Heirarchical Basis Multigrid/ILU Algorithms for Solving Finite Element Equations

求解有限元方程的分层基础Multigrid/ILU算法Randolph E. Bank加州大学圣地亚哥分校数学系La Jolla， CA 92093该提案有两个主要组成部分。首先，我们将研究代数分层基多网格算法（HBMG/ILU）。这些方法求解由偏微分方程的有限差分、有限体积和有限元离散引起的线性方程的稀疏集。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。这允许应用程序解决几何复杂域的问题，这些域需要许多元素来进行几何定义，以及自适应来自移动网格点而不是来自细化的问题。初步的数值实验表明，该方法具有很强的鲁棒性。提案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年代末以来，这个程序的各种版本已经进入公共领域，它被广泛用于教育和研究环境。PLTMG在平面的一般区域内求解标量、参数相关的非线性椭圆偏微分方程。该方法的主要特点是自适应网格生成、后检误差估计、线性方程组的HBMG（即将成为HBMG/ILU）迭代、非线性方程组的牛顿法和参数依赖性的延拓。代码还包括一个初始网格生成器、一个骨架生成器和几个图形例程。尽管PLTMG的名称保持不变，但通常在每个新版本中，软件包的80%或更多内容都会被修改。在许多用偏微分方程建模的系统中，临界现象只发生在物理域的一小部分，并且可能作为时间的函数而移动（例如，作为火焰锋）。即使在硬件方面取得了巨大的进步，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；该问题的需求要求将计算资源集中在最感兴趣的区域。自适应网格算法的动机是，算法本身能够并且应该识别这些关键区域，并在很少或没有人为干预的情况下以适当的网格响应。自适应算法的“大脑”是一个后验误差指示器，它既可以估计当前的误差，也可以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，如多重网格或提出的HBMG/ILU，以有效和可靠地求解生成的方程组。总的来说，这个领域提供了一系列重要的和相互关联的科学问题，从数学分析中的难题到在现代计算机体系结构中实现这些程序的困难计算挑战。

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.