Multiscale Methods for Partial Differential Equations

DMS奖摘要奖#：0209497PI：Xu，金超研究所：宾夕法尼亚州立大学项目：计算数学项目经理：Catherine Mavriplis标题：偏微分方程组的多尺度方法本工作的重点是一种双尺度离散化技术的发展和应用，即基于单位分解的有限元方法。一个主要的应用是为非匹配(重叠或非重叠)网格设计有效的离散化。非匹配网格的主要思想是将一个物理区域划分为一组重叠或不重叠的子区域，这些子区域可以容纳平滑、简单、容易生成的网格。在这种方法中，可以简化复杂几何形状的网格生成，可以在不改变其他网格的情况下添加或删除细化网格，可以在不同的网格上使用不同的方程/数值方法，可以使用高效的结构化网格求解器。此外，重叠网格非常适合并行化和矢量化。基于单位分解的广义有限元方法为这类网格提供了一个通用的、强大的离散化框架。另一项主要任务是开发一种多重网格迭代方法来求解这些新的离散化格式的代数系统。作为分而治之的技术，所提出的多尺度算法适用于并行和高性能计算机。提出了一类新的多尺度技术来研究偏微分方程组的有效数值解。一般情况下，多尺度方法被证明是研究由偏微分方程描述的广泛模型的最强大的数学工具之一。它们在设计快速、可靠和健壮的数值方法以解决各种问题方面发挥着关键作用，使它们成为近年来应用数学中最重要的研究领域之一。由于这些方法在某种意义上与问题无关，它们有望在科学和工程中有许多重要的应用，如复合材料和环境应用中的地下流动。日期：2002年5月28日

DMS Award AbstractAward #: 0209497PI: Xu, JinchaoInstitution: Pennsylvania State University Program: Computational MathematicsProgram Manager: Catherine MavriplisTitle: Multiscale Methods for Partial Differential EquationsThe focus of this work is on the development and applications of a two-scale discretization technique, namely the finite element method based on partition of unity. One main application is on the design of efficient discretization for nonmatching (either overlapping or nonoverlapping) grids. The main idea of nonmatching grids is to divide a physical domain into a set of overlapping or nonoverlapping subregions which can accommodate smooth, simple, easily generated grids. In this approach, a grid generation for complex geometries can be made simple, refinement grids can be added or removed without changing other grids, different equations/numerical methods may be used on different grids, efficient structured grid solvers may be used. Furthermore, overlapping grids are well suited for parallelization and vectorization. The proposed generalized finite element method based on partition of unity provides a general and powerful discretization framework for this type of grids. Another major task is the development of a multigrid iterative method for solving the resulting algebraic systems for these new discretization schemes. As divide and conquer techniques, the proposed multiscale algorithms are suitable for parallel and high-performance computers. A class of new multiscale techniques are proposed to study for efficient numerical solution of partial differential equations. Multiscale methods in general are proven to be among the most powerful mathematical tools for the investigation of a broad range of models that are described by partial differential equations. Their pivotal role in the design of fast, reliable, and robust numerical methods for the solution of various problems places them among the most important research areas in the applied mathematics in the recent years. Since these methods are in some sense problem-independent, they are expected to have many important applications in science and engineering such as composite materials and subsurface flows in environmental applications.Date: May 28, 2002