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

这个项目的主要目标是开发一个引导多重网格算法框架，有可能使一个可观的和广泛的影响数值求解偏微分方程（PDE）的计算方法。 该项目的智力价值来自于它的潜力，使几个不同的理论进步，在设计和分析的几何和代数多重网格方法，并将这些进步纳入算法和软件的大规模科学应用，需要解决耦合PDE系统和PDE特征值问题。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中的各种问题来实现。参与该项目的研究生和博士后将参与PI领导的跨学科研究，并将有机会访问并与来自行业和DOE实验室的同事合作。本研究项目建立在PI在发展多重网格求解器和PDE本征值问题方面的最新进展之上：（1）成功地发展了自适应和自举代数多重网格，作为格子量子色动力学（QCD）中耦合Dirac PDE的Wilson-Dirac和Wilson-Clover离散的快速求解器;（2）Laplace-Beltrami特征值问题的鲁棒Bootstrap MG求解器的设计与分析。具体而言，项目组将专注于两个相互关联的研究目标：（1）将目前用于求解Dirac PDE各种离散化的Bootstrap代数MG方法扩展为求解耦合PDE和广义代数特征值问题的通用方法;（2）设计和分析用于求解曲面PDE特征值问题的新有限元Bootstrap MG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.