Collaborative Research: Numerical Methods for Nonlinear Diffusion Problems

研究了复杂域上非线性椭圆型和抛物型偏微分方程系统的近似解。这些问题被用来模拟工程、数学和科学各个领域的各种物理现象。在这个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和半导体物理中。这些给计算科学带来障碍的非线性问题的共同特征包括：多域和时间依赖域，多尺度和多物理场，以及潜在的规律性损失和爆炸。在这个项目中，研究者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。这个项目的成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，研究者专注于几何分析的应用，用于解决几何中的基本问题，以及在现代生物学，化学和物理学中出现的高度复杂的数学模型的应用。然而，在许多情况下，与复杂、多维域和多尺度、多物理场微分方程相关的计算困难需要更新的计算方法。本文所开发的方法将有助于复杂三维非线性动力学模拟数值方法的发展。我们生产的仿真技术将为探索物理、化学和生物模型以及一些纯数学领域（如几何分析）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.