Multiscale Finite Element Methods

by Efendiev, Yalchin; Hou, Thomas Y.

• ISBN13: 9780387094953
• ISBN10: 0387094954
• Format: Paperback
• Copyright: 2009-02-02
• Publisher: Springer Nature

Summary

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.

Author Biography

Yalchin Efendiev is a Professor at Texas AM University in College Station, Texas and Thomas Hou is the Charles Lee Powell Professor at California Institute of Technology in Pasadena, California.

Table of Contents

Introduction	p. 1
Challenges and motivation	p. 1
Literature review	p. 6
Overview of the content of the book	p. 10
Multiscale finite element methods for linear problems and overview	p. 13
Summary	p. 13
Introduction to multiscale finite element methods	p. 13
Reducing boundary effects	p. 20
Motivation	p. 20
Oversampling technique	p. 22
Generalization of MsFEM: A look forward	p. 23
Brief overview of various global couplings of multiscale basis functions	p. 25
Multiscale finite volume (MsFV) and multiscale finite volume element method (MsFVEM)	p. 25
Mixed multiscale finite element method	p. 27
MsFEM for problems with scale separation	p. 31
Extension of MsFEM to parabolic problems	p. 33
Comparison to other multiscale methods	p. 34
Performance and implementation issues	p. 38
Cost and performance	p. 39
Convergence and accuracy	p. 40
Coarse-grid choice	p. 41
An application to two-phase flow	p. 41
Discussions	p. 45
Multiscale finite element methods for nonlinear equations	p. 47
MsFEM for nonlinear problems. Introduction	p. 47
Multiscale finite volume element method (MsFVEM)	p. 52
Examples of Ph	p. 53
Relation to upscaling methods	p. 54
Multiscale finite element methods for nonlinear parabolic equations	p. 55
Summary of convergence of MsFEM for nonlinear partial differential equations	p. 58
Numerical results	p. 59
Discussions	p. 65
Multiscale finite element methods using limited global information	p. 67
Motivation	p. 67
A motivating numerical example	p. 69
Mixed multiscale finite element methods using limited global information	p. 71
Elliptic equations	p. 71
Parabolic equations	p. 73
Numerical results	p. 75
Galerkin multiscale finite element methods using limited global information	p. 84
A special case	p. 84
General case	p. 85
Numerical results	p. 86
The use of approximate global information	p. 89
Iterative MsFEM	p. 90
The use of approximate global information	p. 91
Discussions	p. 92
Applications of multiscale finite element methods	p. 95
Introduction	p. 95
Multiscale methods for transport equation	p. 96
Governing equations	p. 96
Adaptive multiscale algorithm for transport equation	p. 96
The coarse-to-fine grid interpolation operator	p. 99
Numerical results	p. 100
Results for a two-dimensional test case	p. 101
Three-dimensional test cases	p. 104
Discussion on local boundary conditions	p. 107
Other approaches for coarsening the transport equation	p. 107
Summary	p. 112
Applications to Richards' equation	p. 112
Problem statement	p. 112
MsFVEM for Richards' equations	p. 113
Numerical results	p. 115
Summary	p. 118
Applications to fluid-structure interaction	p. 119
Problem statement	p. 119
Multiscale numerical formulation	p. 120
Numerical examples	p. 122
Discussions	p. 124
Applications of mixed MsFEMs to reservoir modeling and simulation	p. 124
Multiscale method for the three-phase black oil model	p. 126
Adaptive coarsening of the saturation equations	p. 129
Utilization of multiscale methods for operational decision support	p. 133
Summary	p. 136
Multiscale finite volume method for black oil systems	p. 136
Governing equations and discretized formulation	p. 137
Multiscale finite volume formulation	p. 138
Sequential fully implicit coupling and adaptive computation	p. 142
Numerical examples	p. 142
Remarks	p. 144
Applications of multiscale finite element methods to stochastic flows in heterogeneous media	p. 146
Multiscale methods for stochastic equations	p. 148
The applications of MsFEMs to uncertainty quantification in inverse problems	p. 160
Discussions	p. 163
Analysis	p. 165
Analysis of MsFEMs for linear problems (from Chapter 2)	p. 166
Analysis of conforming multiscale finite element methods	p. 166
Analysis of nonconforming multiscale finite element methods	p. 171
Analysis of mixed multiscale finite element methods	p. 173
Analysis of MsFEMs for nonlinear problems (from Chapter 3)	p. 178
Analysis of MsFEMs with limited global information (from Chapter 4)	p. 187
Mixed finite element methods with limited global information	p. 187
Glaerkin finite element methods with limited global information	p. 198
Basic notations	p. 203
Review of homogenization	p. 205
Linear problems	p. 205
Special case: One-dimensional problem	p. 206
Multiscale asymptotic expansions	p. 207
Justification of formal expansions	p. 209
Boundary corrections	p. 209
Nonlocal memory effect of homogenization	p. 210
Convection of microstructure	p. 210
Nonlinear problems	p. 212
References	p. 217
Index	p. 233
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