Multiscale Finite Element Methods
by Efendiev, Yalchin; Hou, Thomas Y.Format: Paperback
Pub. Date: 2009-02-02
Publisher(s): Springer Nature
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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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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