Multiscale Finite Element Methods: Theory and Applications

✍ Scribed by Yalchin Efendiev, Thomas Y. Hou

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 242
• Series: Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 4
• Edition: 2009
• Category: Library

✦ Synopsis

This expository softcover book surveys the main concepts and recent advances in multiscale finite element methods. This monograph is intended for the broader audiences including engineers, applied scientists and those who are interested in multiscale simulations. Each chapter of the book starts with a simple introduction and the description of the proposed methods as with motivating examples. Numerical examples demonstrating the significance of the proposed methods are presented in each chapter. Yalchin Efendiev is a professor at Texas A/M University in College Station, Texas and Thomas Hou is a professor at California Institute of Technology in Pasadena, California.

✦ Table of Contents

Contents......Page 9
1.1 Challenges and motivation......Page 13
1.2 Literature review......Page 18
1.3 Overview of the content of the book......Page 22
2.2 Introduction to multiscale finite element methods......Page 25
2.3.1 Motivation......Page 32
2.3.2 Oversampling technique......Page 34
2.4 Generalization of MsFEM: A look forward......Page 35
2.5.1 Multiscale finite volume (MsFV) and multiscale finite volume element method (MsFVEM)......Page 37
2.5.2 Mixed multiscale finite element method......Page 39
2.6 MsFEM for problems with scale separation......Page 43
2.7 Extension of MsFEM to parabolic problems......Page 45
2.8 Comparison to other multiscale methods......Page 46
2.9 Performance and implementation issues......Page 50
2.9.1 Cost and performance......Page 51
2.9.2 Convergence and accuracy......Page 52
2.10 An application to two-phase flow......Page 53
2.11 Discussions......Page 57
3.1 MsFEM for nonlinear problems. Introduction......Page 59
3.2 Multiscale finite volume element method (MsFVEM)......Page 64
3.3 Examples of P[sub(h)]......Page 65
3.4 Relation to upscaling methods......Page 66
3.5 Multiscale finite element methods for nonlinear parabolic equations......Page 67
3.6 Summary of convergence of MsFEM for nonlinear partial differential equations......Page 70
3.7 Numerical results......Page 71
3.8 Discussions......Page 77
4.1 Motivation......Page 79
4.1.1 A motivating numerical example......Page 81
4.2.1 Elliptic equations......Page 83
4.2.2 Parabolic equations......Page 85
4.2.3 Numerical results......Page 87
4.3.1 A special case......Page 96
4.3.2 General case......Page 97
4.3.3 Numerical results......Page 98
4.4 The use of approximate global information......Page 101
4.4.1 Iterative MsFEM......Page 102
4.4.2 The use of approximate global information......Page 103
4.5 Discussions......Page 104
5.1 Introduction......Page 107
5.2.2 Adaptive multiscale algorithm for transport equation......Page 108
5.2.3 The coarse-to-fine grid interpolation operator......Page 111
5.2.4 Numerical results......Page 112
5.2.5 Results for a two-dimensional test case......Page 113
5.2.6 Three-dimensional test cases......Page 116
5.2.8 Other approaches for coarsening the transport equation......Page 119
5.3.1 Problem statement......Page 124
5.3.2 MsFVEM for Richards' equations......Page 125
5.3.3 Numerical results......Page 127
5.3.4 Summary......Page 130
5.4.1 Problem statement......Page 131
5.4.2 Multiscale numerical formulation......Page 132
5.4.3 Numerical examples......Page 134
5.5 Applications of mixed MsFEMs to reservoir modeling and simulation (by J. E. Aarnes)......Page 136
5.5.1 Multiscale method for the three-phase black oil model......Page 138
5.5.2 Adaptive coarsening of the saturation equations......Page 141
5.5.3 Utilization of multiscale methods for operational decision support......Page 145
5.6 Multiscale finite volume method for black oil systems (by S. H. Lee, C. Wolfsteiner and H. A. Tchelepi)......Page 148
5.6.1 Governing equations and discretized formulation......Page 149
5.6.2 Multiscale finite volume formulation......Page 150
5.6.4 Numerical examples......Page 154
5.6.5 Remarks......Page 156
5.7 Applications of multiscale finite element methods to stochastic flows in heterogeneous media......Page 158
5.7.1 Multiscale methods for stochastic equations......Page 160
5.7.2 The applications of MsFEMs to uncertainty quantification in inverse problems......Page 172
5.8 Discussions......Page 175
6 Analysis......Page 176
6.1.1 Analysis of conforming multiscale finite element methods......Page 177
6.1.2 Analysis of nonconforming multiscale finite element methods......Page 182
6.1.3 Analysis of mixed multiscale finite element methods......Page 184
6.2 Analysis of MsFEMs for nonlinear problems (from Chapter 3)......Page 189
6.3.1 Mixed finite element methods with limited global information......Page 198
6.3.2 Galerkin finite element methods with limited global information......Page 209
A. Basic notations......Page 213
B.1 Linear problems......Page 214
B.1.1 Special case: One-dimensional problem......Page 215
B.1.2 Multiscale asymptotic expansions......Page 216
B.1.4 Boundary corrections......Page 218
B.1.6 Convection of microstructure......Page 219
B.2 Nonlinear problems......Page 221
References......Page 225
P......Page 241
W......Page 242