Nonlocal Modeling, Analysis, and Computation

174 Pages·2019·2.771 MB·English

by  Qiang Du

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CBMS-NSF REGIONAL CONFERENCE SERIES
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c R. Tyrrell Rockafellar, Conjugate Duality and Optimization
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e Cathleen S. Morawetz, Notes on Time Decay and Scattering for Some Hyperbolic Problems
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p Tai-Ping Liu, Hyperbolic and Viscous Conservation Laws
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c Michael Renardy, Mathematical Analysis of Viscoelastic Flows
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CB94_DU_FM_V6.indd 3 1/15/2019 3:28:31 PM
p
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a. Qiang Du
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al Columbia University
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u New York, New York
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SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
PHILADELPHIA
CB94_DU_FM_V6.indd 5 1/15/2019 3:28:32 PM
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h Copyright © 2019 by the Society for Industrial and Applied Mathematics
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Trademarked names may be used in this book without the inclusion of a trademark symbol.
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ct t Library of Congress Cataloging-in-Publication Data
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bj Names: Du, Qiang, 1964- author.
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s Title: Nonlocal modeling, analysis, and computation / Qiang Du (Columbia
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o University, New York, New York).
buti Description: Philadelphia : Society for Industrial and Applied Mathematics,
stri [2019] | Series: CBMS-NSF regional conference series in applied
di mathematics ; 94 | Includes bibliographical references and index.
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Identifiers: LCCN 2018056825 (print) | LCCN 2018061231 (ebook) | ISBN
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3 9781611975628 | ISBN 9781611975611 (print)
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0. Subjects: LCSH: Continuum (Mathematics) | Mathematical models. | Mathematical
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1 analysis.
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5 Classification: LCC QA611.28 (ebook) | LCC QA611.28 .D8 2019 (print) | DDC
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8. 514/.32--dc23
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1 LC record available at https://lccn.loc.gov/2018056825
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is a registered trademark.
CB94_DU_FM_V6.indd 6 1/15/2019 3:28:32 PM
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u Contents
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p:// Foreword xi
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e Preface xiii
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ht; 1 Nonlocalmodeling: Anewperspectiveandoverview 1
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p 2 Nonlocalmodels: Anintroductionandexamples 5
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c 2.1 Modelingchoicesandemergenceofnonlocalmodeling . . . 5
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e o 2.2 Nonlocalmodels: Simpleillustrationsandcomparisons . . . 7
ns 2.3 Nonlocalityviamodelreductionandcoarsegraining . . . . . 12
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c 2.4 Nonlocality,diffusion,andsomehistoricalperspective . . . . 15
M li 2.5 Nonlocaloperatorsascontinuumlimitsofdiscreteoperators 18
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2.6 Nonlocalcontinuummechanics: Peridynamics . . . . . . . . 22
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o 2.7 Nonlocalfluxandnonlocalbalancelaws . . . . . . . . . . . 27
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bje 3 Nonlocalvectorcalculusandvariationalproblems 31
u 3.1 Aconcisereformulationofnonlocalmodels . . . . . . . . . 31
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on 3.2 Nonlocalgradientandnonlocaldivergenceoperators . . . . . 38
uti 3.3 Propertiesofnonlocalspacesofvectorfields . . . . . . . . . 44
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ri 3.4 NonlocalPoincaré–Korntypeinequality . . . . . . . . . . . 52
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di 3.5 Nonlocalvariationalproblemsonboundeddomains . . . . . 56
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R 3.6 Nonlocalcorrespondencemodelsandnonlocalrelaxation . . 66
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2 4 Numericaldiscretizationofnonlocalmodels 73
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0 4.1 Reviewofnumericaldiscretizationofnonlocalmodels . . . . 74
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9. 4.2 Asymptoticallycompatibleschemes: Anabstractframework 79
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1 4.3 Finiteelementapproximationsoflinearnonlocalproblems. . 84
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7 4.4 Approximation of the fractional Laplacian via the nonlocal
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o Laplacian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
19 t 4.5 Multidimensionalquadraturedifferenceapproximations . . . 98
9/ 4.6 Spectralapproximationsofnonlocalmodels . . . . . . . . . 103
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d 5 Nonlocalandlocalcoupling 107
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d 5.1 Anewtracetheoremfornonlocalfunctionspaces . . . . . . 108
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nl 5.2 NonlocalHardyinequalityanddirectionalderivative
w estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
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D 5.3 Nonlocalvariationalproblemswithheterogeneous
localization. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Acouplednonlocal-localmodelwithheterogeneoushorizon 119
ix
x Contents
6 Nonlocal-in-timedynamics 121
p 6.1 Nonlocal-in-timederivativeoperators . . . . . . . . . . . . . 122
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a.p 6.2 Analysisofthenonlocal-in-timeequation. . . . . . . . . . . 126
ojs 6.3 Nonlocal-in-timedynamicsforanomalousdiffusion . . . . . 130
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n 7 Epilogue: Aninvitationtononlocalmodeling 137
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ou 7.1 Nonlocalmodelingandapplications . . . . . . . . . . . . . 137
g/j 7.2 Nonlocalanalysisandcomputation . . . . . . . . . . . . . . 140
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m.o 7.3 Thinkingnonlocally,actinglocally . . . . . . . . . . . . . . 142
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w.si Bibliography 145
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Index 165
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u Foreword
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w NonlocalModeling,Analysis,andComputationisaconciseintroductiontonon-
p:// local continuum models and their mathematical foundation and numerical dis-
htt cretization. To offer a balanced discussion of theoretical and practical issues,
e the monograph includes some motivational examples of nonlocal models; basic
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yri numerical approximations; and various applications, such as nonlocal dynamics
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o of anomalous diffusion and nonlocal peridynamic models of elasticity and frac-
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r turemechanics. Aparticularfocusisonnonlocalsystemswithafiniterangeof
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e interaction so as to draw close connections with traditional local systems repre-
s
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e sentedbypartialdifferentialequations. Themonographintendstoofferapplied
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M li mathematicians, computational scientists, young researchers, and graduate stu-
A dentsinmanydifferentapplicationareasanillustrationofthebroadapplicability
SI andrichmathematicsofnonlocalmodels.
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w Astheworldbecomesincreasinglyconnectedandthecapacityofcomputingtech-
p:// nologyskyrockets,nonlocalcontinuummodelsaresteadilygainingpopularity.
htt Nonlocalmodelscanaccountfornonlocalinteractionandtakeonmoregen-
e eral forms than discrete and local continuum models. They can be effective al-
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yri anomalousbehavior,theyarewellsuitedtosimulationsofcomplexprocessesand
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o multiscalephenomenawheremodelcouplingandreductionsareoftennecessary.
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r With successful application to a wide variety of scientific domains and en-
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e gineering systems, there has also been much progress on the development of a
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e rigorousandsystematicmathematicalfoundationofnonlocalmodels,alongwith
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M li their effective and robust numerical solution. Owing to our increasing desire to
A understand and control complex processes commonly involving anomalies and
SI singularities, nonlocal models can have a central role in the future. It should be
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ect t nstoiltledin,haonwasecveenr,ttshtaagtec,owmipthremheuncshivmeomreatfhuetmuraetidceavlesltoupdmieesnotftonocnolmocea.lMmeoadnewlshialree,
ubj ongoing studies concerning nonlocal models have been not only provoking the
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b models as well as their connections. This monograph thus intends to provide
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2.
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0.2 Acknowledgments
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9. The content of the monograph is based on materials presented in the principal
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1 lecturesattheNSF-CBMSRegionalResearchConferenceonNonlocalDynamics
8.
7 deliveredatChicagointhesummerof2017.IwouldliketothanktheNSF/CBMS
1
o fortheirsupport;IITforhostingtheCBMSconference;andespeciallytheorga-
9 t nizers,ProfessorsJinqiaoDuanandXiaofanLi,fortheirinitiationoftheconfer-
1
9/ ence and their hard work to make it possible. Thanks also to other conference
2
4/ invitedspeakersandattendees.
0
d The materials of the principal lectures are based on joint works with many
e
d collaborators whom I have benefited greatly from, including Max Gunzburger
a
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nl andRichardLehoucq(whointroducedperidynamicstomeinitiallyandinspired
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much of the later development); Tadele Mengesha and Xiaochuan Tian (whose
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D contributions are heavily featured in the book, with most of the technical dis-
cussionsonnonlocalvariationalproblemsandnumericalapproximationscoming
fromtheirworkandwhohavehelpedmetremendouslywithsomeoftheCBMS
xiii