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

该项目的主要目标是开发一个自引导多网格算法框架，该框架有可能对数值求解偏微分方程（PDEs）的计算方法产生可观而广泛的影响。该项目的智力价值源于其在几何和代数多重网格方法的设计和分析方面取得若干不同理论进展的潜力，并将这些进展整合到需要解决耦合PDE系统和PDE特征值问题的大规模科学应用的算法和软件中。通过将这些新算法应用于科学和工程中的各种问题，将实现该项目更广泛的影响。参与该项目的研究生和博士后将参与由PI领导的跨学科研究，并将有机会访问和与工业界和美国能源部实验室的同事一起工作。本研究项目建立在PI在开发多网格解算器和PDE特征值问题系统方面的最新进展的基础上：(1)成功开发了自适应和自举代数多网格作为晶格量子色动力学（QCD）中耦合Dirac PDE的Wilson-Dirac和Wilson-clover离散化的快速解算器；(2)设计并分析了拉普拉斯-贝尔特拉米特征值问题的鲁棒自举MG求解器。具体而言，项目团队将专注于两个相互关联的研究目标：(1)将目前用于解决Dirac PDE的各种离散化的自举代数MG方法扩展到解决耦合PDE系统和广义代数特征值问题的一般方法；(2)设计并分析了求解曲面PDE特征值问题的有限元自举MG新方法。

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.