Algebraic Multigrid Methods and Their Application to Generalized Finite Element Methods

本文研究的是用广义有限元方法离散二阶偏微分方程组所产生的求解线性代数系统的高效代数多重网格法的研究和应用。建议的研究将集中在构建嵌套空间层次结构和近似子空间解算器方面的自适应技术的开发和分析，这些技术导致了适用于大范围广义有限元离散的高效和健壮的多重网格法。当今超级计算机的能力的迅速增加使得科学界可以使用数值模拟来模拟物理现象来产生有意义的结果。能够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。这种方法已被证明是一种非常稳健的离散化工具，适用于工程和科学的各个分支，例如，模拟和确定非均匀材料的弹性、电磁和其他重要的物理性质。与大多数其他离散化技术一样，这类模拟中的大多数计算通常都致力于求解所产生的线性方程组。因此，为这些系统开发有效的解是非常重要的。因此，该研究的结果将会产生广泛而显著的影响，为许多应用中的数值模型所产生的离散线性系统提供急需的迭代多层解技术。该研究还有望产生教育影响，因为它将为培养研究生在科学和工程问题的数值方法的现代理论和实践方面提供坚实的基础。

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.