作者：A.Toselli，O.Widlund[
页数：450
出版社：科学出版社
出版日期：2006
ISBN：9787030166906
电子书格式：pdf/epub/txt
内容简介
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目录
1 Introduction
1.1 Basic Ideas of Domain Decomposition
1.2 Matrix and Vector Representations
1.3 Nonoverlapping Methods
1.3.1 An Equation for ur: the Schur Complement System
1.3.2 An Equation for the Flux
1.3.3 The Dirichlet—Neumann Algorithm
1.3.4 The Neumann—Neumann Algorithm
1.3.5 A Dirichlet—Dirichlet Algorithm or a FETI Method
1.3.6 The Case of Many Subdomains
1.4 The Schwarz Alternating Method
1.4.1 Description of the Method
1.4.2 The Schwarz Alternating Method as a Richardson Method
1.5 Block Jacobi Preconditioners
1.6 Some Results on Schwarz Alternating Methods
1.6.1 Analysis for the Case of Two Subdomains
1.6.2 The Case of More than Two Subdomains
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
2.5.1 Coloring Techniques
2.5.2 A Hybrid Method
2.5.3 Comparison 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
3.10.1 Convergence Results
3.10.2 Smoothed Aggregation Techniques
3.10.3 Partition of Unity Coarse Spaces
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
4.6.1 Interpolation into Coarse Spaces
4.6.2 Inequalities for Edges
4.6.3 Inequalities for Faces
4.6.4 Inequalities for Vertices and Auxiliary Results
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
5.4.1 Vertex Based Methods
5.4.2 Wire Basket Based Algorithms
5.4.3 Face Based Algorithms
6 Neumann—Neumann and FETI Methods
6.1 Introduction
6.2 Balancing Neumann—Neumann Methods
6.2.1 Definition of the Algorithm
6.2.2 Matrix Form of the Algorithm
6.2.3 Condition Number Bounds
6.3 One—Level FETI Methods
6.3.1 A Review of the One—Level FETI Methods
6.3.2 The Case of Nonredundant Lagrange Multipliers
6.4 Dual—Primal FETI Methods
6.4.1 FETI—DP Methods in Two Dimensions
6.4.2 A Fanuly of FETI—DP Algorithms in Three Dimensions
6.4.3 Analysis of Three FETI—DP Algorithms
6.4.4 Implementation of FETI—DP Methods
6.4.5 Computational Results
7 Spectral Element Methods
7.1 Jntroduction
7.2 Deville—Mund Preconditroners
7.3 Two—Level Overlapping Schwarz Methods
7.4 Iterative Substructuring Methods
7.4.1 Technical Tools
7.4.2 Algorithms and Condition Number Bounds
7.5 Remarks on p and hp Appro）amations
7.5.1 More General p Approximations
7.5.2 Extensions to 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
8.4.1 An Extension from the Interface
8.4.2 An Extension from the Wire Basket
8.4.3 A Wire Basket Preconditioner for Linear Elasticity
8.5 Neumann—Neumann and FETI Methods
8.5.1 A Neumann—Neumann Algorithm for Linear Elasticity
8.5.2 One—Level FETI Algorithms for Linear Elasticity
8.5.3 FETI—DP Algorithms for Linear Elasticity
9 Preconditioners for Saddle Point Problems
9.1 Introduction
9.2 Block Preconditioners
9.3 Flows in Porous Media
9.3.1 Iterative Substructuring Methods
9.3.2 Hybrid—Mixed Formulations and Spectral Equivalencies with Crouzeix—Raviart， Approximations
9.3.3 A Balancing Neumann—Neumann Method
9.3.4 Overlapping Methods
9.4 The Stokes Problem and Almost Incompressible Elasticity
9.4.1 Block Preconditioners
9.4.2 Iterative Substructuring Methods
9.4.3 Computational R，esults
10 Problems in H（div；Ω） and H（curl；Ω）
10.1 Overlapping Methods
10.1.1 Problems in H（curl；Ω）
10.1.2 Problems in H （div；Ω）
10.1.3 Final Remarks on Overlapping Methods and Numerical Results
10.2 Iterative Substructuring Methods
10.2.1 Technical Tools
10.2.2 A Face—Based Method
10.2.3 A Neumann—Neumann Method
10.2.4 Remarks on Two—Dimensional Problems and Numerical Results
10.2.5 Iterative Substructuring for Nedelec Approximations in Three Dimensions
11 Indefinute and Nonsymmetric Problems
11.1 Introduction
11.2 Algorithms on Overlapping Subregions
11.3 An Iterative Substructuring Method
11.4 Numerical Results
11.4.1 A Nonsymmetric Problem
11.4.2 The Helmholtz Equation
11.4.3 A Variable—Coefficient， Nonsymmetric Indefinite Problem
11.5 Additional Topics
11.5.1 Convection—Diffusion Problems
11.5.2 The Helmholtz Equation
11.5.3 Optimized Interface Conditions
11.5.4 Nonlinear and Eigenvalue Problems
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.5.1 The Space H（div；Ω）
A.5.2 The Space H（curl；Ω） in Two Dimensions
A.5.3 The Space H（curl；Ω） in Three Dimensions
A.5.4 The Kernel and R，ange of the Curl and Divergence Operators
A.6 Positive Definite Problems
A.6.1 Scalar Problems
A.6.2 Linear Elasticity
A.6.3 Problems in H （div；Ω） and H （curl；Ω）
A.7 Non—Symmetric and Indefinite Problems
A.7.1 Generalizations of the Lax—Milgram Lemma
A.7.2 Saddle—Point Problems
A.8 Regularity Results
B Galerkin Approximations
B.1 Finite Element Approximations
B.1.1 Triangulations
B.1.2 Finite Element Spaces
B.1.3 Symmetric， Positive Definite Problems
B.1.4 Non—Symmetric and Indefinite Problems
B.2 Spectral Element Approximations
B.3 Divergence and Curl Conforming Finite Elements
B.3.1 Raviart—Thomas Elements
B.3.2 Nedelec Elements in Two Dimensions
B.3.3 Nedelec Elements in Three Dimensions
B.3.4 The Kernel and Range of the Curl and Divergence Operators
B.4 Saddle—Point Problems
B.4.1 Finite Element Approximations for the Stokes Problem
B.4.2 Spectral Element Approximations for the Stokes Problem
B.4.3 Firute Element Approximations for Flows in Porous Media
B.5 Inverselnequalities
B.6 Matrix Representation and Condition Number
C Solution of Algebraic Linear Systems
C.1 Eigenvalues and Condition Number
C.2 Direct Methods
C.2.1 Factorizations
C.2.2 Fill—in
C.3 Richardson Method
C.4 Steepest Descent
C.5 Conjugate Gradient Method
C.6 Methods for Non—Symmetric and Indefinite Systems
C.6.1 The Generalized Minimal Residual Method
C.6.2 The Conjugate Residual Method
References
Index
1.1 Basic Ideas of Domain Decomposition
1.2 Matrix and Vector Representations
1.3 Nonoverlapping Methods
1.3.1 An Equation for ur: the Schur Complement System
1.3.2 An Equation for the Flux
1.3.3 The Dirichlet—Neumann Algorithm
1.3.4 The Neumann—Neumann Algorithm
1.3.5 A Dirichlet—Dirichlet Algorithm or a FETI Method
1.3.6 The Case of Many Subdomains
1.4 The Schwarz Alternating Method
1.4.1 Description of the Method
1.4.2 The Schwarz Alternating Method as a Richardson Method
1.5 Block Jacobi Preconditioners
1.6 Some Results on Schwarz Alternating Methods
1.6.1 Analysis for the Case of Two Subdomains
1.6.2 The Case of More than Two Subdomains
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
2.5.1 Coloring Techniques
2.5.2 A Hybrid Method
2.5.3 Comparison 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
3.10.1 Convergence Results
3.10.2 Smoothed Aggregation Techniques
3.10.3 Partition of Unity Coarse Spaces
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
4.6.1 Interpolation into Coarse Spaces
4.6.2 Inequalities for Edges
4.6.3 Inequalities for Faces
4.6.4 Inequalities for Vertices and Auxiliary Results
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
5.4.1 Vertex Based Methods
5.4.2 Wire Basket Based Algorithms
5.4.3 Face Based Algorithms
6 Neumann—Neumann and FETI Methods
6.1 Introduction
6.2 Balancing Neumann—Neumann Methods
6.2.1 Definition of the Algorithm
6.2.2 Matrix Form of the Algorithm
6.2.3 Condition Number Bounds
6.3 One—Level FETI Methods
6.3.1 A Review of the One—Level FETI Methods
6.3.2 The Case of Nonredundant Lagrange Multipliers
6.4 Dual—Primal FETI Methods
6.4.1 FETI—DP Methods in Two Dimensions
6.4.2 A Fanuly of FETI—DP Algorithms in Three Dimensions
6.4.3 Analysis of Three FETI—DP Algorithms
6.4.4 Implementation of FETI—DP Methods
6.4.5 Computational Results
7 Spectral Element Methods
7.1 Jntroduction
7.2 Deville—Mund Preconditroners
7.3 Two—Level Overlapping Schwarz Methods
7.4 Iterative Substructuring Methods
7.4.1 Technical Tools
7.4.2 Algorithms and Condition Number Bounds
7.5 Remarks on p and hp Appro）amations
7.5.1 More General p Approximations
7.5.2 Extensions to 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
8.4.1 An Extension from the Interface
8.4.2 An Extension from the Wire Basket
8.4.3 A Wire Basket Preconditioner for Linear Elasticity
8.5 Neumann—Neumann and FETI Methods
8.5.1 A Neumann—Neumann Algorithm for Linear Elasticity
8.5.2 One—Level FETI Algorithms for Linear Elasticity
8.5.3 FETI—DP Algorithms for Linear Elasticity
9 Preconditioners for Saddle Point Problems
9.1 Introduction
9.2 Block Preconditioners
9.3 Flows in Porous Media
9.3.1 Iterative Substructuring Methods
9.3.2 Hybrid—Mixed Formulations and Spectral Equivalencies with Crouzeix—Raviart， Approximations
9.3.3 A Balancing Neumann—Neumann Method
9.3.4 Overlapping Methods
9.4 The Stokes Problem and Almost Incompressible Elasticity
9.4.1 Block Preconditioners
9.4.2 Iterative Substructuring Methods
9.4.3 Computational R，esults
10 Problems in H（div；Ω） and H（curl；Ω）
10.1 Overlapping Methods
10.1.1 Problems in H（curl；Ω）
10.1.2 Problems in H （div；Ω）
10.1.3 Final Remarks on Overlapping Methods and Numerical Results
10.2 Iterative Substructuring Methods
10.2.1 Technical Tools
10.2.2 A Face—Based Method
10.2.3 A Neumann—Neumann Method
10.2.4 Remarks on Two—Dimensional Problems and Numerical Results
10.2.5 Iterative Substructuring for Nedelec Approximations in Three Dimensions
11 Indefinute and Nonsymmetric Problems
11.1 Introduction
11.2 Algorithms on Overlapping Subregions
11.3 An Iterative Substructuring Method
11.4 Numerical Results
11.4.1 A Nonsymmetric Problem
11.4.2 The Helmholtz Equation
11.4.3 A Variable—Coefficient， Nonsymmetric Indefinite Problem
11.5 Additional Topics
11.5.1 Convection—Diffusion Problems
11.5.2 The Helmholtz Equation
11.5.3 Optimized Interface Conditions
11.5.4 Nonlinear and Eigenvalue Problems
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.5.1 The Space H（div；Ω）
A.5.2 The Space H（curl；Ω） in Two Dimensions
A.5.3 The Space H（curl；Ω） in Three Dimensions
A.5.4 The Kernel and R，ange of the Curl and Divergence Operators
A.6 Positive Definite Problems
A.6.1 Scalar Problems
A.6.2 Linear Elasticity
A.6.3 Problems in H （div；Ω） and H （curl；Ω）
A.7 Non—Symmetric and Indefinite Problems
A.7.1 Generalizations of the Lax—Milgram Lemma
A.7.2 Saddle—Point Problems
A.8 Regularity Results
B Galerkin Approximations
B.1 Finite Element Approximations
B.1.1 Triangulations
B.1.2 Finite Element Spaces
B.1.3 Symmetric， Positive Definite Problems
B.1.4 Non—Symmetric and Indefinite Problems
B.2 Spectral Element Approximations
B.3 Divergence and Curl Conforming Finite Elements
B.3.1 Raviart—Thomas Elements
B.3.2 Nedelec Elements in Two Dimensions
B.3.3 Nedelec Elements in Three Dimensions
B.3.4 The Kernel and Range of the Curl and Divergence Operators
B.4 Saddle—Point Problems
B.4.1 Finite Element Approximations for the Stokes Problem
B.4.2 Spectral Element Approximations for the Stokes Problem
B.4.3 Firute Element Approximations for Flows in Porous Media
B.5 Inverselnequalities
B.6 Matrix Representation and Condition Number
C Solution of Algebraic Linear Systems
C.1 Eigenvalues and Condition Number
C.2 Direct Methods
C.2.1 Factorizations
C.2.2 Fill—in
C.3 Richardson Method
C.4 Steepest Descent
C.5 Conjugate Gradient Method
C.6 Methods for Non—Symmetric and Indefinite Systems
C.6.1 The Generalized Minimal Residual Method
C.6.2 The Conjugate Residual Method
References
Index
