区域分解算法:算法与理论(影印版)(精)/国外数学名著系列

本书的目的是全面讲述偏微分方程中的各种最成功、最通用的区域分解方法，有限元逼近和谱元素逼近预条件算子，内容上做到白包含，而主要侧重点在于算法和数学方面。本书详细阐述的一些重要的方法如FETI、平衡Neumann-Neumann方法、谱元素方法等，都是第一次在数学专著中出现。

　　本书的目的是全面讲述偏微分方程中的各种最成功、最通用的区域分解方法，有限元逼近和谱元素逼近预条件算子，内容上做到白包含，而主要侧重点在于算法和数学方面。本书详细阐述的一些重要的方法如FETI、平衡Neumann-Neumann方法、谱元素方法等，都是第一次在数学专著中出现。

1 Introduction
1.1 Basic Ideas of Domain Decomposition
1.2 Matrix and Vector Representations
1.3 Nonoverlapping Methods
1.4 The Schwarz Alternating Method
1.5 Block Jacobi Preconditioners
1.6 Some Results on Schwarz Alternating Methods
2 Abstract Theory of Schwarz Methods
2.1 Introduction
2.2 Schwarz Methods
2.3 Convergence Theory
2.4 Historical Remarks
2.5 Additional Results
2.6 Remarks on the Implementation
3 Two-Level Overlapping Methods
3.1 Introduction
3.2 Local Solvers
3.3 A Coarse Problem
3.4 Scaling and Quotient Space Arguments
3.5 Technical Tools
3.6 Convergence Results
3.7 Remarks on the Implementation
3.8 Numerical Results
3.9 Restricted Schwarz Algorithms
3.10 Alternative Coarse Problems
4 Substructruing Methods: Introduction
4.1 Introduction
4.2 Problem Setting and Geometry
4.3 Schur Complement Systems
4.4 Discrete Harmonic Extensions
4.5 Condition Number of the Schur Complement
4.6 Technical Tools
5 Primal Iterative Substructuring Methods
5.1 Introduction
5.2 Local Design and Analysia
5.3 Local Solvers
5.4 Coarse Spaces and Condition Number Estimates
6 Neumann-Neumann and FETI Methods
6.1 Introduction
6.2 Balancing Neumann-Neumann Methods
6.3 One-Level FETI Methods
6.4 Dual-Primal FETI Methods
7 Spectral Element Methods
7.1 Introduction
7.2 Deville-Mund Preconditioners
7.3 Two-Level Overlapping Schwarz Methods
7.4 Iterative Substructuring Methods
7.5 Remarks on p and hp Approximations
8 Linear Elasticity
8.1 Introduction
8.2 A Two-Level Overlapping Method
8.3 Iterative Substructuring Methods
8.4 A Wire Basket Based Method
8.5 Neumann-Neumann and FETI Methods
9 Preconditioners for Saddle Point Problems
9.1 Introduction
9.2 Bolck Preconditioners
9.3 Flows in Porous Media
9.4 The Stokes Problem and Almost Incompressible Elasticity
10 Problems in H(div; )and H (curl;)
10.1 Overlapping Methods
10.2 Iterative Substructuring Methods
11 Indefinite and Nonsymmetric Problems
11.1 Introduction
11.2 Algorithms on Overlapping Subregions
11.3 An Iterative Substructuring Method
11.4 Numerical Results
11.5 Additional Topics
A Elliptic Problems and Sobolev Spaces
A.1 Sobolev Spaces
A.2 Trace Spaces
A.3 Linear Operators
A.4 Poincare and Friedrichs Type Inequalities
A.5 Spaces of Vector-Valued Functions
A.6 Positive Definite Problems
A.7 Non-Symmetric and Indefinite Problems
A.8 Regularity Results
B Galerkin Approximations
B.1 Finite Element Approximations
B.2 Spectral Element Approximations
B.3 Divergence and Curl Conforming Finite Elements
B.4 Saddle-Point Problems
B.5 Inverse Inequalities
B.6 Matrix Representation and Condition Number
C Solution of Algebraic Linear Systemse
C.1 Eigenvalues and Condition Number
C.2 Direct Methods
C.3 Richardson Method
C.4 Steepest Descent
C.5 Conjugate Gradient Method
C.6 Methods for Non-Symmetric and Indefinite Systems
References
Index

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