Multiscale Methods in Science and Engineering (eBook)

Björn Engquist, Per Lötstedt, Olof Runborg (Herausgeber)

eBook Download: PDF

2006 | 2005
XI, 289 Seiten
Springer Berlin (Verlag)
978-3-540-26444-6 (ISBN)

Infos

Multiscale problems naturally pose severe challenges for computational science and engineering. The smaller scales must be well resolved over the range of the larger scales. Challenging multiscale problems are very common and are found in e.g. materials science, fluid mechanics, electrical and mechanical engineering. Homogenization, subgrid modelling, heterogeneous multiscale methods, multigrid, multipole, and adaptive algorithms are examples of methods to tackle these problems. This volume is an overview of current mathematical and computational methods for problems with multiple scales with applications in chemistry, physics and engineering.

Inhalt

Preface 5
Contents 7
List of Contributors 9
Multiscale Discontinuous Galerkin Methods for Elliptic Problems with Multiple Scales 12
1 Introduction 12
2 Mathematical Formulations 14
3 Multiscale Methods for Elliptic Problems 19
4 Numerical Results 24
5 Concluding Remarks 28
References 30
Discrete Network Approximation for Highly-Packed Composites with Irregular Geometry in Three Dimensions 32
1 Introduction 32
2 Formulation of the Problem 34
3 Discrete Network 36
4 Variational Error Estimates 46
5 Numerical Illustration 56
6 Appendices 59
7 Acknowledgments 67
References 68
Adaptive Monte Carlo Algorithms for Stopped Diffusion 69
1 Introduction 69
2 Error Expansion 72
3 Adaptive Algorithms for Stopped Diffusion 85
4 Numerical Experiments 91
Acknowledgments 98
References 98
The Heterogeneous Multi-Scale Method for Homogenization Problems 99
1 Introduction 99
2 Variational Problems 101
3 Dynamic Problems 106
4 Stability and Accuracy 111
5 How Can HMM Fail? 115
6 Conclusion 117
References 118
A Coarsening Multigrid Method for Flow in Heterogeneous Porous Media 121
1 Introduction 121
2 Mathematical Statement and Definitions 123
3 Numerical Coarse Graining 127
4 Coarsening Multigrid Method 131
5 Numerical Results 133
6 Summary 140
References 141
On the Modeling of Small Geometric Features in Computational Electromagnetics 143
1 Introduction 143
2 Governing Equations 144
3 ModelingWires and Slots in FETD 146
4 Stability Analysis 151
5 Numerical Results 153
6 Conclusions 156
7 Acknowledgments 157
References 157
Coupling PDEs and SDEs: The Illustrative Example of the Multiscale Simulation of Viscoelastic Flows 159
1 A Prototypical System 159
2 Modeling Dilute Solutions of Flexible Polymers 161
3 Modeling Various Fluids 169
4 An Example Outside Fluid Mechanics: Photon Transport 174
References 176
Adaptive Submodeling for Linear Elasticity Problems with Multiscale Geometric Features 179
1 Introduction 179
2 Linear Elasticity and Finite Element Method 181
3 Adaptive Submodeling 182
4 Conclusions 188
References 189
Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems 191
1 Introduction 191
2 The Variational Multiscale Method 193
3 A Posteriori Error Estimates 197
4 Adaptive Algorithm 198
5 Numerical Examples 199
6 Conclusions and FutureWork 201
References 202
Multipole Solution of Electromagnetic Scattering Problems with Many, Parameter Dependent Incident Waves 204
1 Introduction 204
2 Minimal Residual Interpolation (MRI) 206
3 Numerical Results 210
References 212
Introduction to Normal Multiresolution Approximation 213
1 Introduction 213
2 Basic Normal Multiresolution Analysis 214
3 Higher Order Generalizations 225
References 230
Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics 233
1 Introduction 233
2 Problem Statement 236
3 The Gap-Tooth Scheme 237
4 Patch Dynamics 240
5 Convergence Results 241
6 Conclusions 245
Acknowledgments 245
References 246
Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem 248
1 Introduction 248
2 HMM Strategy 251
3 Main Example 253
4 Generalizations 259
5 Conclusion 267
Acknowledgment 268
References 268
Multiscale Homogenization of the Navier–Stokes Equation 269
1 Introduction 269
2 Scaling and Expansions 270
3 Reynolds Stress Tensor and Eddy Viscosity 272
4 Stationary Flow in Porous Media, Homogenization of the Navier– Stokes Equations 274
5 Appendix: Two-Scale Compensated Compactness 276
References 279
Numerical Simulations of the Dynamics of Fiber Suspensions 280
1 Background and Introduction 280
2 Mathematical Formulation 282
3 The Numerical Method 287
4 The Dynamics of Fiber Suspensions 289
5 Concluding Remarks 293
References 294
Editorial Policy 295
General Remarks 296
Lecture Notes in Computational Science and Engineering 297

Produkt-Details

• Erscheint lt. Verlag: 30.3.2006
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Zusatzinfo: XI, 289 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Bauwesen
• Schlagworte: algorithm • algorithms • differential equation • Finite Element Method • fluid mechanics • homogenization • mechanical engineering • Mechanics • Modeling • multiscale methods • Natur • Physics • Science
• ISBN-10: 3-540-26444-2 / 3540264442
• ISBN-13: 978-3-540-26444-6 / 9783540264446