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Surveys and Tutorials in the Applied Mathematical Sciences Volume 4 Editors S.S. Antman, J.E. Marsden, L. Sirovich

Surveys and Tutorials in the Applied Mathematical Sciences Volume 4 Editors S.S. Antman, J.E. Marsden, L. Sirovich Mathematics is becoming increasingly interdisciplinary and developing stronger interactions with ﬁelds such as biology, the physical sciences, and engineering. The rapid pace and development of the research frontiers has raised the need for new kinds of publications: short, up-to-date, readable tutorials and surveys on topics covering the breadth of the applied mathematical sciences. The volumes in this series are written in a style accessible to researchers, professionals, and graduate students in the sciences and engineering. They can serve as introductions to recent and emerging subject areas and as advanced teaching aids at universities. In particular, this series provides an outlet for material less formally presented and more anticipatory of needs than ﬁnished texts or monographs, yet of immediate interest because of the novelty of their treatments of applications, or of the mathematics being developed in the context of exciting applications. The series will often serve as an intermediate stage of publication of materials which, through exposure here, will be further developed and reﬁned to appear later in one of Springer’s more formal series in applied mathematics.

Yalchin Efendiev • Thomas Y. Hou Multiscale Finite Element Methods Theory and Applications 123

Yalchin Efendiev Department of Mathematics Texas A & M University College Station, TX 77843 USA efendiev@math.tamu.edu Editors: S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park MD 20742-4015 USA ssa@math.umd.edu Thomas Y. Hou Applied and Computational Mathematics, 217-50 California Institute of Technology Pasadena, CA 91125 USA hou@acm.caltech.edu J.E. Marsden Control and Dynamical System, 107-81 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu L. Sirovich Laboratory of Applied Mathematics Department of Bio-Mathematical Sciences Mount Sinai School of Medicine New York, NY 10029-6574 USA Lawrence.Sirovich@mssm.edu ISBN 978-0-387-09495-3 DOI 10.1007/978-0-387-09496-0 e-ISBN 978-0-387-09496-0 Library of Congress Control Number: 2008943964 Mathematics Subject Classiﬁcation (2000): 65N99, 76S05, 35B27 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

Dedicated to my parents, Raﬁk and Ziba, my wife, Denise, and my son, William Yalchin Efendiev Dedicated to my parents, Sum-Hing and Sau-Ying, my wife, Yu-Chung, and my children, George and Anthony Thomas Y. Hou

Preface The aim of this monograph is to describe the main concepts and recent ad- vances in multiscale ﬁnite element methods. This monograph is intended for the broader audience including engineers, applied scientists, and for those who are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale computa- tions. It combines a practical introduction, numerical results, and analysis of multiscale ﬁnite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justiﬁed later in the last chapter. Numerical examples demonstrating the signiﬁcance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The ﬁrst chapter gives a general introduction to multiscale methods and an outline of each chapter. The second chapter discusses the main idea of the multiscale ﬁnite element method and its extensions. This chapter also gives an overview of multiscale ﬁnite element methods and other related methods. The third chapter discusses the exten- sion of multiscale ﬁnite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information. This is motivated by porous media applications where some type of nonlocal information is needed in upscaling as well as multiscale simulations. The ﬁfth chapter of the book is devoted to applications of these methods. Finally, in the last chapter, we present analyses of some representative multiscale methods from Chapters 2, 3, and 4.

VIII Preface Acknowledgments We are grateful to J. E. Aarnes, C. C. Chu, P. Dostert, L. Durlofsky, V. Ginting, O. Iliev, L. Jiang, S. H. Lee, W. Luo, P. Popov, H. Tchelepi, and X. H. Wu for many helpful comments, discussions, and collaborations. The partial support of NSF and DOE is greatly appreciated. College Station & Pasadena August 2008 Yalchin Efendiev Thomas Y. Hou

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Challenges and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Overview of the content of the book . . . . . . . . . . . . . . . . . . . . . . . 10 2 Multiscale ﬁnite element methods for linear problems and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Introduction to multiscale ﬁnite element methods . . . . . . . . . . . . 13 2.3 Reducing boundary eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Oversampling technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Generalization of MsFEM: A look forward . . . . . . . . . . . . . . . . . . 23 2.5 Brief overview of various global couplings of multiscale basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Multiscale ﬁnite volume (MsFV) and multiscale ﬁnite volume element method (MsFVEM) . . . . . . . . . . . . . . . . . 25 2.5.2 Mixed multiscale ﬁnite element method . . . . . . . . . . . . . . 27 2.6 MsFEM for problems with scale separation . . . . . . . . . . . . . . . . . 31 2.7 Extension of MsFEM to parabolic problems . . . . . . . . . . . . . . . . . 33 2.8 Comparison to other multiscale methods . . . . . . . . . . . . . . . . . . . 34 2.9 Performance and implementation issues . . . . . . . . . . . . . . . . . . . . 38 2.9.1 Cost and performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9.2 Convergence and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9.3 Coarse-grid choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.10 An application to two-phase ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.11 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Multiscale ﬁnite element methods for nonlinear equations . 47 3.1 MsFEM for nonlinear problems. Introduction . . . . . . . . . . . . . . . 47 3.2 Multiscale ﬁnite volume element method (MsFVEM) . . . . . . . . . 52

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