Heirarchical Basis Multigrid/ILU Algorithms for Solving Finite Element Equations

用于求解有限元方程的分层基础多重网格/ILU 算法

• 批准号: 9706090
• 负责人: Randolph Bank
• 金额: $ 17.4万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706090&HistoricalAwards=false
• 关键词: Heirarchical; Basis; Multigrid; ILU; Algorithms

9706090 Bank Hierarchical Basis Multigrid/ILU Algorithms for Solving Finite Element Equations Randolph E. Bank Department of Mathematics University of California at San Diego La Jolla, CA 92093 This proposal has two main components. First, we will study algebraic hierarchical basis multigrid algorithms (HBMG/ILU). These methods solve sparse sets of linear equations arising from finite difference, finite volume and finite element discretizations of partial differential equations. They are differentiated from classical HBMG and MG methods in that they do not require a coarse grid and sequence of mesh refinements. This allows application to problems with geometrically complex domains that require many elements just for the geometric definition, and problems where the adaptivity comes from moving the mesh points rather than from refinement. Preliminary numerical experiments indicate that the methods are potentially very powerful and robust. The second component of the of proposal concerns the continuing software development of the finite element program PLTMG. Various versions of this program have been in the public domain since the late 1970's, and it is widely used in education and research environments. PLTMG solves scalar, parameter dependent, nonlinear elliptic PDE's in general regions of the plane. The principle features are adaptive mesh generation, a posteriori error estimation, HBMG (soon to be HBMG/ILU) iteration for linear systems of equations, Newton's method for nonlinearities, and continuation for parameter dependencies. The code also includes an initial mesh generator, a skeleton generator, and several graphics routines. Although the name PLTMG has remained the same, typically 80% or more of the package is revised with each new release. In many systems modeled by partial differential equations, the critical phenomena occur only in a small part of the physical domain, and may move as a function of time ( e.g. as a flame front). Even with the great advances in hardware, it is not adequate to address difficult grand-challenge class problems of this type using software based on simple uniform meshes; the demands of the problem require that computing resources be focused on the regions of most interest. The motivation for adaptive mesh algorithms is that the algorithm itself can and should identify these critical regions and respond with an appropriate mesh with little or no human intervention. The ``brains'' of adaptive algorithms are a posterior error indicators, which both estimate the current error, and indicate where additional resources should be focused. The very nonuniform and unstructured meshes resulting from adaptive algorithms require sophisticated methods, such as multigrid or the proposed HBMG/ILU, to efficiently and reliably solve the resulting systems of equations. Overall, this field provides a mosaic of important and interrelated scientific questions ranging from difficult problems in mathematical analysis to difficult computational challenges in implementing these procedures on modern computer architectures.

9706090银行层次基础跨国基础/ILU算法解决有限元方程Randolph E.加利福尼亚州圣地亚哥La Jolla的数学银行数学系，加利福尼亚州92093，该提案具有两个主要组成部分。首先，我们将研究代数层次基础多族算法（HBMG/ILU）。 这些方法解决了由有限差，有限体积和部分微分方程的有限元离散化引起的线性方程组的稀疏集。 它们与经典的HBMG和MG方法有区别，因为它们不需要粗网格和网状细化序列。 这允许将仅用于几何定义需要许多元素的几何复杂域的问题应用，以及适应性来自移动网格点而不是细化的问题。 初步数值实验表明，这些方法可能非常强大且健壮。提案的第二部分涉及有限元程序PLTMG的持续软件开发。自1970年代后期以来，该计划的各种版本一直在公共领域中，并且广泛用于教育和研究环境中。 PLTMG在平面的一般区域中求解标量，参数依赖性，非线性椭圆PDE。 原理特征是自适应网格生成，后验误差估计，HBMG（即将成为HBMG/ILU）的线性方程系统，牛顿的非线性方法以及参数依赖性的延续。该代码还包括初始网格发电机，骨架发生器和几个图形例程。尽管PLTMG的名称保持不变，但每个新版本通常会修订80％或更多包装。 在许多由部分微分方程建模的系统中，关键现象仅出现在物理领域的一小部分中，并且可能会作为时间的函数移动（例如，作为火焰前部）。即使硬件方面取得了巨大进步，也不足以使用基于简单统一网格的软件来解决此类困难的大挑战类问题；问题的需求要求计算资源集中在最感兴趣的地区。 自适应网格算法的动机是算法本身可以并且应该识别这些关键区域，并以适当的网格反应，而几乎没有人类干预。 自适应算法的``大脑''是后误指标，既估计当前错误，又指示应将其他资源集中在哪里。由自适应算法产生的非常不均匀和非结构化的网格需要复杂的方法，例如Multigrid或拟议的HBMG/ILU，以有效且可靠地求解所得的方程系统。总体而言，该领域提供了重要且相互关联的科学问题的镶嵌，从数学分析中的困难问题到在现代计算机体系结构上实施这些程序时的艰难计算挑战。