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CHAPMAN

DIFFERENCE METHODS

SINGULAR PERTURBATION PROBLEMS

CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics

Main Editors

H. Brezis, Université de Paris

R.G. Douglas, Texas A&M University

A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board

R. Aris, University of Minnesota

G.I. Barenblatt, University of California at Berkeley

H. Begehr, Freie Universität Berlin

P. Bullen, University of British Columbia

R.J. Elliott, University of Alberta

R.P. Gilbert, University of Delaware

R. Glowinski, University of Houston

D. Jerison, Massachusetts Institute of Technology

K. Kirchgässner, Universität Stuttgart

B. Lawson, State University of New York

B. Moodie, University of Alberta

L.E. Payne, Cornell University

D.B. Pearson, University of Hull

G.F. Roach, University of Strathclyde

I. Stakgold, University of Delaware

W.A. Strauss, Brown University

J. van der Hoek, University of Adelaide

DIFFERENCE

METHODS

FOR SINGULAR PERTURBATION PROBLEMS

Grigory I. Shishkin

Lidia P. Shishkina

Chapman & Hall/CRC

Taylor & Francis Group

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Library of Congress Cataloging-in-Publication Data

Shishkin, G. I.

Difference methods for singular perturbation problems / Grigory I. Shishkin, Lidia P. Shishkina. p. cm. -- (Chapman & Hall/CRC monographs and surveys in pure and applied mathematics)

Includes bibliographical references and index. ISBN 978-1-58488-459-0 (hardback : alk. paper)

1. Singular perturbations (Mathematics) 2. Difference equations--Numerical solutions. 3. Algebra, Abstract. I. Shishkina, Lidia P. II. Title. III. Series.

QC20.7.P47S55 2008 515’.392--dc22 2008025636

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Dedication

Dedicatedtothememoryofacademicians

AlexandrAndreevichSamarskiiand

NikolaiSergeevichBakhvalov

Preface xiii

IGridapproximationsofsingularperturbation partialdifferentialequations 1

1Introduction 3

1.1Thedevelopmentofnumericalmethodsforsingularly perturbedproblems......................3

1.2Theoreticalproblemsintheconstructionofdifferenceschemes6

1.3Themainprinciplesintheconstructionofspecialschemes8

1.4Moderntrendsinthedevelopmentofspecialdifference schemes.............................10

1.5Thecontentsofthepresentbook...............11

1.6Thepresentbook.......................12

1.7Theaudienceforthisbook..................16

2Boundaryvalueproblemsforellipticreaction-diffusion equationsindomainswithsmoothboundaries 17

2.1Problemformulation.Theaimoftheresearch.......17

2.2Estimatesofsolutionsandderivatives............19

2.3Conditionsensuring ε-uniformconvergenceofdifference schemesfortheproblemonaslab..............26

2.3.1Sufficientconditionsfor ε-uniformconvergenceof differenceschemes...................26

2.3.2Sufficientconditionsfor ε-uniformapproximationof theboundaryvalueproblem..............29

2.3.3Necessaryconditionsfordistributionofmeshpoints for ε-uniformconvergenceofdifferenceschemes. Constructionofcondensingmeshes..........33

2.4Monotonefinitedifferenceapproximationsoftheboundary valueproblemonaslab. ε-uniformlyconvergentdifference schemes.............................38

2.4.1Problemsonuniformmeshes.............38

2.4.2Problemsonpiecewise-uniformmeshes........44

2.4.3Consistentgridsonsubdomains............51

2.4.4 ε-uniformlyconvergentdifferenceschemes......57

2.5 Boundaryvalueproblemsindomainswithcurvilinear boundaries...........................58

2.5.1Adomain-decomposition-baseddifferenceschemefor theboundaryvalueproblemonaslab........58

2.5.2Adifferenceschemefortheboundaryvalueproblem inadomainwithcurvilinearboundary........67

3Boundaryvalueproblemsforellipticreaction-diffusion equationsindomainswithpiecewise-smoothboundaries75

3.1Problemformulation.Theaimoftheresearch.......75

3.2Estimatesofsolutionsandderivatives............76

3.3Sufficientconditionsfor ε-uniformconvergenceofadifference schemefortheproblemonaparallelepiped.........85

3.4Adifferenceschemefortheboundaryvalueproblemon aparallelepiped........................89

3.5Consistentgridsonsubdomains...............97

3.6Adifferenceschemefortheboundaryvalueproblemin adomainwithpiecewise-uniformboundary.........102

4Generalizationsforellipticreaction-diffusionequations109

4.1MonotonicityofcontinualanddiscreteSchwartzmethods.109

4.2Approximationofthesolutioninaboundedsubdomainfor theproblemonastrip.....................112

4.3Differenceschemesofimprovedaccuracyfortheproblemon aslab..............................120

4.4Domain-decompositionmethodforimprovediterative schemes.............................125

5Parabolicreaction-diffusionequations

133

5.1Problemformulation......................133

5.2Estimatesofsolutionsandderivatives............134

5.3 ε-uniformlyconvergentdifferenceschemes.........145

5.3.1Gridapproximationsoftheboundaryvalueproblem146

5.3.2Consistentgridsonaslab...............147

5.3.3Consistentgridsonaparallelepiped.........154

5.4Consistentgridsonsubdomains...............158

5.4.1Theproblemonaslab.................158

5.4.2Theproblemonaparallelepiped...........161

6Ellipticconvection-diffusionequations 165

6.1Problemformulation......................165

6.2Estimatesofsolutionsandderivatives............166

6.2.1Theproblemsolutiononaslab............166

6.2.2Theproblemonaparallelepiped...........169

6.3 Onconstructionof ε-uniformlyconvergentdifferenceschemes undertheirmonotonicitycondition..............176

6.3.1Analysisofnecessaryconditionsfor ε-uniform convergenceofdifferenceschemes...........177

6.3.2Theproblemonaslab.................180

6.3.3Theproblemonaparallelepiped...........183

6.4Monotone ε-uniformlyconvergentdifferenceschemes....185

7Parabolicconvection-diffusionequations 191

7.1Problemformulation......................191

7.2Estimatesoftheproblemsolutiononaslab.........192

7.3Estimatesoftheproblemsolutiononaparallelepiped...199

7.4Necessaryconditionsfor ε-uniformconvergenceof differenceschemes.......................206

7.5Sufficientconditionsfor ε-uniformconvergenceofmonotone differenceschemes.......................210

7.6Monotone ε-uniformlyconvergentdifferenceschemes....213

IIAdvancedtrendsin ε-uniformlyconvergent differencemethods 219

8Gridapproximationsofparabolicreaction-diffusion equationswiththreeperturbationparameters 221

8.1Introduction..........................221

8.2Problemformulation.Theaimoftheresearch.......222

8.3 Apriori estimates.......................224

8.4Gridapproximationsoftheinitial-boundaryvalueproblem230

9Applicationofwidthsforconstructionofdifference schemesforproblemswithmovingboundarylayers235

9.1Introduction..........................235

9.2Aboundaryvalueproblemforasingularlyperturbed parabolicreaction-diffusionequation.............237

9.2.1Problem(9.2),(9.1)..................237

9.2.2Somedefinitions....................238

9.2.3Theaimoftheresearch................240

9.3 Apriori estimates.......................241

9.4Classicalfinitedifferenceschemes..............243

9.5Constructionof ε-uniformandalmost ε-uniform approximationstosolutionsofproblem(9.2),(9.1).....246

9.6Differenceschemeonagridadaptedinthemoving boundarylayer.........................251

9.7Remarksandgeneralizations.................254

10

High-orderaccuratenumericalmethodsforsingularly perturbedproblems 259

10.1Introduction..........................259

10.2Boundaryvalueproblemsforsingularlyperturbedparabolic convection-diffusionequationswithsufficientlysmoothdata261

10.2.1Problemwithsufficientlysmoothdata........261

10.2.2Afinitedifferenceschemeonanarbitrarygrid....262

10.2.3Estimatesofsolutionsonuniformgrids.......263

10.2.4Special ε-uniformconvergentfinitedifferencescheme263

10.2.5Theaimoftheresearch................264

10.3 Apriori estimatesforproblemwithsufficientlysmoothdata265

10.4Thedefectcorrectionmethod.................266

10.5TheRichardsonextrapolationscheme............270

10.6Asymptoticconstructs.....................273

10.7Aschemewithimprovedconvergenceforfinitevaluesof ε .275

10.8Schemesbasedonasymptoticconstructs..........277

10.9Boundaryvalueproblemforsingularlyperturbedparabolic convection-diffusionequationwithpiecewise-smoothinitial data...............................280

10.9.1Problem(10.56)withpiecewise-smoothinitialdata.280

10.9.2Theaimoftheresearch................281

10.10 Apriori estimatesfortheboundaryvalueproblem(10.56) withpiecewise-smoothinitialdata..............282

10.11Classicalfinitedifferenceapproximations..........285

10.12Improvedfinitedifferencescheme...............287

11Afinitedifferenceschemeon apriori adaptedgridsfor asingularlyperturbedparabolicconvection-diffusion equation 289

11.1Introduction..........................289

11.2Problemformulation.Theaimoftheresearch.......290

11.3Gridapproximationsonlocallyrefinedgridsthatareuniform insubdomains.........................293

11.4Differenceschemeon apriori adaptedgrid.........297

11.5Convergenceofthedifferenceschemeon apriori adaptedgrid303

11.6Appendix............................307

12Onconditioningofdifferenceschemesandtheirmatrices forsingularlyperturbedproblems 309

12.1Introduction..........................309

12.2Conditioningofmatricestodifferenceschemesonpiecewiseuniformanduniformmeshes.ModelproblemforODE...311

12.3Conditioningofdifferenceschemesonuniformandpiecewiseuniformgridsforthemodelproblem.............316

12.4 Onconditioningofdifferenceschemesandtheirmatricesfor aparabolicproblem......................323

13Approximationofsystemsofsingularlyperturbedelliptic reaction-diffusionequationswithtwoparameters 327

13.1Introduction..........................327

13.2Problemformulation.Theaimoftheresearch.......328

13.3Compatibilityconditions.Some apriori estimates.....330

13.4Derivationof apriori estimatesfortheproblem(13.2)under thecondition(13.5)......................333

13.5 Apriori estimatesfortheproblem(13.2)undertheconditions (13.4),(13.6)..........................341

13.6Theclassicalfinitedifferencescheme.............343

13.7Thespecialfinitedifferencescheme.............345

13.8Generalizations.........................348

14Survey 349

14.1Applicationofspecialnumericalmethodstomathematical modelingproblems.......................349

14.2Numericalmethodsforproblemswithpiecewise-smoothand nonsmoothboundaryfunctions................351

14.3Ontheapproximationofsolutionsandderivatives.....352

14.4Ondifferenceschemesonadaptivemeshes..........354

14.5Onthedesignofconstructivedifferenceschemesforanelliptic convection-diffusionequationinanunboundeddomain..357

14.5.1Problemformulationinanunboundeddomain.The taskofcomputingthesolutioninaboundeddomain357

14.5.2Domainofessentialdependenceforsolutionsofthe boundaryvalueproblem................359

14.5.3Generalizations.....................363

14.6Compatibilityconditionsforaboundaryvalueproblemona rectangleforanellipticconvection-diffusionequationwitha perturbationvectorparameter................364

14.6.1Problemformulation..................365

14.6.2Compatibilityconditions................366

References 371

Preface

The presentbookisdevotedtothe developmentofdifferenceschemes that converge ε-uniformly inthemaximumnormforarepresentativeclassof singularlyperturbedproblems.Italsodealswiththe justification oftheir convergence,and surveysnewdirectionsandapproachesdevelopedrecently,which areofimportanceforfurtherprogressinnumericalmethods.

ThebookwasintendedtobeanEnglishtranslationoftheRussianbook [138] ShishkinG.I.(1992). DiscreteApproximationsofSingularlyPerturbed EllipticandParabolicEquations.RussianAcademyofSciences,UralSection, Ekaterinburg(inRussian) thatwasinitiatedbyJohnJ.H.Miller.ThetranslationwasmadebyZoraUzelac,butwedecidednottopublishthisversionof thebook.Theverydensenatureofthisbook,thatallowedustocoveralarge classofsingularlyperturbedboundaryvalueproblemsinlittlespace,wastoo difficultformostreadersandalsocreatedproblemsintheimplementationof theresults.Sincetheappearanceofthebook[138],newresultsandideas haveappearedthataredealtwithinthepresentbook.

First,Iwouldliketothankmyteachers.Myscientificinterestsincomputationalmathematicswereformedandmaturedundertheinfluenceofthe scientificschoolsoftheAcademiciansoftheRussianAcademyofScience. A.M.Il’in,A.A.Samarskii,N.S.Bakhvalov,G.I.Marchukandtheirinfluence ledtotheappearanceofmyseconddoctoralthesis.Thisthesisbecamethe basisof[138],andisacontinuinginfluenceonmywork.

ItiswithpleasurethatInotethelong-termandfruitfulcollaborationwith theIrishandDutchmathematiciansinthegroupsofJ.MillerandP.Hemker. Thiscollaborationbeganin1990,andyieldedprogressinthedevelopmentof numericalmethodsforproblemswithboundarylayers,andledtonewresults thatwerepublishedinnumerousjointpapersandintwobooks[87]and[33].

TheRussianscientistsK.V.Emelianov,V.D.Liseikin,P.N.Vabishchevich, V.B.Andreev,V.F.Butuzov,A.V.Gulin,I.G.Belukhina,N.V.Kopteva, V.V.Shaidurov,B.M.Bagaev,E.D.Karepova,M.M.Lavrentiev,Jr,Yu.M. Laevsky,A.I.Zadorin,A.D.LjashkoandI.B.Badrievalsoinfluencedmuchof thedetailoftheapproachesinitiatedin[138].

TheideatotranslateintoEnglishthebook[138]beganduringmycollaborationoverthelastdozenyearswithmathematiciansandtheirstudents, namely,J.J.H.Miller,E.O’Riordan,A.F.Hegarty,M.Stynes,A.Ansari(Ireland),P.W.Hemker,J.Maubach,P.Wesseling(theNetherlands),P.A.Farrell (USA),F.Lisbona,C.Clavero,J.L.Gracia,J.C.Jorge(Spain),D.Creamer, LinPin(Singapore),andthroughdiscussionsofpapers(basedonideasfrom

[138]) oninternationalconferenceswith,amongothers,I.P.Boglaev(New Zealand),R.E.O’Malley,R.B.Kellogg(USA),L.Tobiska,H.-G.Roos,G. Lube,T.Linß(Germany),WangSong(Australia),L.G.Vulkov,I.A.Braianov(Bulgaria),R. ˘ Ciegis(Lithuania),andP.P.Matus(Belarus).Numerous ideasfrom[138]wereextendedandpublishedinmanypapers.

MythanksespeciallytoL.P.Shishkina,mybetterhalf,andmainassistantcolleagueandmathematicianforparticipationasco-authorinwritingthis book,forenormousscientificandtechnicalsupport.Shehaspreparedthe presentbookincludingallstages:theclarificationofresultsbynumerous discussions,preparationinLaTeX,thetranslation,compilingtheIndex,and reviewingthepage-proofs.

SignificantassistanceinthepreparationoftheEnglishversionofthepresent book,inthetranslationfromRussian-EnglishtoidiomaticEnglish,wasmade byM.Stynes(PartI,andfragmentsof PartII)andM.Mortell(thePreface andtheIntroduction)towhomIwouldliketoexpressmydeepestthanks.

Mythankstoourassistant-colleagueI.V.Tselishchevaforsupportinthe processofpreparingthebook,participationinthetranslationofsomechapters fromPartII,oftheIntroduction,oftheSurvey,andmanyothertasks.

Iamgratefulforfinancialandmaterialsupport(scientificbooks,computationaltechnique)tothe

InstituteofMathematicsandMechanics,UralBranchoftheRussian AcademyofSciences,Yekaterinburg,Russia;

InstituteforNumericalComputationandAnalysis,Dublin(INCA), Ireland;

DepartmentofMathematicsatTrinityCollegeDublin,Ireland; CWI(ResearchInstituteoftheStichtingMathematischCentrum), Amsterdam,theNetherlands;

SchoolofMathematicalSciences,DublinCityUniversity,Ireland;

DepartmentofMathematicsandStatisticsattheUniversityof Limerick,Ireland;

NationalResearchInstituteforMathematicsandComputerScience(NUS),Singapore;

SchoolofMathematicalSciencesattheNationalUniversityofIrelandinCork(UCC),Ireland;

BooleCentreforResearchinInformaticsattheUCC,Ireland;

MathematicsApplicationsConsortiumforScienceandIndustry inIreland(MACSI)undertheScienceFoundationIreland(SFI) MathematicsInitiative; andtoChapman&Hallforfriendlycooperation.

In particular,theresearchworkofG.I.ShishkinandL.P.Shishkinawas supportedbythe

RussianFoundationforBasicResearchundergrants No.01-01-01022,04-01-00578,07-01-00729; grantRFBR-NWO(RFFI-NWO)04-01-89007-NWO a);

DutchResearchOrganisationNWOundergrants No.047.008.007,047.016.008.

TheresearchworkofG.I.Shishkinwasalsosupportedbythe InternationalCollaborationProgrammeofForbairt,Dublin,IrelandNo.IC/97/057; EnterpriseIrelandBasicResearchgrants SC–98–612,SC–2000–070.

PartI

Gridapproximationsof singularperturbation partialdifferential equations

Chapter 1

Introduction

1.1 The developmentofnumericalmethodsforsingularlyperturbedproblems

Thewideuseofcomputingtechniques,combinedwiththedemandsofscientificandtechnicalpractices,hasstimulatedthedevelopmentofnumerical methodstoagreatextent,andinparticular,methodsforsolvingdifferentialequations.Theefficiencyofsuchmethodsisgovernedbytheiraccuracy, simplicityincomputingthediscretesolutionandalsotheirrelativeinsensitivitytoparametersintheproblem.Atpresent,numericalmethodsforsolvingpartialdifferentialequations,inparticular,finitedifferenceschemes,are welldevelopedforwideclassesofboundaryvalueproblems(see,forexample, [79, 108, 100, 214, 91, 216]).

Amongboundaryvalueproblems,aconsiderableclassincludesproblemsfor singularlyperturbedequations,i.e.,differentialequationswhosehighest-order derivativesaremultipliedbya(perturbation)parameter ε.Theperturbation parameter ε maytakearbitraryvaluesintheopen-closedinterval(0, 1](see, e.g.,[211, 210, 57, 94, 62]).Solutionsofsingularlyperturbedproblems,unlike regularproblems,haveboundaryand/orinteriorlayers,thatis,narrowsubdomainsspecifiedbytheparameter ε onwhichthesolutionsvarybyafinite value.Thederivativesofthesolutioninthesesubdomainsgrowwithout boundas ε tendstozero.

Inthecaseofsingularlyperturbedproblems,theuseofnumericalmethods developedforsolvingregularproblemsleadstoerrorsinthesolutionthatdependonthevalueoftheparameter ε.Errorsofthenumericalsolutiondepend onthedistributionofmeshpointsandbecomesmallonlywhentheeffective mesh-sizeinthelayerismuchlessthanthevalueoftheparameter ε (see,e.g., [138, 87, 106, 33]).Suchnumericalmethodsturnouttobeinapplicablefor singularlyperturbedproblems.

Duetothis,thereisaninterestinthedevelopmentofspecialnumerical methodswheresolutionerrorsareindependentoftheparameter ε anddefinedonlybythenumberofnodesinthemeshesused,i.e., numericalmethods (inparticular, finitedifferenceschemes)that converge ε-uniformly.When thesolutionsbysuchmethodsare ε-uniformlyconvergent,wewillcallthese methodsandsolutions robust (asin[33]).Atpresent,onlyseveralbooks aredevotedtothe development ofnumericalmethodsforsolvingsingularly

perturbedproblems.Gridmethodsforboundaryvalueproblemsforpartial differentialequationsareconsideredinthebooks[138,87,33,75];seealso [26, 13, 14, 76]forordinarydifferentialequations.Inthebook[106],the authorsgiveanumberofresultsandalsoacomprehensivebibliographyon numericalmethodsforsolvingsingularlyperturbedproblemsforpartialdifferentialequationsandforordinarydifferentialequations.

ThepresentbookwasintendedtobeanEnglishtranslationofthebook [138].Avarietyofideasandapproachesfrom[138]havesincebeenfurther developed.Newapproachesandtrendsappear,whichrequirefurtherinvestigation.Inthepresentbook,weelaborateonapproachestothedevelopmentof ε-uniformlyconvergentnumericalmethodsforseveralboundaryvalueproblemsfrom[138]anddiscusssomenewtrendsinthedevelopmentofother methods,whichhaveappearedrecently.

Quiteoftensolutionsofboundaryvalueproblems,theirgridsolutions,and alsotheirconvergenceareconsideredusingmaximumnorms.Theuseofeither theenergynormor L1, L2-normsisinadequatetodescribethesolutionsof singularlyperturbedproblemsandtheirapproximations.Forexample,inthe caseofproblemswithaparabolicboundarylayer,theboundary-layerfunction (thatisfiniteinthemaximumnorm)tendstozerointhenormsmentioned aboveas ε → 0[87,33].Inthisbook, maximumnorms areconsistently used.Asarule,weavoidreferencetoworkswhereproblemsforsingularly perturbedordinarydifferentialequationsareconsideredsincesuchresults andtechniquescannot,ingeneral,becarriedovertoproblemsforpartial differentialequations.

Thefirst ε-uniformlyconvergentdifferenceschemesconstructedforsingularlyperturbedproblemsusedtwomainapproaches: fittedoperatormethod and condensingmesh(grid)method/fittedmesh(grid)method.Schemesbased onthefittedoperatormethodwereconstructedin[2]and,independently, constructedandjustifiedin[56](forordinarydifferentialequations);in[15],a schemeforthecondensingmeshmethodwasconstructedandjustified(foran ellipticequation).Forschemesusingcondensingmeshes, ε-uniformconvergenceofthesolutionofadifferenceschemetothesolutionoftheboundary valueproblemisguaranteedbyaspecialchoiceofthedistributionofmesh points(forthegivennumberofnodes).Restrictionsonthechoiceofdifference equationsapproximatingsingularlyperturbedproblems(forensuringthe εuniformconvergenceofthescheme)are,ingeneral,notimposed.Infitted operatormethods, ε-uniformconvergenceofthesolutionofthedifference schemeisachievedbyaspecialchoiceofcoefficientsofthedifferenceequations approximatingthedifferentialproblem.Restrictionsonthedistributionof meshpointsforensuringthe ε-uniformconvergenceoftheschemearenot imposed.

Wementionalsoanapproachrelatedto additivesplittingofasingularity suggestedforsingularlyperturbedproblemsin[12](seealso[11]).Inthis method,basicfunctionsincludespecialfunctionsapproximatingthesingular componentofthesolutionoftheproblem.Inthecaseofsingularlyperturbed

problemsforpartialdifferentialequations,thisapproachwasnotwidelyused becausethesingularcomponentsofsolutionsofboundaryvalueproblemshave theformtoocomplicatedfortheeffectiveconstructionofasystemofbasis functions.Thedifferenceschemesbasedonthemethodofadditivesplitting ofasingularityandconstructedin[12,11]converge ε-uniformlyintheenergy norm.

Afterthepublications[15,56],therewasalargeefforttodevelopfittedoperatormethods.Thefirstbook[26]iscompletelydevotedtothedevelopment ofsuchmethodsforordinarydifferentialequations.Later,fittedoperator methodscontinuedtobeintensivelydeveloped(see,forexample,aseries ofvariantsofthefittedoperatorschemesforellipticequationsin[103]).A comprehensivebibliographyonnumericalmethodsforsingularlyperturbed problemsisgivenin[106].After[15,56],inthecaseofpartialdifferential equations,thefirstfinitedifferenceschemesthatconverge ε-uniformlyinthe maximumnormareconstructedin[29](seealso[30, 1]forthefittedoperator scheme,and[74, 118]fortheschemesoncondensingmeshes).

Notethatfittedoperatormethods(seetheirdescription,e.g.,in[26, 87, 33, 106])haveanadvantageinsimplicitybecausemeshesusedareuniform, andthiscontributedtotheirmorerapidprogresscomparedwithcondensing meshmethods.However,fittedoperatormethodshavearestricteddomainof applicabilityforconstructing ε-uniformlyconvergentnumericalmethods.

Itwasfirstestablishedin[124]thatthereare no ε-uniformlyconvergent schemesbasedonthefittedoperatormethod inthecaseforsingularlyperturbedellipticconvection-diffusionequationsindomainswherepartsofthe boundaryarecharacteristicsofareducedequationand parabolicboundary layers appear.Inthesamepaper,aschemewasconstructedthatconverges ε-uniformly,usingboththefittedoperatormethodfortheapproximationof derivativesalongcharacteristicsofthereducedequationandthecondensing meshmethodfortheapproximationofderivativesinthedirectionorthogonal tothecharacteristics.Theresultingdiscretesolutionsalsomakeitpossible toapproximatethenormalizedderivatives ε-uniformly.

Forparabolicequationswith paraboliclayers itisprovedthatthere existno schemes usingthefittedoperatormethodthatconverge ε-uniformlyinanyof thepapers:

[130]inthecaseofa parabolicboundarylayer and [148]inthecaseofa parabolicinitiallayer

Whenconstructingschemesfor nonlinearproblems,thesituationismuch morecomplicated.In[137](forreaction-diffusionequations)and[139](for convection-diffusionequations),itwasestablishedthatevenforsemilinear ordinarydifferentialequationsthere existnoschemesbasedonthefittedoperatormethod thatconverge ε-uniformly.Similardifficultiesrelatedtothe useoffittedoperatormethodsinnumericalmethodsarediscussedinlater

32] for semilinear ordinary differential equations). Numerical exp eriments that publications (see, e.g., [138, 87, 86] for partial differential equations and [84,

demonstratetheinconsistencyofthefittedoperatormethodforsemilinear

ordinarydifferentialequationareconsideredin[33].Thus,forawideclassof boundaryvalueproblemstherearenoschemesusingfittedoperatormethods thatconverge ε-uniformlyinthemaximumnorm;independentlyofthese, schemesmaybeconstructedusingclassicalfinitedifferenceapproximations orfiniteelementorfinitevolumemethods.

Havingsummarizingtheapproachestotheconstructionof ε-uniformlyconvergentnumericalmethods,wemakeonemoreremark.Inworks[134,140] (seealsodiscussionsin[87, 83]),aclassofproblemsisdistinguishedforellipticandparabolicequationsthatdegenerateontheboundaryofthedomain whosesolutionscontaininitialandparabolicboundarylayers.Itisshown thatforsuchproblems therearenoschemesofthecondensingmeshmethod (forrectangularmeshes)that converge ε-uniformly.Buttheapplicationof bothapproaches—fittedmeshandfittedoperatormethods— makesitpossible toconstructschemesthatconverge ε-uniformly.

Uptonow,practicallyallsingularlyperturbedpartialdifferentialequations, forwhichdifferenceschemesthatconverge ε-uniformlyinthemaximumnorm havebeenconstructed,donotcontainmixedderivatives.Boundaryvalue problemshavebeenconsideredonlyforthesimplestsubdomainsofdimension nothigherthantwo,andtheellipticoperatorinthedifferentialequationsis theLaplaceoperator.Inthecaseofanellipticequationwithmixedderivatives consideredonlyonarectangle,the ε-uniformlyconvergentschemeobtained appearstobetoocomplexandcannotbeextendedtootherdimensionsin geometry[28].Problemsindomainswithcurvilinearboundariesareconsideredonlyinfewpublications.

Itisoneofthegoalsofthepresentbooktoovercomesuchanexisting unsatisfactorystateintheareaofdevelopmentof ε-uniformlyconvergentdifferenceschemes.Anothergoalofthebookis,onmodelproblems,toconsider somemoderntrendsinthedevelopmentofnumericalmethodsforsingularly perturbedproblemsthatrequirefurtherinvestigation.

1.2 Theoretical problemsintheconstructionofdifferenceschemes

Wediscusssome basicprinciples relatedtothefoundationsofthetheory offinitedifferenceschemes,whichariseinthedevelopmentof ε-uniformly convergentnumericalmethods.

Thebehaviourofderivativesofasolutiontosingularlyperturbedboundary valueproblemsmotivatesthetypeofgridandthedifferenceschemesonthat. Thederivativesofthesolutionoftheboundaryvalueprobleminadomainwith asufficientlysmoothboundaryare ε-uniformlyboundedinmostofthedomain andgrowwithoutboundonlyinanarrowsubdomain(theboundarylayer) whosewidthtendstozerowith ε.Theessential anisotropyofthedirectional

derivatives isexhibited.Themaximalvaluesofderivativesofthesolution inaneighborhoodoftheboundarylayerarefoundonlyforthederivative alongthenormaltotheboundaryofthedomain;the k-thnormalderivative intheneighborhoodoftheboundaryisoforder O ε k ,andmoreover,the derivativedecreasesexponentiallytofinitevaluesasthevalue ε k ρ grows, where ρ isthedistancetotheboundary.Thebehaviourofthesolutionofthe boundaryvalueproblemisparticularlycomplicatedinthecaseofdomains withpiecewise-smoothboundaries.

Theobservedanisotropyinthebehaviourofthederivativesofthesolution andthe unboundedgrowthofthederivatives for ε → 0motivatestheuseof essentiallyanisotropicmeshes condensinginaneighborhoodoftheboundary layer.

Notethatthetermsofadifferentialconvection-diffusionequation,involving spatialderivativesacrosstheboundarylayer,becomeunboundedas ε → 0. So,veryclosetotheboundary,inordertoapproximatethesolutionofthe boundaryvalueproblembyasolutionofafinitedifferencescheme,itisnecessarytousemesheswhosestep-sizealongthenormaltotheboundaryisof order o(ε)[138].Underthisrequirementonthemesh-size, errors resultingin theapproximationofthetermsofthedifferentialequation,whenthederivativesintheequationarereplacedbydifferencederivatives,alsogrowwithout bound[138].Thedifferenceschemenolongerapproximatestheboundary valueproblem ε-uniformly.Thisunboundedbehaviouroftermsinthedifferentialequationandtheirdifferenceapproximationsgivesrisetodifficultiesin theproofof ε-uniformconvergenceoftherelevantschemes.

Theviolationofthe monotonicityproperty (see[108])ofaboundaryvalue problemwhengridapproximationsareconstructedevenforthesimplestproblemscanleadtobothlargeerrorsandnonphysicalresults.Forexample,inthe caseofanordinarydifferentialconvection-diffusionequation,whencentered differenceapproximationsonuniformmeshesareused,theerrorinthediscrete solutionoscillatesandgrowswithoutboundiftheparameter ε ismuchless thanthemesh-size[33].

Inthepresenceofmixedderivativesindifferentialequationsandalsointhe caseofproblemsindomainswithcurvilinearboundaries,significantdifficulties ariseintheconstructionofmonotonediscreteapproximationswhen meshes thatcondensegraduallyintheboundarylayer areused(forexample,Bakhvalov’smeshes[15]).Also,fordifferenceschemesthatdonotapproximatethe boundaryvalueproblems ε-uniformly,additionaldifficultiesappearwhenjustifyingthe ε-uniformconvergenceofsolutionsinthemaximumnorm.

Conventionalapproachestoovercomingthespecificproblemsmentioned abovethatariseintheconstructionof ε-uniformlyconvergentnumericalmethodsturnouttobeineffective.

So,inconvection-diffusionproblemstheuseofanartificialviscosityina numericalmethodinordertosuppressoscillationsinthegridsolutionleads toessentialerrorsinaneighborhoodoftheboundarylayer[214,55].

Inthecaseofsingularlyperturbedproblemsthemethodsforthederivation

of aprioriestimates developed forregularproblems (see,e.g.,[67, 68, 69, 37]) resultinestimatesforthederivativesthatdonotdistinguishthedifferent behavioursofderivativesindifferentdirections.Therequirementsimposed onthedataofaregularproblemforobtaining apriori estimatesbythis approachareclosetobeingnecessary.Butestimatesobtainedinthisway forsingularlyperturbedproblemsturnouttobeunabletojustify ε-uniform convergence.

Methodsbasedon asymptoticexpansionsinpowersoftheparameter ε allow onetoconstructapproximationstothesolutionswithanaccuracyuptoa sufficientlyhighorderoftheparameter ε (see,e.g.,[211, 210, 208, 59, 62, 209, 94, 57]andreferencestherein).Butthedetailedbehaviourofthederivativesof thesolutioninvariousdirectionsis,ingeneral,notconsidered.Toconstruct asymptoticexpansionsitisrequiredthatthedataoftheboundaryvalue problempossesssufficientsmoothness.Thisthenrestrictstheclassofboundaryvalueproblemsforwhich ε-uniformlyconvergentnumericalmethodscan bedevelopedandjustified.Thus,theaboveapproachesturnouttobeof littleusefortheconstructionof apriori estimatesrequiredfordeveloping ε-uniformlyconvergentnumericalmethods.

1.3 The mainprinciplesintheconstructionofspecialschemes

Wenowoutlinethe mainprinciples underlyingthedevelopmentofthe εuniformlyconvergentdifferenceschemeswhichareusedinthisbook.Even whennumericalmethodsareconstructedforregularproblemsthathavea complicatedsolution,itisnecessarytousegridapproximationsthatinherit themonotonicitypropertyoftheboundaryvalueproblem[108]inorderto preventtheappearanceofnonphysicaleffectinthesolution.Inthecaseof singularlyperturbedproblemssuchanaturalrequirementleadstocomplicateddifferenceschemes(inthecasewhenmixedderivativesarepresentin theequations,see,e.g.,schemesin[112]forregularproblemsandin[138]for singularlyperturbedboundaryvalueproblems).Fortheclassesofboundary valueproblemsstudiedheretheproblemsthatappeararetofindappropriate classesofgridapproximationsandalsotoobtainsufficientconditions,close tonecessaryones,forthe ε-uniformconvergenceoftheschemesconstructed.

Thestudyofsufficientlywideclassesofboundaryvalueproblemsrequires thedevelopmentofnewapproachestotheconstructionofnumericalmethods and,ingeneral,leadstomorecomplicatedschemes.Primarily,weneed a priori estimatesforthesolutionsthatadequatelyreflectthebehaviourofsuch singularitiesasaboundarylayerinaneighborhoodofdifferentpartsofthe domainboundary.Toconstructspecialschemes,weusethesimpleststandard ε-uniformlymonotoneapproximationstotheboundaryvalueproblem.The

schemesareconstructedonthesimplestpiecewiseuniformmeshesthatensure the ε-uniformconvergenceofthegridsolution.

Thetechniqueforconstructing aprioriestimatesbasedonadecomposition ofthesolution firstconceivedin[118]anddevelopedin[138](seealso[129, 131, 87]andthediscussions,forexample,in[105, 96, 95]foratwo-dimensional problem)wascalledtheShishkindecomposition(see,e.g.,[72, 73]).This techniqueuses thedecompositionofthesolutionintoregularandsingular components and,inthecaseofproblemsindomainswithpiecewisesmooth boundaries,the singularcomponent,inturn,is representedintermsofits componentsasthesumofregularandcornerlayers ofdifferentdimension. Thespecificbehaviouroftheregularcomponent,aswellasofconstituentsof thesingularcomponents,isrevealedusingthe firsttermsoftheirasymptotic expansions withrespecttotheparameter ε.Suchatechniqueallowsusto detectthedistinctionsinthebehaviourofderivativesofthesingularcomponentindifferentdirections,andtoobtainestimatesthatarenecessaryto provethe ε-uniformconvergenceofthedifferenceschemesbeingconstructed. Inthemethodofcondensinggridsconceivedin[128],whichuses piecewise uniformmeshes,thedistributionofmeshpointsinthedirectionacrossthe boundarylayer,normaltotheboundary,isdefinedbyonetransitionparameter,thatis,bythepointatwhichthemeshsizechanges.Intheliterature,such meshesarecalledShishkin’smeshes(see,e.g.,[83]).Fordifferenceschemeson thesemeshes,thedifficultiesinthedevelopmentof ε-uniformlyconvergentgrid methodsareessentiallyreducedthatallowstheconstructionof ε-uniformly monotonedifferenceschemes whichconverge ε-uniformly forrepresentative classesofsingularlyperturbedboundaryvalueproblems.Thefirstnumerical resultsonpiecewiseuniformmesheswereobtainedin[88]foranordinarydifferentialequation,in[89]foraparabolicpartialdifferentialequationwitha parabolicboundarylayer,andin[40]foranellipticconvection-diffusionequationonarectanglewithaparabolicboundarylayerthatisgeneratedonthe characteristicpartsoftheboundary.

An ε-uniformlyconvergentschemeforaprobleminatwo-dimensionaldomainwithacurvilinearboundaryandanequationwithmixedderivatives wasconstructedin1989forthefirsttimein[130]inthecaseofalinear parabolicreaction-diffusionequation.Forsomeboundarylayerproblemsin n-dimensionaldomains,suchsimilarschemeswereconstructedin[131],and in[136,137]forquasilinearequationsinan n-dimensionalslab(seealso[139] (1992)).The ε-uniformlyconvergentschemesinallthesecaseswereconstructedonpiecewiseuniformmeshes.

Theappearanceofpiecewiseuniformmeshes(see,e.g.,[127, 128])that simplifiedtheconstructionandstudyof ε-uniformlyconvergentgridmethods andalsotheprogressintechniquesforobtaining apriori estimates,aswellas theidentificationofalargeclassofproblemsforwhichtheapproachbasedon afittedoperatormethodisrestricted,contributedtothedevelopmentof εuniformlyconvergentdifferenceschemesforpartialdifferentialequations(see someresultsrelatedtothedevelopmentof ε-uniformlyconvergentschemes,

forexample,inthebooks[138, 87, 106, 33],inthesurvey[204],andinthe referencestherein).

1.4 Moderntrendsinthedevelopmentofspecialdifferenceschemes

Inthebookwepointoutsomeproblemsrelatedtonumericalmethodsfor singularlyperturbedproblemsthatrequirefurtherinvestigation.

• Asarule,theorderof ε-uniformconvergenceofdifferenceschemesislow; thereforetheseschemesaretooinefficientforpracticaluse.Thus,the developmentof schemeswithahighconvergenceorder isanimportant problem.

• Apriori estimatesforsolutionsofboundaryvalueproblemslargely“dictatethestructure”ofspecialrobustschemes.Therefore,thedevelopmentofatechniqueforobtaining apriori estimatesfornewclassesof boundaryvalueproblems,inparticular, problemsthathaveseveralperturbationparameters,or problemsforsystemsofequations isanimportanttask.

• Inanumberofboundaryvalueproblems,forexample,forparabolic equationswithmovingboundaryand/orinteriorlayers(see,e.g.,[159]), thecomplicatedsolutionsbringtoocomplicated ε-uniformlyconvergent schemesundertheconstruction.Aproblemofcurrentinterestistofind necessaryandsufficientconditionsfor ε-uniformconvergence ofsuch specialschemes.

• Eveninregularproblems,theconditioningnumberofclassicaldifference schemesgrowswithoutboundwhenthemesh-sizetendstozero.Inthe condensinggridmethod,themesh-sizeinthelayerforafixednumber ofnodescanbearbitrarilysmalldependingontheparameter ε.Thus, thestudyofconditioning,aswellas ε-uniformconditioningofspecial schemes,isanimportantproblem.

Theseproblemsrelatetothefoundationsofthetheoryofnumericalmethods(differenceschemes)forsingularlyperturbedproblemswithcomplicated singularities.

1.5 The contentsofthepresentbook

Thecontentsofthebookcanbedividedintotwoparts.PartIis “translation” ofcertainpartsofthebook[138]publishedin1992inRussian,and Part II isa survey ofsomerecentresultsinthedevelopmentofrobustmethods forboundaryvalueproblemswithboundarylayersandothersingularities. Theworkinvolvedinthe “translation” resultedinthedevelopmentindetail ofprinciplesforconstructingmonotonedifferenceschemesforsomeclassesof singularlyperturbedproblems.

InPartI(Chapters2–7)boundaryvalueproblemsareconsideredforelliptic andparabolicreaction-diffusionandconvection-diffusionequations in n-dimensionaldomainswithsmoothandpiecewisesmoothboundaries inthecase whenthe differentialequationscontainmixedderivatives.Hereourgoalis todevelopatechniqueforconstructingandjustifying ε-uniformlyconvergent differenceschemes,basedonclassicaldiscreteapproximations,forboundary valueproblemswithlessrestrictionsontheproblemdata.Inthissituation wearenottooconcernedabouttherateof ε-uniformconvergenceofschemes constructed.Significantattentionisgiventothestudyof conditions thatare necessaryandsufficient forthe ε-uniformconvergence oftherelevantschemes. Whenconstructing specialdifferenceschemesthatconverge ε-uniformly,we usethetechniquesdevelopedfor canonicalproblemsonaslabandaparallelepiped.

PartIIcontainsmaterialpublishedmainlyinthelastfouryears.These areproblemswith boundarylayers,and additionalsingularities generatedby nonsmoothdata,unboundednessofthedomain,andalsobythepresenceof theperturbationvectorparameter.Anotheraspectoftheconsiderationsin thispartisthatwe findboththesolutionanditsderivatives witherrorsthat areindependentoftheperturbationparameters.

In Chapters8 and 13,boundaryvalueproblemsforequationswithperturbationvectorparametersarestudied.InChapter8,problemsareconsidered forascalarparabolicreaction-diffusionequationwiththreescalarparameters, andinChapter13,aproblemisstudiedforasystemoftwoellipticreactiondiffusionequationswithtwoscalarparameters.Intheseproblems,layersof differenttypesarisedependingontherelationbetweenthescalarparameters.

Forgridapproximationsofboundaryvalueproblems withamovingboundarylayer,necessaryandsufficientconditionsfortheschemestoberobust areobtainedin Chapter9.Takingintoaccounttheseconditions,difference schemesareconstructedforaparabolicreaction-diffusionequationinadomainwithmovingboundaries.

Methodsof improvingtheaccuracyofgridsolutions whilepreservingtheir robustnessareexaminedin Chapter10. Conditioning and correctness (or well-posedness)ofadifferenceschemethatisconstructedbasedonaclassical gridapproximationofaboundaryvalueproblemforaparabolicconvection-

diffusionequationonpiecewise-uniformmeshesarediscussedin Chapter12. In Chapter11,differenceschemeson apriori adaptedmeshesaredeveloped.

Promisingapproachestothedevelopmentofrobustmethodsbasedon a posterioriadaptedtechniques arediscussedin Chapter14.Inthecaseof singularlyperturbedproblems,theapproachesknownforregularproblems improvetheconvergenceonlyforfinitevaluesoftheparameter ε,and do notprovidefortherobustnessofthemethod.Here,differenceschemesare discussedthatconverge“almost ε-uniformly”.

InthesurveyChapter14wegivecommentstosomeapproachesintheconstructionofspecialdifferenceschemesandresearchdirections.Inparticular, thefollowingproblemsarediscussed:

• some“applied”problemswhosesolutionsrequirerobustnumerical methods;

• approachestotheconstructionofrobustnumericalmethodsforparabolicequationswithsmoothandpiecewisesmoothboundaryconditions;

• approximationofsolutionsandderivatives;

• specificboundaryvalueproblemsforellipticconvection-diffusionequationssuchasaproblemonanunboundeddomainforanequationwith ascalarparameterandaproblemonarectangleforanequationwitha vectorparameter;

• inthecaseofanunboundeddomain,theconceptofthedomainof essentialdependenceofthesolutionthatallowsrobustschemestobe constructedwhosesolutionsconvergeonboundeddomainsbyincreasing thenumberofnodesinthemeshesused;

• compatibilityconditionsforaproblemonarectanglethatguaranteethe smoothnessofthesolutionanditsregularandsingularcomponentsthat arerequiredfortheconstructionandjustificationofrobustschemes.

1.6 The presentbook

Wediscuss theexistingstateoftheart intheareaofthedevelopmentof differenceschemesthatconverge ε-uniformlyinthemaximumnorm.

Thebooks[138,87,33],aswellasthepresentbook,arefocussedonthe developmentofa technologyforconstructingfinitedifferenceschemesthat converge ε-uniformlyinthemaximumnorm forawideclassofsingularlyperturbedboundaryvalueproblems.In[106], ε-uniformlyconvergentnumerical methods areconsideredthatusevarioustechniquesandmethodsdevelopedfor

regularandsingularlyperturbedproblems.Theconvergenceofdiscretemethodsisconsideredin normscorrespondingtotheappliednumericalmethod In[106]thebasicresultsdevelopedin[138]aregiveninacondensedform,and problemsforequationswithmixedderivatives and multi-dimensionalproblems arenotdiscussed

Thebooks[138,87,33]aredevotedtothedevelopmentoffinitedifference schemesforsingularlyperturbedellipticandparabolicequations.Difference schemesareconstructedbasedon standardmonotonefinitedifferenceapproximationsofdifferentialproblemsonpiecewiseuniformmeshes.Thebooks [138,87]emphasizethetheory,and[33]isstrictlynumerical.

Appearingin1988,piecewise-uniformmeshes[127,128]allowustoovercomeanumberofessentialdifficultiesinthedevelopmentandjustification of ε-uniformlyconvergentfinitedifferenceschemesforrepresentativeclasses ofsingularlyperturbedequationswithpartialderivatives.Thedevelopment oftechniquestoderive apriori estimatesonthebasisofthedecomposition methodappliedtothesolutionofboundaryvalueproblems,andalsotechniquestoconstructmonotonediscreteapproximations(ofboundaryvalueproblems)constructedonpiecewiseuniformmeshes,allowthedevelopmentofspecialfinitedifferenceschemesthatareconvergent ε-uniformly.Asaresultof suchaprogress,thebook[138]waswritten.

In[138] ratherwideclassesofellipticandparabolicequations asreactiondiffusion,andconvection-diffusion,havebeenconsideredin n-dimensionaldomainswithsmoothandpiecewisesmoothboundaries,andalsoincomposite domainsinthepresenceofconcentratedsources.Fortheseproblems, techniquestoconstruct ε-uniformlyconvergentschemeshavebeendeveloped,such schemeshavebeenconstructedandtheirconvergencehasbeenstudied.

Thepresentationofresultsderivedin[138]hasbeengiveninaconcentrated formasinareferencebook,sothatitwaspossibletocoverlargeclasses ofboundaryvalueproblems.However,suchaformofexpositionmakesit difficulttousetheminthedevelopmentofnumericalmethodsforequations withpartialderivatives.Itcouldbethatthiscircumstanceisoneofthe reasonsforthedelayinusingresultsfrom[138],aswellasaratherlong delayinthedevelopmentofrobustnumericalmethodsformultidimensional problems.

Thebook [87]isathoroughintroductiontorobustfinitedifferenceschemes. In[87], basicideasandthetechniques thatareusedtoconstructandjustify ε-uniformlyconvergentdifferenceschemeswereconsideredusingmodel problemsasexamples.Here,difference schemes basedon thefittedoperator method and thecondensingmeshmethod havebeenconstructedthatconverge ε-uniformly.Difficultiesthatarisewhenconstructingrobustnumerical methodsforparabolicequationswithparabolicboundarylayershavebeen considered.

Inthebook [33],foranumberofmodelproblemswithboundarylayers, extensive numericalstudies havebeenperformedforrobustdifferenceschemes, andinparticular,forthosethathavebeenconsideredin[138,87].Forthe

quantitativeanalysisofnumericalmethods,anexperimentaltechniquehas beendevelopedwhichallowsustojustify ε-uniformconvergence,andalsoto findtheparametersintheestimatesoftheconvergencerate.A techniquefor experimentalstudy ofspecialdifferenceschemeshasbeendevelopedin[33]. Thistechniquewastestedbynumericalexperimentscarriedoutforlinearand nonlinearproblems,inparticularforellipticconvection-diffusionequations whosesolutionshaveparabolicboundarylayersthatdegenerateonthedomain boundary.Detailsofthe solvers thatareusedinthecomputationshavealso beengiven.

Intheconcludingchaptersofthebook[33],theefficiencyofthetechniques thathavebeendevelopedinthefirsttenchaptersisdemonstratedwhensolvingthe classicalPrandtlproblem forflowofanincompressiblefluidpasta semi-infiniteflatplate[113].ForthenonlinearPrandtlproblem,adifference schemewasconstructedusinganonlinearsolver;inthisproblem, ε = Re 1 isaperturbationparameter,and Re istheReynoldsnumber.Thenumerical methodgives theproblemsolution (thatiscomponentsoftheflowvelocity) and thenormalizedfirst-orderderivatives withanaccuracyclosetoorderone. Computationalexpansesforthesolution (uptoalogarithmicmultiplier,are proportionaltothenumberofmeshpointsused) areindependentoftheparameter ε.

Inthebook [106],numericalmethodshavebeenconsideredforsingularly perturbedboundaryvalueproblemstoordinarydifferentialequationsand alsotoone-dimensionalparabolicandtwo-dimensionalellipticequations(with theLaplaceoperatorasthemainpartofthedifferentialoperator).Various ideasandtechniquesthatareusedinthenumericalanalysisforsingularly perturbeddifferentialequations,andalsoapproachesdevelopedforregular problems,arediscussed. Finitedifferencemethods and finiteelementmethods havealsobeenconsideredforsingularlyperturbedboundaryvalueproblems. Approximationsofsolutionstoboundaryvalueproblemsin variousnorms, inparticularin themaximum and theenergynorms,havebeendiscussed. Significantattentionhasbeendevotedto ε-uniformlyconvergentnumerical methodsinwhich thefittedoperatormethod and thefittedmeshmethod are appliedusing Bakhvalovmeshes and piecewise-uniformmeshes.Aseparate chapterisdevotedto incompressibleNavier-Stokesequations.Thebookcontainsanextensivebibliographyonnumericalmethodsforsingularlyperturbed boundaryvalueproblems.

Thepresentbook isdevotedto asystematicdetaileddevelopmentofapproaches totheconstructionof ε-uniformlyconvergentfinitedifferenceschemes forsomeclassesofsingularlyperturbed boundaryvalueproblemsconsidered in[138].Problemsfor multi-dimensionalellipticandparabolicequationsthat arereaction-diffusionandconvection-diffusion arestudiedinthecasewhen mixedderivativesareinvolvedinthedifferentialequations.Theproblems areconsideredin domainswithpiecewisesmoothandcurvilinearboundaries Boundaryvalueproblemsfrom[138]suchas problemsforequationswithdis-

continuouscoefficientsinthepresenceofconcentratedsources,andalso problemsforconvection-diffusionequationswithcharacteristicpartsontheboundary,arenotconsidered.

The approach presentedhereincludes:

• atechniquetoconstructaprioriestimatesforsolutions ofboundary valueproblemswhichare necessary toconstructandjustifyfinitedifferenceschemes ε-uniformlyconvergentinthemaximumnorm,

• atechniquetoconstructfinitedifferenceschemes basedonstandard discreteapproximationsofboundaryvalueproblems,preservingmonotonicityofdifferentialproblems,

• atechniquetoconstructspecialpiecewiseuniformmeshes(grids) providing ε-uniformconvergenceoftheschemesunderconstruction.

Furthermore,significantattentionisdevotedtotheconsiderationof modern trends inthedevelopmentof ε-uniformlyconvergentnumericalmethodsfor singularlyperturbedproblems.Theinvestigationsofaseriesofmodelproblemscanbeconsideredasrepresentativemodelsofmoregeneralclasses,for whichapproachesfromthefirstpartofthepresentbookcouldbeappliedto thedevelopmentofappropriatenumericalmethods.

Approachesdevelopedinthebook,andresultsderived,shouldpromote furtherdevelopmentofefficientnumericalmethodsfor singularlyperturbed problemswithdifferenttypesofsingularities,andatthesametime,preserve thesamequalitiesalreadywell-establishedinnumericalmethodsforregular boundaryvalueproblems.

Inthepresentbook,aswellinthebooks[138,87,33], ε-uniformlyconvergentschemesareconstructedon simplest piecewiseuniformmeshes,i.e.,mesheshavingonlyone transitionpoint inaneighborhoodoftheboundarylayer. SchemesonBakhvalovmeshes,aswellasonmesheswithfew transitionpoints inaneighborhoodoftheboundarylayer(see,e.g.,[169]),arenotconsidered here.

Thepresentbookandthebooks[138,87,106,33],consideredasonepackage,coverawide spectrumofproblems thatarisewhenconstructingspecial numericalmethodsforproblemswithboundarylayers.Nevertheless,there isalarge gap betweencomplicatedsystemsofnonlinearsingularlyperturbed equationswithpartialderivativeswhich needtobesolved,e.g.,frommathematicalmodelling,andthosesingularlyperturbedproblemsforwhich theoreticallyjustifiednumericalmethodsexist.Wehopethattheresultsgivenhere couldbehelpful inthedevelopmentandanalysisofnewnumericalmethods forproblemsthathaveyettobesolved.

1.7 The audienceforthisbook

Thepresentbook,togetherwiththebooks[138,87,106,33],couldbe usefulforscientists-researchers(fromstudentsuptoprofessionals)inthedevelopmentofnumericalmethodsforsingularlyperturbedproblemsandalso forspecialist-researcherswhoareinterestinginmathematicalmodellingand whereproblemswithboundaryandinteriorlayersarisenaturally.

Inthefirstpartofthebookdevotedto multi-dimensionalproblems,techniquesandideasexploitedto deriveaprioriestimatesforthesolutions andto constructthesimplestdiscreteconstructions,i.e.,piecewiseuniformmeshes andstandardmonotonefinitedifferenceoperatorsonthem,arecarefullydiscussedforeachtypeofboundaryvalueproblem.Conditionsimposedon discreteconstructionsarediscussed,anditisshownthatsuch conditionsare closetonecessaryconditions.Techniquesandprinciplesgiveninthebook allowthereadertoconstructindependentlyrobustfinitedifferenceschemes fornewclassesofboundaryvalueproblems,inparticular,forproblemsfrom [138]thatarenottreatedinthepresentbook.

Inthesecondpartofthebook,aconsiderationofmoderntrendsarecarried outonmodelproblems,whichallowtocheckonideasandtechniquesused intheconstruction.Here,wegivereferencestooriginalsourceswheredetails aregivenintheexposedbibliography.Thediscussionofmoderntrendsallows thereadertobemoreconfidentinthisfastgrowingareaofresearch.Thefirst andsecondpartsofthebookcanbestudiedindependently.Thebookcanbe alsohelpfulforresearchersstudyingsingularlyperturbedordinarydifferential equations.

Ifthereaderwishestodevelopnewdifferencemethodsforparticularsingularlyperturbedproblems,itisonlynecessarytoreadPartI.

Ifthereaderwishestoapplydifferencemethodsalreadydevelopedforsingularlyperturbedproblems,thereadershouldpassto PartII.

Chapter 2

Boundary valueproblemsforellipticreaction-diffusion equationsindomainswithsmoothboundaries

Inthischapterboundaryvalueproblemsareconsideredforellipticreactiondiffusionequationsindomainswithsmoothboundaries.Theconstruction offinitedifferenceschemesandthejustificationoftheirconvergenceiscarriedoutusingananalogofthewell-knownsufficientconditionforconvergenceofschemesforregularboundaryvalueproblemsthatisaconsequence ofapproximationbyastablefinitedifferencescheme(itsdescriptionsee,e.g., [79, 108, 100]).Whenappliedtosingularlyperturbedboundaryvalueproblems,this sufficientconditionfor ε-uniformconvergence canbestatedinthe followingway:

ε-uniformconvergenceofafinitedifferenceschemefollowsfrom

ε-uniformapproximationoftheboundaryvalueproblembyan

ε-uniformlystablefinitedifferencescheme.

Ourfinitedifferenceschemesareconstructedbasedonclassicaldifferenceapproximationsofdifferentialequations. ε-uniformapproximationofboundary valueproblemsbyfinitedifferenceschemesisobtainedbyaspecialdistributionofmeshpoints. ε-uniformmonotonicity(and,hence,stability)ofthe finitedifferenceschemesisachievedbymeansofspatialmesheswhosedistributionsofnodesineachcoordinatedirectionareconsistentwitheachother.

2.1 Problem formulation.Theaimoftheresearch

Onan n-dimensionaldomain D withsufficientlysmoothboundaryΓ= Γ(D)and D = D Γ, (2.1) where D is, ingeneral,anunboundeddomain,weconsidertheDirichletproblemforelliptic reaction-diffusionequations

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Title: The star-stealers

Author: Edmond Hamilton

Illustrator: Hugh Rankin

Release date: April 21, 2024 [eBook #73442]

Language: English

Original publication: Indianapolis, IN: Popular Fiction Publishing Company, 1929

Credits: Greg Weeks, Mary Meehan and the Online Distributed Proofreading Team at http://www.pgdp.net

*** START OF THE PROJECT GUTENBERG EBOOK THE STARSTEALERS ***

THE STAR-STEALERS

[Transcriber's Note: This etext was produced from Weird Tales May, February 1929.

Extensive research did not uncover any evidence that the U.S. copyright on this publication was renewed.]

As I stepped into the narrow bridgeroom the pilot at the controls there turned toward me, saluting.

"Alpha Centauri dead ahead, sir," he reported.

"Turn thirty degrees outward," I told him, "and throttle down to eighty light-speeds until we've passed the star."

Instantly the shining levers flicked back under his hands, and as I stepped over to his side I saw the arrows of the speed-dials creeping backward with the slowing of our flight. Then, gazing through the broad windows which formed the room's front side, I watched the interstellar panorama ahead shifting sidewise with the turning of our course.

The narrow bridgeroom lay across the very top of our ship's long, cigar-like hull, and through its windows all the brilliance of the heavens around us lay revealed. Ahead flamed the great double star of Alpha Centauri, two mighty blazing suns which dimmed all else in the heavens, and which crept slowly sidewise as we veered away from them. Toward our right there stretched along the inky skies the far-flung powdered fires of the Galaxy's thronging suns, gemmed with the crimson splendors of Betelgeuse and the clear brilliance of Canopus and the hot white light of Rigel. And straight ahead, now, gleaming out beyond the twin suns we were passing, shone the clear yellow star that was the sun of our own system.

It was the yellow star that I was watching, now, as our ship fled on toward it at eighty times the speed of light; for more than two years had passed since our cruiser had left it, to become a part of that great navy of the Federation of Stars which maintained peace over all the Galaxy. We had gone far with the fleet, in those two years, cruising with it the length and breadth of the Milky Way, patrolling the space-lanes of the Galaxy and helping to crush the occasional pirate ships which appeared to levy toll on the interstellar commerce. And now that an order flashed from the authorities of our own solar system had recalled us home, it was with an unalloyed eagerness

that we looked forward to the moment of our return. The stars we had touched at, the peoples of their worlds, these had been friendly enough toward us, as fellow-members of the great Federation, yet for all their hospitality we had been glad enough to leave them. For though we had long ago become accustomed to the alien and unhuman forms of the different stellar races, from the strange brainmen of Algol to the birdlike people of Sirius, their worlds were not human worlds, not the familiar eight little planets which swung around our own sun, and toward which we were speeding homeward now.

While I mused thus at the window the two circling suns of Alpha Centauri had dropped behind us, and now, with a swift clicking of switches, the pilot beside me turned on our full speed. Within a few minutes our ship was hurtling on at almost a thousand light-speeds, flung forward by the power of our newly invented de-transforming generators, which could produce propulsion-vibrations of almost a thousand times the frequency of the light-vibrations. At this immense velocity, matched by few other craft in the Galaxy, we were leaping through millions of miles of space each second, yet the gleaming yellow star ahead seemed quite unchanged in size.

Abruptly the door behind me clicked open to admit young Dal Nara, the ship's second-officer, descended from a long line of famous interstellar pilots, who grinned at me openly as she saluted.

"Twelve more hours, sir, and we'll be there," she said.

I smiled at her eagerness. "You'll not be sorry to get back to our little sun, will you?" I asked, and she shook her head.

"Not I! It may be just a pin-head beside Canopus and the rest, but there's no place like it in the Galaxy. I'm wondering, though, what made them call us back to the fleet so suddenly."

My own face clouded, at that. "I don't know," I said, slowly. "It's almost unprecedented for any star to call one of its ships back from the Federation fleet, but there must have been some reason——"

"Well," she said cheerfully, turning toward the door, "it doesn't matter what the reason is, so long as it means a trip home. The crew is worse than I am—they're scrapping the generators down in the engine-room to get another light-speed out of them."

I laughed as the door clicked shut behind her, but as I turned back to the window the question she had voiced rose again in my mind, and I gazed thoughtfully toward the yellow star ahead. For as I had told Dal Nara, it was a well-nigh unheard-of thing for any star to recall one of its cruisers from the great fleet of the Federation. Including as it did every peopled star in the Galaxy, the Federation relied entirely upon the fleet to police the interstellar spaces, and to that fleet each star contributed its quota of cruisers. Only a last extremity, I knew, would ever induce any star to recall one of its ships, yet the message flashed to our ship had ordered us to return to the solar system at full speed and report at the Bureau of Astronomical Knowledge, on Neptune. Whatever was behind the order, I thought, I would learn soon enough, for we were now speeding over the last lap of our homeward journey; so I strove to put the matter from my mind for the time being.

With an odd persistence, though, the question continued to trouble my thoughts in the hours that followed, and when we finally swept in toward the solar system twelve hours later, it was with a certain abstractedness that I watched the slow largening of the yellow star that was our sun. Our velocity had slackened steadily as we approached that star, and we were moving at a bare one light-speed when we finally swept down toward its outermost, far-swinging planet, Neptune, the solar system's point of arrival and departure for all interstellar commerce. Even this speed we reduced still further as we sped past Neptune's single circling moon and down through the crowded shipping-lanes toward the surface of the planet itself.

Fifty miles above its surface all sight of the planet beneath was shut off by the thousands of great ships which hung in dense masses above it—that vast tangle of interstellar traffic which makes the great planet the terror of all inexperienced pilots. From horizon to

horizon, it seemed, the ships crowded upon each other, drawn from every quarter of the Galaxy. Huge grain-boats from Betelgeuse, vast, palatial liners from Arcturus and Vega, ship-loads of radium ores from the worlds that circle giant Antares, long, swift mail-boats from distant Deneb—all these and myriad others swirled and circled in one great mass above the planet, dropping down one by one as the official traffic-directors flashed from their own boats the brilliant signals which allowed a lucky one to descend. And through occasional rifts in the crowded mass of ships could be glimpsed the interplanetary traffic of the lower levels, a swarm of swift little boats which darted ceaselessly back and forth on their comparatively short journeys, ferrying crowds of passengers to Jupiter and Venus and Earth, seeming like little toy-boats beside the mighty bulks of the great interstellar ships above them.

As our own cruiser drove down toward the mass of traffic, though, it cleared away from before us instantly; for the symbol of the Federation on our bows was known from Canopus to Fomalhaut, and the cruisers of its fleet were respected by all the traffic of the Galaxy. Arrowing down through this suddenly opened lane we sped smoothly down toward the planet's surface, hovering for a moment above its perplexing maze of white buildings and green gardens, and then slanting down toward the mighty flat-roofed building which housed the Bureau of Astronomical Knowledge. As we sped down toward its roof I could not but contrast the warm, sunny green panorama beneath with the icy desert which the planet had been until two hundred thousand years before, when the scientists of the solar system had devised the great heat-transmitters which catch the sun's heat near its blazing surface and fling it out as high-frequency vibrations to the receiving-apparatus on Neptune, to be transformed back into the heat which warms this world. In a moment, though, we were landing gently upon the broad roof, upon which rested scores of other shining cruisers whose crews stood outside them watching our arrival.

Five minutes later I was whirling downward through the building's interior in one of the automatic little cone-elevators, out of which I

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