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

NochettoDMS-1109325 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根本的重要性。 这使得基本机制的理解，设计和实施有效的数值模拟和控制随后的过程中，模型和算法的分析方法。 生物物理学（如流体和凝胶状态的生物膜），材料科学（如晶体表面形态和双层致动器）和形状优化（包括电介质上的电润湿）中的数学模型是研究人员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：点和线奇点（界面），薄层，和大的域变形，可能导致拓扑结构的变化。 该项目的目标是建模和控制这样的多尺度现象，并设计，测试和分析可靠和有效的自适应有限元方法与时空误差控制的基础上的后验误差估计。 理解微观和纳米尺度上非线性现象的机制在科学和工程的许多领域都是必不可少的。 研究人员开发数学模型和可靠的计算方法，用于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（如新的有效数值方法的设计）。 这是一个合作的奋进，涉及一些科学家在美国和国外，以及一些研究生和博士后。 在教育和人力资源开发方面投入了大量努力。

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.