Preface......Page 6
Contents......Page 10
Part I Space Discretisation Methods for PDEs......Page 18
1 Introduction......Page 19
2 The Equations of Elasticity......Page 20
3 Mixed Finite Element Methods......Page 24
4 Finite Element Families......Page 27
References......Page 32
1 Introduction......Page 35
2 Re-entrant Corners......Page 36
2.1 A Nested Newton Algorithm......Page 38
2.2 Numerical Examples with Second Order Elements......Page 39
3 Eigenvalue Problems......Page 41
3.1 Convergence Analysis......Page 42
3.2 Numerical Computation of Eigenvalues......Page 44
4 Jumping Coefficients......Page 45
4.1 Numerical Results......Page 47
Conclusion......Page 50
References......Page 51
1 Introduction......Page 53
2 Stabilized Galerkin......Page 55
3 Previous Results......Page 56
4 H(Div,Ω)-Conforming Approximation......Page 57
References......Page 60
1 Introduction......Page 62
2 Poisson-Nernst-Planck Model......Page 64
3 Weak Formulation and Discretization......Page 66
Results and Conclusion......Page 68
References......Page 71
1 Introduction......Page 72
2 Model Problem......Page 73
3 The Proposed Method......Page 74
3.1 A Priori Analysis......Page 75
4 Observations on the Method......Page 77
References......Page 78
1 Introduction......Page 79
2 Unified Framework, Stability and Robustness......Page 81
3 Local Computation of Multipliers and Fluxes......Page 83
4 Numerical Experiment......Page 86
References......Page 87
1 Problem Formulation and Notation......Page 88
2 Discontinuous Galerkin (DG) Formulation......Page 89
3 Reconstructed Discontinuous Galerkin (RDG) Formulation......Page 90
3.1 Construction of the Reconstruction Operator......Page 92
3.2 Relation Between RDG and Standard DG......Page 93
4 Numerical Experiments......Page 94
References......Page 96
1 Introduction......Page 97
2 DG Discretization......Page 99
3 Adaptivity......Page 100
4 Efficient Solution of Linear Systems......Page 101
5 Numerical Results......Page 102
References......Page 104
1 Introduction......Page 106
2 Numerical Approximation of the Scalar Flux......Page 108
3 Extension to Systems of Conservation Laws......Page 109
4 Numerical Example......Page 112
Concluding Remarks......Page 113
References......Page 114
1 Continuous Problem......Page 115
2 Space-Time Discretization......Page 116
3 Error Estimates......Page 118
4 Unconditional Stability of the Space-Time DGM......Page 119
5 Numerical Experiments......Page 121
References......Page 123
1 Introduction......Page 124
3 Discretization......Page 125
3.1 Notation......Page 126
3.2 Space Discretization......Page 127
3.3 Time Discretization......Page 129
4 Numerical Experiments......Page 130
Conclusion......Page 131
References......Page 132
1 Introduction......Page 133
2 MFD Method Applied to Shape Optimization Problems......Page 134
2.1 Elliptic Problem......Page 135
2.2 Drag Minimization......Page 136
2.3 Free-Boundary Problem......Page 137
3 Adaptive Strategy......Page 139
References......Page 140
1 Introduction......Page 141
2.1 The Truncation Error......Page 143
3 The Time-Dependent Case uxxt=uxxxx + b uxx +c ux+d u+f(x,t)......Page 144
4 The Time-Dependent Case ut=-uxxxx + b uxx +c ux+d u+f(x,t)......Page 147
5 Numerical Results......Page 149
References......Page 150
1 Motivation......Page 151
2 Problem Formulation......Page 152
3.1 Regularization and Augmented Lagrangian Functional......Page 153
3.2 Iterative Algorithm......Page 154
3.3 Numerical Solution of the Constrained Nonlinear Problem......Page 155
4 Finite Element Approximation......Page 156
5 Numerical Validation......Page 157
References......Page 159
1 Introduction......Page 160
1.1 Weak Solution......Page 162
2 Finite Volume Approximation......Page 163
3.1 First Example......Page 164
3.2 Second Example......Page 166
References......Page 168
Part II Time Integration Schemes......Page 169
Stability of Explicit Runge-Kutta Methods for High Order Finite Element Approximation of Linear Parabolic Equations......Page 170
1 Introduction......Page 171
2 Stability Condition for Explicit Time Stepping......Page 172
3 Summary and Conclusion......Page 177
References......Page 178
1 Introduction......Page 179
2.1 The Static Problem......Page 180
2.2 The Dynamic Problem......Page 181
3.1 Benchmark with Analytical Solution......Page 183
3.2 Swinging Beam......Page 185
References......Page 187
1 Hamiltonian Problems......Page 188
2 Multi-value Methods......Page 189
3 G-Symplecticity......Page 191
4 Control of Parasitism......Page 192
5 Long-Term Behaviour......Page 193
References......Page 195
1 Introduction......Page 197
2 Parareal......Page 198
3 Linear Stability Analysis......Page 199
4 Numerical Results for Driven Cavity Flow......Page 200
References......Page 203
1 Introduction......Page 205
2 The Main Results......Page 206
References......Page 213
Part III A Posteriori Error Estimation and Adaptive Methods......Page 214
1 Introduction......Page 215
2 Reduction to Spectral Problems......Page 216
3 Errors Generated by Simplification of a Model......Page 219
4 A Posteriori Estimates of Approximation Errors......Page 221
References......Page 222
1 Introduction......Page 224
2 Functional Type a Posteriori Error Estimates......Page 226
3 Numerical Results......Page 229
References......Page 231
1 Introduction......Page 232
2 BEM-Based FEM for the Model Problem......Page 233
3 Residual Based Error Estimate and Adaptivity......Page 236
4 Numerical Experiment......Page 237
References......Page 239
1 Introduction and General Idea......Page 241
2.1 Stable Hybrid Formulation......Page 243
2.2 Local Computation of Fluxes......Page 245
3 Estimator and Stopping Criterion......Page 246
4 Numerical Experiments......Page 247
References......Page 249
1 Introduction......Page 250
2 The Discrete Augmented Formulation......Page 252
2.1 Well-Posedness and a Priori Error Estimate......Page 253
3 A Ritz Projection-Based a Posteriori Error Analysis......Page 254
4 Numerical Examples......Page 255
References......Page 258
1 Introduction......Page 259
2 The Augmented Mixed Finite Element Method......Page 260
3 A Posteriori Error Analysis......Page 262
4 Numerical Experiments......Page 265
References......Page 266
1 Introduction......Page 268
2 Dynamic Signorini Contact Problem......Page 269
3 Discretization in Space and Time......Page 271
4 Numerical Results......Page 274
References......Page 276
1 Introduction......Page 277
2.1 Linearisation of the Continuous Problem (1)......Page 279
2.2 Discretisation of the Sequence of Linear PDEs (9)......Page 280
3 Numerical Experiments......Page 282
3.1 Benchmarking and Convergence: Classical Solution......Page 283
3.2 A Known Viscosity Solution to the Homogeneous Problem......Page 284
References......Page 285
1 Introduction to the Problem......Page 286
1.1 The Modified Ambrosio-Tortorelli (MAT) Functional......Page 287
1.2 The Discretized MAT Functional......Page 288
2 The Anisotropic a Posteriori Error Estimator......Page 289
3 An Optimize-and-Adapt Algorithm......Page 290
4 Sensitivity Assessment......Page 291
References......Page 294
Part IV Numerical Linear Algebra......Page 295
1 Introduction......Page 296
2 Overdamped Problems......Page 297
3 Nonoverdamped Problems......Page 299
References......Page 303
1 Introduction......Page 305
2 Preliminary Definitions and Results......Page 306
3 Statement and Proof of the Main Result......Page 309
References......Page 312
1 Introduction......Page 314
2 M-Matrices and Diagonal Dominance......Page 315
3 Totally Positive Matrices and Bidiagonal Factorizations......Page 316
References......Page 321
1 Introduction......Page 323
2.1 Which Vector is Targeted: p=Ax vs. z=Ax-QA(x)x......Page 326
2.2 What Is Approximated: Subspace vs. Vector......Page 327
2.3 How the New Direction Is Used: Averaging vs. Enrichment......Page 328
3 Numerical Example......Page 329
References......Page 331
1 Introduction......Page 332
1.1 Class of Test Problems......Page 333
2 The Incomplete Orthogonalization Method......Page 334
2.1 Polynomial Approximation Property......Page 335
3.1 Bounds for the Field of Values and the Norm of Hk,2......Page 336
4 A Posteriori Error Estimate......Page 338
5 Numerical Examples......Page 339
References......Page 340
1 Introduction......Page 341
2 Convergence of Inexact Iterative Methods......Page 342
3 Convergence of Inexact Newton Methods......Page 343
4 Numerical Experiments......Page 347
References......Page 349
Part V Multiscale Modeling and Simulation......Page 350
1 Introduction......Page 351
2 Model Problem......Page 352
3 FE-HMM for Flow in Porous Media......Page 353
4 Adaptive Method......Page 355
5 Numerical Experiments......Page 356
5.1 2D Experiment......Page 357
References......Page 358
1 Introduction......Page 360
2 The Physical Problem and Elementary Properties......Page 361
3 Homogenized Results and Their Proof......Page 363
4 Numerical Examples......Page 366
References......Page 367
1 Introduction......Page 369
2.1 Simple Linear Kinetic Equations......Page 370
2.2 A Kinetic Semiconductor Equation......Page 371
3.1 Inner Integrator......Page 372
3.2 Outer Integrator......Page 373
3.3 Stability Analysis......Page 374
4 Numerical Experiments......Page 375
References......Page 376
1 Introduction......Page 378
2 RB-FE-HMM for the Wave Equation......Page 380
2.1 RB-FE-HMM: Online Stage......Page 381
2.2 RB-FE-HMM: Offline Stage......Page 382
3 Numerical Examples......Page 383
References......Page 385
Part VI Reduced Order Modeling......Page 387
1 Introduction......Page 388
2 The Optimal Control Problem......Page 389
4 Reduced-Order Modelling Using POD......Page 390
5 Numerical Results......Page 393
References......Page 395
1 Introduction......Page 397
2 Stabilized Reduced Basis Method......Page 398
3 Numerical Test: Advection-Diffusion Problem with a Boundary Layer......Page 400
Conclusions......Page 403
References......Page 404
1 Introduction......Page 405
2 Reduced Basis Method and Empirical Interpolation......Page 406
3 Adaptive Snapshot Selection......Page 408
4.1 Performance of the Adaptive Snapshot Selection......Page 410
4.2 ROM-Based Optimization......Page 411
References......Page 412
1 Introduction......Page 414
2 Problem Definition......Page 415
3 Reduced Basis Scheme......Page 416
3.1 Smoothed Solutions......Page 418
4 Error Estimation......Page 419
5 Numerical Results......Page 420
References......Page 422
1 Introduction......Page 423
2 A Transverse Average Model......Page 425
2.1 A Geometrical Multiscale Approach......Page 426
3 Hi-Mod Reduction......Page 427
3.1 Piecewise Hi-Mod Reduction......Page 429
4 A Numerical Comparison......Page 430
References......Page 431
1 The Mathematical Challenges of Large Data Sets......Page 432
2 Grassmann, Stiefel and Flag Manifolds......Page 433
3.1 An Algorithm for Computing a Flag from a Collection of Subspaces......Page 434
4 Numerical Experiments......Page 435
4.2 Video Sequences......Page 436
4.3 Pose Flags......Page 438
Conclusions......Page 439
References......Page 440
Part VII Optimal Control......Page 441
1 Introduction......Page 442
2 The Optimal Control Problem......Page 443
3 Discontinuous Galerkin Discretization......Page 444
4 Primal-Dual Active Set (PDAS) Strategy......Page 445
5 Moreau-Yosida (MY) Regularization Approach......Page 447
6 Implementation Details......Page 448
References......Page 451
1 The Model Problem......Page 452
2 Transformation to Two Systems with a Real Matrix......Page 454
3 Preconditioning......Page 455
3.1 The Practical Preconditioner......Page 456
3.2 Numerical Results......Page 457
4 Alternative Stopping Criteria......Page 459
References......Page 460
1 Introduction......Page 461
2 Model Problems and Building Blocks......Page 463
3 The Accelerated Scheme for HJB Equations......Page 465
4 Numerical Tests......Page 467
References......Page 469
1 Introduction......Page 470
2 The Differential Formulation of a Problem......Page 471
3 A Problem of Optimal Control......Page 472
4 Numerical Tests for Boundary Function Recovery......Page 475
References......Page 478
1 Introduction......Page 479
2 Computing Extremal Points with an Interior Point Method......Page 481
3 Numerical Results......Page 483
References......Page 486
Part VIII Uncertainty, Stochastic Modeling and Applications......Page 487
1 Introduction......Page 488
2 A Viable ``All Events Method''-Implementation......Page 489
2.2 Path-Wise Analysis of Perturbations......Page 490
3 Sample Applications......Page 492
3.1 Spatial Stochastic Focusing......Page 493
3.2 Enzymatic Control......Page 494
Conclusions......Page 495
References......Page 496
1 Introduction......Page 497
2.1 Standard Approach......Page 498
2.2 Divergence-Free Approach......Page 499
3.1 Computational Cost......Page 500
4 Results......Page 501
4.1 2D Synthetic Test Case......Page 502
4.2 3D Experimental Data Set......Page 503
References......Page 504
1 Introduction......Page 505
2 Multilevel Monte Carlo Method for Stiff SDEs......Page 507
3 Improved Stabilized Multilevel Monte Carlo Method for Stiff SDEs......Page 508
4 Numerical Experiments......Page 511
References......Page 513
1 Random Discrete Least Squares......Page 514
2 Adaptive Random Discrete Least Squares......Page 517
3 Numerical Results......Page 519
References......Page 521
Part IX Solvers, High Performance Computing and Software Libraries......Page 522
1 Introduction......Page 523
2 The PWDG Method for the Helmholtz Problem......Page 525
3.1 Local and Coarse Spaces, Prolongation and Restriction Operators......Page 529
3.2 Schwarz Operators......Page 531
4 Numerical Results......Page 532
5 The Issue of GMRES Convergence......Page 535
References......Page 537
1 Introduction......Page 539
2 The Deflation Method......Page 540
3 A New Coarse Space for Almost Incompressible Linear Elasticity......Page 541
4 Numerical Results......Page 544
References......Page 546
Simulation Software......Page 548
The Challenge......Page 549
Levels of Parallelism......Page 550
Data Access......Page 551
4 Performance Evaluation......Page 552
Summary and Conclusions......Page 554
References......Page 555
1 Introduction......Page 556
2 Software Engineering......Page 557
3 System Architecture......Page 558
3.2 Software Architecture Model......Page 559
4.1 Volcanic Ash Dispersal......Page 561
References......Page 564
1.1 Software Frameworks......Page 565
1.3 Hardware Development......Page 566
2 Design and Implementation......Page 567
2.2 Containers: Matrices and Vectors......Page 568
2.3 Block Matrices......Page 569
3 Experimental Evaluation......Page 570
References......Page 573
1 Introduction......Page 574
2 Background......Page 576
3.1 Kernels......Page 577
3.2 Expansions......Page 578
3.3 Tree and Traversals......Page 579
3.4 Optimizations......Page 580
4.1 Preliminary Benchmark......Page 581
References......Page 582
Part X Computational Fluid and Structural Mechanics......Page 584
1 Introduction......Page 585
2 Optimal Incremental Projection Scheme for Open Boundary Problem......Page 587
3 Variable Time Stepping......Page 590
References......Page 593
1 Motivation......Page 594
2 Equations......Page 595
3 A 1D Averaged Model......Page 597
4 ALE Based Method for Full Thin-Strip Computations and Discretization......Page 598
6 Numerical Tests......Page 599
References......Page 601
1 Introduction......Page 603
2 Mathematical Model......Page 605
3 Numerical Method......Page 606
4 Design of Computational Domain......Page 607
5 Simulation of Hydrodynamic Events......Page 608
References......Page 610
Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations......Page 611
1 Introduction......Page 612
2.1 First-Order Stress-Velocity-Pressure System......Page 613
2.3 Newton Linearization......Page 614
2.4 Variational Formulation......Page 615
2.6 Discrete Least-Squares Principle......Page 616
3 Numerical Results and Discussions......Page 617
4 Summary......Page 618
References......Page 619
1 Introduction......Page 620
3 Computational Domain and Boundary Conditions......Page 621
4 Numerical Solution......Page 622
References......Page 625
1 Introduction......Page 626
2.1 Case 1: Five Pipe Sections Connected at a Junction......Page 629
2.2 Case 2: A Closed System of Three Pipe Sections and Two Junctions......Page 630
References......Page 633
1 Governing Equations......Page 635
1.1 EARSM Model of Turbulence......Page 636
2 Calibration of Model Constants......Page 637
4.1 Subsonic Flow Around the Flat Plate......Page 640
4.3 Transonic Flow Through the SE 1050 Turbine Cascade......Page 642
Conclusion......Page 644
References......Page 645
1 Introduction......Page 646
2 DRBEM Application......Page 648
3 Numerical Results......Page 650
References......Page 653
2 The Model......Page 655
3 Numerical Method......Page 657
4 Numerical Results......Page 658
References......Page 663
1 Introduction......Page 664
2 Membrane Model, Viscoelasticity and Temperature Coupling......Page 665
3 Multiphase Treatment......Page 666
4 Numerical Treatment......Page 667
5 Numerical Results......Page 668
6 Summary......Page 670
References......Page 671
1 Introduction......Page 672
2 Governing Equations......Page 673
3 Numerical Methods......Page 675
4.1 Lid-Driven Cavity with Flexible Bottom......Page 676
4.2 Valveless Micropump......Page 678
Conclusion......Page 679
References......Page 680
1 Introduction......Page 681
2 Model Set Up......Page 682
3 Numerical Approximation......Page 684
4 Error Analysis, Numerical Results and Discussion......Page 686
References......Page 688
The Interaction of Compressible Flow and an Elastic Structure Using Discontinuous Galerkin Method......Page 689
2 Mathematical Model......Page 690
3.1 Space Discretization......Page 692
4 Implementation Remarks......Page 693
5 Numerical Results......Page 695
Conclusion......Page 696
References......Page 697
1 Introduction......Page 698
2.1 Variational Formulation in Eulerian Coordinates......Page 700
2.2 The Initial Point Set Method......Page 701
3 Finite Element Discretization......Page 702
4 Outlook: Accurate Temporal Discretization......Page 704
References......Page 705
1 Introduction......Page 707
2 Numerical Validation: Benchmark Problems......Page 708
3 Large Motion: 360 Rotation......Page 709
4 Touching the Boundary......Page 710
5 Locally Modified Finite Element Scheme......Page 711
6 Growing Structures and Clogging Phenomena......Page 712
Conclusion......Page 713
References......Page 714
Part XI Computational Electromagnetics......Page 715
1 Introduction......Page 716
2 Stability of the Proposed Method......Page 718
3 Error Analysis for Smooth Solutions......Page 720
4 Improved Error Estimates......Page 723
References......Page 725
1.1 The Ferromagnetic Screening Effect in General......Page 726
1.2 The Industrial Application: An Aluminum Electrolysis Cell......Page 727
2 Modelling of Ferromagnetism......Page 728
2.1 Static Maxwell Equations Without and With Magnetization......Page 729
2.2 A Theorem of Existence and Uniticity......Page 730
3.1 The Domain Decomposition Algorithm......Page 731
4.1 The Academic Case......Page 732
4.2 The Industrial Case......Page 733
References......Page 734
1 Introduction......Page 735
2 Continuous Problem Formulation......Page 737
3 Discrete Problem Formulation......Page 738
4 Numerical Results and Analogy with Linear Elasticity......Page 741
References......Page 742
1 Introduction......Page 744
2 The Finite Element Method......Page 746
4 Error Estimates......Page 748
References......Page 750
Editorial Policy......Page 752
Lecture Notes in Computational Science and Engineering......Page 754
Texts in Computational Science and Engineering......Page 759