| Decomposition Frameworks | p. 1 |
| Hybrid Formulations | p. 2 |
| Schwarz Framework | p. 9 |
| Steklov-Poincare Framework | p. 16 |
| Lagrange Multiplier Framework | p. 27 |
| Least Squares-Control Framework | p. 36 |
| Schwarz Iterative Algorithms | p. 47 |
| Background | p. 48 |
| Projection Formulation of Schwarz Algorithms | p. 56 |
| Matrix Form of Schwarz Subspace Algorithms | p. 66 |
| Implementational Issues | p. 72 |
| Theoretical Results | p. 77 |
| Schur Complement and Iterative Substructuring Algorithms | p. 107 |
| Background | p. 108 |
| Schur Complement System | p. 110 |
| FFT Based Direct Solvers | p. 125 |
| Two Subdomain Preconditioners | p. 140 |
| Preconditioners in Two Dimensions | p. 155 |
| Preconditioners in Three Dimensions | p. 162 |
| Neumann-Neumann and Balancing Preconditioners | p. 175 |
| Implementational Issues | p. 185 |
| Theoretical Results | p. 192 |
| Lagrange Multiplier Based Substructuring: FETI Method | p. 231 |
| Constrained Minimization Formulation | p. 232 |
| Lagrange Multiplier Formulation | p. 239 |
| Projected Gradient Algorithm | p. 241 |
| FETI-DP and BDDC Methods | p. 250 |
| Computational Issues and Parallelization | p. 263 |
| Algorithms for Automated Partitioning of Domains | p. 264 |
| Parallelizability of Domain Decomposition Solvers | p. 280 |
| Least Squares-Control Theory: Iterative Algorithms | p. 295 |
| Two Overlapping Subdomains | p. 296 |
| Two Non-Overlapping Subdomains | p. 303 |
| Extensions to Multiple Subdomains | p. 310 |
| Multilevel and Local Grid Refinement Methods | p. 313 |
| Multilevel Iterative Algorithms | p. 314 |
| Iterative Algorithms for Locally Refined Grids | p. 321 |
| Non-Self Adjoint Elliptic Equations: Iterative Methods | p. 333 |
| Background | p. 334 |
| Diffusion Dominated Case | p. 340 |
| Advection Dominated Case | p. 348 |
| Time Stepping Applications | p. 364 |
| Theoretical Results | p. 366 |
| Parabolic Equations | p. 377 |
| Background | p. 378 |
| Iterative Algorithms | p. 381 |
| Non-Iterative Algorithms | p. 384 |
| Parareal-Multiple Shooting Method | p. 401 |
| Theoretical Results | p. 408 |
| Saddle Point Problems | p. 417 |
| Properties of Saddle Point Systems | p. 418 |
| Algorithms Based on Duality | p. 426 |
| Penalty and Regularization Methods | p. 434 |
| Projection Methods | p. 437 |
| Krylov Space and Block Matrix Methods | p. 445 |
| Applications to the Stokes and Navier-Stokes Equations | p. 456 |
| Applications to Mixed Formulations of Elliptic Equations | p. 474 |
| Applications to Optimal Control Problems | p. 489 |
| Non-Matching Grid Discretizations | p. 515 |
| Multi-Subdomain Hybrid Formulations | p. 516 |
| Mortar Element Discretization: Saddle Point Approach | p. 523 |
| Mortar Element Discretization: Nonconforming Approach | p. 551 |
| Schwarz Discretizations on Overlapping Grids | p. 555 |
| Alternative Nonmatching Grid Discretization Methods | p. 559 |
| Applications to Parabolic Equations | p. 564 |
| Heterogeneous Domain Decomposition Methods | p. 575 |
| Steklov-Poincare Heterogeneous Model | p. 576 |
| Schwarz Heterogeneous Models | p. 585 |
| Least Squares-Control Heterogeneous Models | p. 589 |
| [chi]-Formulation | p. 594 |
| Applications to Parabolic Equations | p. 603 |
| Fictitious Domain and Domain Imbedding Methods | p. 607 |
| Background | p. 608 |
| Preconditioners for Neumann Problems | p. 610 |
| Preconditioners for Dirichlet Problems | p. 611 |
| Lagrange Multiplier and Least Squares-Control Solvers | p. 614 |
| Variational Inequalities and Obstacle Problems | p. 621 |
| Background | p. 622 |
| Projected Gradient and Relaxation Algorithms | p. 628 |
| Schwarz Algorithms for Variational Inequalities | p. 633 |
| Monotone Convergence of Schwarz Algorithms | p. 636 |
| Applications to Parabolic Variational Inequalities | p. 644 |
| Maximum Norm Theory | p. 647 |
| Maximum Principles and Comparison Theorems | p. 648 |
| Well Posedness of the Schwarz Hybrid Formulation | p. 659 |
| Convergence of Schwarz Iterative Algorithms | p. 661 |
| Analysis of Schwarz Nonmatching Grid Discretizations | p. 668 |
| Analysis of Schwarz Heterogeneous Approximations | p. 674 |
| Applications to Parabolic Equations | p. 677 |
| Eigenvalue Problems | p. 679 |
| Background | p. 680 |
| Gradient and Preconditioned Gradient Methods | p. 682 |
| Schur Complement Methods | p. 683 |
| Schwarz Subspace Methods | p. 684 |
| Modal Synthesis Method | p. 686 |
| Optimization Problems | p. 689 |
| Traditional Algorithms | p. 690 |
| Schwarz Minimization Algorithms | p. 697 |
| Helmholtz Scattering Problem | p. 699 |
| Background | p. 700 |
| Non-Overlapping and Overlapping Subdomain Methods | p. 701 |
| Fictitious Domain and Control Formulations | p. 704 |
| Hilbert Uniqueness Method for Standing Waves | p. 705 |
| References | p. 711 |
| Index | p. 761 |
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