Preconditioning Techniques for Algebraic Equations Arising from Partial Differential Equations

Elman这个项目涉及数值线性和非线性代数技术的发展、分析和测试，这些技术用于求解一组偏微分方程组的离散化。重点将放在预处理策略及其与线性和非线性系统的Krylov子空间迭代方法的结合上。这些想法将被应用于几个微分方程的例子，主要来自流体动力学模型。我们的重点将集中在两类问题，由不可压缩的Navier-Stokes方程的原始格式的直接离散产生的鞍点问题，以及对流-扩散方程。之所以选择它们，是因为它们被用于许多流体流动的工程模型，而且它们包含具有挑战性的数学和计算性质，使得它们很难求解。特别是，在二维和三维中的精确离散化导致了大型稀疏代数方程组是不对称的，并且对于第一个问题，是非线性的和不确定的。此外，它们还与雷诺数和离散化网格宽度等基本参数有关，对于实际感兴趣的问题的解决，导致许多算法收敛缓慢，需要大型系统才能达到精度。我们的目标是扩展和分析为鞍点问题设计的某些预适应技术。计划包括演示这些预条件在一般混合有限元离散中的有效性，发展用于分析收敛的严格数学技术，展示它们在涉及一般网格和区域的现实环境中的有效性，以及将对流扩散方程的快速算法纳入求解过程。开发计算算法的目的是使数学建模中使用的微分方程的有效数值解成为可能。这些模型的使用正成为制造和设计(例如，航空航天飞行器、汽车、核装置冷却装置)以及加强我们对关键自然现象(例如，血液流动和环境污染物扩散)的理解的一个日益重要的组成部分。通过纯粹的实验技术来理解这些现象是昂贵得令人望而却步的，或者是不可能的，而使用数学模型通过提供对流量和压力等数量的近似来引入对物理学的基本理解。这导致了对有前景的设计特征的识别，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，只有在有可靠和快速的求解算法的情况下，数学模型的准确求解才是可行的。这个项目的目标是开发这样的算法。

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.