Adaptive Finite Element Methods for Multiscale Problems Governed by Geometric PDE

对表现出不同时空尺度的连续统现象进行适当的数值处理是一项艰巨的数学和计算挑战。现代算法应该能够解决某些物理量的精细尺度，而不会过度解决其他物理量，从而优化计算工作并使现实的3d模拟可行。生物物理学中的数学模型（如流体和凝胶状态的生物膜），材料科学中的数学模型（如晶体表面形态学和双层致动器），以及形状优化（包括流体和固体力学）是本项目中典型但非常独特的例子。它们的多尺度结构在时间上表现为快瞬态和慢动状态，在空间上表现为点和线的奇点（界面）和薄层。大域变形可能导致拓扑变化，这是另一个有趣的特征。该项目的目标是设计、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，这些问题大多数由几何偏微分方程控制，并基于后验误差估计进行时空误差控制。这个项目建立在，实际上扩展和增强，以前的NSF拨款DMS-0505454的工作。它被组织在许多小而相互关联的主题中，这些主题贯穿了几何学和适应性的角色，从数值分析和计算到应用。这个项目在几个具有战略重要性的研究领域，如纳米技术、材料和制造、生物技术和高性能计算，融合了相当微妙的分析、计算和建模问题。设计智能机电设备以及了解生物体内脂质膜的关键功能，无论是在纳米还是微观尺度上，都需要能够处理大形状变化的强大而灵活的计算算法。本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。由于在教育和人力资源开发方面投入了大量的精力，大部分资源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.