Collaborative Research: Numerical Methods for Nonlinear Diffusion Problems

合作研究：非线性扩散问题的数值方法

• 批准号: 0411723
• 负责人: Michael Holst
• 金额: $ 23.9万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0411723&HistoricalAwards=false
• 关键词: Collaborative; Research; Numerical; Methods; Nonlinear

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.

该建议与非线性椭圆形和抛物线偏微分方程的系统的近似解决方案有关。 这些问题用于在工程，数学和科学的各个领域中建模各种物理现象。在该项目中考虑的特定应用会出现在几何分析中的Generalfaces的拓扑分类以及半导体物理学中的拓扑分类中。 这种非线性问题出现障碍的共同示意力包括：多域和时间依赖于时间范围的领域，多规模和多物理学以及潜在的规律性和爆炸的潜在丧失。在该项目中，研究人员计划开发一些基本数学工具来处理这些范围广泛的工程。 在该项目中，研究者专注于在几何分析中的应用，这些应用用于解决几何学的基本测量，以及在现代生物学，化学和物理学中引起的高度复杂数学模型的应用。 但是，与复杂的，多维域和多尺度的多物理微分方程相关的计算困难，在许多情况下需要计算方法。 该方法开发的将有助于进步的数值方法，以进行复杂的尺寸非线性动力学模拟。 我们生产的仿真技术将为物理，化学和生物学模型的探索以及纯数学分析（例如几何分析）的缺陷领域提供令人兴奋的强大工具。