Multiscale Methods in Science and Engineering (eBook)

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