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
7 Spectral Element Methods 8 Linear Elasticity 9 Preconditioners for Saddle Point Problems 10 Problems in H(div; )and H (curl;) 11 Indefinite and Nonsymmetric Problems A Elliptic Problems and Sobolev Spaces B Galerkin Approximations C Solution of Algebraic Linear Systemse References Index
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