Geometric and algebraic multigrid solvers for coupled systems of PDEs and PDE eigenvalue problems

用于偏微分方程和偏微分方程特征值问题耦合系统的几何和代数多重网格求解器

• 批准号: 1620346
• 负责人: James Brannick
• 金额: $ 16万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620346&HistoricalAwards=false
• 关键词: Geometric; algebraic; multigrid; solvers; coupled

The primary goal of this project is to develop a bootstrap multigrid algorithmic framework that has the potential to make an appreciable and broad impact on computational methods for numerically solving partial differential equations (PDEs). The intellectual merit of the project derives from its potential to make several distinct theoretical advances in the design and analysis of geometric and algebraic multigrid methods and to integrate those advances into algorithms and software for large-scale scientific applications that require solving coupled PDE systems and PDE eigenvalue problems. The broader impact of the project will be realized by applying these new algorithms to various problems in science and engineering. The graduate student and post doc involved in the project will engage in interdisciplinary research led by the PI and will have opportunities to visit and work with colleagues from industry and DOE labs. This research project builds on recent advances by the PI in the development of multigrid solvers for systems of and PDE eigenvalue problems: (1) the successful development of adaptive and bootstrap algebraic multigrid as fast solvers for the Wilson-Dirac and Wilson-clover discretizations of the coupled Dirac PDE in lattice quantum chromodynamics (QCD); (2) the design and analysis of a robust bootstrap MG solver for the Laplace-Beltrami eigenvalue problem. Specifically, the project team will focus on two interrelated research goals: (1) to extend the bootstrap algebraic MG methods currently being used to solve various discretizations of the Dirac PDE to a general approach for solving systems of coupled PDEs and generalized algebraic eigenvalue problems; (2) to design and analyze new finite element bootstrap MG methods for solving PDE eigenvalue problems on surfaces.

该项目的主要目的是开发一个自举多机算法框架，该框架有可能对数值求解偏微分方程（PDES）的计算方法产生明显和广泛的影响。 该项目的智力优点源于其在几何和代数跨境方法的设计和分析中取得几个不同的理论进步的潜力，并将这些进步整合到大规模科学应用中，以用于大规模的科学应用，这些应用需要解决求解PDE Systems和PDE EigenValue问题。该项目的更广泛影响将通过将这些新算法应用于科学和工程学的各种问题来实现。参与该项目的研究生和文档将参与PI领导的跨学科研究，并将有机会与行业和DOE实验室的同事一起访问和合作。该研究项目的基础是PI在为特征和PDE特征值问题开发的Multigrid Solvers开发中的最新进展，（1）（1）自适应和Bootstrap代数多移民作为Wilson-Dirac和Wilson-Dirac和Wilson-Clover的快速求解器的成功开发。 （2）针对Laplace-Beltrami特征值问题的强大引导MG求解器的设计和分析。具体而言，项目团队将重点介绍两个相互关联的研究目标：（1）扩展目前用于求解DIRAC PDE的各种离散化的Bootstrap代数MG方法到解决耦合PDES和广义代数Eigenvalue问题的一般方法； （2）设计和分析新的有限元引导MG方法，以解决表面上的PDE特征值问题。