Adaptive Multigrid Methods for Partial Differential Equations

摘要(美国国家科学基金会提案：DMS-0074299，PI：徐金超)该项目是关于偏微分方程解的高级方法的研究，这些方法来自于科学和工程应用。本文的研究主题是多重网格法的发展、应用和分析。多重网格法是求解由偏微分方程组离散化而产生的大规模线性和非线性系统的最强大的技术之一。但该方法在实践中没有得到应有的频繁使用(因为它通常不容易编码和使用)，也没有像它们所能达到的那样有效(因为让方法正确工作通常并不容易)。我们的研究一方面是为了开发一种特殊类型的多重网格法，这种方法对于一些标准应用的一般用户来说相对更容易使用，另一方面是为了开发一种为某些特殊的实际有趣的问题而精心定制的多重网格法。该研究的一个主要组成部分是系统地研究了与多重网格方法相关的各种基本理论问题，以及针对几个有实际意义的具体问题而开发的算法的相关理论问题。我们建议开发和研究的多重网格方法有望适用于一大类实际问题，包括电化学功率器件(电池)的数值模拟以及晶格块材料和液晶材料等先进材料的数值模拟。这些多重网格法预计将对这些和相关应用产生重大影响，特别是使以其他传统方法可能不可行的方式在三维中模拟这些问题成为可能。例如，拟议的模拟电化学电池的先进数值方法的研究已经并将对先进的电池技术和制造做出重大贡献，这些技术和制造在我们的日常生活中至关重要，从手表和相机闪光灯到电动汽车、现代航天器和信息技术中的无线通信。由于这些问题的实际背景，提出者及其研究助理和研究生有望与来自行业的物理学家、工程师、计算科学家和实践者积极互动和合作。

ABSTRACT(NSF proposal: DMS-0074299, PI: Jinchao Xu)The proposed project is on the study of advanced solution methods for partial differential equations that arise from scientific and engineering applications. The theme of research is on the development, application and analysis of multigrid methods. The multigrid method is among the most powerful techniques for solving large scale linear and nonlinear systems arising from the discretization of partial differential equations. But the method has not been used in practice as often as they should be (because it is often not easy to code and to use), nor as efficiently as they could be (because it is often not easy to get the method work correctly). Our research is, on one hand, to develop special type of multigrid methods that are relatively easier to use for general users for some standard applications and is, on the other hand, to develop multigrid methods that are carefully tailored for some special class of practically interesting problems. One major component of the proposed research is a systematical investigation on various fundamental theoretical issues related to multigrid methods in general and also theoretical questions related to the algorithms to be developed for several specific problems of practical interests.The multigrid methods we propose to develop and study are expected to be applicable to a large class of practical problems including numerical simulations for electrochemical power devices (batteries) and advanced materials such as lattice block materials and liquid crystalline materials. These multigrid methods are expected to make a major impact in these and related applications and in particular to make it possible to simulate these problems in three dimensions in such a way that other traditional approaches may not be feasible. For example, the proposed study of advanced numerical methods for simulating electrochemical batteries has been and will be making a significant contribution to advanced battery technologies and manufactures that are vitally important in our everyday life, from the watch and the camera flash to electromobilies, modern space vehicles and wireless communications in information technology. Because of the practical backgrounds of these problems, the proposers and their research associates and graduate students are expected to actively interact and collaborate with physicists, engineers, computational scientists and practitioners from industries.