2 Multiscale finite element methods for linear pro

2Multiscalefiniteelementmethodsforlinearproblemsandoverview2.1SummaryInthissection,themainconceptofmultiscalefiniteelementmethods(Ms-FEM)ispresented.Wekeepthepresentationsimpletomakeitaccessibletoabroaderaudience.TwomainingredientsofMsFEMsaretheglobalformula-tionofthemethodandtheconstructionofbasisfunctions.Wediscussglobalformulationsusingvariousfiniteelement,finitevolume,andmixedfiniteele-mentmethods.Asformultiscalebasisfunctions,thesubgridcapturingerrorsarediscussed.Wepresentsimplifiedcomputationsofbasisfunctionsforcaseswithscaleseparation.Wealsodiscusstheimprovementofsubgridcapturingerrorsviaoversamplingtechniques.Finally,wepresentsomerepresentativenumericalexamplesanddiscussthecomputationalcostofMsFEMs.AnalysisofsomerepresentativecasesispresentedinChapter6.2.2IntroductiontomultiscalefiniteelementmethodsWestartourdiscussionwiththeMsFEMforlinearellipticequationsLp=finΩ,(2.1)whereΩisadomaininRd(d=2,3),Lp:=−div(k(x)∇p),andk(x)isaheterogeneousfieldvaryingovermultiplescales.WenotethatMsFEMcanbeeasilyextendedtosystemssuchaselasticityequations,aswellastononlinearproblems(seeSection2.4andChapter3).Thechoiceofthenotationsk(x)andp(x)in(2.1)isusedbecauseoftheapplicationsofthemethodtoporousmediaflowslaterinthebook.Wenotethatthetensork(x)=(kij(x))isassumedtobesymmetricandsatisfiesα|ξ|2≤kijξiξj≤β|ξ|2,forallξ∈Rdandwith0αβ.Weomitxdependencewhenthereisnoambiguityandassumethesummationoverrepeatedindices(Einsteinsummationconvention)unlessotherwisestated.Y.Efendiev,T.Y.Hou,MultiscaleFiniteElementMethods:TheoryandApplications,13SurveysandTutorialsintheAppliedMathematicalSciences4,DOI10.1007/978-0-387-09496-02,cSpringerScience+BusinessMediaLLC2009142MsFEMsforlinearproblemsandoverviewMsFEMsconsistoftwomajoringredients:multiscalebasisfunctionsandaglobalnumericalformulationthatcouplesthesemultiscalebasisfunctions.Basisfunctionsaredesignedtocapturethemultiscalefeaturesofthesolu-tion.Importantmultiscalefeaturesofthesolutionareincorporatedintotheselocalizedbasisfunctionswhichcontaininformationaboutthescalesthataresmaller(aswellaslarger)thanthelocalnumericalscaledefinedbythebasisfunctions.Aglobalformulationcouplesthesebasisfunctionstoprovideanaccurateapproximationofthesolution.Next,wediscusssomebasicchoicesformultiscalebasisfunctionsandglobalformulations.Basisfunctions.First,wediscussthebasisfunctionconstruction.LetThbeausualpartitionofΩintofiniteelements(triangles,quadrilaterals,andsoon).Wecallthispartitionthecoarsegridandassumethatthecoarsegridcanberesolvedviaafinerresolutioncalledthefinegrid.Forclarityofthisexposition,weplotrectangularcoarseandfinegridsinFigure2.1(leftfigure).LetxibetheinteriornodesofthemeshThandφ0ibethenodalbasisofthestandardfiniteelementspaceWh=span{φ0i}.Forsimplicity,onecanassumethatWhconsistsofpiecewiselinearfunctionsifThisatriangularpartition.DenotebySi=supp(φ0i)(thesupportofφ0i)anddefineφiwithsupportinSiasfollowsLφi=0inK,φi=φ0ion∂K,∀K∈Th,K⊂Si;(2.2)thatismultiscalebasisfunctionscoincidewithstandardfiniteelementbasisfunctionsontheboundariesofacoarse-gridblockK,andareoscillatoryintheinteriorofeachcoarse-gridblock.Throughout,Kdenotesacoarse-gridblock.Notethateventhoughthechoiceofφ0icanbequitearbitrary,ourmainassumptionisthatthebasisfunctionssatisfytheleading-orderhomo-geneousequationswhentheright-handsidefisasmoothfunction(e.g.,L2integrable).WewouldliketoremarkthatMsFEMformulationallowsonetotakeadvantageofscaleseparationwhichisdiscussedlaterinthebook.Inparticular,KcanbechosentobeadomainsmallerthanthecoarsegridasillustratedinFigure2.1(rightfigure)ifthesmallregioncanbeusedtorepresenttheheterogeneitieswithinthecoarse-gridblock.Inthiscase,thebasisfunctionhastheformulation(2.2),exceptKisreplacedbyasmallerregion,Kloc,L(φi)=0inKloc,φi=φ0ion∂Kloc,wherethevaluesofφ0iinsideKareusedinimposingboundaryconditionson∂Kloc.Ingeneral,onesolves(2.2)onthefinegridtocomputebasisfunctions.Insomecases,thecomputationsofbasisfunctionscanbeperformedanalytically.Toillustratethebasisfunctions,wedepicttheminFigure2.2.Ontheleft,thebasisfunc-tionisconstructedwhenKisacoarsepartitionelement,andontheright,thebasisfunctionisconstructedbytakingKtobeanelementsmallerthanthecoarse-gridblocksize.NotethatabilinearfunctioninFigure2.2(rightfigure)isusedtodemonstrateboundaryconditionsonasmallcomputationaldomainandthisbilinearfunctionisnotapartofthebasisfunction.Inthiscase,theassemblyofthestiffnessmatrixusesonlytheinformationinsmallcomputationalregionsandthebasisfunctioncanbe“periodically”extended2.2Introductiontomultiscalefiniteelementmethods15Coarse−gridFine−griddomainCoarse−gridFine−gridlocalcomputationalFig.2.1.Schematicdescriptionofacoarsegrid.00.10.20.30.40.50.60.70.80.910.10.20.30.40.50.60.70.80.9KbilinearFig.2.2.Exampleofbasisfunctions.Left:basisfunctionwithKbeingacoarseelement.Right:basisfunctionwithKbeingRVE(bilinearfunctiondemonstratesonlytheboundaryconditionsonRVEandisnotapartofthebasisfunction).tothecoarse-gridblock,ifneeded(seelaterdiscussionsandSection2.6).Computationalregionssmallerthanthecoarse-gridblockareusedifonecanusesmallerregionstocharacterizethelocalhet