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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