Adaptive Finite Element Methods for Multiscale Problems Governed by Geometric PDE

NochettoDMS-0807811 充分的数值处理的连续现象表现出不同的时空尺度是一个可怕的数学和计算的挑战。 现代算法应该能够在不过度解析其他物理量的情况下解析某些物理量的精细尺度，从而优化计算工作并使真实的3d模拟可行。 生物物理学（如流体和凝胶状态下的生物膜）、材料科学（如晶体表面形态和双层致动器）和形状优化（包括流体和固体力学）中的数学模型是本项目中解决的典型但相当独特的例子。 它们的多尺度结构表现在时间上的快速瞬变和慢运动状态，以及空间上的点和线奇点（界面）和薄层。 大磁畴变形，可能导致拓扑结构的变化，是另一个有趣的特征。 该项目的目标是设计，测试和分析可靠和有效的自适应有限元方法，这些多尺度问题，大多数由几何偏微分方程，与时空误差控制的基础上的后验误差估计。 这个项目建立在，事实上扩展和增强了先前NSF资助DMS-0505454的工作。 它是组织在一些小的，但相互关联的主题渗透的几何和适应性的角色贯穿始终，从问题的数值分析和计算的应用。 这个项目融合了相当微妙的分析，计算和建模问题，在几个研究领域的战略重要性，如纳米技术，材料和制造，生物技术和高性能计算。设计智能机电设备以及了解生物体中脂质膜的关键功能，无论是在纳米级还是微米级，都需要能够处理大形状变化的强大而灵活的计算算法。 该项目开发了这样的工具。 这是一个跨学科的合作项目，涉及美国和国外的一些科学家（其中四位是工程师和物理学家），以及一些学生和博士后。 因为大量的精力都投入到了教育和人力资源开发上，所以大部分资源都用于部分支持学生和博士后。

NochettoDMS-0807811 The adequate numerical treatment of continuum phenomenaexhibiting disparate space-time scales is a formidablemathematical and computational challenge. Modern algorithmsshould be able to resolve fine scales for certain physicalquantities without overresolving others, thereby optimizing thecomputational effort and making realistic 3d simulationsfeasible. Mathematical models in biophysics (such asbiomembranes in both fluid and gel state), in materials science(such as crystal surface morphologies and bilayer actuators), andin shape optimization (including both fluid and solid mechanics)are typical yet quite distinct examples addressed in thisproject. Their multiscale structure manifests in rapidtransients and slow motion regimes in time, as well as point andline singularities (interfaces) and thin layers in space. Largedomain deformations, perhaps leading to topology changes, isanother intriguing feature. The goal of this project is todesign, test, and analyze reliable and efficient adaptive finiteelement methods for these multiscale problems, most governed bygeometric partial differential equations, with space-time errorcontrol based on a posteriori error estimation. This projectbuilds upon, and in fact extends and enhances, the work of theprior NSF Grant DMS-0505454. It is organized in a number ofsmall but interrelated topics permeated by the roles of geometryand adaptivity throughout, from issues in numerical analysis andcomputation to applications. This project blends quite delicate analytical,computational, and modeling issues in several areas of researchof strategic importance such as nanotechnology, materials andmanufacturing, biotechnology, and high performance computing. Designing smart electro-mechanical devices as well asunderstanding key functions of lipid membranes in livingorganisms, both at the nano and microscales, require powerful andflexible computational algorithms capable of dealing with largeshape variations. This project develops such tools. It is acollaborative and interdisciplinary project involving a number ofscientists in the US (four of them engineers and physicists) andabroad, as well as several students and postdocs. Because asubstantial effort is devoted to education and human resourcedevelopment, most resources are used to partially supportstudents and postdocs.