多重网格（英文版）

多重网格（英文版）
出版时间：2015年版
内容简介
目次：导论；多重网格基础?；多重网格理论基础；局部傅里叶分析；多重网格基础Ⅱ；实践中的平行多重网格；更多高等多重网格；方程系统的多重网格；自适应多重网格；更多多重网格应用；附录：代数多重网格导论；子空间关联方法和多重网格应用；计算流体力学动力学中的多重网格有效性最新进展。
　　这是一部旨在为数学、物理、化学、气象学、流体和连续力学等众多领域的专家和学者介绍多重网格方法的教程。多重网格法在金融和经济学中也占据着越来越重要的地位，有读者认为这是多重网格法最好的书，没有之一，相信这样的评价是客观的。书中介绍了解决偏微分方程的多重网格法，和多重网格法的更高层次的最新研究以及在实践中的应用。本书三大特色：
　　1、囊括了多重网格这个领域的从基础到高等的所有；
　　2、风格简洁但数学严谨；
　　3、没有一本书能够想这本书做到既基础又专业，适于学生和科研人员。
　　读者对象：应用数学和力学等相关专业的科研人员。
目录
Preface
1 Introduction
1.1 Types of PDEs
1.2 Grids and Discretization Approaches
1.2.1 Grids
1.2.2 Discretization Approaches
1.3 Some Notation
1.3.1 Continuous Boundary Value Problems
1.3.2 Discrete Boundary Value Problems
1.3.3 Inner Products and Norms
1.3.4 Stencil Notation
1.4 Poisson's Equation and Model Problem 1
1.4.1 Matrix Terminology
1.4.2 Poisson Solvers
1.5 A First Glance at Multigrid
1.5.1 The Two Ingredients of Multigrid
1.5.2 High and Low Frequencies， and Coarse Meshes
1.5.3 From Two Grids to Multigrid
1.5.4 Multigrid Features
1.5.5 Multigrid History
1.6 Intermezzo： Some Basic Facts and Methods
1.6.1 Iterative Solvers， Splittings and Preconditioners
2 Basic Multigrid I
2.1 Error Smoothing Procedures
2.1.1 Jacobi—type Iteration （Relaxation）
2.1.2 Smoothing Properties of ω—Jacobi Relaxation
2.1.3 Gauss—Seidel—type Iteration （Relaxation）
2.1.4 Parallel Properties of Smoothers
2.2 Introducing the Two—grid Cycle
2.2.1 Iteration by Approximate Solution of the Defect Equation
2.2.2 Coarse Grid Correction
2.2.3 Structure of the Two—grid Operator
2.3 Multigrid Components
2.3.1 Choices of Coarse Grids
2.3.2 Choice of the Coarse Grid Operator
2.3.3 Transfer Operators： Restriction
2.3.4 Transfer Operators： Interpolation
2.4 The Multigrid Cycle
2.4.1 Sequences of Grids and Operators
2.4.2 Recursive Definition
2.4.3 Computational Work
2.5 Multigrid Convergence and Efficiency
2.5.1 An Efficient 2D Multigrid Poisson Solver
2.5.2 How to Measure the Multigrid Convergence Factor in Practice
2.5.3 Numerical Efficiency
2.6 Full Multigrid
2.6.1 Structure of Full Multigrid
2.6.2 Computational Work
2.6.3 FMG for Poisson's Equation
2.7 Further Remarks on Transfer Operators
2.8 First Generalizations
2.8.1 2D Poisson—like Differential Equations
2.8.2 Time—dependent Problems
2.8.3 Cartesian Grids in Nonrectangular Domains
2.8.4 Multigrid Components for Cell—centered Discretizations
2.9 Multigrid in 3D
2.9.1 The 3D Poisson Problem
2.9.2 3D Multigrid Components
2.9.3 Computational Work in 3D
3 Elementary Multigrid Theory
3.1 Survey
3.2 Why it is Sufficient to Derive Two—grid Convergence Factors
3.2.1 h—Independent Convergence of Multigrid
3.2.2 A Theoretical Estimate for Full Multigrid
3.3 How to Derive Two—grid Convergence Factors by Rigorous Fourier Analysis
3.3.1 Asymptotic Two—grid Convergence
3.3.2 Norms of the Two—grid Operator
3.3.3 Results for Multigrid
3.3.4 Essential Steps and Details of the TNo—grid Analysis
3.4 Range of Applicability of the Rigorous Fourier Analysis， Other Approaches
3.4.1 The 3D Case
3.4.2 Boundary Conditiops
3.4.3 List of Applications and Limitations
3.4.4 Towards Local Fourier Analysis
3.4.5 Smoothing and Approximation Property： a Theoretical Overview
4 Local Fourier Analysis
4.1 Background
4.2 Terminology
4.3 Smoothing Analysis 1
4.4 Two—grid Analysis
4.5 Smoothing Analysis 2
4.5.1 Local Fourier Analysis for GS—RB
4.6 Some Results， Remarks and Extensions
4.6.1 Some Local Fourier Analysis Results for Model Problem 1
4.6.2 Additional Remarks
4.6.3 Other Coarsening Strategies
4.7 h—Ellipticity
4.7.1 The Concept of h—Ellipticity
4.7.2 Smoothing and h—Ellipticity
5 Basic Multigrid 2
5.1 Anisotropic Equations in 2D
5.1.1 Failure of Pointwise Relaxation and Standard Coarsening
5.1.2 Semicoarsening
5.1.3 Line Smoothers
5.1.4 Strong Coupling of Unknowns in Two Directions
5.1.5 An Example with Varying Coefficients
5.2 Anisotropic Equations in 3D
5.2.1 Standard Coarsening for 3D Anisotropic Problems
5.2.2 Point Relaxation for 3D Anisotropic Problems
5.2.3 Further Approaches， Robust Variants
5.3 Nonlinear Problems， the Full Approximation Scheme
5.3.1 Classical Numerical Methods for Nonlinear PDEs： an Example
5.3.2 Local Linearization
5.3.3 Linear Multigrid in Connection with Global Linearization
5.3.4 Nonlinear Multigrid： the Full Approximation Scheme
5.3.5 Smoothing Analysis： a Simple Example
5.3.6 FAS for the Full Potential Equation
5.3.7 The （h， H）—Relative Truncation Error and τ—Extrapolation
5.4 Higher Order Discretizations
5.4.1 Defect Correction
5.4.2 The Mehrstellen Discretization for Poisson's Equation
5.5 Domains with Geometric Singularities
5.6 Boundary Conditions and Singular Systems
5.6.1 General Treatment of Boundary Conditions in Multigrid
5.6.2 Neumann Boundary Conditions
5.6.3 Periodic Boundary Conditions and Global Constraints
5.6.4 General Treatment of Singular Systems
5.7 Finite Volume Discretization and Curvilinear Grids
5.8 General Grid Structures
6 Parallel Multigrid in Practice
6.1 Parallelism of Multigrid Components
6.1.1 Parallel Components for Poisson's Equation
6.1.2 Parallel Complexity
6.2 Grid Partitioning
6.2.1 Parallel Systems， Processes and Basic Rules for Parallelization
6.2.2 Grid Partitioning for Jacobi and Red—Black Relaxation
6.2.3 Speed—up and Parallel Efficiency
6.2.4 A Simple Communication Model
6.2.5 Scalability and the Boundary—volume Effect
6.3 Grid Partitioning and Multigrid
6.3.1 Two—grid and Basic Multigrid Considerations
6.3.2 Multigrid and the Very Coarse Grids
6.3.3 Boundary—volume Effect and Scalability in the Multigrid Context
6.3.4 Programming Parallel Systems
6.4 Parallel Line Smoothers
6.4.11D Reduction （or Cyclic Reduction） Methods
6.4.2 Cyclic Reduction and Grid Partitioning
6.4.3 Parallel Plane Relaxation
6.5 Modifications of Multigrid and Related Approaches
6.5.1 Domain Decomposition Methods： a Brief Survey
6.5.2 Multigrid Related Parallel Approaches
……
7 More Advanced Multigrid
8 Multigrid for Systems of Equations
9 Adaptive Multigrid
10 Some More Multigrid Applications
Appendixes
References
Index