Numerical Mathematics and Advanced Applications - ENUMATH 2013 : Proceedings of ENUMATH 2013, the 10th European Conference on Numerical Mathematics and Advanced Applications, Lausanne, August 2013

This Book Gathers A Selection Of Invited And Contributed Lectures From The European Conference On Numerical Mathematics And Advanced Applications (enumath) Held In Lausanne, Switzerland, August 26-30, 2013. It Provides An Overview Of Recent Developments In Numerical Analysis, Computational Mathematics And Applications From Leading Experts In The Field. New Results On Finite Element Methods, Multiscale Methods, Numerical Linear Algebra And Discretization Techniques For Fluid Mechanics And Optics Are Presented. As Such, The Book Offers A Valuable Resource For A Wide Range Of Readers Looking For A State-of-the-art Overview Of Advanced Techniques, Algorithms And Results In Numerical Mathematics And Scientific Computing. Preface -- Part I: Space Discretisation Methods For Pdes -- Part Ii: Time Integration Schemes -- Part Iii: a Posteriori Error Estimation And Adaptive Methods -- Part Iv: numerical Linear Algebra -- Part V: Multiscale Modeling And Simulation -- Part Vi: Reduced Order Modeling -- Part Vii: Optimal Control -- Part Viii: Uncertainty, Stochastic Modeling And Applications -- Part Ix: solvers, High Performance Computing And Software Libraries -- Part X: Computational Fluid And Structural Mechanics -- Part Xi: Computational Electromagnetics. Edited By Assyr Abdulle, Simone Deparis, Daniel Kressner, Fabio Nobile, Marco Picasso. 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