Engquist / Lötstedt / Runborg Multiscale Methods in Science and Engineering

1. Auflage 2006
ISBN: 978-3-540-26444-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 289 Seiten

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.

Autoren/Hrsg.

Engquist, Björn

Lötstedt, Per

Runborg, Olof

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;List of Contributors;9
4;Multiscale Discontinuous Galerkin Methods for Elliptic Problems with Multiple Scales;12
4.1;1 Introduction;12
4.2;2 Mathematical Formulations;14
4.3;3 Multiscale Methods for Elliptic Problems;19
4.4;4 Numerical Results;24
4.5;5 Concluding Remarks;28
4.6;References;30
5;Discrete Network Approximation for Highly-Packed Composites with Irregular Geometry in Three Dimensions;32
5.1;1 Introduction;32
5.2;2 Formulation of the Problem;34
5.3;3 Discrete Network;36
5.4;4 Variational Error Estimates;46
5.5;5 Numerical Illustration;56
5.6;6 Appendices;59
5.7;7 Acknowledgments;67
5.8;References;68
6;Adaptive Monte Carlo Algorithms for Stopped Diffusion;69
6.1;1 Introduction;69
6.2;2 Error Expansion;72
6.3;3 Adaptive Algorithms for Stopped Diffusion;85
6.4;4 Numerical Experiments;91
6.5;Acknowledgments;98
6.6;References;98
7;The Heterogeneous Multi-Scale Method for Homogenization Problems;99
7.1;1 Introduction;99
7.2;2 Variational Problems;101
7.3;3 Dynamic Problems;106
7.4;4 Stability and Accuracy;111
7.5;5 How Can HMM Fail?;115
7.6;6 Conclusion;117
7.7;References;118
8;A Coarsening Multigrid Method for Flow in Heterogeneous Porous Media;121
8.1;1 Introduction;121
8.2;2 Mathematical Statement and Definitions;123
8.3;3 Numerical Coarse Graining;127
8.4;4 Coarsening Multigrid Method;131
8.5;5 Numerical Results;133
8.6;6 Summary;140
8.7;References;141
9;On the Modeling of Small Geometric Features in Computational Electromagnetics;143
9.1;1 Introduction;143
9.2;2 Governing Equations;144
9.3;3 ModelingWires and Slots in FETD;146
9.4;4 Stability Analysis;151
9.5;5 Numerical Results;153
9.6;6 Conclusions;156
9.7;7 Acknowledgments;157
9.8;References;157
10;Coupling PDEs and SDEs: The Illustrative Example of the Multiscale Simulation of Viscoelastic Flows;159
10.1;1 A Prototypical System;159
10.2;2 Modeling Dilute Solutions of Flexible Polymers;161
10.3;3 Modeling Various Fluids;169
10.4;4 An Example Outside Fluid Mechanics: Photon Transport;174
10.5;References;176
11;Adaptive Submodeling for Linear Elasticity Problems with Multiscale Geometric Features;179
11.1;1 Introduction;179
11.2;2 Linear Elasticity and Finite Element Method;181
11.3;3 Adaptive Submodeling;182
11.4;4 Conclusions;188
11.5;References;189
12;Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems;191
12.1;1 Introduction;191
12.2;2 The Variational Multiscale Method;193
12.3;3 A Posteriori Error Estimates;197
12.4;4 Adaptive Algorithm;198
12.5;5 Numerical Examples;199
12.6;6 Conclusions and FutureWork;201
12.7;References;202
13;Multipole Solution of Electromagnetic Scattering Problems with Many, Parameter Dependent Incident Waves;204
13.1;1 Introduction;204
13.2;2 Minimal Residual Interpolation (MRI);206
13.3;3 Numerical Results;210
13.4;References;212
14;Introduction to Normal Multiresolution Approximation;213
14.1;1 Introduction;213
14.2;2 Basic Normal Multiresolution Analysis;214
14.3;3 Higher Order Generalizations;225
14.4;References;230
15;Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics;233
15.1;1 Introduction;233
15.2;2 Problem Statement;236
15.3;3 The Gap-Tooth Scheme;237
15.4;4 Patch Dynamics;240
15.5;5 Convergence Results;241
15.6;6 Conclusions;245
15.7;Acknowledgments;245
15.8;References;246
16;Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem;248
16.1;1 Introduction;248
16.2;2 HMM Strategy;251
16.3;3 Main Example;253
16.4;4 Generalizations;259
16.5;5 Conclusion;267
16.6;Acknowledgment;268
16.7;References;268
17;Multiscale Homogenization of the Navier–Stokes Equation;269
17.1;1 Introduction;269
17.2;2 Scaling and Expansions;270
17.3;3 Reynolds Stress Tensor and Eddy Viscosity;272
17.4;4 Stationary Flow in Porous Media, Homogenization of the Navier– Stokes Equations;274
17.5;5 Appendix: Two-Scale Compensated Compactness;276
17.6;References;279
18;Numerical Simulations of the Dynamics of Fiber Suspensions;280
18.1;1 Background and Introduction;280
18.2;2 Mathematical Formulation;282
18.3;3 The Numerical Method;287
18.4;4 The Dynamics of Fiber Suspensions;289
18.5;5 Concluding Remarks;293
18.6;References;294
19;Editorial Policy;295
20;General Remarks;296
21;Lecture Notes in Computational Science and Engineering;297