Adaptive Finite Element Methods for Multiscale Problems Governed by Geometric PDE

注意到DMS-0807811对表现出不同时空尺度的连续统现象的适当数值处理是一个巨大的数学和计算挑战。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逼真的3D模拟成为可能。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)以及形状优化(包括流体力学和固体力学)中的数学模型是本项目所涉及的典型而又截然不同的例子。它们的多尺度结构在时间上表现为快瞬变和慢动作状态，在空间上表现为点、线奇异性(界面)和薄层。另一个耐人寻味的特征是大变形，可能会导致拓扑变化。这个项目的目标是设计、测试和分析这些多尺度问题的可靠和有效的自适应有限元方法，这些问题大多由几何偏微分方程组控制，时空误差控制基于后验误差估计。这个项目建立在之前的美国国家科学基金会拨款DMS-0505454的基础上，实际上是对其工作的扩展和加强。它被组织在一些小但相互关联的主题中，从数值分析和计算到应用，贯穿始终的是几何和适应性的角色。该项目融合了几个具有战略重要性的研究领域中相当微妙的分析、计算和建模问题，如纳米技术、材料和制造、生物技术和高性能计算。设计智能的机电设备，以及在纳米和微米尺度上理解活体生物体中类脂膜的关键功能，都需要强大而灵活的计算算法来处理大的形状变化。这个项目开发了这样的工具。这是一个合作和跨学科的项目，涉及美国和国外的许多科学家(其中四人是工程师和物理学家)，以及几名学生和博士后。由于实质性的努力致力于教育和人力资源开发，大多数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.