Adaptive Finite Element Methods for Multiscale Geometric PDE: Modeling, Analysis, and Computation

用最简单和最粗糙的模型捕捉微观和纳米尺度上非线性现象的基本行为在科学和工程中具有重要意义。这允许理解基本机制，设计和实施有效的数值方法来模拟和控制随后的过程，以及模型和算法的分析。生物物理学（如流体和凝胶状态下的生物膜）、材料科学（如晶体表面形态学和双层致动器）和形状优化（包括电介质上的电润湿）中的数学模型是研究者在该项目中研究的典型但非常独特的例子。控制偏微分方程是几何的，表现出不同的时空尺度：点和线的奇点（界面）、薄层和大的域变形，可能导致拓扑变化。该项目的目标是对这种多尺度现象进行建模和控制，并设计、测试和分析基于后验误差估计的时空误差控制的可靠、有效的自适应有限元方法。了解微观和纳米尺度的非线性现象的机制在科学和工程的许多领域是必不可少的。研究者开发数学模型和可靠的计算方法来研究广泛的此类问题。该项目涉及联邦战略利益的应用，如纳米和微技术（如微机电系统（MEMS）的设计和控制），生物技术（如生物膜的研究）和高性能计算（如新型高效数值方法的设计）。这是一项合作的努力，涉及美国和国外的一些科学家，以及一些研究生和博士后。在教育和人力资源发展方面作出了大量努力。

NochettoDMS-1109325 Capturing the essential behavior of nonlinear phenomena at the micro- and nanoscales with the simplest and crudest models is of fundamental importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of ensuing processes, and the analysis of both models and algorithms. Mathematical models in biophysics (such as biomembranes in both fluid and gel state), in materials science (such as crystal surface morphologies and bilayer actuators), and in shape optimization (including electro-wetting on dielectric) are typical yet quite distinct examples that the investigator studies in the project. The governing partial differential equations are geometric and exhibit disparate space-time scales: point and line singularities (interfaces), thin layers, and large domain deformations, perhaps leading to topology changes. The goal of the project is to model and control such multiscale phenomena, and to design, test, and analyze reliable and efficient adaptive finite element methods for them with space-time error control based on a posteriori error estimation. Understanding the mechanisms of nonlinear phenomena at micro- and nanoscales is essential in many areas of science and engineering. The investigator develops mathematical models and reliable computational methods for studying a wide range of such problems. This project deals with applications of Federal strategic interest such as nano and microtechnology (such as the design and control of micro electro-mechanical system (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. A substantial effort is devoted to education and human resource development.