Numerical mathematics and advanced applications 2011 proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011

Cover......Page 1
Numerical Mathematics and Advanced Applications 2011......Page 4
Hierarchical Model Reduction: Three Different Approaches......Page 6
Contents......Page 10
1 The Obstacle Problem......Page 18
2 The Mimetic Discretization......Page 19
3.1 Mesh Refinement......Page 22
3.2 Hierachical Error Indicators......Page 23
3.3 Numerical Results......Page 25
References......Page 26
1 Introduction......Page 28
2 Unified Framework: The Main Ideas......Page 29
3.1 Discontinuous Galerkin Method......Page 31
3.2 Nonconforming Method......Page 32
3.3 Conforming Method......Page 33
4 Numerical Results......Page 34
References......Page 36
1 Introduction......Page 37
2 Notations......Page 38
4 An Error Estimator Based on Higher-Order Residuals......Page 39
4.1 Local Equivalence of the Error Estimators......Page 40
4.2 A Cheaper Variant on a Coarser Mesh......Page 42
5 Numerical Results......Page 43
References......Page 45
1 Introduction......Page 46
2 Discretization and the Mesh Adaptation Algorithm......Page 48
3.1 Accuracy of the Numerical Solution......Page 50
3.2 Condition Number of the Stiffness Matrix......Page 52
4 Conclusion......Page 53
References......Page 54
1 Introduction......Page 56
2.1 The Anisotropic Setting......Page 57
2.2 An Error Estimator for the H1-Seminorm......Page 58
2.3 A Goal-Oriented Error Estimator......Page 59
3.1 The Adaptive Procedure......Page 61
3.2 The ``Arrow\'\' Test Case......Page 62
References......Page 64
1 Introduction......Page 65
2.1 Linear Elliptic Equations......Page 66
2.2 Linear Elasticity......Page 68
3 Errors of Defeatured Models......Page 69
4 Errors of Homogenized Models......Page 71
References......Page 72
1 Introduction......Page 74
2 Governing Equations and the Finite Element Discretization......Page 75
2.1 Governing Equations......Page 76
3 Anisotropic Error Estimation......Page 77
3.1 Error Estimation and Anisotropy Detection......Page 78
4 Numerical Results......Page 79
References......Page 80
1 Introduction......Page 82
2 A Posteriori Error Estimates......Page 83
3.1 Equidistribution Strategy (Refinement)......Page 86
References......Page 91
1 Introduction......Page 92
2 The Magnetic Field......Page 93
3 Boundary Conditions for Hartmann Flow......Page 95
4 Numerical Experiments......Page 96
References......Page 98
1 Introduction......Page 100
2.1 The Continuous Maxwell\'s Problem......Page 101
2.2 DGTD-PpQk Space and Time Discretization......Page 102
3.2 A-Priori Convergence Analysis......Page 104
4 2D Numerical Test Case......Page 106
5 Conclusion and Future Research Tracks......Page 107
References......Page 108
1 Model Problem and Introduction......Page 109
2 Degeneracy and the Penalty Method......Page 111
3.1 Minimum Norm Solution and Stabilized Energy Functional......Page 113
3.2 Convergence......Page 114
References......Page 116
1 Introduction......Page 118
2 Maxwell\'s Equations......Page 119
3 Numerical Scheme......Page 120
4 Computational Results......Page 121
References......Page 125
1 Introduction......Page 126
2.1 Time-Harmonic Maxwell\'s Equations in 3D......Page 127
2.2 Notations......Page 128
3.1 Principles......Page 129
3.2 Characterization of the Reduced Problem......Page 130
3.3 On Numerical Traces......Page 131
4 Numerical Results......Page 132
References......Page 134
1 Introduction......Page 136
2.1 Component Splitting......Page 138
2.2 The Locally Implicit Method from Dol10, Pip06......Page 139
2.4 Convergence......Page 140
3 Numerical Results......Page 141
4 Fourth-Order Methods (Preliminary Investigation)......Page 143
References......Page 144
1 Introduction......Page 145
2.1 The Stefan Problem......Page 146
2.2 Level Set Method......Page 147
3.2 Material Derivative......Page 148
4.2 Simulation of Particle Growth......Page 149
5 Conclusions......Page 151
References......Page 152
1 Introduction......Page 153
2 Accelerated Simultaneous Method......Page 154
3 Convergence Theorem......Page 156
4 Computational Aspects......Page 157
References......Page 160
1 Introduction......Page 161
2 Multilevel Sparse Kernel-Based Interpolation......Page 162
3 Numerical Experiments......Page 164
References......Page 167
1 Introduction......Page 169
2 Dynamic Contact Problems for Non-linear Material Laws......Page 170
3 A Time Integration Scheme for Non-linear Elastic Problems......Page 172
4 Results......Page 174
References......Page 176
1 Introduction......Page 178
2 Problem Formulation......Page 179
3 Graph Case......Page 180
5 Numerical Results......Page 182
6 Conclusion......Page 184
References......Page 185
1 Introduction......Page 186
2.1 Tensorization of Univariate Bases......Page 188
2.2 Construction of the Special Univariate Multiwavelet Bases......Page 189
3 Selected Numerical Results......Page 191
References......Page 193
1 Motivation......Page 194
2 Image Processing Issues for a Model Situation......Page 195
3 Physically Based Image Processing and Optimal Control......Page 197
4 Numerical Method......Page 198
5 Numerical Results......Page 199
6 Conclusion......Page 201
References......Page 202
1 Introduction......Page 203
2 Regularity of the Solution......Page 204
4 Numerical Method......Page 205
5 Numerical Example......Page 208
References......Page 210
1 Introduction......Page 211
2.1 Navier-Stokes Equations for Two-Phase Flow......Page 212
3 Numerical Methods......Page 213
4 Curvature Discretizations......Page 214
5.1 A Static Disc Above a Rectangle......Page 215
5.2 Drop Collision in Shear Flow......Page 216
References......Page 218
1 Introduction......Page 220
2 H2-Optimal Model Reduction......Page 222
4 Numerical Results on the Advection–Diffusion Equation......Page 224
References......Page 227
1 Introduction......Page 228
2.1 Existence of Trajectories......Page 229
2.2 The Algorithm......Page 230
2.3 Potential Pitfalls, and Adaptivity as Partial Remedy......Page 231
3 Numerical Illustrations......Page 232
References......Page 235
1 Introduction......Page 236
2 Assumptions......Page 238
3 A Local Projection Discretization......Page 239
4 Well-Posedness of the Nonlinear Discrete Problem......Page 240
5 Error Estimate......Page 241
6 Numerical Results......Page 243
References......Page 244
1 Introduction......Page 245
2 The Modified bold0mu mumu BDM1BDM1equationBDM1BDM1BDM1BDM1 Scheme......Page 247
3.1 Stationary Advection-Diffusion Equation......Page 251
References......Page 252
1 Introduction......Page 254
2 Asymptotic Behavior of the Exact Solution......Page 256
3 A Finite Difference Scheme on a Special Mesh......Page 258
4 Numerical Results......Page 259
References......Page 262
1 Introduction......Page 263
2 Numerical Method......Page 264
3 Selection of the Artificial Diffusion Parameter......Page 265
4 An Exact Solution and Convergence Rates......Page 266
5 Eikonal Equation......Page 268
6 A Second-Order Fully Nonlinear Equation......Page 269
References......Page 270
1 Introduction......Page 271
2 Linear Stabilized Methods......Page 272
3 Nonlinear Stabilized Methods......Page 273
4 A Posteriori Optimization of Stabilization Parameters......Page 274
5 Choice of the Functional Ih......Page 275
References......Page 279
1 Introduction and Problem Formulation......Page 280
2 A Priori Estimates......Page 282
3 Numerical Discretization......Page 283
5 Discussion......Page 284
References......Page 286
1 Introduction......Page 288
2 Classical Difference Scheme for a Convection-Diffusion Problem......Page 289
3 Conditioning of Difference Scheme (4), (3)......Page 290
4 Stability of Difference Scheme (4), (3) to Perturbation in the Data......Page 294
References......Page 296
1 Introduction......Page 297
2 Problem Formulation. Aim of Research......Page 298
3 Difference Scheme on Uniform and Piecewise-Uniform Grids for Problems (1)......Page 299
4 Scheme of the Solution Decomposition Method for Problem (1)......Page 300
References......Page 304
1 Introduction......Page 306
2 Finite Element Formulation......Page 307
3.1 Implementation in deal.ii......Page 309
3.2 Numerical Examples......Page 311
References......Page 313
1 Introduction......Page 314
2 A Posteriori Estimators on Simple Domains......Page 316
2.1 Extension to a Discontinuous Galerkin Scheme......Page 317
3.1 Weighted Spaces......Page 318
References......Page 321
1 Introduction......Page 323
2 Detecting Steady Bifurcation Points in the Presence of O(2) Symmetry......Page 324
3 A Posteriori Error Estimation......Page 325
4 Numerical Experiment......Page 327
References......Page 330
1 Introduction......Page 331
2 Two-Grid hp–Version DGFEM......Page 332
3 A Posteriori Error Estimates......Page 334
4 Two-Grid hp–Adaptive Mesh Refinement Algorithm......Page 335
5 Numerical Experiment......Page 336
References......Page 339
1 Introduction......Page 340
2 Model Problem and DG Discretisation......Page 341
3.1 DWR Approach for Functionals......Page 342
4 Anisotropic Adaptive Strategy......Page 344
5 Numerical Experiment......Page 345
References......Page 348
1 Flow Problem......Page 349
2 Elasticity Problem......Page 351
3 Fluid-Structure Interaction Coupling and the Construction of the ALE Mapping......Page 352
5 Numerical Results......Page 353
References......Page 356
1 Continuous Problem......Page 357
2 Discretization......Page 358
3 Some Necessary Results......Page 359
4 Error Analysis for the Method of Lines......Page 360
5 Error Estimates for a Fully Implicit Scheme......Page 363
References......Page 365
1 Introduction......Page 367
2.1 Time and Space Discretization......Page 368
3 A Posteriori Error Analysis: Upper and Lower Error Bound......Page 369
4 Numerical Example......Page 373
References......Page 375
1 Introduction......Page 376
2 The Optimal Control Problem......Page 377
3.1 Discretization of State Equation......Page 378
3.3 Optimize-Then-Discretize Approach......Page 380
4 Numerical Results......Page 381
References......Page 383
1 Introduction......Page 385
2 An Immersed Boundary Model for Drug Release......Page 386
3 Computational Analysis of Dedicated Stents to Coronary Bifurcations......Page 389
4 Results and Conclusions......Page 391
References......Page 392
1 Introduction......Page 394
2 Model Problem and Variational Formulations......Page 395
3 Numerical Experiments......Page 397
References......Page 400
1 Introduction......Page 401
2 Operators Between Unrelated Finite Element Spaces......Page 402
3 Uniform Boundedness of the Linear NodalInterpolation in 1D......Page 404
4 Counterexamples for Higher Order Nodal Interpolation......Page 407
References......Page 408
1 Introduction......Page 410
2.1 Elemental Stiffness Matrix......Page 412
2.2 Elemental Mass Matrix......Page 414
3.1 Dispersion and Stability Analysis on a Rectangular Mesh......Page 415
3.2 Eliminating Dispersion and Anisotropy in the Low Frequency Limit......Page 416
4 Numerical Experiments......Page 418
References......Page 419
1 Introduction and Problem Setting......Page 421
2 Discretisation......Page 423
3 Numerical Experiments......Page 425
References......Page 428
1 Geodesic Finite Elements......Page 429
2 Spaces of Zero Curvature......Page 431
3 Interpolation Error......Page 432
4 Nonlinear Scaling and the Richards Equation......Page 435
References......Page 437
1 Introduction and Background......Page 438
2.1 Specifying Anisotropy......Page 439
3 Numerical Solution......Page 440
3.1 Flux Approximation......Page 441
4 Convergence Studies......Page 442
6 Conclusion......Page 444
References......Page 445
1 Introduction......Page 447
2.1 Geometrical Configuration......Page 448
2.2 Weak Formulation of the Model......Page 449
3.1 Notation......Page 450
3.3 Semi-discrete Finite Element Approximation of (6d)......Page 452
4 Numerical Results......Page 453
References......Page 455
1 Introduction......Page 456
3 Numerical Schemes......Page 457
4 Computational Setup......Page 458
5 Numerical Results......Page 459
6 Conclusion......Page 462
References......Page 463
1 Introduction......Page 464
2 Governing Equations......Page 466
3 Numerical Methods......Page 467
5.1 Rising Bubble Benchmark in Newtonian Fluids......Page 468
6.1 Rising Bubble in Newtonian Fluids......Page 469
6.2 Rising Bubble in Viscoelastic Fluids......Page 470
References......Page 471
1 Introduction......Page 473
2 Governing Equations......Page 474
3 Reduced Model for the Advection-Diffusion Problem......Page 475
4 Numerical Discretization......Page 477
5 Results......Page 478
References......Page 480
1 Introduction......Page 482
2 Galerkin Time Stepping for the Navier-Stokes Equations......Page 483
2.1 The cGP(2)-Method......Page 485
3 Numerical Results......Page 487
5 Conclusion......Page 489
References......Page 490
1 Introduction......Page 491
2 The Density-Enthalpy Model......Page 492
3 The Solution Strategy......Page 493
3.1  Numerical Treatment of the Mass Equation......Page 494
4 Numerical Experiments......Page 495
4.1 Flow Due to the Gravity Term......Page 496
4.2 Flow Due to the Gravity and Temperature Gradient......Page 497
References......Page 498
1 Mathematical Model......Page 500
1.1 Stress Tensor......Page 501
2 Numerical Solution......Page 503
3 Numerical Results......Page 505
References......Page 507
1 Introduction......Page 509
2 Mathematical Model and Solution Method......Page 510
2.1 Boundary Conditions......Page 511
3 Numerical Results......Page 512
4 Conclusions......Page 516
References......Page 517
1 Introduction and the Geometry of the Problem......Page 518
2.2 Transformation to the Rotating Coordinates......Page 520
3.1 Weak Formulation of the Problem (8)–(11)......Page 523
3.2 Auxiliary Estimates......Page 524
References......Page 526
1 Introduction......Page 527
2 Mathematical Model......Page 528
3 Numerical Scheme......Page 529
4 Experimental Analysis of Convergence......Page 532
5 Conclusion......Page 534
References......Page 535
1 Introduction......Page 536
2 Mathematical Models......Page 537
4 Numerical Solution......Page 538
5 Numerical Results......Page 539
6 Discussion and Conclusions......Page 540
References......Page 544
1 Mathematical Model......Page 545
1.1 Constitutive Models......Page 546
3 Numerical Results......Page 547
3.1 Conclusion......Page 551
References......Page 552
1 Introduction......Page 553
2 Variational Multiscale Model of Navier-Stokes Problem......Page 554
3 Applications with Layer-Adapted Meshes......Page 556
4 Applications with Weak Dirichlet Boundary Conditions......Page 558
References......Page 561
1 Introduction......Page 562
2 Mathematical Model......Page 563
3 Numerical Approximation......Page 565
4 Numerical Results......Page 567
References......Page 569
1 Introduction......Page 571
2 The Visco–Elastic Model......Page 572
3 Stability of the Steady–State Equilibrium......Page 573
4 Analytic Solutions to the One-Dimensional Visco–Elasticity Model......Page 575
References......Page 579
1 Introduction......Page 580
2 Derivation......Page 581
3 Numerical Verification......Page 585
4 Conclusions......Page 586
References......Page 587
1 Introduction......Page 588
2 The Equations......Page 589
3 Stability of the Time-Discretized ALE Fluid Problem and the FSI Problem......Page 590
3.1 Discretization of the ALE Convection Term......Page 592
4 Numerical Tests and Observations......Page 593
References......Page 595
1 Introduction......Page 596
2 The Mathematical Model......Page 597
3 Strategy and Numerical Methods......Page 600
4 Results......Page 601
References......Page 603
1 Introduction......Page 605
1.1 Governing Equations......Page 606
2.1 A Library for Fast PDE Solver Proto-Typing......Page 607
2.2 Improving Defect Correction with Mixed Precision......Page 609
3 Concluding Remarks......Page 611
References......Page 612
1 Introduction......Page 613
2 Helmholtz Equation and Solver......Page 614
3 Multi-GPU Implementation......Page 615
3.1 Split of the Algorithm......Page 616
4.1 Vector- and Sparse Matrix-Vector Operations......Page 617
4.2 Bi-CGSTAB and Gauss-Seidel on Multi-GPU......Page 618
5 Numerical Experiments for the Wedge Problem......Page 619
6 Conclusions......Page 620
References......Page 621
1 Introduction......Page 622
2 Bouyancy Driven Fluid Flow......Page 623
3 Preconditioning Strategies......Page 625
5 Numerical Results......Page 627
6 Conclusion and Outlook......Page 629
References......Page 630
1 Introduction......Page 631
2 Shallow Water Simulation......Page 632
3 Sparse Cell Updates......Page 633
4 Results......Page 637
References......Page 638
2 BDDC Preconditioner with Two and More Levels......Page 640
3 Parallel Implementation......Page 642
4 Numerical Results......Page 643
References......Page 646
1 Introduction......Page 648
2 Physical and Mathematical Modeling......Page 649
3 Parametrization of Uncertainty......Page 650
4 Bayesian Inference......Page 651
5 Simulation Study......Page 652
References......Page 656
1 Introduction......Page 657
2.1 The Numerical Scheme......Page 659
2.2 Convergence......Page 660
3 The Mixed Variational Formulation......Page 661
3.2 Convergence......Page 662
4 A Numerical Example......Page 663
References......Page 664
2 Preliminaries......Page 667
3 A Variational Multiscale Decomposition of Poisson\'s Equation in Mixed Form......Page 668
4 A Posteriori Error Estimate......Page 669
5 Numerical Example......Page 672
References......Page 674
1 Introduction......Page 675
2 The Shallow Water Equations......Page 676
3 The Adaptive Geometrical Multiscale Solver......Page 678
3.1 Heuristic Modeling Error Indicator for the SWE......Page 679
4 A Numerical Example......Page 681
References......Page 682
1 Introduction......Page 683
2 Review and Preliminaries......Page 684
3 Known Bounds for the Residual Norm......Page 685
4 A New A-posteriori Bound......Page 686
References......Page 690
1 Introduction......Page 691
2 Problem Setup......Page 692
3 Analysis of the Fixed-Point Iteration......Page 694
4 Preconditioning of the Iteration......Page 696
5 Numerical Example and Conclusion......Page 697
References......Page 698
1 Introduction......Page 700
2.1 Eigenvectors......Page 702
2.2 Interpolation Matrices......Page 704
3 Does Eigenvector-Based Transfer Operator Lead to a Better Convergence?......Page 705
4 Singular but Consistent Systems......Page 706
References......Page 708
1 Introduction......Page 710
2 Interior Penalty Methods......Page 712
3 Preconditioning......Page 714
4 Numerical Tests......Page 716
References......Page 717
1 Introduction......Page 719
2 The Optimal Control Problem......Page 720
3 A Robust Preconditioning Technique......Page 722
4 Numerical Experiments......Page 725
References......Page 726
1 Introduction......Page 728
1.2 Tensor Train Matrix Format......Page 729
2 Preconditioned Inverse Iteration......Page 730
3 Computing Inner Eigenvalues by Folded Spectrum Method......Page 731
4 Numerical Experiments......Page 733
References......Page 735
1 Introduction and Motivation......Page 736
2.1 Definition......Page 739
2.2 Metastability Along a Reaction Coordinate......Page 741
2.3 A First Example: The Adaptive Biasing Force Technique......Page 742
2.4 A Second Example: Obtaining an Effective Dynamics on ξ(Xt)......Page 745
3.1 Definition and Two Basic Properties of the Quasi-stationary Distribution......Page 747
3.2 A First Application: Analysis of the Parallel Replica Dynamics......Page 750
3.3 A Second Application: Going from Continuous State Space Dynamics to Kinetic Monte Carlo Models......Page 751
References......Page 753
1 Introduction......Page 756
2 Problem Definition and Notation......Page 757
3 Error Indication......Page 758
4 Numerical Results and Conclusions......Page 762
References......Page 763
1 Introduction......Page 765
2 American Option Model......Page 766
3 Reduced Basis Method......Page 767
4 Reduced Basis Construction......Page 768
5 Numerical Results......Page 770
References......Page 772
1 Introduction......Page 774
2 Overview of SAVU......Page 775
3 Application Examples......Page 778
4 The Axially Travelling Panel Submerged in Ideal Fluid......Page 779
References......Page 781
1 Introduction......Page 783
2 Singularity of Fisher Information......Page 784
2.2 Fractional Brownian Motion......Page 785
2.3 Stable Lévy Process......Page 786
2.4 Meixner Lévy Process......Page 787
3 Speed of Convergence to Singularity......Page 788
References......Page 790
1 Motivations......Page 792
2.1 Uniform Hi-Mod Reduction......Page 793
2.2 Piecewise Hi-Mod Reduction......Page 794
2.3 Nodewise Hi-Mod Reduction......Page 796
3 Numerical Assessment......Page 797
References......Page 800