| Introduction | p. 1 |
| Basic Ideas of Domain Decomposition | p. 1 |
| Matrix and Vector Representations | p. 2 |
| Nonoverlapping Methods | p. 5 |
| An Equation for u¿: the Schur Complement System | p. 5 |
| An Equation for the Flux | p. 6 |
| The Dirichlet-Neumann Algorithm | p. 8 |
| The Neumann-Neumann Algorithm | p. 10 |
| A Dirichlet-Dirichlet Algorithm or a FETI Method | p. 12 |
| The Case of Many Subdomains | p. 15 |
| The Schwarz Alternating Method | p. 21 |
| Description of the Method | p. 21 |
| The Schwarz Alternating Method as a Richardson Method | p. 22 |
| Block Jacobi Preconditioners | p. 24 |
| Some Results on Schwarz Alternating Methods | p. 27 |
| Analysis for the Case of Two Subdomains | p. 27 |
| The Case of More than Two Subdomains | p. 29 |
| Abstract Theory of Schwarz Methods | p. 35 |
| Introduction | p. 35 |
| Schwarz Methods | p. 35 |
| Convergence Theory | p. 39 |
| Historical Remarks | p. 46 |
| Additional Results | p. 46 |
| Coloring Techniques | p. 46 |
| A Hybrid Method | p. 47 |
| Comparison Results | p. 51 |
| Remarks on the Implementation | p. 52 |
| Two-Level Overlapping Methods | p. 55 |
| Introduction | p. 55 |
| Local Solvers | p. 56 |
| A Coarse Problem | p. 59 |
| Scaling and Quotient Space Arguments | p. 60 |
| Technical Tools | p. 62 |
| Convergence Results | p. 67 |
| Remarks on the Implementation | p. 70 |
| Numerical Results | p. 73 |
| Restricted Schwarz Algorithms | p. 75 |
| Alternative Coarse Problems | p. 75 |
| Convergence Results | p. 76 |
| Smoothed Aggregation Techniques | p. 81 |
| Partition of Unity Coarse Spaces | p. 84 |
| Substructuring Methods: Introduction | p. 87 |
| Introduction | p. 87 |
| Problem Setting and Geometry | p. 88 |
| Schur Complement Systems | p. 94 |
| Discrete Harmonic Extensions | p. 96 |
| Condition Number of the Schur Complement | p. 97 |
| Technical Tools | p. 99 |
| Interpolation into Coarse Spaces | p. 99 |
| Inequalities for Edges | p. 101 |
| Inequalities for Faces | p. 105 |
| Inequalities for Vertices and Auxiliary Results | p. 111 |
| Primal Iterative Substructuring Methods | p. 113 |
| Introduction | p. 113 |
| Local Design and Analysis | p. 113 |
| Local Solvers | p. 115 |
| Coarse Spaces and Condition Number Estimates | p. 117 |
| Vertex Based Methods | p. 118 |
| Wire Basket Based Algorithms | p. 123 |
| Face Based Algorithms | p. 126 |
| Neumann-Neumann and FETI Methods | p. 131 |
| Introduction | p. 131 |
| Balancing Neumann-Neumann Methods | p. 133 |
| Definition of the Algorithm | p. 133 |
| Matrix Form of the Algorithm | p. 137 |
| Condition Number Bounds | p. 139 |
| One-Level FETI Methods | p. 143 |
| A Review of the One-Level FETI Methods | p. 144 |
| The Case of Nonredundant Lagrange Multipliers | p. 150 |
| The Case of Redundant Lagrange Multipliers | p. 156 |
| Dual-Primal FETI Methods | p. 160 |
| FETI-DP Methods in Two Dimensions | p. 161 |
| A Family of FETI-DP Algorithms in Three Dimensions | p. 167 |
| Analysis of Three FETI-DP Algorithms | p. 175 |
| Implementation of FETI-DP Methods | p. 185 |
| Computational Results | p. 187 |
| Spectral Element Methods | p. 193 |
| Introduction | p. 193 |
| Deville-Mund Preconditioners | p. 196 |
| Two-Level Overlapping Schwarz Methods | p. 198 |
| Iterative Substructuring Methods | p. 200 |
| Technical Tools | p. 202 |
| Algorithms and Condition Number Bounds | p. 206 |
| Remarks on p and hp Approximations | p. 210 |
| More General p Approximations | p. 210 |
| Extensions to hp Approximations | p. 214 |
| Linear Elasticity | p. 217 |
| Introduction | p. 217 |
| A Two-Level Overlapping Method | p. 219 |
| Iterative Substructuring Methods | p. 220 |
| A Wire Basket Based Method | p. 221 |
| An Extension from the Interface | p. 222 |
| An Extension from the Wire Basket | p. 222 |
| A Wire Basket Preconditioner for Linear Elasticity | p. 224 |
| Neumann-Neumann and FETI Methods | p. 225 |
| A Neumann-Neumann Algorithm for Linear Elasticity | p. 225 |
| One-Level FETI Algorithms for Linear Elasticity | p. 227 |
| FETI-DP Algorithms for Linear Elasticity | p. 227 |
| Preconditioners for Saddle Point Problems | p. 231 |
| Introduction | p. 231 |
| Block Preconditioners | p. 235 |
| Flows in Porous Media | p. 239 |
| Iterative Substructuring Methods | p. 241 |
| Hybrid-Mixed Formulations and Spectral Equivalencies with Crouzeix-Raviart Approximations | p. 246 |
| A Balancing Neumann-Neumann Method | p. 250 |
| Overlapping Methods | p. 255 |
| The Stokes Problem and Almost Incompressible Elasticity | p. 257 |
| Block Preconditioners | p. 258 |
| Iterative Substructuring Methods | p. 261 |
| Computational Results | p. 269 |
| Problems in H(div ; ¿) and H(curl; ¿) | p. 271 |
| Overlapping Methods | p. 274 |
| Problems in H(curl; ¿) | p. 276 |
| Problems in H(div; ¿) | p. 283 |
| Final Remarks on Overlapping Methods and Numerical Results | p. 286 |
| Iterative Substructuring Methods | p. 288 |
| Technical Tools | p. 291 |
| A Face-Based Method | p. 299 |
| A Neumann-Neumann Method | p. 301 |
| Remarks on Two-Dimensional Problems and Numerical Results | p. 305 |
| Iterative Substructuring for Nédélec Approximations in Three Dimensions | p. 308 |
| Indefinite and Nonsymmetric Problems | p. 311 |
| Introduction | p. 311 |
| Algorithms on Overlapping Subregions | p. 314 |
| An Iterative Substructuring Method | p. 320 |
| Numerical Results | p. 321 |
| A Nonsymmetric Problem | p. 322 |
| The Helmholtz Equation | p. 324 |
| A Variable-Coefficient, Nonsymmetric Indefinite Problem | p. 324 |
| Additional Topics | p. 326 |
| Convection-Diffusion Problems | p. 326 |
| The Helmholtz Equation | p. 330 |
| Optimized Interface Conditions | p. 333 |
| Nonlinear and Eigenvalue Problems | p. 334 |
| Elliptic Problems and Sobolev Spaces | p. 337 |
| Sobolev Spaces | p. 337 |
| Trace Spaces | p. 341 |
| Linear Operators | p. 343 |
| Poincaré and Friedrichs Type Inequalities | p. 343 |
| Spaces of Vector-Valued Functions | p. 346 |
| The Space H(div; ¿) | p. 347 |
| The Space H(curl; ¿) in Two Dimensions | p. 348 |
| The Space H(curl; ¿) in Three Dimensions | p. 349 |
| The Kernel and Range of the Curl and Divergence Operators | p. 350 |
| Positive Definite Problems | p. 353 |
| Scalar Problems | p. 355 |
| Linear Elasticity | p. 357 |
| Problems in H(div; ¿) and H(curl; ¿) | p. 360 |
| Non-Symmetric and Indefinite Problems | p. 362 |
| Generalizations of the Lax-Milgram Lemma | p. 362 |
| Saddle-Point Problems | p. 364 |
| Regularity Results | p. 369 |
| Galerkin Approximations | p. 371 |
| Finite Element Approximations | p. 371 |
| Triangulations | p. 371 |
| Finite Element Spaces | p. 372 |
| Symmetric, Positive Definite Problems | p. 374 |
| Non-Symmetric and Indefinite Problems | p. 375 |
| Spectral Element Approximations | p. 376 |
| Divergence and Curl Conforming Finite Elements | p. 380 |
| Raviart-Thomas Elements | p. 380 |
| Nédélec Elements in Two Dimensions | p. 382 |
| Nédélec Elements in Three Dimensions | p. 383 |
| The Kernel and Range of the Curl and Divergence Operators | p. 384 |
| Saddle-Point Problems | p. 386 |
| Finite Element Approximations for the Stokes Problem | p. 387 |
| Spectral Element Approximations for the Stokes Problem | p. 388 |
| Finite Element Approximations for Flows in Porous Media | p. 389 |
| Inverse Inequalities | p. 389 |
| Matrix Representation and Condition Number | p. 390 |
| Solution of Algebraic Linear Systems | p. 395 |
| Eigenvalues and Condition Number | p. 395 |
| Direct Methods | p. 397 |
| Factorizations | p. 397 |
| Fill-in | p. 398 |
| Richardson Method | p. 399 |
| Steepest Descent | p. 402 |
| Conjugate Gradient Method | p. 403 |
| Methods for Non-Symmetric and Indefinite Systems | p. 407 |
| The Generalized Minimal Residual Method | p. 407 |
| The Conjugate Residual Method | p. 409 |
| References | p. 413 |
| Index | p. 447 |
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