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解有限元方程的Bank分层基多重网格/ILU算法Randolph E.加州大学圣迭戈分校数学系，CA 92093这项建议有两个主要部分。首先，我们将研究代数分层基多重网格算法(HBMG/ILU)。这些方法求解由偏微分方程组的有限差分、有限体积和有限元离散所产生的稀疏线性方程组。它们不同于经典的HBMG和MG方法，因为它们不需要粗网格和网格细化序列。这允许应用于具有几何复杂区域的问题，该问题仅需要用于几何定义的许多元素，以及其中自适应来自移动网格点而不是来自精化的问题。初步的数值实验表明，该方法具有很强的计算能力和较强的鲁棒性。建议的第二部分涉及有限元程序PLTMG的持续软件开发。自20世纪70年代末S以来，该程序的各种版本一直处于公共领域，并在教育和研究环境中广泛使用。PLTMG求解平面上一般区域的标量、参数相关的非线性椭圆型偏微分方程组。其主要特点是自适应网格生成，后验误差估计，线性方程组的HBMG(即将成为HBMG/ILU)迭代，非线性的牛顿方法，以及参数依赖的连续性。该代码还包括初始网格生成器、骨架生成器和几个图形例程。尽管PLTMG的名称保持不变，但通常每个新版本都会修改包中80%或更多的内容。在许多用偏微分方程建模的系统中，临界现象只出现在物理域的一小部分，并且可能作为时间的函数移动(例如，作为火焰锋面)。即使硬件有了很大的进步，使用基于简单统一网格的软件来解决这类困难的大挑战类问题也是不够的；问题的要求要求计算资源集中在最感兴趣的区域上。自适应网格算法的动机是，算法本身能够而且应该识别这些关键区域，并在很少或根本没有人工干预的情况下使用适当的网格进行响应。自适应算法的“大脑”是一个后验误差指示器，它既估计当前的误差，又指示应该将额外的资源集中在哪里。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，如多重网格或建议的HBMG/ILU，以高效和可靠地求解所产生的方程组。总体而言，该领域提供了一系列重要且相互关联的科学问题，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过程的困难计算挑战。