Algebraic Multigrid Methods and Their Application to Generalized Finite Element Methods

本课题的研究内容是研究和应用代数多重网格法求解二阶偏微分方程的广义有限元离散化问题。 本研究的重点是发展和分析嵌套空间层次结构中的自适应技术，以及选择近似子空间求解器，从而形成适用于各种广义有限元离散化的高效和鲁棒的多重网格方法。当今超级计算机的快速增长使科学界使用数值模拟成为可能对物理现象进行建模以产生有意义的结果。其中一个现代技术，可以提供定量的结果，通过这样的模拟是广义有限元法。 这种方法已被证明是一种非常强大的离散化工具，适用于工程和科学的各个分支，例如，在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。 像大多数其他离散化技术，最经常的大部分计算在这样的simulationsis致力于解决所产生的线性系统的方程。 因此，开发有效的求解器对于这些系统是非常重要的。本文的研究成果将为离散线性系统的数值模型的求解提供一种新的迭代多层求解方法，具有广泛而显著的应用前景，同时也将为现代理论与应用领域的研究生培养提供一个坚实的基础。科学与工程问题的数值方法的实用方面。

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.