi-physics, and potential loss of regularity and blowup.In this project, the investigator plans to develop some fundamental mathematical tools for dealing with these difficulties that will have a wide range ofapplicability.The results of this project will have a broad impact in mathematicsas well as in science and engineering. In this project, the investigator is focused on applications in geometric analysis that are used to address fundamentalquestions in geometry, and on applications to highly complex mathematicalmodels arising in modern biology, chemistry, and physics. However, thecomputational difficulties associated with complicated, multidimensionaldomains and multi-scaled, multi-physics differential equations that requirenew computational approaches arise in many contexts. The methods developedhere will contribute to the advancement of numerical methods for complexthree dimensional nonlinear dynamical simulations. The simulationtechnology we produce will provide exciting and powerful tools for theexploration of models in physics, chemistry, and biology, as well as insome areas of pure mathematics such as geometric analysis.