Algebraic Multigrid Methods and Their Application to Generalized Finite Element Methods

代数多重网格方法及其在广义有限元方法中的应用

• 批准号: 0511800
• 负责人: Ludmil Zikatanov
• 金额: $ 12万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0511800&HistoricalAwards=false
• 关键词: Algebraic; Multigrid; Methods; Their; Application

The research in this proposal is on the study and applications ofefficient algebraic multigrid methods for the solution of linearalgebraic systems arising from the discretization of second orderpartial differential equations by the generalized finite elementmethod. The proposed research will focus on the development andanalysis of adaptive techniques in the construction of hierarchy ofnested spaces and the choice of approximate subspace solvers that leadto the efficient and robust multigrid methods applicable to wide rangeof generalized finite element discretizations.The rapid increase in the power of today's supercomputers has made itfeasible for the scientific community to use numerical simulations tomodel physical phenomena to produce meaningful results. One of themodern techniques that can deliver quantitative results via suchsimulations is the generalized finite element method. This method hasproved to be a very robust discretization tool, applicable in variousbranches of engineering and sciences, for example, in simulating anddetermining the elastic, electromagnetic and other important physicalproperties of heterogeneous materials. Like most other discretizationtechniques, most often the majority of computation in such simulationsis devoted to the solution of the resulting linear systems ofequations. Hence, it is very important to develop efficient solversfor these systems. The results from the proposed research are thusexpected to have a broad and noticeable impact by providing the muchneeded iterative multilevel solution techniques for the discretelinear systems arising from numerical models in many applications.The proposed research is also expected to have an educational impactas it will provide a solid base for training of graduate students inthe modern theoretical and practical aspects of numerical methods forproblems in science and engineering.

本提案的研究是研究和应用高效代数多重网格方法来求解由广义有限元方法离散二阶偏微分方程所产生的线性代数系统。 拟议的研究将重点关注嵌套空间层次结构构建中的自适应技术的开发和分析以及近似子空间求解器的选择，从而产生适用于广泛的广义有限元离散化的高效且鲁棒的多重网格方法。当今超级计算机能力的快速增长使得科学界可以使用数值模拟来模拟物理现象以产生有意义的结果。 结果。可以通过这种模拟提供定量结果的现代技术之一是广义有限元方法。 该方法已被证明是一种非常强大的离散化工具，适用于工程和科学的各个分支，例如，模拟和确定异质材料的弹性、电磁和其他重要物理性质。 与大多数其他离散化技术一样，此类模拟中的大部分计算通常致力于求解所得线性方程组。 因此，为这些系统开发高效的求解器非常重要。因此，通过为许多应用中由数值模型产生的离散线性系统提供急需的迭代多级求解技术，预计该研究的结果将产生广泛而显着的影响。该研究还预计将产生教育影响，因为它将为研究生在科学和工程问题数值方法的现代理论和实践方面的培训提供坚实的基础。