Preface......Page 6
Contents......Page 9
Part I Space Discretization Methods for PDEs......Page 15
1 Introduction......Page 16
2 The Physical Problem and Mathematical Formulation......Page 17
3 DRBEM Application......Page 19
4 Numerical Results......Page 20
5 Conclusion......Page 22
References......Page 23
1 Introduction......Page 25
2 Governing Equations......Page 26
3 Application of the DRBEM......Page 28
4 Numerical Results......Page 29
References......Page 32
1 Introduction......Page 34
2 Modification of the Homogeneous Flux......Page 36
3 Modification of the Inhomogeneous Flux......Page 38
4 Numerical Example......Page 40
5 Concluding Remarks......Page 41
References......Page 42
1 Introduction......Page 43
2.1 Finite Element Method......Page 44
2.2 Finite Volume Method with Flux-Limiting......Page 45
2.3 Discontinuous Galerkin Method......Page 46
4 Numerical Examples......Page 47
4.1 Transport Equation......Page 48
4.2 Buckley-Leverett Equation......Page 49
References......Page 50
1 Introduction......Page 52
2 Finite Volume Method......Page 53
3 Integral Representation of the Fluxes......Page 55
3.1 Closure of the Scheme......Page 57
4 Numerical Results......Page 58
References......Page 60
1 Introduction......Page 61
2.1 Reconstruction Procedure......Page 62
2.2 Fully Discrete Scheme......Page 64
3.1 A Sufficient Condition in the Linear Case......Page 65
3.2 Failure of q>2: Numerical Evidence......Page 66
3.3 Failure of q>2: Analytical Evidence......Page 67
References......Page 69
2 The Boundary Value Problem......Page 71
2.1 The Equations......Page 72
2.2 Boundary and Matching Conditions......Page 73
3 Singular Equations......Page 74
4 Discretization......Page 76
6 The Output Power......Page 78
References......Page 80
Part II Finite Element Methods......Page 81
1 Introduction......Page 82
2 Description of the Scheme......Page 84
2.1 Formulation for Unsteady Problems......Page 85
3.2 Simulations......Page 89
References......Page 92
1 Introduction......Page 94
2 Approximation Spaces in H(div,Ω)......Page 95
3 Application to Mixed Finite Element Formulation......Page 96
References......Page 101
1 Introduction......Page 102
2 Variational Formulation......Page 103
3 Finite Element Discretization......Page 104
4 A Priori Estimate......Page 105
5 Numerical Results......Page 108
References......Page 110
1 Introduction......Page 111
2 Petrov-Galerkin Framework and Quasi-Optimality......Page 112
3 Two Weak Formulations of Linear Parabolic Problems......Page 113
3.1 Standard Weak Formulation......Page 114
3.2 Ultra-Weak Formulation......Page 115
4 Galerkin Approximation in Space and Quasi-Optimality Constants......Page 116
4.1 Standard Weak Formulation......Page 117
4.2 Ultra-Weak Formulation......Page 118
References......Page 119
1 Introduction......Page 120
2 Numerical Scheme......Page 121
3.1 Numerical Experiment 1: Convergence Rate Verification......Page 123
3.2 Numerical Experiment 2: Marsigli's Flow with Re=2,000......Page 124
References......Page 127
1 Introduction......Page 128
2 Finite Element Heterogeneous Multiscale Method for Linear Elasticity......Page 130
3 A Priori Error Estimates......Page 131
4 Numerical Experiments......Page 132
References......Page 135
1 Substructuring, Trace Spaces, and Minimization......Page 137
2 Weakly Conforming Approximation......Page 140
3 Minimal Residuals......Page 142
References......Page 145
1 Introduction......Page 146
2 Iterative Coupling Scheme and Space-Time Approximation of the Subproblems......Page 148
3 Numerical Experiments......Page 151
References......Page 153
Part III Discontinuous Galerkin Methods for PDEs......Page 155
1 Description of the Dynamic Elasticity Problem......Page 156
2 Discrete Problem......Page 158
3.1 A Benchmark Problem......Page 160
3.2 Example of Fluid-Structure Interaction......Page 161
References......Page 163
1 Introduction......Page 165
2 Notation......Page 166
3 Stability of Lifted Gradients......Page 168
4 Counterexample to Stability for Equal-Order Liftings......Page 170
5 A Modified LDG Method Without Penalty Parameters......Page 171
References......Page 173
1 Introduction......Page 174
2 Time-Space Discretization......Page 175
3 Adaptive Method of Time Layers (AMOT)......Page 177
4.1 Sheering Flow......Page 179
4.2 Expanding Flow......Page 180
References......Page 181
1 Continuous Problem......Page 183
2 Discretization......Page 185
2.2 Numerical Flux......Page 186
2.3 Time Discretization......Page 187
2.3.1 Linearization of the Source Terms lh......Page 188
3 Numerical Experiment......Page 189
References......Page 191
1 Introduction......Page 192
2 The Phase-Field Fracture Model......Page 193
3 A Quasi-monolithic Incremental Formulation......Page 194
4 Spatial Discretization with DG and EG......Page 195
5 A Numerical Test: Single Edge Notched Tension......Page 197
References......Page 200
1 Introduction......Page 201
3 Discretization......Page 203
3.2 Discretization of the Navier-Stokes Equations......Page 204
4 Numerical Experiment......Page 206
5 Conclusion......Page 208
References......Page 209
1 Formulation of the Continuous Problem......Page 210
2 Space-Time Semidiscretization......Page 212
3 Analysis of the Stability......Page 214
References......Page 218
1 Introduction......Page 219
3 Discretization......Page 220
4.1 Error Measure......Page 222
4.3 Reconstruction of the Solution with Respect to Space......Page 224
4.4 Upper Error Bound......Page 225
4.5 Asymptotic Lower Error Bound......Page 226
References......Page 227
Part IV Numerical Linear Algebra and High Performance Computing......Page 228
1 Introduction......Page 229
2 Multi-level Methods......Page 231
3 Fault Model......Page 233
4 Lyapunov Exponents......Page 235
5 Summary of Results on Convergence......Page 239
References......Page 244
1 Introduction......Page 246
2 Inexact Nonlinear FETI-DP......Page 247
3 Model Problems and Numerical Results......Page 249
References......Page 254
1 Introduction......Page 256
2 Governing Equations......Page 257
3 Discretization......Page 258
4 Solution Strategy......Page 259
5 Numerical Experiments......Page 261
References......Page 263
1 Discretization and Iterative Solution......Page 265
2.1 A Three-Dimensional Flow Problem......Page 269
2.2 Parallel Scalability......Page 270
References......Page 272
1 Introduction......Page 274
2 Proposal Outline......Page 275
2.1 Proposed Approach......Page 276
3 Convection-Diffusion Equation......Page 278
4 Numerical Results......Page 279
References......Page 281
1 Introduction......Page 283
2 Krylov Methods for Solving Systems of Linear Equations......Page 284
3 Numerical Experiments......Page 285
3.2 IDR(s), GMRES, and Restarted GMRES......Page 288
3.3 IDR(s) and Bi-CGSTAB......Page 289
References......Page 290
1 Introduction......Page 292
2 The Block COCG and Block COCR Methods......Page 293
2.1 Derivation of the Block COCG and Block COCR Methods......Page 294
2.2 Improving the Numerical Stability of the Block COCG and Block COCR Methods by Residual Orthonormalization......Page 295
3 Numerical Experiments......Page 296
4 Conclusions......Page 298
References......Page 299
Part V Reduced Order Modeling......Page 301
1 Introduction......Page 302
2.1 A Pore-Scale Lithium-Ion Battery Model......Page 303
2.2 Reduced Basis Method and Empirical Interpolation......Page 305
2.3 Numerical Experiments......Page 307
3.1 A Battery: Heat Conduction Model with Resolved Electrode Geometry......Page 310
3.2 Localization of Reduced Basis Methods: LRBMS......Page 311
3.3 Numerical Experiments......Page 313
4 Conclusion......Page 314
References......Page 315
1 Introduction......Page 317
2 Multiscale Porous Media and Flow Models......Page 319
2.1 Two-Scale Porous Media......Page 320
2.2 Three-Scale Porous Media......Page 321
3 Numerical Multiscale Methods......Page 323
3.1 Cell Problems Transformation and Discretization......Page 324
3.2 General form of a Cell Problem......Page 326
4 Model-Order Reduction......Page 327
4.1 Petrov–Galerkin RB Method......Page 328
4.2 RB Method at the Micro Scale......Page 329
4.3 RB Method at the Meso Scale......Page 330
5 Numerical Experiments......Page 331
References......Page 332
1 Time-Harmonic Maxwell's Equations......Page 334
2 Reduced Basis Method......Page 336
3 Numerical Results......Page 338
References......Page 341
1 Introduction......Page 342
2.1 Residual-Based Error Estimators......Page 343
2.2 (Theoretical) Computing via Affine Decomposition and Riesz Representation......Page 344
2.3 The ``Truth''......Page 345
3.1 Wavelet Bases......Page 346
3.2 Wavelet-Based Residual Expansion......Page 347
4 Numerical Experiments......Page 348
References......Page 349
1 Introduction......Page 351
2 Full Order Model......Page 352
3 Reduced Order Model......Page 354
4 Discrete Empirical Interpolation Method (DEIM)......Page 355
5 Numerical Results......Page 356
6 Conclusions and Outlook......Page 358
References......Page 359
1 Motivation: Parametric Model Reduction......Page 360
2 A Taylor-Like Expansion for Stiefel and Grassmann Curves......Page 362
3 Numerical Illustration......Page 365
References......Page 368
1 Introduction......Page 369
2 Problem Formulation and Pareto Optimality......Page 370
3 The Scalar-Valued Optimal Control Problem......Page 372
4 Numerical Solution Strategy......Page 373
5 Numerical Example......Page 374
References......Page 376
Part VI Problems with Singularities......Page 378
1 Introduction......Page 379
2 Notation, Spaces, Equations......Page 380
2.1 Variational-Monolithic ALE Fluid-Structure Interaction......Page 381
2.2 Variational Phase-Field for Dynamic Pressurized-Fractures......Page 382
2.3 The Final System......Page 383
3 Aspects of Discretization and the Solution Algorithm......Page 384
4 A Prototype Numerical Example......Page 385
References......Page 386
1 Introduction......Page 388
2 Notation: Auxiliary Statements......Page 389
3 Problem Statement: Rv-Generalized Solution......Page 390
4 Weighted Finite Element Method......Page 391
5 Results of Numerical Experiments......Page 393
References......Page 395
1 Introduction......Page 397
2 Main Result......Page 398
3 Proof of Lemma 1......Page 400
4 Numerical Experiments......Page 403
References......Page 404
1 Introduction......Page 406
2 Model Equations, Discretization and Splitting Algorithm......Page 407
2.1 Mixed Variational Formulation......Page 408
3.1 Undrained Split Multirate Algorithm......Page 409
4.1 Weak Formulation of Difference of Two Successive Iterates......Page 410
Step 2: Flow Equation......Page 411
Step 3: Combining Mechanics and Flow......Page 412
5 Main Result: Contraction......Page 413
References......Page 414
Part VII Computational Fluid Dynamics......Page 416
1 Introduction......Page 417
2.1 Governing System of Equations......Page 418
2.2 Description of the Profile Motion......Page 419
3.3 Mesh Modification for the ALE Computation......Page 420
3.4 Boundary Conditions......Page 421
4 Numerical Results......Page 422
References......Page 425
1 Introduction......Page 426
2 Governing Equations......Page 427
3 Numerical Implementation......Page 428
4 Results and Discussions......Page 430
5 Conclusion......Page 433
References......Page 434
1 Introduction......Page 435
2.1 Basis Functions and Projection......Page 436
4 Results and Conclusion......Page 439
References......Page 441
1 Introduction......Page 443
2.1.2 Turbulence......Page 444
2.1.4 Vegetation......Page 445
2.2.1 CFD Solver......Page 446
3.1 Case Settings......Page 447
3.2 Results......Page 448
4 Discussion......Page 449
References......Page 450
1 Introduction......Page 452
2 Non-standard Saddle Point Problem Formulation......Page 454
3 Robust Nonlinear Solver......Page 457
4 Summary......Page 460
References......Page 461
1 Introduction......Page 462
2 Mathematical Model......Page 463
4 Numerical Results......Page 465
5 Conclusions......Page 468
References......Page 469
1 Introduction......Page 470
2 Methods......Page 471
2.1 Patient-Specific 1D Coronary Network......Page 473
2.2 Numerical Implementation......Page 474
3 Results......Page 476
4 Discussion......Page 477
References......Page 478
1 Introduction......Page 479
2 The Simplified Oldroyd-B Problem......Page 481
3.1 Setting up the Numerical Model......Page 483
3.2 Numerical Implementation......Page 484
3.3 Results......Page 485
References......Page 486
Part VIII Computational Methods for Multi-physics Phenomena......Page 488
1 The Two-Level Overlapping Schwarz Preconditioner......Page 489
2.1 Model Description......Page 490
2.2 Monolithic Coupling in FSI......Page 491
3 Numerical Results for Fluid-Structure Interaction......Page 492
3.1 Time to Solution Using Different Preconditioners for the Structure Block......Page 494
3.2 Strong Scaling for the Fluid-Structure Interaction Problem......Page 496
References......Page 497
1 Introduction......Page 499
2 Mathematical Model......Page 500
3 Numerical Solution......Page 501
4 Numerical Test......Page 504
References......Page 506
1 Introduction......Page 507
2 Parametric Method......Page 508
4 Numerical Solution......Page 510
5 Computational Studies......Page 511
6 Conclusion......Page 513
References......Page 514
1 Introduction......Page 515
2 T,ϕ-ϕ Formulation of the Eddy Current Problem......Page 516
4 Finite Element Discretization......Page 519
5 Numerical Results......Page 522
References......Page 523
1 Introduction......Page 524
2 Continuous Magnetic Induction Equation......Page 525
3 Discretization of the Induction Problem......Page 527
4 Numerical Simulations......Page 530
References......Page 532
1 Introduction......Page 533
2 Constitutive Models and Finite Element Model......Page 534
3 Results......Page 536
References......Page 540
1 Introduction......Page 542
2.1 Elastic Structure......Page 543
2.3 Fluid Flow......Page 544
2.4 Coupling Conditions......Page 545
3.2 Fluid Flow......Page 546
4 Numerical Results......Page 548
References......Page 550
Part IX Miscellaneous Topics......Page 551
1 Introduction......Page 552
2.1 Boundary Conditions......Page 555
3 Perturbation Expansions......Page 558
3.1.2 OperatorLc, x IL, c=(-1+δ,1-δ)......Page 559
3.2 Operator M......Page 560
3.2.2 OperatorMc, x IM, c=(-1+δ, 1-δ)......Page 561
3.3 Operator N......Page 562
3.3.2 Operator Nc, x IN, c=(-δ, δ)......Page 563
3.3.4 Values of N at Boundary and Transition Points......Page 564
4.1 Comparison in the Bulk......Page 565
4.2 Comparison of Higher Order Approximations in the Bulk......Page 566
4.3 Comparison at the Boundary and Transition Points......Page 567
References......Page 568
1 Introduction......Page 570
2 Dimension Reduction......Page 571
3 Adaptive Finite Differences in Space......Page 572
4 Discontinuous Galerkin in Time......Page 574
5 Numerical Results......Page 575
References......Page 577
1 Introduction......Page 579
2.1 Notation......Page 580
2.2 Discrete Problem......Page 581
2.3 The Three-Layer Finite Difference Scheme......Page 582
2.4 The Two-Layer Finite Difference Scheme......Page 583
3.1 The Eigenstructure of the Matrix Λ......Page 584
3.2 The Structure of the Spectrum of the Matrix S......Page 585
References......Page 587
1 Introduction......Page 589
2 Notation......Page 590
3 Riemannian BFGS Method with Cautious Update......Page 591
4 Convergence Analysis......Page 592
5.1 Problem, Retraction, Vector Transport and Step Size......Page 593
5.2 Tests and Results......Page 594
6 Conclusion......Page 595
References......Page 596
1 Introduction......Page 597
2 Properties of the Lie Derivative......Page 598
3 Derivations at a Point......Page 599
4 The Discrete Lie Derivative of a 1-Cochain......Page 602
References......Page 604