Gómez / Quintela / Salgado Numerical Mathematics and Advanced Applications

Proceedings of ENUMATH 2005 the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 2005
1. Auflage 2007
ISBN: 978-3-540-34288-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 1232 Seiten

The European Conference on Numerical Mathematics and Advanced App- cations (ENUMATH) is a series of meetings held every two years to provide a forum for discussion on recent aspects of numerical mathematics and their applications. They seek to convene leading experts and young scientists with special emphasis on contributions from Europe. The ?rst ENUMATH me- ing held in Paris (1995), and the series continued by the ones in Heidelberg (1997), Jyv ¨ askyla ¨ (1999), Ischia (2001) and Prague (2003). This book collects the major part of the lectures given at ENUMATH 2005, that took place in Santiago de Compostela, Spain, from July 18 to 22, 2005. It contents texts of invited speakers, and a selection of papers presented in minisymposia and works communicated within the sessions. The importance of numerical methods has increased dramatically in s- ence and engineering, re?ecting today’s unprecedented use of computers. The increasing importance of modeling in addition to numerical simulation was again evident in ENUMATH 2005. Indeed, nodaways mathematics is g- erally accepted as a technology, playing a crucial role in many branches of industrial activity. Recent results and new trends in the analysis of numerical algorithms as well as their application to challenging scienti?c and industrial problems were discussed during the meeting. Apart from the theoretical - pects, a major part of the conference was devoted to numerical methods for interdisciplinary applications, with emphasis on showing the potential of new computational methods for solving practical multidisciplinary problems.

Autoren/Hrsg.

Gómez, Dolores

Quintela, Peregrina

Salgado, Pilar

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;PLENARY LECTURES;20
3.1;Compatible Discretizations in Two Dimensions;21
3.1.1;1 Introduction;21
3.1.2;2 Construction of the dual complex;23
3.1.3;3 Applications;30
3.1.4;References;37
3.2;Finite Element Approximation of the Three Field Formulation of the Elasticity Problem Using Stabilization;39
3.2.1;1 Introduction;39
3.2.2;2 Problem statement and Galerkin finite element discretization;41
3.2.3;3 Finite element approximation using subscales;42
3.2.4;4 Numerical analysis of the original formulation;45
3.2.5;5 A modified stabilized problem;49
3.2.6;6 Concluding remarks;54
3.2.7;References;55
3.3;Convergence of Adaptive Wavelet Methods for Goal– Oriented Error Estimation*;57
3.3.1;1 Introduction;57
3.3.2;2 Goal–oriented error estimation;59
3.3.3;3 Adaptive error estimation;64
3.3.4;4 Numerical experiments;72
3.3.5;References;77
3.4;Quadratic Programming and Scalable Algorithms for Variational Inequalities;80
3.4.1;1 Introduction;80
3.4.2;2 Bound constrained problems;81
3.4.3;3 Bound and equality constrained problems;83
3.4.4;4 Model problem;85
3.4.5;5 FETI and total FETI domain decomposition;87
3.4.6;6 FETI–DP domain decomposition and discretization;89
3.4.7;7 Numerical scalability;90
3.4.8;8 Numerical experiments;91
3.4.9;9 Comments and conclusions;93
3.4.10;Acknowledgements;93
3.4.11;References;93
3.5;Discontinuous Galerkin Methods for Friedrichs’ Systems;97
3.5.1;1 Introduction;97
3.5.2;2 Friedrichs’ systems;99
3.5.3;3 Design and analysis of DG methods;101
3.5.4;4 DG approximation of two-field Friedrichs’ systems;105
3.5.5;5 Examples;108
3.5.6;6 Concluding remarks;113
3.5.7;References;113
3.6;Highly Oscillatory Quadrature: The Story so Far;115
3.6.1;1 The challenge of high oscillation;115
3.6.2;2 Asymptotic expansion in the absence of critical points;117
3.6.3;3 Asymptotic, filon and levin methods;120
3.6.4;4 Critical points;129
3.6.5;5 Conclusions and pointers for further research;133
3.6.6;References;136
3.7;The 3D Inverse Electromagnetic Scattering Problem for a Coated Dielectric;137
3.7.1;1 Introduction;137
3.7.2;2 Formulation of the direct and inverse scattering problem;138
3.7.3;3 Analysis of the inverse problem;141
3.7.4;4 Numerical example;149
3.7.5;5 Conclusion;150
3.7.6;Acknowledgment;151
3.7.7;References;151
3.8;Functional Approach to Locally Based A Posteriori Error Estimates for Elliptic and Parabolic Problems;153
3.8.1;1 Introduction;153
3.8.2;2 Functional a posteriori estimates for elliptic problems;155
3.8.3;3 Functional a posteriori estimates for a model evolutionary problem;164
3.8.4;References;167
3.9;Finite Element Approximation of 2D Parabolic Optimal Design Problems;169
3.9.1;1 Introduction;169
3.9.2;2 Preliminaries;173
3.9.3;3 Preliminaries on the convergence of the numerical scheme;175
3.9.4;4 Convergence of discrete optimal shapes;179
3.9.5;5 Gradient calculations: A numerical approach;181
3.9.6;6 Conclusions;191
3.9.7;Acknowledgements;192
3.9.8;References;192
4;CONTRIBUTED LECTURES;195
4.1;3D Free Surface Flows Simulations Using a Multilayer Saint- Venant Model. Comparisons with Navier- Stokes Solutions;197
4.1.1;1 Introduction;197
4.1.2;2 Navier-Stokes equations and hydrostatic approximation;197
4.1.3;3 A Multilayer saint-venant system;199
4.1.4;4 Numerical method;201
4.1.5;5 Numerical results;202
4.1.6;References;204
4.2;Some Well-Balanced Shallow Water-Sediment Transport Models;206
4.2.1;1 Sediment transport model;206
4.2.2;2 Finite volume method for non conservative hyperbolic systems;209
4.2.3;3 High order schemes based on state reconstruction;211
4.2.4;4 Numerical test: comparison with an analytical solution;212
4.2.5;References;213
4.3;Highly Accurate Conservative Finite Difference Schemes and Adaptive Mesh Refinement Techniques for Hyperbolic Systems of Conservation Laws*;214
4.3.1;1 Introduction;214
4.3.2;2 Finite-difference Shu-Osher schemes;215
4.3.3;3 Adaptive mesh refinement for Shu-Osher schemes;217
4.3.4;4 Numerical examples;219
4.3.5;5 Conclusions;220
4.3.6;References;221
4.4;Finite Volume Solvers for the Shallow Water Equations Using Matrix Radial Basis Function Reconstruction;223
4.4.1;1 Finite volume methods for shallow water models;223
4.4.2;2 Matrix valued Radial Basis Functions for vector field reconstruction;225
4.4.3;3 Applications to environmental modelling;226
4.4.4;References;230
4.5;On Numerical Schemes for a Hierarchy of Kinetic Equations;232
4.5.1;1 Introduction;232
4.5.2;2 Hexagonal kinetic models;233
4.5.3;3 Hexagonal hierarchy;235
4.5.4;4 Coupling of two kinetic models;236
4.5.5;References;239
4.6;Computational Aspects of the Mesh Adaptation for the Time Marching Procedure;240
4.6.1;1 Euler equations;240
4.6.2;2 Adaptive algorithm;241
4.6.3;3 Anisotropic mesh adaptation;242
4.6.4;4 Geometric mass conservation law;244
4.6.5;5 Numerical example;246
4.6.6;Acknowledgement;247
4.6.7;References;247
4.7;On the Use of Slope Limiters for the Design of Recovery Based Error Indicators;248
4.7.1;1 Introduction;248
4.7.2;2 A posteriori error indication;249
4.7.3;3 Limited gradient averaging;250
4.7.4;4 Limited gradient reconstruction;251
4.7.5;5 Adaptation strategy;253
4.7.6;6 Numerical examples;254
4.7.7;7 Conclusions;254
4.7.8;References;255
4.8;On a Superconvergence Result for Mixed Approximation of Eigenvalue Problems;257
4.8.1;1 Introduction;257
4.8.2;2 Statement of the problem and its discretization;258
4.8.3;3 Main results;259
4.8.4;4 Numerical results;260
4.8.5;References;264
4.9;Comparative Study of the a Posteriori Error Estimators for the Stokes Problem;266
4.9.1;1 Introduction;266
4.9.2;2 The stokes problem and its approximation;267
4.9.3;3 Estimation of the deviation from the exact solution;268
4.9.4;4 Numerical experiments;269
4.9.5;References;273
4.10;Error Control for Discretizations of Electromagnetic- Mechanical Multifield Problem;274
4.10.1;1 Introduction;274
4.10.2;2 Electromagnetic forming;275
4.10.3;3 Error control for coupled and mixed problems;278
4.10.4;References;281
4.11;A Safeguarded Zienkiewicz-Zhu Estimator;283
4.11.1;1 Introduction;283
4.11.2;2 Error estimator and theoretical results;284
4.11.3;3 Numerical results;287
4.11.4;References;290
4.12;Some Remarks on a Model for the Atmospheric Pressure in Ocean Dynamics;292
4.12.1;1 Introduction;292
4.12.2;2 Modelling of non-hydrostatic free-surface flows;293
4.12.3;3 A linearised model for the free surface equation;296
4.12.4;References;299
4.13;Computational Time Improvement for Some Shallow Water Finite Volume Models Applying Parallelization and Optimized Small Matrix Computations.;301
4.13.1;1 Introduction;301
4.13.2;2 Equations;302
4.13.3;3 Numerical scheme;303
4.13.4;4 Parallel SIMD implementation;303
4.13.5;References;308
4.14;Discretization Error Estimates for an Optimal Control Problem in a Nonconvex Domain;311
4.14.1;1 Introduction;311
4.14.2;2 Theory;312
4.14.3;3 Numerical results;315
4.14.4;References;319
4.15;A Posteriori Estimates for Cost Functionals of Optimal Control Problems;320
4.15.1;1 Introduction;320
4.15.2;2 Majorants for the cost functional;322
4.15.3;3 Properties of majorants;323
4.15.4;4 Practical implementation;324
4.15.5;5 Numerical experiments;325
4.15.6;References;327
4.16;Optimization of a Duality Method for the Compressible Reynolds Equation;330
4.16.1;1 The mathematical model;330
4.16.2;2 Numerical solution;331
4.16.3;3 Optimization of the duality algorithm;333
4.16.4;4 Numerical examples;335
4.16.5;5 Conclusions;337
4.16.6;Aknowledgments;337
4.16.7;References;337
4.17;Time-Space & Space-Time Elements for Unsteady Advection- Dominated Problems;339
4.17.1;1 Introduction;339
4.17.2;2 Stabilization techniques for the stationary problem;340
4.17.3;3 Stabilized methods for the non-stationary problem;341
4.17.4;4 Numerical Experiments;345
4.17.5;Acknowledgements;346
4.17.6;References;346
4.18;On Discontinuity–Capturing Methods for Convection– Diffusion Equations;347
4.18.1;1 Introduction;347
4.18.2;2 Galerkin’s finite element discretization;348
4.18.3;3 The SUPG method;348
4.18.4;4 Methods diminishing spurious oscillations in layers;349
4.18.5;5 Conclusions;353
4.18.6;Acknowledgements;354
4.18.7;References;354
4.19;Algebraic Flux Correction for Finite Element Approximation of Transport Equations;356
4.19.1;1 Introduction;356
4.19.2;2 Flux decomposition;356
4.19.3;3 Algebraic flux correction;357
4.19.4;4 Numerical example;362
4.19.5;5 Conclusions;363
4.19.6;References;363
4.20;A Parallel Multiparametric Gauss-Seidel Method*;365
4.20.1;1 Introduction;365
4.20.2;2 The local modified extrapolated Gauss-Seidel ( LMEGS) method;366
4.20.3;3 The eigenvalue relationship;366
4.20.4;4 Determination of good values;368
4.20.5;5 Numerical results and conclusions;370
4.20.6;Acknowledgement;372
4.20.7;References;372
4.21;A Numerical Scheme for the Micro Scale Dissolution and Precipitation in Porous Media;373
4.21.1;1 Introduction;373
4.21.2;2 The numerical scheme;376
4.21.3;3 A fixed point iteration;377
4.21.4;4 A numerical example;379
4.21.5;Acknowledgment;379
4.21.6;References;380
4.22;Discrete Kinetic Methods for a Degenerate Parabolic Equation in Dimension Two.;383
4.22.1;1 Introduction;383
4.22.2;2 Kinetic models;384
4.22.3;3 The numerical schemes;386
4.22.4;4 Numerical experiments;389
4.22.5;References;390
4.23;Anisotropic Doubly Nonlinear Degenerate Parabolic Equations;391
4.23.1;1 Introduction;391
4.23.2;2 Entropy solution;392
4.23.3;3 Renormalized entropy solution;394
4.23.4;Acknowledgement;396
4.23.5;References;396
4.24;A Multiresolution Method for the Simulation of Sedimentation- Consolidation Processes;397
4.24.1;1 Introduction;397
4.24.2;2 The multiresolution scheme;398
4.24.3;3 Sedimentation-consolidation processes;402
4.24.4;4 Numerical results;402
4.24.5;Acknowledgements;404
4.24.6;References;404
4.25;Diffusive Relaxation Limit for Hyperbolic Systems;406
4.25.1;1 Introduction;406
4.25.2;2 BGK approximation of strongly parabolic systems;409
4.25.3;3 One–dimensional semilinear model in viscoelasticity;410
4.25.4;4 Multidimensional viscoelasticity and modulated energy;411
4.25.5;References;413
4.26;Parallel Algorithms for Nonlinear Diffusion by Using Relaxation Approximation;414
4.26.1;1 Relaxation approximation of nonlinear diffusion;414
4.26.2;2 The numerical scheme;415
4.26.3;3 Numerical results;417
4.26.4;4 Concluding remarks;421
4.26.5;References;421
4.27;On a Degenerated Parabolic-Hyperbolic Problem Arising From Stratigraphy;422
4.27.1;1 Introduction and presentation of the model;422
4.27.2;2 A locally hyperbolic behaviour;423
4.27.3;3 Definition of a solution and existence results for a discretized problem;424
4.27.4;4 Some numerical illustrations;428
4.27.5;5 Conclusion and open problems;428
4.27.6;References;429
4.28;Schwarz Domain Decomposition Preconditioners for Interior Penalty Approximations of Elliptic Problems;432
4.28.1;1 Introduction;432
4.28.2;2 Discontinuous Galerkin methods for elliptic problems;433
4.28.3;3 Non-overlapping Schwarz methods;434
4.28.4;4 Convergence analysis;436
4.28.5;5 Numerical results;437
4.28.6;References;439
4.29;Higher Order Semi-Implicit Discontinuous Galerkin Finite Element Schemes for Nonlinear Convection- Diffusion Problems*;441
4.29.1;1 Introduction;441
4.29.2;2 Scalar equation;442
4.29.3;3 Numerical results;446
4.29.4;4 Conclusion;448
4.29.5;References;448
4.30;On Some Aspects of the Discontinuous Galerkin Method*;449
4.30.1;1 Continuous problem;449
4.30.2;2 Discretization of the problem;450
4.30.3;3 Error estimates;453
4.30.4;4 Application of the DGFEM to compressible flow with a wide range of mach numbers;455
4.30.5;5 Conclusion;455
4.30.6;References;456
4.31;Mixed Discontinuous Galerkin Methods with Minimal Stabilization;457
4.31.1;1 DG methods;457
4.31.2;2 Minimal stabilization;460
4.31.3;3 Numerical results;462
4.31.4;References;464
4.32;Discontinuous Galerkin Finite Element Method for a Fourth- Order Nonlinear Elliptic Equation Related to the Two- Dimensional Navier– Stokes Equations;466
4.32.1;1 Introduction;466
4.32.2;2 Mathematical formulation;468
4.32.3;3 DGFEM for a 4th-order advective PDE;469
4.32.4;4 Numerical results;471
4.32.5;References;473
4.33;Fourier Method with Nitsche-Mortaring for the Poisson Equation in 3D;475
4.33.1;1 Introduction;475
4.33.2;2 Fourier decomposition and mortaring in 2D;477
4.33.3;3 Fourier-nitsche-finite-element approximation in 3D;479
4.33.4;References;481
4.34;Substructuring Preconditioners for the Bidomain Extracellular Potential Problem;483
4.34.1;1 Introduction;483
4.34.2;2 Mortar method;484
4.34.3;3 Substructuring preconditioners;485
4.34.4;References;490
4.35;A Face Penalty Method for the Three Fields Stokes Equation Arising from Oldroyd- B Viscoelastic Flows;493
4.35.1;1 Introduction;493
4.35.2;2 A finite element formulation;495
4.35.3;3 The inf-sup condition;496
4.35.4;4 A priori error estimates;496
4.35.5;5 A stable iterative algorithm;497
4.35.6;6 Preliminary numerical results;497
4.35.7;References;499
4.36;Anisotropic H1-Stable Projections on Quadrilateral Meshes;501
4.36.1;1 Introduction;501
4.36.2;2 Anisotropic H1-stable projectors;503
4.36.3;3 General result;508
4.36.4;References;509
4.37;Continuous Interior Penalty hp-Finite Element Methods for Transport Operators;510
4.37.1;1 Introduction;510
4.37.2;2 Continuous interior penalty finite element methods;511
4.37.3;3 Technical results;512
4.37.4;4 Convergence analysis;514
4.37.5;5 Numerical results;514
4.37.6;References;517
4.38;A Nonconforming Finite Element Method with Face Penalty for Advection– Diffusion Equations;518
4.38.1;1 Introduction;518
4.38.2;2 The nonconforming finite element scheme with face penalty;519
4.38.3;3 A posteriori error estimates;522
4.38.4;4 Numerical results;523
4.38.5;Acknowledgment;524
4.38.6;References;525
4.39;Efficient Multigrid and Data Structures for Edge- Oriented FEM Stabilization;526
4.39.1;1 Introduction;526
4.39.2;2 Sparsity of the matrix;527
4.39.3;3 Local pressure Schur complement approach;530
4.39.4;4 Numerical example;532
4.39.5;References;533
4.40;Adaptive Methods for Dynamical Micromagnetics;535
4.40.1;1 Introduction;535
4.40.2;2 Numerical methods;536
4.40.3;3 Adaptive algorithm;538
4.40.4;4 Numerical experiment;539
4.40.5;References;541
4.41;Stability for Walls in Ferromagnetic Nanowire;543
4.41.1;1 Model for ferromagnetic nanowires;543
4.41.2;2 Landau-Lifschitz Equation in the mobile frame;545
4.41.3;3 A new system of coordinates;546
4.41.4;4 Estimates for the perturbations;548
4.41.5;References;550
4.42;Continuous Galerkin Methods for Solving Maxwell Equations in 3D Geometries;551
4.42.1;1 Introduction and notations;551
4.42.2;2 Variational formulations and discretization;553
4.42.3;3 Numerical results and conclusion;555
4.42.4;References;558
4.43;On the Use of the Gautschi-Type Exponential Integrator for Wave Equations;560
4.43.1;1 Introduction;560
4.43.2;2 Sine–Gordon equation;561
4.43.3;3 Discretisation;562
4.43.4;4 The Gautschi-type exponential integrator;564
4.43.5;5 Numerical example;565
4.43.6;References;566
4.44;Positivity of Exponential Multistep Methods;567
4.44.1;1 Introduction;567
4.44.2;2 Analytical framework;568
4.44.3;3 Exponential multistep methods;570
4.44.4;4 Positivity and order barrier;571
4.44.5;References;574
4.45;Stability Results and Algorithmic Strategies for the Finite Element Approach to the Immersed Boundary Method;576
4.45.1;1 Introduction;576
4.45.2;2 The finite element immersed boundary method;577
4.45.3;3 Time discretization by finite differences;579
4.45.4;4 Stability analysis by energy estimates;580
4.45.5;5 Numerical results;581
4.45.6;6 Conclusions;583
4.45.7;References;583
4.46;A Comparison of Enthalpy and Temperature Methods for Melting Problems on Composite Domains;585
4.46.1;1 Introduction;585
4.46.2;2 Problem description;586
4.46.3;3 Numerical methods;588
4.46.4;4 Numerical experiments;590
4.46.5;5 Conclusions;592
4.46.6;References;592
4.47;Qualitative Properties of a Numerical Scheme for the Heat Equation;593
4.47.1;1 Introduction;593
4.47.2;2 Proof of the results;595
4.47.3;Acknowledgements;600
4.47.4;References;600
4.48;Modeling Radiation and Moisture Content in Fire Spread;601
4.48.1;1 Introduction;601
4.48.2;2 Physical model;602
4.48.3;3 Governing equations;602
4.48.4;4 Numerical method;604
4.48.5;5 Numerical results;607
4.48.6;Acknowledgement;608
4.48.7;References;608
4.49;Fast Multipole Method for Solving the Radiosity Equation;609
4.49.1;1 Introduction;609
4.49.2;2 Radiosity equation and numerical solution;610
4.49.3;3 FMM and kernel expansion;611
4.49.4;4 A new fast method;612
4.49.5;5 Numerical results;615
4.49.6;6 Conclusion;616
4.49.7;References;616
4.50;Numerical Modelling of Kinetic Equations;618
4.50.1;1 Introduction;618
4.50.2;2 Asymptotic method;619
4.50.3;3 Numerical examples;622
4.50.4;References;625
4.51;On a Subclass of H¨ older Continuous Functions with Applications to Signal Processing;627
4.51.1;1 Introduction;627
4.51.2;2 H¨ older continuous spaces;628
4.51.3;3 Generalized Harten’s Subcell resolution technique;630
4.51.4;4 Numerical experiments;631
4.51.5;Acknowledgments;634
4.51.6;References;634
4.52;Modelisation and Simulation of Static Grain Deep- Bed Drying;636
4.52.1;1 Introduction;636
4.52.2;2 Models;637
4.52.3;3 The numerical schemes;638
4.52.4;4 Numerical experiments and comparisons;642
4.52.5;References;643
4.53;Hybrid Godunov-Glimm Method for a Nonconservative Hyperbolic System with Kinetic Relations;644
4.53.1;1 Introduction;644
4.53.2;2 The PDE model and main properties;645
4.53.3;3 Godunov-Glimm Hybrid method;647
4.53.4;4 The nonlinear projection;649
4.53.5;References;651
4.54;Cell-Average Multiwavelets Based on Hermite Interpolation*;652
4.54.1;1 Introduction;652
4.54.2;2 Harten’s framework for multiresolution analysis;652
4.54.3;3 Vector multiresolution analysis for cell-averaged data;654
4.54.4;4 Numerical experiments;657
4.54.5;References;659
4.55;On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws;660
4.55.1;1 Introduction;660
4.55.2;2 Choice of paths;662
4.55.3;3 Godunov’s method;664
4.55.4;4 Application to linear balance laws;665
4.55.5;Acknowledgements;667
4.55.6;References;667
4.56;Sequential Flux-Corrected Remapping for ALE Methods;669
4.56.1;1 Introduction;669
4.56.2;2 Sequential FCR method;671
4.56.3;3 Numerical examples;675
4.56.4;Acknowledgments;676
4.56.5;References;676
4.57;Orthogonal hp-FEM for Elliptic Problems Based on a Non- Affine Concept;679
4.57.1;1 Introduction and historical remarks;679
4.57.2;2 Preliminaries;680
4.57.3;3 Numerical example;681
4.57.4;4 Construction of basis functions;682
4.57.5;5 Numerical example continued;685
4.57.6;Acknowledgment;686
4.57.7;References;686
4.58;On Some Aspects of the hp-FEM for Time- Harmonic Maxwell’s Equations;687
4.58.1;1 Introduction;687
4.58.2;2 Formulation of the problem;688
4.58.3;3 Shape functions;689
4.58.4;4 Numerical experiments;692
4.58.5;5 Conclusion and outlook;692
4.58.6;Acknowledgment;694
4.58.7;References;694
4.59;Numerical Simulation of Phase-Transition Front Propagation in Thermoelastic Solids;697
4.59.1;1 Introduction;697
4.59.2;2 Formulation of the problem;698
4.59.3;3 Conservative wave propagation algorithm;699
4.59.4;4 Contact quantities and numerical fluxes;701
4.59.5;5 Conclusions;703
4.59.6;Acknowledgment;704
4.59.7;References;704
4.60;The Level Set Method for Solid-Solid Phase Transformations;706
4.60.1;1 Introduction;706
4.60.2;2 The physical problem;708
4.60.3;3 The computational method;709
4.60.4;4 Numerical results;710
4.60.5;5 Conclusions;712
4.60.6;References;713
4.61;A Non-Monotone Fast Marching Scheme for a Hamilton- Jacobi Equation Modelling Dislocation Dynamics*;715
4.61.1;1 Introduction;715
4.61.2;2 The FMM algorithm for unsigned velocity;717
4.61.3;3 Numerical tests;719
4.61.4;Acknowledgments;722
4.61.5;References;722
4.62;A Time–Adaptive Semi–Lagrangian Approximation to Mean Curvature Motion;724
4.62.1;1 Introduction;724
4.62.2;2 General requirements on the adaptation strategy for geometric equations;726
4.62.3;3 A strategy based on local truncation error;727
4.62.4;4 Numerical tests;729
4.62.5;References;731
4.63;Heterogeneous Multiscale Methods with Quadrilateral Finite Elements;733
4.63.1;1 Introduction;733
4.63.2;2 HMM with quadrilaterals finite elements;734
4.63.3;3 Error analysis;736
4.63.4;4 Numerical experiments;738
4.63.5;Acknowledgment;739
4.63.6;References;740
4.64;Stabilizing the P1/P0 Element for the Stokes Problem via Multiscale Enrichment;742
4.64.1;1 Introduction;742
4.64.2;2 The model problem and the general framework;743
4.64.3;3 Application to the;745
4.64.4;pair;745
4.64.5;4 Numerical validations;748
4.64.6;References;749
4.65;Adaptive Multiresolution Methods for the Simulation of Shocks/ Shear Layer Interaction in Confined Flows;751
4.65.1;1 Introduction;751
4.65.2;2 Numerical results;752
4.65.3;3 Conclusion and perspectives;756
4.65.4;References;758
4.66;Local Projection Stabilization for the Stokes System on Anisotropic Quadrilateral Meshes;760
4.66.1;1 Introduction;760
4.66.2;2 Local projection stabilization on isotropic meshes;761
4.66.3;3 Anisotropic affine linear meshes;762
4.66.4;4 Local projection stabilization on anisotropic meshes;764
4.66.5;References;767
4.67;An Interior Penalty Variational Multiscale Method for High Reynolds Number Flows;769
4.67.1;1 Introduction;769
4.67.2;2 The equations of incompressible flow;770
4.67.3;3 Separation of scales and stabilized finite element methods;770
4.67.4;4 A posteriori error estimation;774
4.67.5;5 A numerical result;775
4.67.6;References;775
4.68;Variational Multiscale Large Eddy Simulation of Turbulent Flows Using a Two- Grid Finite Element or Finite Volume Method;778
4.68.1;1 Introduction;778
4.68.2;2 Multiscale formulation;779
4.68.3;3 Numerical results for turbulent flow in a diffuser;782
4.68.4;4 Conclusions;784
4.68.5;Acknowledgements;785
4.68.6;References;785
4.69;Issues for a Mathematical Definition of LES;786
4.69.1;1 Introduction;786
4.69.2;2 Suitable approximations;788
4.69.3;3 Review of existing pre–LES–models;789
4.69.4;4 Discretization;791
4.69.5;References;793
4.70;Stabilized FEM with Anisotropic Mesh Refinement for the Oseen Problem;795
4.70.1;1 Introduction;795
4.70.2;2 Stabilized FEM for linearized Navier-Stokes problem;796
4.70.3;3 Stability and convergence on hybrid meshes;798
4.70.4;4 Error estimates and design of stabilization parameters;799
4.70.5;5 Application to channel flow;801
4.70.6;Acknowledgment;802
4.70.7;References;802
4.71;Semi-Implicit Multiresolution for Multiphase Flows;804
4.71.1;1 Introduction;804
4.71.2;2 Semi implicit scheme on uniform grid;806
4.71.3;3 Multiscale analysis of the explicit-implicit scheme;807
4.71.4;References;811
4.72;Numerical Simulation of Vortex-Dipole Wall Interactions Using an Adaptive Wavelet Discretization with Volume Penalisation;812
4.72.1;1 Introduction;812
4.72.2;2 Adaptive wavelet discretization with volume penalisation;813
4.72.3;3 Vortex-dipole wall interactions;816
4.72.4;4 Conclusions;818
4.72.5;Acknowledgements;819
4.72.6;References;819
4.73;Inviscid Flow on Moving Grids with Multiscale Space and Time Adaptivity;821
4.73.1;1 Introduction;821
4.73.2;2 The ALE formulation of the Euler equations;822
4.73.3;3 Finite volume discretization;823
4.73.4;4 Adaptive multiscale method;825
4.73.5;5 Numerical results;827
4.73.6;References;828
4.74;A Relaxation Method for a Two Phase Flow with Surface Tension;831
4.74.1;1 Introduction;831
4.74.2;2 Numerical approximation;833
4.74.3;3 Relaxation method for the simplified model with surface tension forces;834
4.74.4;4 Numerical results;837
4.74.5;5 Conclusion;838
4.74.6;References;838
4.75;Extension of Interface Coupling to General Lagrangian Systems;840
4.75.1;1 Introduction;840
4.75.2;2 Coupling two p- systems;843
4.75.3;3 Coupling two Euler systems in Lagrangian coordinates;844
4.75.4;4 Coupling Lagrangian systems of different dimensions;846
4.75.5;5 Conclusion;847
4.75.6;References;847
4.76;A Numerical Scheme for the Modeling of Condensation and Flash Vaporization in Compressible Multi- Phase Flows;849
4.76.1;1 Introduction;849
4.76.2;2 The thermodynamics of phase transition;849
4.76.3;3 The Riemann problem with phase transition at equilibrium;850
4.76.4;4 The Riemann problem with out of equilibrium EOS;852
4.76.5;5 Numerical scheme;854
4.76.6;6 Numerical results;854
4.76.7;7 Conclusions;856
4.76.8;Acknowledgments;856
4.76.9;References;856
4.77;An Adaptive Operator Splitting of Higher Order for the Navier- Stokes Equations;858
4.77.1;1 Introduction;858
4.77.2;2 The Taylor based gradient recovery technique;859
4.77.3;3 The stabilized base splitting;860
4.77.4;4 The multi-grid postprocessing;861
4.77.5;5 Numerical results;862
4.77.6;6 Conclusions;865
4.77.7;Acknowledgment;865
4.77.8;References;865
4.78;The POD Technique for Computing Bifurcation Diagrams: A Comparison among Different Models in Fluids;867
4.78.1;1 Introduction;867
4.78.2;2 The POD technique;868
4.78.3;3 Description of the examples;870
4.78.4;4 Computation of the bifurcation diagrams of the example models;872
4.78.5;5 Conclusions;873
4.78.6;Acknowledgement;874
4.78.7;References;874
4.79;Filtering of Singularities in a Marangoni Convection Problem;876
4.79.1;1 Introduction;876
4.79.2;2 Formulation of the problem;877
4.79.3;3 Basic state;878
4.79.4;4 Linear stability;880
4.79.5;5 Conclusions;883
4.79.6;Acknowledgments;883
4.79.7;References;883
4.80;On Application of Stabilized Higher Order Finite Element Method on Unsteady Incompressible Flow Problems;884
4.80.1;1 Mathematical model;884
4.80.2;2 Time-spatial discretization;887
4.80.3;3 Numerical solution and results;889
4.80.4;References;891
4.81;Numerical Simulation of Coupled Fluid-Solid Systems by Fictitious Boundary and Grid Deformation Methods;893
4.81.1;1 Introduction;893
4.81.2;2 Grid deformation method;895
4.81.3;3 Numerical solution of the fluid-solid system;896
4.81.4;4 Verification of the numerical techniques;898
4.81.5;5 Conclusions;899
4.81.6;References;900
4.82;An Iterative Method for Solving Non-Linear Hydromagnetic Equations;903
4.82.1;1 Introduction. Statement of the problem;903
4.82.2;2 An iterative method for the magnetostatic system;905
4.82.3;3 Finite element discretization;906
4.82.4;4 Computational tests;909
4.82.5;References;910
4.83;Mathematical and Numerical Analysis of a Class of Non- linear Elliptic Equations in the Two Dimensional Case;912
4.83.1;1 Introduction;912
4.83.2;2 Statement of the main result;913
4.83.3;3 Proof of theorem 1;914
4.83.4;4 Numerical method;916
4.83.5;References;919
4.84;A s-step Variant of the Double Orthogonal Series Algorithm;922
4.84.1;1 Introduction;922
4.84.2;2 The algorithm of the Double Orthogonal Series;923
4.84.3;3 s-Step methods;924
4.84.4;4 The s-step variant of the double orthogonal series;924
4.84.5;5 Numerical results;926
4.84.6;6 Conclusions;929
4.84.7;Acknowledgment;929
4.84.8;References;929
4.85;Linear Equations in Quaternions;930
4.85.1;1 Basic properties and definitions for quaternions;930
4.85.2;2 Linear equations in quaternions;932
4.85.3;Acknowledgment;937
4.85.4;References;937
4.86;Computing the Analytic Singular Value Decomposition via a Pathfollowing;939
4.86.1;1 Introduction;939
4.86.2;2 Formulation of the problem;940
4.86.3;3 Solving defining equations;942
4.86.4;4 Experiments;943
4.86.5;5 Conclusions;945
4.86.6;Acknowledgments;946
4.86.7;References;946
4.87;A Jacobi-Davidson Method for Computing Partial Generalized Real Schur Forms;948
4.87.1;1 Introduction;948
4.87.2;2 The RJDQZ method for real matrix pencils;949
4.87.3;3 Numerical comparison;953
4.87.4;4 Conclusions;955
4.87.5;References;955
4.88;Pricing Multi-Asset Options with Sparse Grids and Fourth Order Finite Differences;958
4.88.1;1 Introduction;958
4.88.2;2 Model;958
4.88.3;3 Discretization;959
4.88.4;4 Sparse grid combination technique;961
4.88.5;5 Numerical results;963
4.88.6;6 Conclusions;965
4.88.7;References;965
4.89;A Third Order Linearly Implicit Fractional Step Method for Semilinear Parabolic Problems;968
4.89.1;1 Introduction;968
4.89.2;2 A new third order linearly implicit FSRK method;970
4.89.3;3 Numerical tests;973
4.89.4;References;976
4.90;Numerical Solution of Optimal Control Problems with Sparse SQP- Methods;977
4.90.1;1 Introduction;977
4.90.2;2 Optimal control problem;978
4.90.3;3 SQP-methods;979
4.90.4;4 Equality constrained quadratic subproblem;980
4.90.5;5 Approximation of the Hessian;981
4.90.6;6 Example;982
4.90.7;References;984
4.91;Semi-Deterministic Recursive Optimization Methods for Multichannel Optical Filters;986
4.91.1;1 Introduction;986
4.91.2;2 Global optimization methods;987
4.91.3;3 Application to multichannel optical filters design;990
4.91.4;4 Conclusions;993
4.91.5;References;993
4.92;A Multigrid Method for Coupled Optimal Topology and Shape Design in Nonlinear Magnetostatics*;994
4.92.1;1 Introduction;994
4.92.2;2 Topology optimization for magnetostatics;995
4.92.3;3 Sequential coupling of topology and shape optimization;997
4.92.4;4 Multilevel shape optimization;997
4.92.5;5 Numerical results;999
4.92.6;6 Conclusion;1000
4.92.7;References;1001
4.93;Nonsmooth Optimization of Eigenvalues in Topology Optimization;1002
4.93.1;1 Introduction;1002
4.93.2;2 Topology optimization and eigenproblems;1003
4.93.3;3 Nonsmooth analysis;1005
4.93.4;4 Numerical results;1007
4.93.5;5 Conclusion;1009
4.93.6;References;1009
4.94;Derivative Free Optimization of Stirrer Configurations;1010
4.94.1;1 Introduction;1010
4.94.2;2 Flow solver and numerical optimization tool;1011
4.94.3;3 Results;1013
4.94.4;4 Conclusion;1016
4.94.5;Acknowledgment;1017
4.94.6;References;1017
4.95;Mathematical Modelling and Numerical Optimization in the Process of River Pollution Control;1019
4.95.1;1 Introduction;1019
4.95.2;2 Mathematical formulation;1020
4.95.3;3 Numerical solution;1022
4.95.4;4 Numerical results;1025
4.95.5;Acknowledgements;1026
4.95.6;References;1026
4.96;A Family of C0 Finite Elements for Kirchhoff Plates with Free Boundary Conditions;1029
4.96.1;1 Introduction;1029
4.96.2;2 Kirchhoff plate bending problem;1030
4.96.3;3 Finite element formulation;1031
4.96.4;4 A-priori error estimates;1032
4.96.5;5 A-posteriori error estimates;1033
4.96.6;6 Computational results;1034
4.96.7;References;1036
4.97;A Postprocessing Method for the MITC Plate Elements;1037
4.97.1;1 Introduction;1037
4.97.2;2 The Reissner–Mindlin plate model;1037
4.97.3;3 MITC finite element methods;1038
4.97.4;4 Superconvergence;1039
4.97.5;5 Postprocessing method;1040
4.97.6;6 Benchmark computations;1041
4.97.7;References;1045
4.98;A Uniformly Stable Finite Difference Space Semi- Discretization for the Internal Stabilization of the Plate Equation in a Square;1046
4.98.1;1 Statement of the main result;1046
4.98.2;2 Proof of Theorem 1;1048
4.98.3;References;1053
4.99;An e-Uniform Hybrid Scheme for Singularly Perturbed 1- D Reaction- Diffusion Problems;1056
4.99.1;1 Introduction;1056
4.99.2;2 The cubic spline-cum-finite difference scheme;1057
4.99.3;3 Uniform convergence analysis on a Shishkin mesh;1059
4.99.4;4 Numerical experiments;1062
4.99.5;5 Conclusions;1062
4.99.6;References;1063
4.100;A Dynamic Frictional Contact Problem of a Viscoelastic Beam;1066
4.100.1;1 Introduction;1066
4.100.2;2 The variational formulation;1067
4.100.3;3 Numerical approximation;1068
4.100.4;4 Numerical results;1070
4.100.5;Acknowledgements;1072
4.100.6;References;1073
4.101;Numerical Analysis of a Frictional Contact Problem for Viscoelastic Materials with Long- Term Memory;1074
4.101.1;1 Introduction;1074
4.101.2;2 The model and its well-posedness;1075
4.101.3;3 Fully discrete approximation;1078
4.101.4;4 Numerical simulations;1079
4.101.5;Acknowledgments;1081
4.101.6;References;1081
4.102;A Suitable Numerical Algorithm for the Simulation of the Butt Curl Deformation of an Aluminium Slab;1083
4.102.1;1 Introduction;1083
4.102.2;2 Mathematical model;1084
4.102.3;3 Weak formulation;1085
4.102.4;4 Numerical solution;1086
4.102.5;5 Numerical results;1089
4.102.6;References;1091
4.103;An Efficient Solution Algorithm for Elastoplasticity and its First Implementation Towards Uniform h- and p- Mesh Refinements;1092
4.103.1;1 Introduction;1092
4.103.2;2 The Model of Elastoplasticity;1093
4.103.3;3 The algorithm;1095
4.103.4;4 Numerical experiments;1097
4.103.5;5 Conclusions and future work;1098
4.103.6;Acknowledgment;1098
4.103.7;References;1099
4.104;A LDG-BEM Coupling for a Class of Nonlinear Exterior Transmission Problems;1102
4.104.1;1 Introduction;1102
4.104.2;2 An exterior transmission problem;1103
4.104.3;3 LDG-BEM coupling;1105
4.104.4;Acknowledgements;1108
4.104.5;References;1109
4.105;High Order Boundary Integral Methods for Maxwell’s Equations: Coupling of Microlocal Discretization and Fast Multipole Methods;1110
4.105.1;1 Introduction;1110
4.105.2;2 The integral equations of Despr ´ es (EID) and MLFMM;1111
4.105.3;3 Finite elements of higher order and MLFMM;1112
4.105.4;4 Microlocal discretization (MD) and MLFMM;1113
4.105.5;5 Numerical results;1115
4.105.6;6 Conclusion;1117
4.105.7;References;1117
4.106;Indirect Methods with Brakhage–Werner Potentials for Helmholtz Transmission Problems;1119
4.106.1;1 Problem;1119
4.106.2;2 Boundary integral formulation;1120
4.106.3;3 Petrov–Galerkin methods;1122
4.106.4;4 Numerical approximation in two dimensions;1123
4.106.5;5 A numerical example;1125
4.106.6;Acknowledgements;1126
4.106.7;References;1126
4.107;A FEM–BEM Formulation for a Time– Dependent Eddy Current Problem;1128
4.107.1;1 Introduction;1128
4.107.2;2 Model problem;1129
4.107.3;3 Semi–discrete problem;1134
4.107.4;References;1136
4.108;Mixed Boundary Element–Finite Volume Methods for Thermohydrodynamic Lubrication Problems*;1137
4.108.1;1 Introduction;1137
4.108.2;2 Thermohydrodynamic mathematical model;1138
4.108.3;3 Numerical solution of hydrodynamic and fluid thermal models;1141
4.108.4;4 Numerical solution in the bush and the shaft;1142
4.108.5;5 Numerical solution of the global problem;1142
4.108.6;6 Extension to the evolution problem;1143
4.108.7;References;1144
4.109;Numerical Modelling for Leaching of Pesticides in Soils Modified by a Cationic Surfactant;1147
4.109.1;1 Introduction;1147
4.109.2;2 Laboratory experiments;1148
4.109.3;3 Models describing leaching of solutes in soils;1149
4.109.4;4 Numerical methods;1150
4.109.5;5 Parameter adjustment;1152
4.109.6;6 Examples;1153
4.109.7;Acknowledgement;1153
4.109.8;References;1154
4.110;Formulation of Mixed-Hybrid FE Model of Flow in Fractured Porous Medium*;1156
4.110.1;1 Rock Massif environment;1156
4.110.2;2 Linear steady Darcy’s flow;1158
4.110.3;3 Mathematical formulation of the problem;1158
4.110.4;4 Mixed-hybrid formulation;1159
4.110.5;5 Finite element approximation;1162
4.110.6;6 Conclusions;1163
4.110.7;References;1163
4.111;Newton–Type Methods for the Mixed Finite Element Discretization of Some Degenerate Parabolic Equations;1164
4.111.1;1 Introduction;1164
4.111.2;2 The fully discrete problem;1165
4.111.3;3 The Newton scheme;1167
4.111.4;4 Conclusions;1171
4.111.5;References;1171
4.112;Domain Decomposition Methods for Wave Propagation in Heterogeneous Media;1174
4.112.1;1 Formulation of the problem;1174
4.112.2;2 Time discretization;1176
4.112.3;3 Fully discrete scheme;1177
4.112.4;4 Energy inequality;1179
4.112.5;5 Numerical experiments;1180
4.112.6;References;1181
4.113;Galbrun’s Equation Solved by a First Order Characteristics Method;1183
4.113.1;1 Galbrun’s equation;1183
4.113.2;2 Characteristic curves;1186
4.113.3;3 Numerical approximation;1187
4.113.4;Acknowledgment;1190
4.113.5;References;1190
4.114;Open Subsystems of Conservative Systems;1191
4.114.1;1 Overview;1191
4.114.2;2 Open systems within conservative extensions;1193
4.114.3;3 Discussion;1198
4.114.4;References;1198
5;Author Index;1199