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自举多重网格算法框架，该框架有可能对数值求解偏微分方程组(PDE)的计算方法产生显著和广泛的影响。该项目的智力优势来自于它有潜力在几何和代数多重网格法的设计和分析方面取得几个明显的理论进步，并将这些进步整合到大规模科学应用中的算法和软件中，这些应用需要求解耦合的PDE系统和PDE特征值问题。通过将这些新算法应用于科学和工程中的各种问题，该项目将实现更广泛的影响。参与该项目的研究生和博士后将从事由PI领导的跨学科研究，并将有机会访问来自行业和能源部实验室的同事并与其合作。这一研究项目建立在PI在发展多重网格求解器方面的最新进展的基础上：(1)成功地开发了自适应和引导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离散化的快速求解器；(2)设计和分析了Laplace-Beltrami特征值问题的稳健的Bootstrap MG求解器。具体地说，项目团队将专注于两个相互关联的研究目标：(1)将目前用于求解Dirac偏微分方程组各种离散化的Bootstrap代数MG方法扩展到求解耦合偏微分方程组和广义代数特征值问题的通用方法；(2)设计和分析求解曲面上PDE特征值问题的新的有限元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.