Numerical Mathematics and Advanced Applications (eBook)

Preface 5
Contents 7
PLENARY LECTURES 20
Compatible Discretizations in Two Dimensions 21
1 Introduction 21
2 Construction of the dual complex 23
3 Applications 30
References 37
Finite Element Approximation of the Three Field Formulation of the Elasticity Problem Using Stabilization 39
1 Introduction 39
2 Problem statement and Galerkin finite element discretization 41
3 Finite element approximation using subscales 42
4 Numerical analysis of the original formulation 45
5 A modified stabilized problem 49
6 Concluding remarks 54
References 55
Convergence of Adaptive Wavelet Methods for Goal– Oriented Error Estimation* 57
1 Introduction 57
2 Goal–oriented error estimation 59
3 Adaptive error estimation 64
4 Numerical experiments 72
References 77
Quadratic Programming and Scalable Algorithms for Variational Inequalities 80
1 Introduction 80
2 Bound constrained problems 81
3 Bound and equality constrained problems 83
4 Model problem 85
5 FETI and total FETI domain decomposition 87
6 FETI–DP domain decomposition and discretization 89
7 Numerical scalability 90
8 Numerical experiments 91
9 Comments and conclusions 93
Acknowledgements 93
References 93
Discontinuous Galerkin Methods for Friedrichs’ Systems 97
1 Introduction 97
2 Friedrichs’ systems 99
3 Design and analysis of DG methods 101
4 DG approximation of two-field Friedrichs’ systems 105
5 Examples 108
6 Concluding remarks 113
References 113
Highly Oscillatory Quadrature: The Story so Far 115
1 The challenge of high oscillation 115
2 Asymptotic expansion in the absence of critical points 117
3 Asymptotic, filon and levin methods 120
4 Critical points 129
5 Conclusions and pointers for further research 133
References 136
The 3D Inverse Electromagnetic Scattering Problem for a Coated Dielectric 137
1 Introduction 137
2 Formulation of the direct and inverse scattering problem 138
3 Analysis of the inverse problem 141
4 Numerical example 149
5 Conclusion 150
Acknowledgment 151
References 151
Functional Approach to Locally Based A Posteriori Error Estimates for Elliptic and Parabolic Problems 153
1 Introduction 153
2 Functional a posteriori estimates for elliptic problems 155
3 Functional a posteriori estimates for a model evolutionary problem 164
References 167
Finite Element Approximation of 2D Parabolic Optimal Design Problems 169
1 Introduction 169
2 Preliminaries 173
3 Preliminaries on the convergence of the numerical scheme 175
4 Convergence of discrete optimal shapes 179
5 Gradient calculations: A numerical approach 181
6 Conclusions 191
Acknowledgements 192
References 192
CONTRIBUTED LECTURES 195
3D Free Surface Flows Simulations Using a Multilayer Saint- Venant Model. Comparisons with Navier- Stokes Solutions 197
1 Introduction 197
2 Navier-Stokes equations and hydrostatic approximation 197
3 A Multilayer saint-venant system 199
4 Numerical method 201
5 Numerical results 202
References 204
Some Well-Balanced Shallow Water-Sediment Transport Models 206
1 Sediment transport model 206
2 Finite volume method for non conservative hyperbolic systems 209
3 High order schemes based on state reconstruction 211
4 Numerical test: comparison with an analytical solution 212
References 213
Highly Accurate Conservative Finite Difference Schemes and Adaptive Mesh Refinement Techniques for Hyperbolic Systems of Conservation Laws* 214
1 Introduction 214
2 Finite-difference Shu-Osher schemes 215
3 Adaptive mesh refinement for Shu-Osher schemes 217
4 Numerical examples 219
5 Conclusions 220
References 221
Finite Volume Solvers for the Shallow Water Equations Using Matrix Radial Basis Function Reconstruction 223
1 Finite volume methods for shallow water models 223
2 Matrix valued Radial Basis Functions for vector field reconstruction 225
3 Applications to environmental modelling 226
References 230
On Numerical Schemes for a Hierarchy of Kinetic Equations 232
1 Introduction 232
2 Hexagonal kinetic models 233
3 Hexagonal hierarchy 235
4 Coupling of two kinetic models 236
References 239
Computational Aspects of the Mesh Adaptation for the Time Marching Procedure 240
1 Euler equations 240
2 Adaptive algorithm 241
3 Anisotropic mesh adaptation 242
4 Geometric mass conservation law 244
5 Numerical example 246
Acknowledgement 247
References 247
On the Use of Slope Limiters for the Design of Recovery Based Error Indicators 248
1 Introduction 248
2 A posteriori error indication 249
3 Limited gradient averaging 250
4 Limited gradient reconstruction 251
5 Adaptation strategy 253
6 Numerical examples 254
7 Conclusions 254
References 255
On a Superconvergence Result for Mixed Approximation of Eigenvalue Problems 257
1 Introduction 257
2 Statement of the problem and its discretization 258
3 Main results 259
4 Numerical results 260
References 264
Comparative Study of the a Posteriori Error Estimators for the Stokes Problem 266
1 Introduction 266
2 The stokes problem and its approximation 267
3 Estimation of the deviation from the exact solution 268
4 Numerical experiments 269
References 273
Error Control for Discretizations of Electromagnetic- Mechanical Multifield Problem 274
1 Introduction 274
2 Electromagnetic forming 275
3 Error control for coupled and mixed problems 278
References 281
A Safeguarded Zienkiewicz-Zhu Estimator 283
1 Introduction 283
2 Error estimator and theoretical results 284
3 Numerical results 287
References 290
Some Remarks on a Model for the Atmospheric Pressure in Ocean Dynamics 292
1 Introduction 292
2 Modelling of non-hydrostatic free-surface flows 293
3 A linearised model for the free surface equation 296
References 299
Computational Time Improvement for Some Shallow Water Finite Volume Models Applying Parallelization and Optimized Small Matrix Computations. 301
1 Introduction 301
2 Equations 302
3 Numerical scheme 303
4 Parallel SIMD implementation 303
References 308
Discretization Error Estimates for an Optimal Control Problem in a Nonconvex Domain 311
1 Introduction 311
2 Theory 312
3 Numerical results 315
References 319
A Posteriori Estimates for Cost Functionals of Optimal Control Problems 320
1 Introduction 320
2 Majorants for the cost functional 322
3 Properties of majorants 323
4 Practical implementation 324
5 Numerical experiments 325
References 327
Optimization of a Duality Method for the Compressible Reynolds Equation 330
1 The mathematical model 330
2 Numerical solution 331
3 Optimization of the duality algorithm 333
4 Numerical examples 335
5 Conclusions 337
Aknowledgments 337
References 337
Time-Space & Space-Time Elements for Unsteady Advection- Dominated Problems
1 Introduction 339
2 Stabilization techniques for the stationary problem 340
3 Stabilized methods for the non-stationary problem 341
4 Numerical Experiments 345
Acknowledgements 346
References 346
On Discontinuity–Capturing Methods for Convection– Diffusion Equations 347
1 Introduction 347
2 Galerkin’s finite element discretization 348
3 The SUPG method 348
4 Methods diminishing spurious oscillations in layers 349
5 Conclusions 353
Acknowledgements 354
References 354
Algebraic Flux Correction for Finite Element Approximation of Transport Equations 356
1 Introduction 356
2 Flux decomposition 356
3 Algebraic flux correction 357
4 Numerical example 362
5 Conclusions 363
References 363
A Parallel Multiparametric Gauss-Seidel Method* 365
1 Introduction 365
2 The local modified extrapolated Gauss-Seidel ( LMEGS) method 366
3 The eigenvalue relationship 366
4 Determination of good values 368
5 Numerical results and conclusions 370
Acknowledgement 372
References 372
A Numerical Scheme for the Micro Scale Dissolution and Precipitation in Porous Media 373
1 Introduction 373
2 The numerical scheme 376
3 A fixed point iteration 377
4 A numerical example 379
Acknowledgment 379
References 380
Discrete Kinetic Methods for a Degenerate Parabolic Equation in Dimension Two. 383
1 Introduction 383
2 Kinetic models 384
3 The numerical schemes 386
4 Numerical experiments 389
References 390
Anisotropic Doubly Nonlinear Degenerate Parabolic Equations 391
1 Introduction 391
2 Entropy solution 392
3 Renormalized entropy solution 394
Acknowledgement 396
References 396
A Multiresolution Method for the Simulation of Sedimentation- Consolidation Processes 397
1 Introduction 397
2 The multiresolution scheme 398
3 Sedimentation-consolidation processes 402
4 Numerical results 402
Acknowledgements 404
References 404
Diffusive Relaxation Limit for Hyperbolic Systems 406
1 Introduction 406
2 BGK approximation of strongly parabolic systems 409
3 One–dimensional semilinear model in viscoelasticity 410
4 Multidimensional viscoelasticity and modulated energy 411
References 413
Parallel Algorithms for Nonlinear Diffusion by Using Relaxation Approximation 414
1 Relaxation approximation of nonlinear diffusion 414
2 The numerical scheme 415
3 Numerical results 417
4 Concluding remarks 421
References 421
On a Degenerated Parabolic-Hyperbolic Problem Arising From Stratigraphy 422
1 Introduction and presentation of the model 422
2 A locally hyperbolic behaviour 423
3 Definition of a solution and existence results for a discretized problem 424
4 Some numerical illustrations 428
5 Conclusion and open problems 428
References 429
Schwarz Domain Decomposition Preconditioners for Interior Penalty Approximations of Elliptic Problems 432
1 Introduction 432
2 Discontinuous Galerkin methods for elliptic problems 433
3 Non-overlapping Schwarz methods 434
4 Convergence analysis 436
5 Numerical results 437
References 439
Higher Order Semi-Implicit Discontinuous Galerkin Finite Element Schemes for Nonlinear Convection- Diffusion Problems* 441
1 Introduction 441
2 Scalar equation 442
3 Numerical results 446
4 Conclusion 448
References 448
On Some Aspects of the Discontinuous Galerkin Method* 449
1 Continuous problem 449
2 Discretization of the problem 450
3 Error estimates 453
4 Application of the DGFEM to compressible flow with a wide range of mach numbers 455
5 Conclusion 455
References 456
Mixed Discontinuous Galerkin Methods with Minimal Stabilization 457
1 DG methods 457
2 Minimal stabilization 460
3 Numerical results 462
References 464
Discontinuous Galerkin Finite Element Method for a Fourth- Order Nonlinear Elliptic Equation Related to the Two- Dimensional Navier– Stokes Equations 466
1 Introduction 466
2 Mathematical formulation 468
3 DGFEM for a 4th-order advective PDE 469
4 Numerical results 471
References 473
Fourier Method with Nitsche-Mortaring for the Poisson Equation in 3D 475
1 Introduction 475
2 Fourier decomposition and mortaring in 2D 477
3 Fourier-nitsche-finite-element approximation in 3D 479
References 481
Substructuring Preconditioners for the Bidomain Extracellular Potential Problem 483
1 Introduction 483
2 Mortar method 484
3 Substructuring preconditioners 485
References 490
A Face Penalty Method for the Three Fields Stokes Equation Arising from Oldroyd- B Viscoelastic Flows 493
1 Introduction 493
2 A finite element formulation 495
3 The inf-sup condition 496
4 A priori error estimates 496
5 A stable iterative algorithm 497
6 Preliminary numerical results 497
References 499
Anisotropic H1-Stable Projections on Quadrilateral Meshes 501
1 Introduction 501
2 Anisotropic H1-stable projectors 503
3 General result 508
References 509
Continuous Interior Penalty hp-Finite Element Methods for Transport Operators 510
1 Introduction 510
2 Continuous interior penalty finite element methods 511
3 Technical results 512
4 Convergence analysis 514
5 Numerical results 514
References 517
A Nonconforming Finite Element Method with Face Penalty for Advection– Diffusion Equations 518
1 Introduction 518
2 The nonconforming finite element scheme with face penalty 519
3 A posteriori error estimates 522
4 Numerical results 523
Acknowledgment 524
References 525
Efficient Multigrid and Data Structures for Edge- Oriented FEM Stabilization 526
1 Introduction 526
2 Sparsity of the matrix 527
3 Local pressure Schur complement approach 530
4 Numerical example 532
References 533
Adaptive Methods for Dynamical Micromagnetics 535
1 Introduction 535
2 Numerical methods 536
3 Adaptive algorithm 538
4 Numerical experiment 539
References 541
Stability for Walls in Ferromagnetic Nanowire 543
1 Model for ferromagnetic nanowires 543
2 Landau-Lifschitz Equation in the mobile frame 545
3 A new system of coordinates 546
4 Estimates for the perturbations 548
References 550
Continuous Galerkin Methods for Solving Maxwell Equations in 3D Geometries 551
1 Introduction and notations 551
2 Variational formulations and discretization 553
3 Numerical results and conclusion 555
References 558
On the Use of the Gautschi-Type Exponential Integrator for Wave Equations 560
1 Introduction 560
2 Sine–Gordon equation 561
3 Discretisation 562
4 The Gautschi-type exponential integrator 564
5 Numerical example 565
References 566
Positivity of Exponential Multistep Methods 567
1 Introduction 567
2 Analytical framework 568
3 Exponential multistep methods 570
4 Positivity and order barrier 571
References 574
Stability Results and Algorithmic Strategies for the Finite Element Approach to the Immersed Boundary Method 576
1 Introduction 576
2 The finite element immersed boundary method 577
3 Time discretization by finite differences 579
4 Stability analysis by energy estimates 580
5 Numerical results 581
6 Conclusions 583
References 583
A Comparison of Enthalpy and Temperature Methods for Melting Problems on Composite Domains 585
1 Introduction 585
2 Problem description 586
3 Numerical methods 588
4 Numerical experiments 590
5 Conclusions 592
References 592
Qualitative Properties of a Numerical Scheme for the Heat Equation 593
1 Introduction 593
2 Proof of the results 595
Acknowledgements 600
References 600
Modeling Radiation and Moisture Content in Fire Spread 601
1 Introduction 601
2 Physical model 602
3 Governing equations 602
4 Numerical method 604
5 Numerical results 607
Acknowledgement 608
References 608
Fast Multipole Method for Solving the Radiosity Equation 609
1 Introduction 609
2 Radiosity equation and numerical solution 610
3 FMM and kernel expansion 611
4 A new fast method 612
5 Numerical results 615
6 Conclusion 616
References 616
Numerical Modelling of Kinetic Equations 618
1 Introduction 618
2 Asymptotic method 619
3 Numerical examples 622
References 625
On a Subclass of H¨ older Continuous Functions with Applications to Signal Processing 627
1 Introduction 627
2 H¨ older continuous spaces 628
3 Generalized Harten’s Subcell resolution technique 630
4 Numerical experiments 631
Acknowledgments 634
References 634
Modelisation and Simulation of Static Grain Deep- Bed Drying 636
1 Introduction 636
2 Models 637
3 The numerical schemes 638
4 Numerical experiments and comparisons 642
References 643
Hybrid Godunov-Glimm Method for a Nonconservative Hyperbolic System with Kinetic Relations 644
1 Introduction 644
2 The PDE model and main properties 645
3 Godunov-Glimm Hybrid method 647
4 The nonlinear projection 649
References 651
Cell-Average Multiwavelets Based on Hermite Interpolation* 652
1 Introduction 652
2 Harten’s framework for multiresolution analysis 652
3 Vector multiresolution analysis for cell-averaged data 654
4 Numerical experiments 657
References 659
On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws 660
1 Introduction 660
2 Choice of paths 662
3 Godunov’s method 664
4 Application to linear balance laws 665
Acknowledgements 667
References 667
Sequential Flux-Corrected Remapping for ALE Methods 669
1 Introduction 669
2 Sequential FCR method 671
3 Numerical examples 675
Acknowledgments 676
References 676
Orthogonal hp-FEM for Elliptic Problems Based on a Non- Affine Concept 679
1 Introduction and historical remarks 679
2 Preliminaries 680
3 Numerical example 681
4 Construction of basis functions 682
5 Numerical example continued 685
Acknowledgment 686
References 686
On Some Aspects of the hp-FEM for Time- Harmonic Maxwell’s Equations 687
1 Introduction 687
2 Formulation of the problem 688
3 Shape functions 689
4 Numerical experiments 692
5 Conclusion and outlook 692
Acknowledgment 694
References 694
Numerical Simulation of Phase-Transition Front Propagation in Thermoelastic Solids 697
1 Introduction 697
2 Formulation of the problem 698
3 Conservative wave propagation algorithm 699
4 Contact quantities and numerical fluxes 701
5 Conclusions 703
Acknowledgment 704
References 704
The Level Set Method for Solid-Solid Phase Transformations 706
1 Introduction 706
2 The physical problem 708
3 The computational method 709
4 Numerical results 710
5 Conclusions 712
References 713
A Non-Monotone Fast Marching Scheme for a Hamilton- Jacobi Equation Modelling Dislocation Dynamics* 715
1 Introduction 715
2 The FMM algorithm for unsigned velocity 717
3 Numerical tests 719
Acknowledgments 722
References 722
A Time–Adaptive Semi–Lagrangian Approximation to Mean Curvature Motion 724
1 Introduction 724
2 General requirements on the adaptation strategy for geometric equations 726
3 A strategy based on local truncation error 727
4 Numerical tests 729
References 731
Heterogeneous Multiscale Methods with Quadrilateral Finite Elements 733
1 Introduction 733
2 HMM with quadrilaterals finite elements 734
3 Error analysis 736
4 Numerical experiments 738
Acknowledgment 739
References 740
Stabilizing the P1/P0 Element for the Stokes Problem via Multiscale Enrichment 742
1 Introduction 742
2 The model problem and the general framework 743
3 Application to the 745
pair 745
4 Numerical validations 748
References 749
Adaptive Multiresolution Methods for the Simulation of Shocks/ Shear Layer Interaction in Confined Flows 751
1 Introduction 751
2 Numerical results 752
3 Conclusion and perspectives 756
References 758
Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes 760
1 Introduction 760
2 Local projection stabilization on isotropic meshes 761
3 Anisotropic affine linear meshes 762
4 Local projection stabilization on anisotropic meshes 764
References 767
An Interior Penalty Variational Multiscale Method for High Reynolds Number Flows 769
1 Introduction 769
2 The equations of incompressible flow 770
3 Separation of scales and stabilized finite element methods 770
4 A posteriori error estimation 774
5 A numerical result 775
References 775
Variational Multiscale Large Eddy Simulation of Turbulent Flows Using a Two- Grid Finite Element or Finite Volume Method 778
1 Introduction 778
2 Multiscale formulation 779
3 Numerical results for turbulent flow in a diffuser 782
4 Conclusions 784
Acknowledgements 785
References 785
Issues for a Mathematical Definition of LES 786
1 Introduction 786
2 Suitable approximations 788
3 Review of existing pre–LES–models 789
4 Discretization 791
References 793
Stabilized FEM with Anisotropic Mesh Refinement for the Oseen Problem 795
1 Introduction 795
2 Stabilized FEM for linearized Navier-Stokes problem 796
3 Stability and convergence on hybrid meshes 798
4 Error estimates and design of stabilization parameters 799
5 Application to channel flow 801
Acknowledgment 802
References 802
Semi-Implicit Multiresolution for Multiphase Flows 804
1 Introduction 804
2 Semi implicit scheme on uniform grid 806
3 Multiscale analysis of the explicit-implicit scheme 807
References 811
Numerical Simulation of Vortex-Dipole Wall Interactions Using an Adaptive Wavelet Discretization with Volume Penalisation 812
1 Introduction 812
2 Adaptive wavelet discretization with volume penalisation 813
3 Vortex-dipole wall interactions 816
4 Conclusions 818
Acknowledgements 819
References 819
Inviscid Flow on Moving Grids with Multiscale Space and Time Adaptivity 821
1 Introduction 821
2 The ALE formulation of the Euler equations 822
3 Finite volume discretization 823
4 Adaptive multiscale method 825
5 Numerical results 827
References 828
A Relaxation Method for a Two Phase Flow with Surface Tension 831
1 Introduction 831
2 Numerical approximation 833
3 Relaxation method for the simplified model with surface tension forces 834
4 Numerical results 837
5 Conclusion 838
References 838
Extension of Interface Coupling to General Lagrangian Systems 840
1 Introduction 840
2 Coupling two p- systems 843
3 Coupling two Euler systems in Lagrangian coordinates 844
4 Coupling Lagrangian systems of different dimensions 846
5 Conclusion 847
References 847
A Numerical Scheme for the Modeling of Condensation and Flash Vaporization in Compressible Multi- Phase Flows 849
1 Introduction 849
2 The thermodynamics of phase transition 849
3 The Riemann problem with phase transition at equilibrium 850
4 The Riemann problem with out of equilibrium EOS 852
5 Numerical scheme 854
6 Numerical results 854
7 Conclusions 856
Acknowledgments 856
References 856
An Adaptive Operator Splitting of Higher Order for the Navier- Stokes Equations 858
1 Introduction 858
2 The Taylor based gradient recovery technique 859
3 The stabilized base splitting 860
4 The multi-grid postprocessing 861
5 Numerical results 862
6 Conclusions 865
Acknowledgment 865
References 865
The POD Technique for Computing Bifurcation Diagrams: A Comparison among Different Models in Fluids 867
1 Introduction 867
2 The POD technique 868
3 Description of the examples 870
4 Computation of the bifurcation diagrams of the example models 872
5 Conclusions 873
Acknowledgement 874
References 874
Filtering of Singularities in a Marangoni Convection Problem 876
1 Introduction 876
2 Formulation of the problem 877
3 Basic state 878
4 Linear stability 880
5 Conclusions 883
Acknowledgments 883
References 883
On Application of Stabilized Higher Order Finite Element Method on Unsteady Incompressible Flow Problems 884
1 Mathematical model 884
2 Time-spatial discretization 887
3 Numerical solution and results 889
References 891
Numerical Simulation of Coupled Fluid-Solid Systems by Fictitious Boundary and Grid Deformation Methods 893
1 Introduction 893
2 Grid deformation method 895
3 Numerical solution of the fluid-solid system 896
4 Verification of the numerical techniques 898
5 Conclusions 899
References 900
An Iterative Method for Solving Non-Linear Hydromagnetic Equations 903
1 Introduction. Statement of the problem 903
2 An iterative method for the magnetostatic system 905
3 Finite element discretization 906
4 Computational tests 909
References 910
Mathematical and Numerical Analysis of a Class of Non- linear Elliptic Equations in the Two Dimensional Case 912
1 Introduction 912
2 Statement of the main result 913
3 Proof of theorem 1 914
4 Numerical method 916
References 919
A s-step Variant of the Double Orthogonal Series Algorithm 922
1 Introduction 922
2 The algorithm of the Double Orthogonal Series 923
3 s-Step methods 924
4 The s-step variant of the double orthogonal series 924
5 Numerical results 926
6 Conclusions 929
Acknowledgment 929
References 929
Linear Equations in Quaternions 930
1 Basic properties and definitions for quaternions 930
2 Linear equations in quaternions 932
Acknowledgment 937
References 937
Computing the Analytic Singular Value Decomposition via a Pathfollowing 939
1 Introduction 939
2 Formulation of the problem 940
3 Solving defining equations 942
4 Experiments 943
5 Conclusions 945
Acknowledgments 946
References 946
A Jacobi-Davidson Method for Computing Partial Generalized Real Schur Forms 948
1 Introduction 948
2 The RJDQZ method for real matrix pencils 949
3 Numerical comparison 953
4 Conclusions 955
References 955
Pricing Multi-Asset Options with Sparse Grids and Fourth Order Finite Differences 958
1 Introduction 958
2 Model 958
3 Discretization 959
4 Sparse grid combination technique 961
5 Numerical results 963
6 Conclusions 965
References 965
A Third Order Linearly Implicit Fractional Step Method for Semilinear Parabolic Problems 968
1 Introduction 968
2 A new third order linearly implicit FSRK method 970
3 Numerical tests 973
References 976
Numerical Solution of Optimal Control Problems with Sparse SQP- Methods 977
1 Introduction 977
2 Optimal control problem 978
3 SQP-methods 979
4 Equality constrained quadratic subproblem 980
5 Approximation of the Hessian 981
6 Example 982
References 984
Semi-Deterministic Recursive Optimization Methods for Multichannel Optical Filters 986
1 Introduction 986
2 Global optimization methods 987
3 Application to multichannel optical filters design 990
4 Conclusions 993
References 993
A Multigrid Method for Coupled Optimal Topology and Shape Design in Nonlinear Magnetostatics* 994
1 Introduction 994
2 Topology optimization for magnetostatics 995
3 Sequential coupling of topology and shape optimization 997
4 Multilevel shape optimization 997
5 Numerical results 999
6 Conclusion 1000
References 1001
Nonsmooth Optimization of Eigenvalues in Topology Optimization 1002
1 Introduction 1002
2 Topology optimization and eigenproblems 1003
3 Nonsmooth analysis 1005
4 Numerical results 1007
5 Conclusion 1009
References 1009
Derivative Free Optimization of Stirrer Configurations 1010
1 Introduction 1010
2 Flow solver and numerical optimization tool 1011
3 Results 1013
4 Conclusion 1016
Acknowledgment 1017
References 1017
Mathematical Modelling and Numerical Optimization in the Process of River Pollution Control 1019
1 Introduction 1019
2 Mathematical formulation 1020
3 Numerical solution 1022
4 Numerical results 1025
Acknowledgements 1026
References 1026
A Family of C0 Finite Elements for Kirchhoff Plates with Free Boundary Conditions 1029
1 Introduction 1029
2 Kirchhoff plate bending problem 1030
3 Finite element formulation 1031
4 A-priori error estimates 1032
5 A-posteriori error estimates 1033
6 Computational results 1034
References 1036
A Postprocessing Method for the MITC Plate Elements 1037
1 Introduction 1037
2 The Reissner–Mindlin plate model 1037
3 MITC finite element methods 1038
4 Superconvergence 1039
5 Postprocessing method 1040
6 Benchmark computations 1041
References 1045
A Uniformly Stable Finite Difference Space Semi- Discretization for the Internal Stabilization of the Plate Equation in a Square 1046
1 Statement of the main result 1046
2 Proof of Theorem 1 1048
References 1053
An e-Uniform Hybrid Scheme for Singularly Perturbed 1- D Reaction- Diffusion Problems 1056
1 Introduction 1056
2 The cubic spline-cum-finite difference scheme 1057
3 Uniform convergence analysis on a Shishkin mesh 1059
4 Numerical experiments 1062
5 Conclusions 1062
References 1063
A Dynamic Frictional Contact Problem of a Viscoelastic Beam 1066
1 Introduction 1066
2 The variational formulation 1067
3 Numerical approximation 1068
4 Numerical results 1070
Acknowledgements 1072
References 1073
Numerical Analysis of a Frictional Contact Problem for Viscoelastic Materials with Long- Term Memory 1074
1 Introduction 1074
2 The model and its well-posedness 1075
3 Fully discrete approximation 1078
4 Numerical simulations 1079
Acknowledgments 1081
References 1081
A Suitable Numerical Algorithm for the Simulation of the Butt Curl Deformation of an Aluminium Slab 1083
1 Introduction 1083
2 Mathematical model 1084
3 Weak formulation 1085
4 Numerical solution 1086
5 Numerical results 1089
References 1091
An Efficient Solution Algorithm for Elastoplasticity and its First Implementation Towards Uniform h- and p- Mesh Refinements 1092
1 Introduction 1092
2 The Model of Elastoplasticity 1093
3 The algorithm 1095
4 Numerical experiments 1097
5 Conclusions and future work 1098
Acknowledgment 1098
References 1099
A LDG-BEM Coupling for a Class of Nonlinear Exterior Transmission Problems 1102
1 Introduction 1102
2 An exterior transmission problem 1103
3 LDG-BEM coupling 1105
Acknowledgements 1108
References 1109
High Order Boundary Integral Methods for Maxwell’s Equations: Coupling of Microlocal Discretization and Fast Multipole Methods 1110
1 Introduction 1110
2 The integral equations of Despr ´ es (EID) and MLFMM 1111
3 Finite elements of higher order and MLFMM 1112
4 Microlocal discretization (MD) and MLFMM 1113
5 Numerical results 1115
6 Conclusion 1117
References 1117
Indirect Methods with Brakhage–Werner Potentials for Helmholtz Transmission Problems 1119
1 Problem 1119
2 Boundary integral formulation 1120
3 Petrov–Galerkin methods 1122
4 Numerical approximation in two dimensions 1123
5 A numerical example 1125
Acknowledgements 1126
References 1126
A FEM–BEM Formulation for a Time– Dependent Eddy Current Problem 1128
1 Introduction 1128
2 Model problem 1129
3 Semi–discrete problem 1134
References 1136
Mixed Boundary Element–Finite Volume Methods for Thermohydrodynamic Lubrication Problems* 1137
1 Introduction 1137
2 Thermohydrodynamic mathematical model 1138
3 Numerical solution of hydrodynamic and fluid thermal models 1141
4 Numerical solution in the bush and the shaft 1142
5 Numerical solution of the global problem 1142
6 Extension to the evolution problem 1143
References 1144
Numerical Modelling for Leaching of Pesticides in Soils Modified by a Cationic Surfactant 1147
1 Introduction 1147
2 Laboratory experiments 1148
3 Models describing leaching of solutes in soils 1149
4 Numerical methods 1150
5 Parameter adjustment 1152
6 Examples 1153
Acknowledgement 1153
References 1154
Formulation of Mixed-Hybrid FE Model of Flow in Fractured Porous Medium* 1156
1 Rock Massif environment 1156
2 Linear steady Darcy’s flow 1158
3 Mathematical formulation of the problem 1158
4 Mixed-hybrid formulation 1159
5 Finite element approximation 1162
6 Conclusions 1163
References 1163
Newton–Type Methods for the Mixed Finite Element Discretization of Some Degenerate Parabolic Equations 1164
1 Introduction 1164
2 The fully discrete problem 1165
3 The Newton scheme 1167
4 Conclusions 1171
References 1171
Domain Decomposition Methods for Wave Propagation in Heterogeneous Media 1174
1 Formulation of the problem 1174
2 Time discretization 1176
3 Fully discrete scheme 1177
4 Energy inequality 1179
5 Numerical experiments 1180
References 1181
Galbrun’s Equation Solved by a First Order Characteristics Method 1183
1 Galbrun’s equation 1183
2 Characteristic curves 1186
3 Numerical approximation 1187
Acknowledgment 1190
References 1190
Open Subsystems of Conservative Systems 1191
1 Overview 1191
2 Open systems within conservative extensions 1193
3 Discussion 1198
References 1198
Author Index 1199