Preconditioning Techniques for Algebraic Equations Arising from Partial Differential Equations

由偏微分方程产生的代数方程的预处理技术

• 批准号: 9972490
• 负责人: Howard Elman
• 金额: $ 13万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9972490&HistoricalAwards=false
• 关键词: Preconditioning; Techniques; Algebraic; Equations; Arising

ElmanThis project concerns the development, analysis and testing of techniquesof numerical linear and nonlinear algebra for solving systems of equationsarising from discretization of a collection of partial differential equations.The emphasis will be on preconditioning strategies and their combination with Krylov subspace iterative methods for linear and nonlinear systems.These ideas will be applied to several examples of differential equations,primarily coming from models of fluid dynamics. Our focus will be on two types of problems, the saddle point problems produced by direct discretization of the primitive formulation of the incompressible Navier-Stokes equations, and the convection-diffusion equation. These have been chosen because they are used in numerous engineering models of fluid flow, and because they containchallenging mathematical and computational qualities that make them difficult to solve. In particular, accurate discretization in two and three dimensions leads to large sparse systems of algebraic equations that are nonsymmetric,and for the first problem, nonlinear and indefinite. In addition, they have associated with them fundamental parameters such as Reynolds numbers and discretization mesh widths that, for the solution of problems of practical interest, cause many algorithms to converge slowly and require large-scale systems to attain accuracy. Our goals are to extend and analyze certain preconditioning techniques designed for saddle point problems. Plans include demonstration of the effectiveness of these preconditioners for general mixed finite element discretizations, development of rigorous mathematical techniquesfor analyzing convergence, demonstration of their usefulness in realistic settings involving general meshes and domains, and incorporation of fastalgorithms for the convection-diffusion equation into the solution process.The purpose of developing computational algorithms is to enable the efficient numerical solution of differential equations used in mathematical modelling. The use of such models is becoming an increasingly importantcomponent in manufacturing and design (for example, of aerospace vehicles,automobiles, cooling devices for nuclear devices) and in enhancing ourunderstanding of critical natural phenomena (for example, blood flows anddispersal of environmental pollutants). Understanding these phenomenathrough purely experimental techniques is prohibitively expensive orimpossible, whereas the use of mathematical models introduces a basic understanding of the physics by providing approximations to quantitiessuch as flow rates and pressures. This leads to the identification of promising design features, such as wing shape in airplane manufacturing. (For example, many of the design decisions for the Boeing 777 jet were made using computational studies.) Accurate solution of the mathematical models is only feasible, however, if reliable and fast solution algorithms are available. The goal of this project is to develop such algorithms.

Elman该项目涉及数值线性和非线性代数技术的开发、分析和测试，用于求解由偏微分方程组离散化产生的方程组。重点是预处理策略及其与线性和非线性的 Krylov 子空间迭代方法的结合这些想法将应用于微分方程的几个例子，主要来自流体动力学模型。 我们的重点将放在两类问题上，即由不可压缩纳维-斯托克斯方程的原始公式直接离散化产生的鞍点问题，以及对流扩散方程。 之所以选择这些模型，是因为它们被用于众多流体流动的工程模型中，并且因为它们包含具有挑战性的数学和计算质量，使得它们难以求解。 特别是，二维和三维的精确离散化会导致非对称的大型稀疏代数方程组，并且对于第一个问题来说，是非线性和不定的。 此外，它们还与雷诺数和离散化网格宽度等基本参数相关联，为了解决实际感兴趣的问题，这些参数导致许多算法收敛缓慢，并且需要大规模系统才能达到精度。 我们的目标是扩展和分析为鞍点问题设计的某些预处理技术。 计划包括演示这些预处理器对于一般混合有限元离散化的有效性，开发用于分析收敛性的严格数学技术，演示它们在涉及一般网格和域的实际设置中的有用性，以及将对流扩散方程的快速算法纳入解决方案开发计算算法的目的是为了能够有效地数值求解数学建模中使用的微分方程。 这些模型的使用正在成为制造和设计（例如，航空航天器、汽车、核装置冷却装置）以及增强我们对重要自然现象（例如，血流和环境污染物的扩散）的理解中越来越重要的组成部分。 通过纯粹的实验技术来理解这些现象是非常昂贵或不可能的，而数学模型的使用通过提供流量和压力等数量的近似值来引入对物理学的基本理解。 这导致了有前途的设计特征的识别，例如飞机制造中的机翼形状。 （例如，波音 777 喷气式飞机的许多设计决策都是通过计算研究做出的。）然而，只有可靠且快速的求解算法可用，数学模型的精确求解才是可行的。 该项目的目标是开发此类算法。