THE NUMERICAL SOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

By MARY THOMPSON

1 Decomposition Frameworks 1 11 Hybrid Formulations 2 12 Schwarz Framework 9 13 Steklov-Poincar´e Framework 16 14 Lagrange Multiplier Framework 27 15 Least Squares-Control Framework 36 2 Schwarz Iterative Algorithms 47 21 Background 48 22 Projection Formulation of Schwarz Algorithms 56 23 Matrix Form of Schwarz Subspace Algorithms 66 24 Implementational Issues 72 25 Theoretical Results 77 3 Schur Complement and Iterative Substructuring Algorithms 107 31 Background 108 32 Schur Complement System 110 33 FFT Based Direct Solvers 125 34 Two Subdomain Preconditioners 140 35 Preconditioners in Two Dimensions 155 36 Preconditioners in Three Dimensions 162 37 Neumann-Neumann and Balancing Preconditioners 175 38 Implementational Issues 185 39 Theoretical Results 192 4 Lagrange Multiplier Based Substructuring: FETI Method 231 41 Constrained Minimization Formulation 232 42 Lagrange Multiplier Formulation 239 43 Projected Gradient Algorithm 241 44 FETI-DP and BDDC Methods 250 5 Computational Issues and Parallelization 263 51 Algorithms for Automated Partitioning of Domains 264 52 Parallelizability of Domain Decomposition Solvers 280 6 Least Squares-Control Theory: Iterative Algorithms 295 61 Two Overlapping Subdomains 296 62 Two Non-Overlapping Subdomains 303 63 Extensions to Multiple Subdomains 310 7 Multilevel and Local Grid Refinement Methods 313 71 Multilevel Iterative Algorithms 314 72 Iterative Algorithms for Locally Refined Grids 321 8 Non-Self Adjoint Elliptic Equations: Iterative Methods 333 81 Background 334 82 Diffusion Dominated Case 340 83 Advection Dominated Case 348 84 Time Stepping Applications 364 85 Theoretical Results 366 9 Parabolic Equations 377 91 Background 378 92 Iterative Algorithms 381 93 Non-Iterative Algorithms 384 94 Parareal-Multiple Shooting Method 401 95 Theoretical Results 408 10 Saddle Point Problems 417 101 Properties of Saddle Point Systems 418 102 Algorithms Based on Duality 426 103 Penalty and Regularization Methods 434 104 Projection Methods 437 105 Krylov Space and Block Matrix Methods 445 106 Applications to the Stokes and Navier-Stokes Equations 456 107 Applications to Mixed Formulations of Elliptic Equations 474 108 Applications to Optimal Control Problems 489 11 Non-Matching Grid Discretizations 515 111 Multi-Subdomain Hybrid Formulations 516 112 Mortar Element Discretization: Saddle Point Approach 523 113 Mortar Element Discretization: Nonconforming Approach 551 114 Schwarz Discretizations on Overlapping Grids 555 115 Alternative Nonmatching Grid Discretization Methods 559 116 Applications to Parabolic Equations 564