Numerical Mathematics and Advanced Applications ENUMATH 2017

Preface......Page 6
 Programme Committee......Page 8
Contents......Page 9
Part I Plenary Lectures......Page 19
 PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach......Page 20
  1 Introduction......Page 21
   2.1 Governing Equations......Page 23
   2.2 Quantities of Interest (QoI)......Page 25
   2.3 Models and Apps......Page 26
   3.1 Library......Page 27
   3.2 Model Synthesis......Page 30
   3.3 Truth Finite Element Approximation......Page 32
   3.4 Static Condensation: Finite Element......Page 33
   3.5 SCRBE Method......Page 34
   3.6 Computational Procedure: Offline-Online Decomposition......Page 38
   3.7 PDE App Architecture......Page 40
  4 PDE Apps: Examples......Page 41
  References......Page 49
  1 Introduction......Page 51
  2 Preconditioner for the Coupled Problem......Page 54
   3.1 Auxiliary Operators......Page 55
   3.2 Discrete Preconditioner for the Coupled Problem......Page 57
  4 Perfusion Experiment......Page 58
   4.1 Discussion of Perfusion Experiment......Page 61
  References......Page 62
 Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations......Page 64
  1 Introduction......Page 65
  2 The Fully Discrete Approximation......Page 66
  3 A Robust Iterative Scheme......Page 68
  4 Iterative Schemes Based on Regularisation......Page 71
  5 Numerical Examples......Page 73
  6 Conclusion......Page 76
  References......Page 77
 Mathematics and Medicine: How Mathematics, Modelling and Simulations Can Lead to Better Diagnosis and Treatments......Page 79
  1 Introduction......Page 80
  2 A Brief Introduction to Compartment Models and Tracer Kinetic......Page 81
   3.1 Conservation of Fluid Mass......Page 84
   3.2 Balance of Forces......Page 85
   3.3 Tracer Mass Balance and Indicator Dilution......Page 86
  4 Parameter Estimation......Page 87
  5 Numerical Example......Page 89
   5.1 Forward Model......Page 90
   5.2 Solution of Inverse Problem......Page 91
  6 Outlook......Page 92
  References......Page 93
Part II Kernel Methods for Large Scale Problems: Algorithms and Applications......Page 95
  1 Introduction......Page 96
  2 Convergence of the Convolution Approximation......Page 98
  3 Iterative Refinement......Page 101
  4 Native Space for Gaussian Approximation......Page 103
  References......Page 105
 Anisotropic Weights for RBF-PU Interpolation with Subdomains of Variable Shapes......Page 106
  1 Introduction......Page 107
  2 The RBF-Based Partition of Unity Method......Page 108
   3.1 Local Error Estimates......Page 109
   3.2 Description of the PU-LOOCV Method......Page 110
   4.1 Experiments with Artificial Data......Page 111
  5 Conclusions......Page 113
  References......Page 114
  1 Introduction......Page 115
  2 Basket Option Pricing Under Jump Diffusion Processes......Page 116
  3 Payoff and Boundary Conditions......Page 117
  4 Radial Basis Function Collocation Schemes......Page 118
   4.1 Radial Basis Function Partition of Unity Method......Page 119
   5.1 Approximation of the Integral Term......Page 120
  6 Numerical Experiments......Page 121
  References......Page 124
  1 Matrix-Valued Kernels......Page 125
  2 Greedy Algorithm......Page 127
   2.2 f-Greedy......Page 128
   2.3 f/P-Greedy......Page 129
  3 Numerical Example......Page 131
  References......Page 133
  1 Introduction......Page 134
  2 Eigenvalue Solver......Page 135
  4 GPU Computing......Page 136
  5 GPU Implementation of ELPA 1......Page 137
  6 Numerical Results......Page 138
  References......Page 141
Part III Advanced Discretization Methods for Computational Wave Propagation......Page 143
  1 Introduction......Page 144
  2 Framework......Page 145
   3.1 Temporal Discretization......Page 147
   3.2 Spatial Discretization......Page 148
  4 Efficiency......Page 149
   4.1 Structure of ∂i,h......Page 150
   4.2 Structure of the Discrete Split Operators......Page 151
  References......Page 152
  1 Introduction......Page 154
  2 Trefftz-DG Formulation for Elastodynamics......Page 155
   2.1 Space-Time Trefftz-DG Formulation......Page 156
   2.2 Well-Posedness of Trefftz-DG Formulation......Page 158
   3.1 Polynomial Basis......Page 159
  4 Conclusion......Page 160
  References......Page 161
Part IV Unfitted Finite Element Methods: Analysis and Applications......Page 163
  1 Introduction......Page 164
  2 The Virtual Element Method (VEM)......Page 165
  3 The FETI-DP Domain Decomposition Method for the VEM......Page 166
  4 Subdomain Partitioning by Conformal Meshing......Page 167
  5 Numerical Results......Page 168
  References......Page 171
  1 Introduction......Page 172
  2 Mathematical Model......Page 173
  3 Hybrid Finite Volume: Finite Element Method......Page 175
  4 Numerical Results......Page 176
   4.2 Steady Analytical Solution for a Triple Fracture Problem......Page 177
  References......Page 179
  1 Introduction......Page 180
   2.2 Admissible Meshes......Page 182
   2.3 Trace Inequalities......Page 183
   3.1 Uncut Cells......Page 184
   3.2 Cut Cells: Fictitious Domain Problem......Page 185
  4 Main Result: Error Estimate......Page 187
  References......Page 188
  1 Introduction......Page 189
  2 The CutFEM for Stokes Equations: Derivation......Page 190
   2.2 The Mesh, Discrete Domains, and Finite Element Spaces......Page 191
   2.3 Numerical Modelling......Page 192
   3.2 A Priori Error Estimate......Page 195
  4 Numerical Example......Page 196
  References......Page 197
  1 Introduction......Page 199
   2.1 A Piecewise Multilinear Approximation of the Geometry......Page 200
   2.2 Improved Geometrical Accuracy with a Parametric Mapping......Page 201
   2.3 Reduction to Integration Problems on the Reference Element......Page 202
  4 Integration on the Unit Square......Page 203
  6 Numerical Example......Page 206
  References......Page 207
Part V Advances in Numerical Linear Algebra Methods and Applications to Partial Differential Equations......Page 209
  1 Introduction......Page 210
  2 Well-Known Basis Functions......Page 211
  3 Computation of Coefficients wi......Page 213
  4 Computation of Basis Functions......Page 214
   5.1 Truncation Bound with Matrix Exponential Remainder Term......Page 215
   5.2 Bounds Using Decay of Elements in the Matrix Exponential......Page 216
  6 Illustrating Example......Page 217
  References......Page 218
  1 Introduction......Page 220
  2 Elastic Wave Equation and Space-Time Discretisation......Page 222
  3 Fully Discrete Systems......Page 224
  4 Numerical Experiments......Page 225
  5 Conclusions......Page 227
  References......Page 228
  1 Introduction......Page 229
   2.1 GLT and the Symbols of a Matrix......Page 231
   2.2 The Symbol of the 3D Anisotropic Laplacian......Page 232
  4 Choosing a Smoother for the GLT-Based AMG......Page 233
  5 Analysis of the Required Computer Resources for GLT Based AMG......Page 234
  6 Numerical Experiments......Page 235
  References......Page 236
Part VI Numerical Methods in Biophysics......Page 238
 Mathematical Modelling of Phenotypic Selection Within SolidTumours......Page 239
  2 Model Description......Page 240
   2.1 Dynamics of Cancer Cells......Page 241
  3 Formal Analysis of Phenotypic Selection......Page 242
  4 Numerical Solutions......Page 244
  References......Page 246
  1 Introduction......Page 248
   2.1 The Cells and Their Forces......Page 249
   2.2 The Cells and Their Migration, Death and Differentiation......Page 251
  3 Numerical Method and Uncertainty Assessment......Page 252
  4 Computer Simulations......Page 254
  References......Page 255
Part VII Structure Preserving Discretizations and High Order Finite Elements for Differential Forms......Page 257
  1 Introduction......Page 258
  2 Moments and Potentials......Page 260
  3 Weights and Potentials......Page 264
  References......Page 266
  1 Introduction......Page 267
  2 Absolute Coordinate Formulation (ACF)......Page 268
  3 MOR of Eq.(4)......Page 271
  4 Pendulum Example......Page 273
  References......Page 274
  1 Introduction......Page 276
  2 Parametric Description and Projection to the Plane......Page 277
  3 Numerical Solution......Page 279
  4 Computational Experiments......Page 280
  References......Page 283
  1 Introduction......Page 285
  2 FFT-Based Homogenization......Page 286
  3 General Homogenization Problem of Order α......Page 287
  4 Numerical Results......Page 291
  5 Conclusion......Page 292
  References......Page 293
Part VIII Monge-Ampère Solvers with Applications to Illumination Optics......Page 294
  1 Introduction......Page 295
  2 Mathematical Formulation......Page 296
  3 The Extended Least-Squares Algorithm......Page 299
  4 Numerical Examples......Page 301
  5 Concluding Remarks......Page 302
  References......Page 303
  1 Introduction......Page 304
  2 Optics as Open-Loop Controllers......Page 306
   2.1 First-Order Necessary Conditions......Page 307
  3 Results......Page 309
  References......Page 311
Part IX Mixed and Nonsmooth Methods in Numerical Solid Mechanics......Page 313
  1 Introduction......Page 314
  2 The Hellinger-Reissner Principle and Stress Finite Element Spaces......Page 315
  3 The H(div)-Nonconforming Space by Augmenting Raviart-Thomas Elements......Page 317
   3.1 The Augmenting Space ΣhΔ......Page 318
  4 A Numerical Comparison......Page 319
  5 Postprocessing for Elementwise Symmetry on Average......Page 323
  References......Page 324
  1 Introduction......Page 325
  2 Stress Based Evolution Model......Page 326
   3.1 Plate with Top-Load......Page 330
   3.2 Beam with Top-Load......Page 331
  References......Page 333
  1 Introduction......Page 334
   2.1 A Brief Description of a Viscous-Plastic Sea Ice Model......Page 335
   2.2 Linearization by Newton......Page 336
   2.3 The Operator Related Damped Newton Method for the Simulation of the Sea Ice Dynamics......Page 337
  3 First Analysis of the Operator Related Newton Scheme......Page 338
   3.1 Numerical Analysis of the Operator Related Newton Method......Page 340
  References......Page 342
Part X A Posteriori Error Estimation, Adaptivity and Approximation......Page 343
  1 Introduction to Best Error Localizations......Page 344
  2 Sobolev-Hilbert Triplet, Meshes, and Piecewise Polynomials......Page 345
  3 Approximating Gradients......Page 347
  4 Approximating Functions......Page 348
  5 Approximating Functionals......Page 350
  6 Approximating Simultaneously......Page 351
  References......Page 352
  1 Introduction......Page 353
  2 Governing Equations......Page 354
  3 Discretization......Page 355
  4 Adaptivity......Page 356
  5 Numerical Simulations......Page 358
   5.1 2d Two-Phase Flow......Page 359
   5.2 3d Two-Phase Flow......Page 360
  References......Page 363
  1 Introduction......Page 365
  2 Enough Diffusion Makes E-Schemes......Page 366
  3 E-Schemes Are Monotone......Page 367
  4 Less Artificial Diffusion Is More Resolution......Page 368
  5 Application to Isentropic Navier–Stokes Equations......Page 370
  References......Page 373
  1 Introduction......Page 374
  2 A New Adaptive Filtered Scheme......Page 375
  3 Construction of the Scheme and Convergence......Page 377
  4 Numerical Tests......Page 380
  5 Conclusions......Page 382
  References......Page 383
  1 Introduction......Page 384
  2 Continuous Problem Setting......Page 385
  3 Finite Element Discretization Using Selective Reduced Integration......Page 386
   4.1 Error Identity......Page 387
   4.2 Numerical Approximation of the Error Identity......Page 388
  5 Numerical Results......Page 389
  References......Page 391
  1 Introduction......Page 392
  2 Problem Statement......Page 393
  3 Stabilised Finite Element Method......Page 395
  4 Nitsche\'s Method......Page 396
  5 Numerical Results......Page 397
  References......Page 399
Part XI Noncommutative Stochastic Differential Equations: Analysis and Simulation......Page 401
  1 Introduction......Page 402
  2 Some Notation, Definitions and Preliminary Results on Stochastic B-Series......Page 403
  3 Main Results......Page 406
  References......Page 410
  1 Introduction......Page 411
  2 Post-Lie Algebra and Examples......Page 412
  3 Post-Lie Algebras and Lie Group Integration......Page 414
  References......Page 418
  1 Introduction......Page 420
  2 Stochastic Exponentials......Page 421
   2.2 Operator-Valued Exponentials in Free Probability......Page 422
  3 Quasi-Shuffle Algebra and Itô Stochastic Integral......Page 423
   3.1 Shuffle Algebra and Stratonovich Stochastic Integral......Page 424
  References......Page 427
Part XII Innovative Numerical Methods and Their Analysis for Elliptic and Parabolic PDEs......Page 429
  1 Introduction......Page 430
   2.1 Galerkin Approximations......Page 431
   2.2 Non-conforming P1 Finite Elements......Page 432
   2.3 Two-Point Flux Approximation Finite Volumes on Cartesian Meshes......Page 434
  3 The Gradient Discretisation Method......Page 436
  References......Page 438
  1 Introduction......Page 439
  2 Face-Connected Meshes of Non-Lipschitz Domains......Page 440
  3 Full Stability and Simplified Averaging......Page 442
  4 Quasi-Optimality by Moment-Preserving Smoothing......Page 444
  References......Page 447
  1 Introduction......Page 448
  2 Adaptive Algorithm with Abstract Marking Strategy......Page 449
  3 Convergence and Marking Strategy......Page 453
  4 Convergent Marking Strategies......Page 455
  References......Page 456
  1 Introduction......Page 457
  2 Discretization......Page 458
   2.1 Neumann Boundary Values on Unfitted Meshes......Page 459
   2.2 Numerical Experiments......Page 461
   3.1 Reference Process......Page 462
   3.2 Simulation Results......Page 463
  References......Page 464
Part XIII Polyhedral Methods and Applications......Page 466
  1 Introduction......Page 467
  2 The PWDG Method for the Helmholtz Problem......Page 468
  3 Conditioning of the Plane Wave Basis......Page 469
   3.2 Dependence on hk and p......Page 470
  4 Orthogonalization of the Plane Wave Basis......Page 471
  References......Page 474
  1 Introduction of the Problem......Page 475
  2 VEM Discretization of the Problem......Page 478
  3 The Virtual Element Method for DFN Simulations......Page 479
  4 An Application of the Method......Page 481
  References......Page 482
  1 Introduction......Page 483
   2.1 Variational Definition of the Basis Functions......Page 485
   2.2 Discretization and Algebraic Realization......Page 486
  3 General Workflow of the msHHO Method......Page 488
  4 Numerical Experiments......Page 490
  References......Page 491
  1 Introduction......Page 492
  3 Virtual Elements Discretization......Page 493
  4 Numerical Results......Page 497
  References......Page 500
Part XIV Recent Advances in Space-Time Galerkin Methods......Page 501
  1 Continuous Elasticity Problems......Page 502
   1.2 Nonlinear Elasticity......Page 503
   2.1 Space DG Discretization......Page 504
   2.3 Realization of the Discrete Dynamic Elasticity Problem......Page 506
   3.1 Choice of the Penalty Coefficient......Page 507
   3.2 Computation of Vocal Fold Vibrations......Page 508
  References......Page 510
  1 Introduction......Page 512
  3 A Fully Discrete Higher Order Numerical Scheme......Page 513
   3.1 Variational Formulation of the Non-linear Biot Model......Page 514
   3.3 Discretization in Time: Discontinuous Galerking dG(r)......Page 515
   3.4 Discretization in Space by cG(p+1)-MFEM(p)......Page 517
  4 Numerical Results......Page 518
  References......Page 519
  1 Introduction and Mathematical Model......Page 521
  2 Iterative Coupling and Space-Time Discretization......Page 522
  3 Convergence of the Iteration Scheme......Page 525
  References......Page 530
  1 Introduction......Page 531
  2 Formulation of the Continuous Problem......Page 532
   3.1 ALE Mappings and Triangulations......Page 533
   3.3 Some Notation and Important Concepts......Page 534
   3.4 Discretization......Page 535
   4.1 Important Estimates......Page 536
   4.2 Discrete Characteristic Function......Page 538
  References......Page 539
Part XV PDE Software Frameworks......Page 541
  1 Introduction......Page 542
  2 Mixed-Dimensional Flow in Fractured Porous Media......Page 543
   3.1 Grid Structure......Page 544
   3.2 Conservative Discretizations......Page 546
  4 Example Simulation......Page 547
  References......Page 549
  1 Introduction......Page 550
  2 Hybridizable Discontinuous Galerkin Discretization......Page 551
   2.2 Formulation in Terms of u and u......Page 553
  3 Sum Factorization Algorithms for HDG......Page 554
   3.2 Matrix-Vector Product for Collocation Basis......Page 555
  4 Numerical Experiments and Outlook......Page 557
  References......Page 558
Part XVI Numerical Methods for Simulating Processes in Porous Media......Page 559
  1 Introduction......Page 560
   1.1 Definition of the Benchmark Configuration......Page 561
   1.2 Short Description of the Used Numerical Techniques......Page 562
   2.1 Validation of the 3D Results via 2.5D Configuration......Page 563
   2.2 3D Benchmark Results......Page 565
  3 Conclusions......Page 567
  References......Page 568
  1 Introduction......Page 569
  2 Problem Description......Page 571
  3 Convergence of the Scheme......Page 574
  4 Conclusion......Page 579
  References......Page 580
 A Numerical Method for an Inverse Problem Arising in Two-Phase Fluid Flow Transport Through a Homogeneous Porous Medium......Page 581
  1 Introduction......Page 582
  2 Direct Problem......Page 583
  3 Inverse Problem......Page 584
  5 Numerical Method......Page 585
  6 Stability and Convergence Results......Page 587
  7 A Numerical Example......Page 588
  References......Page 589
  1 Introduction......Page 590
  2 Two Phase Flow Model in the Averaged Pressure Formulation......Page 591
  3 Discretization......Page 592
   3.1 Convergence Analysis......Page 593
   3.2 Numerical Results......Page 596
  References......Page 598
  1 Introduction......Page 599
   2.1 Coefficients in General Formulation......Page 600
   3.1 Benchmark Problems......Page 601
   3.2 Numerical Analysis......Page 602
  4 Heterogeneous Porous Media......Page 603
   4.1 Mass Lumping......Page 604
  5 Mass Lumping in Homogeneous Porous Media......Page 605
  6 Conclusion......Page 606
  References......Page 607
  1 Introduction......Page 608
  2 Governing Equations......Page 609
  3 Problem Formulation......Page 610
  4 Numerical Results......Page 612
  5 Conclusions......Page 614
  References......Page 615
  1 Introduction......Page 616
  2 Governing Equations......Page 617
   2.2 Rock Deformation......Page 618
   2.3 Fracture Deformation......Page 619
  4 Solution Strategy......Page 620
  5 Numerical Results......Page 621
  References......Page 623
Part XVII Model Reduction Methods for Simulation and (Optimal)Control......Page 624
  1 Introduction......Page 625
  2 The Optimal Control Problem......Page 626
  3 Model Predictive Control (MPC)......Page 628
  4 Numerical Tests......Page 630
  References......Page 632
  1 Introduction......Page 634
   2.1 Parametrized Domain and Biophysical Model......Page 635
   2.2 Parametrized Optimal Control Problem......Page 637
  3 Reduced Basis Approximation......Page 638
  4 Numerical Results and Summary......Page 639
  References......Page 642
  1 Introduction......Page 643
  2 The Stiefel Manifold in Numerical Schemes......Page 646
  3 Quasi-Linear Interpolation of Orthogonal Bases......Page 647
  4 Numerical Experiments......Page 648
  5 Summary and Conclusion......Page 649
  References......Page 651
  1 Introduction......Page 652
   2.1 Oseen-Iteration......Page 653
  3 Spectral Element Discretization......Page 654
  4 Reduced Order Modelling......Page 655
  5 Model and Numerical Results......Page 656
  References......Page 659
  1 Introduction......Page 661
  2 Convective Cahn-Hilliard System......Page 662
  3 Optimal Control of Cahn-Hilliard......Page 663
  5 Numerical Results......Page 666
  References......Page 668
Part XVIII Recent Advances on Polyhedral Discretizations......Page 670
  1 Introduction......Page 671
  2 The Model Problem......Page 672
  3 The Optimization Approach......Page 673
  4 The VEM-Based Approach......Page 674
  5 Numerical Results......Page 677
  References......Page 679
  1 Introduction......Page 680
  2 Preliminaries......Page 681
  3 Virtual Element Method......Page 683
  4 Numerical Experiment......Page 685
  References......Page 687
Part XIX FEM Meshes with Guaranteed Geometric Properties......Page 689
  1 Introduction......Page 690
  2 Denotations and Definitions......Page 691
  3 Main Results......Page 693
  References......Page 694
  1 Introduction......Page 696
  2 Augmented Dual-Mixed Variational Formulation......Page 697
  3 Augmented Mixed Finite Element Method......Page 698
  4 A Posteriori Error Indicator......Page 699
  5 Numerical Experiments......Page 700
  References......Page 704
  1 Introduction......Page 705
  2 The Set of Simplices and Symmetries......Page 706
   3.1 Triangular Shapes from Normalized Positions of Vertices......Page 707
   3.2 Triangular Shapes from Angles......Page 708
   3.3 Mappings Between Models......Page 710
  4 Tetrahedra......Page 711
  References......Page 712
  1 Introduction......Page 713
   2.1 Relations Between Vertices and Normals of Simplices......Page 714
   2.2 The Vertex-to-Normal Mapping......Page 716
   2.3 Vertex-Normal Duality in Higher Dimensions......Page 717
   2.4 Tetrahedra Whose Facets Have Equal Circumradius......Page 718
  References......Page 719
  1 Introduction......Page 721
  2 The Maximum Angle Condition in Higher Dimensions......Page 723
  3 Examples......Page 725
  References......Page 726
Part XX Discretizations and Solvers for Multi-Physics Problems......Page 728
  1 Introduction......Page 729
  2 Biot\'s Poroelasticity Equations......Page 730
   2.1 Numerical Discretization......Page 731
   2.2 Stabilized EbFVM Formulation......Page 732
  3 Numerical Results......Page 733
  4 Conclusions......Page 736
  References......Page 737
  1 Introduction......Page 738
  3 Fixed-Stress Splitting Scheme......Page 740
   3.1 The Tuning Parameter Kdr......Page 741
  4 Numerical Study: Optimal Tuning Parameter Kdr......Page 742
   4.1 Test Case 1a: Effective 1d Deformation......Page 743
   4.3 Test Case 1c: Influence of Flow Parameters......Page 744
  5 Conclusion......Page 745
  References......Page 746
  1 Model Equations......Page 747
   1.1 Boundary and Initial Conditions......Page 750
  2 Implementation......Page 751
  3 Numerical Simulation of Biofilm Formation in a Strip......Page 752
   3.1 Parametric Studies......Page 753
  References......Page 754
  1 Introduction......Page 756
  2 Direct Minimization of Discrete Willmore Energy......Page 758
   2.1 Addressing the Failure......Page 760
  3 Harmonic Energy Regularization......Page 761
  4 Conclusion and Open Issues......Page 762
  References......Page 763
 Heavy Metals Phytoremediation: First Mathematical ModellingResults......Page 765
  1 Introduction......Page 766
  2 Mathematical Modelling......Page 767
  3 Numerical Solution......Page 769
  4 Computational Results......Page 770
  References......Page 773
  1 Introduction......Page 774
  2 Analytical Formulation of the Environmental Problem......Page 775
  3 Numerical Computation of the Optimal Control......Page 777
  4 Numerical Results......Page 778
  References......Page 781
 Nitsche-Based Finite Element Method for Contact with Coulomb Friction......Page 783
  1 Setting and Discretization......Page 784
  2 Existence and Well-Posedness Results......Page 787
  3 Numerical Results......Page 789
  References......Page 791
  1 Introduction......Page 792
  2 Physical Problem and the Mathematical Model......Page 794
  3 The DRBEM Application......Page 795
   4.1 Case 1: (s=0, l=0, Ha Increases)......Page 797
   4.2 Case 2: (l=0, Ha=10, s Varies as s<1, s=1, s>1)......Page 798
   4.4 Case 4: (Ha=10, s Varies as s<1, s=1, s>1 and l Varies as l=0.3, l=0.5)......Page 799
  References......Page 800
  1 Introduction......Page 802
  2 Mathematical Model of the Air Flow......Page 803
  3 Pollution Transport Modelling......Page 805
   3.1 Dry Deposition......Page 806
  4 Numerical Results......Page 807
  References......Page 809
  1 The Kirchhoff Plate Bending Problem......Page 811
  2 New Mixed Formulation......Page 813
   2.1 Coupling Condition as Standard Boundary Conditions for ϕ......Page 814
   3.1 Characterization of Λ......Page 815
   3.2 The Discretization Method......Page 816
  4 Numerical Tests......Page 817
   4.1 Square Plate with Clamped, Simply Supported and Free Boundary......Page 818
  References......Page 819
Part XXI Reduced Order Models for Time-Dependent Problems......Page 820
  1 Introduction......Page 821
  2 Problem Formulation......Page 822
  3 The Reference Point Method......Page 823
  4 Numerical Results......Page 824
  References......Page 828
  1 Problem Setting......Page 829
  2 Kernel Based Surrogates and the VKOGA......Page 831
  3 The Complete Algorithm: VKOGA-IE......Page 832
  4 Experiments......Page 833
  5 Conclusion and Further Work......Page 835
  References......Page 836
Part XXII Limiter Techniques for Flow Problems......Page 837
  1 Introduction......Page 838
  2 Reconstruction with Third-Order Limiting......Page 839
   2.1 Formulation on Equidistant Meshes......Page 840
   2.2 Generalization to Non-equidistant Meshes......Page 841
  3 Numerical Examples......Page 844
  References......Page 846
  1 Introduction......Page 847
  2 An Algebraic Flux Correction Scheme......Page 848
  3 General Condition for the Discrete Maximum Principle......Page 849
  4 Old and New Limiters......Page 850
   4.2 Limiter by Barrenechea et al. [3] (Limiter 2)......Page 851
   4.3 A Linearity Preserving Limiter of Upwind Type (Limiter 3)......Page 853
   4.4 A Linearity Preserving Limiter of Upwind Type Satisfying the DMP on Arbitrary Meshes (Limiter 4)......Page 854
  References......Page 856
Part XXIII New Frontiers in Domain Decomposition Methods: Optimal Control, Model Reduction, and Heterogeneous Problems......Page 857
  1 Introduction......Page 858
  2 Optimization......Page 861
  3 Numerical Experiments......Page 863
  4 Conclusion......Page 865
  References......Page 866
  1 Introduction......Page 867
  2 One- and Two-Level FETI Algorithms......Page 868
  3 Convergence Analysis......Page 870
  4 Numerical Experiments......Page 874
  References......Page 875
  1 Introduction......Page 876
  2 The Optimal Control Problem......Page 877
  3 Reduced-Order Modeling (ROM)......Page 878
  4 Coupling MPC and HJB......Page 879
  5 Numerical Test......Page 882
  References......Page 884
  1 Preliminaries......Page 885
  2 Adaptive Shooting Intervals for Linear Problems......Page 887
  3 Adaptive Shooting Intervals for Nonlinear Problems......Page 889
  4 Conclusions......Page 892
  References......Page 893
Part XXIV Error Analysis for Finite Element Methods for PDEs......Page 894
  1 Continuous Problem......Page 895
   1.1 Time Growth of the Exact Solution......Page 896
  2 Exponential Scaling......Page 897
   2.2 Construction of the Scaling Function μ......Page 899
  3 Discontinuous Galerkin Method......Page 900
   3.1 Error Estimates......Page 901
  References......Page 903
  1 Introduction......Page 904
  2 Energy Corrected Finite Element......Page 906
  3 Maximum Norm Error Estimates......Page 908
  4 Numerical Results......Page 911
  References......Page 912
  1 Introduction......Page 913
   2.1 Digital Calderon–Zygmund Operators......Page 914
   2.2 Digital Pseudo-Differential Operators......Page 915
  3 Discrete Equations in a Half-Space......Page 916
  4 Solvability......Page 917
  5 Discrete Equations and Comparison......Page 918
  References......Page 920
  1 Introduction and Model Problem......Page 922
  2 A Simple Domain Approximation for Domains with Smooth Boundaries......Page 924
   2.1 Analysis of the Semi-discrete System......Page 925
   2.2 Analysis of the Fully-Discrete System......Page 928
  3 Numerical Experiment......Page 929
  References......Page 930
Part XXV Fluid Dynamics......Page 931
  1 Introduction......Page 932
  3 The Stochastic Galerkin Projection......Page 933
  4 The Energy Estimate......Page 934
  5 The Semi-discrete Formulation......Page 936
   5.1 Stability......Page 937
  6 Numerical Experiments......Page 938
  References......Page 940
  1 Introduction......Page 941
  2 Local BVP for the Numerical Flux......Page 942
   3.1 The Case uL ≥uR......Page 944
   3.2 The Case uL < uR......Page 945
   3.3 Choice of the Numerical Flux......Page 946
  4 Numerical Example......Page 948
  References......Page 949
  1 Introduction......Page 950
  3 Numerical Formulation......Page 952
  4 Numerical Procedure......Page 953
  6 Results......Page 956
  References......Page 958
 Conservative Mimetic Cut-Cell Method for Incompressible Navier-Stokes Equations......Page 959
   1.1 The Cut-Cell Primal-Dual Cell-Complex......Page 960
   1.2 The Incidence Matrices......Page 962
   1.3 The Discrete Hodge Operators......Page 963
   1.4 The Numerical Scheme......Page 964
  2 Numerical Results: The Flow Around a Cylinder......Page 965
  References......Page 966
Index......Page 968