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TheWeightedRieszGalerkinMethodforEllipticBoundaryValueProblemsonUnboundedDomainsHaeSooOhy1,BongsooJangandYichungJouDept.ofMathematics,UniversityofNorthCarolinaatCharlotte,Charlotte,NC282230001Email:hso@uncc.edu,bsjang@math.uncc.eduKeyWords:MethodofAuxiliaryMapping,thepVersionoftheFiniteElementMethod,WeightedRieszGalerkinMethod,WeightedSobolevSpace,InfiniteElements.RecentlyBabuˇskaOhintroducedtheMethodofAuxiliaryMapping(MAM)whichefficientlyhandlesellipticboundaryvalueproblemscontainingsingularities.Inthispaper,theWeightedRieszGalerkinMethod(WRGM)isinvestigatedbyintroducingspecialweightfunctions.Togetherwiththismethod,MAMismodifiedtoyieldhighlyaccuratefiniteelementsolutionstogeneralellipticboundaryvalueproblemsontheexteriorofboundeddomainsatlowcost.1.INTRODUCTIONInthispaper,weintroduceanewapproachtodealwithageneralellipticboundaryvalueproblemoftheformnXi;j=1@@xj(aij(x)@u(x)@xi)+b(x)u(x)=f(x)onanunboundeddomainΩwhichisanexteriorofaboundeddomain.Overtheyears,relatedtotheFiniteElementMethod(FEM)([9],[29]),muchworkhasbeendoneonunboundeddomainproblems([1],[9],[11],[12],[14],[15],[22],[28]).Severalnumericalmethodsfortheseproblemsweresuggested.Thefollowingapproachesarethemosttypical:truncatingtheunboundedpartofthedomainandintroducinganartificialboundaryconditionontheresultingartificialboundary;couplingboundaryelementmethodwithFEM([19]);andusinginfiniteelements([6],[7],[8],[31]).ThebasicideaoftheseapproachesistodividethegivenunboundeddomainΩintotwoparts:theboundedpartΩc=fx2Ω:jxjcgandtheunboundedpartΩ1=ΩnΩc.In([16],[17],[18],[20],[22]),undertheassumptionthatf(x)=0onΩ1,artificialboundaryconditionsweresetupfortheartificialboundariesoftheremainingbounded1ThisresearchissupportedinpartbyNSFgrantINT9722699.12OH,JANGANDJOUdomainΩc.Thus,theseapproachesarenotapplicableunlessf(x)hascompactsupport.Moreover,thisisimpracticalifthesupportoff(x)isverylarge.In([6],[7],[8],[31]),Ω1ispartitionedintoafinitenumberofinfiniteelementsincorporatedwiththemeshesonΩc.Thenthespecialdecayshapefunctionsareconstructedforthoseinfiniteelements.Thus,theimplementationofthismethodinaFEMcodeleadstoanalterationofthestructureofthestandardFEMcode.Thenewmethodintroducedinthispaper,similartotheinfiniteelementapproach,doesnothavethesedifficulties.First,weintroduceanauxiliarymappingthatcantransformΩ1ontoaboundeddomainˆΩ1;theunitball.ThenweapplythestandardFEMtoΩc[ˆΩ1(likeonepointcompactificationofΩforFEM).ThenoveltyofourmethodisthatnoartificialboundariesarecreatedandnoalterationofanyexistingFEMcodeisrequiredforitsimplementation.Thenumericalexamplesshowthatourmethodiseffectiveinhandlingunboundeddomainproblems.Moreover,themethodyieldshighlyaccuratenumericalsolutionsatlowcost.Thispaperisorganizedasfollows:Insection2,acutoffweightfunction,whichmakestheweightedSobolevspacebeingtheusualoneontheboundedsubdomainΩc,isintroduced.Also,theWeightedRieszGalerkinMethod(WRGM)isintroducedandtheexistenceofasolutionoftherelatedvariationalproblemisproved.Insection3,anauxiliarymappingisconstructedtodealwiththeunboundedsubdomainΩ1.ThenewapproachdealingwithunboundeddomainproblemsisdescribedintheframeworkofthepversionofFEM.Insection4,theefficiencyofthemethodistestedwithvarioustypesofellipticboundaryvalueproblemsontheexteriorboundeddomainsforthecaseswhensupportsoff(x)arenotbounded.Foraclearerpresentationofthemethod,proofsoftechnicallemmas,usedintheorydevelopmentandnumericalexperiments,wereplacedinappendix.Themethodisalsoapplicabletoelasticityproblemsonunboundeddomains([26]).2.THEWEIGHTEDRIESZGALERKINMETHODThroughoutthispaperΩRndenotesanunboundeddomainwhichistheexteriorofaclosedboundeddomainenclosedbyasimpleclosedcurveΓ(see,Fig.1).jxj=(x21+:::+x2n)12,x2Ω:Theng:Ω![0;1];isasmoothcutofffunctionsatisfyingthefollowingproperties:(i)g(x)=0onΩc=fx2Ω:jxjcg,(ii)g(x)=1onx2ΩnΩc+b=fΩ:jxjc+bg;b0:bisselectedsothatjrgjisassmallaspossible.Forconvenienceofcoding,theconstructionofaspecificcutofffunctionisgiveninappendix1.Definition2.1.LetthespaceHk(Ω;;g),withk0aninteger,betheBanachspaceofallfunctionsu(x)suchthatkuk2Hk(Ω;;g)=ZΩe2g(x)jxjX0jjk[Du(x)]2dx1;whereisthemultiindex,thatis,=(1;2;:::;n)2(N+0)n,N+0:=thesetofnonnegativeintegers.jj=1+2+:::+n;D=@jj(@x1)1:::(@xn)n.HereisELLIPTICBOUNDARYVALUEPROBLEMSONUNBOUNDEDDOMAINS3anonnegativerealnumber,whichis1andwillbedeterminedlater.For=0,onegetstheusualSobolevspace.Inparticular,H0(Ω;;g)=L2(Ω;;g):Itisconvenienttoemploytheoperatorsr,div,andΔ.Thesearedefinedbyru=(@u@x1;:::;@u@xn)T,whereTmeanstranspose,div(f)=Pni=1@fi@xi,wheref=(f1;:::;fn)T;andΔu=div(ru):2.1.WeightedResidueMethodLetusconsiderageneralsecondorderellipticboundaryvalueproblem,X@@xj(aij(x)@u(x)@xi)+b(x)u(x)=f(x)inΩ;(1)u(x)=0onΓ;(2)wheref2L2(Ω;;g);0b(x)forallx2Ω,andthecoefficientmatrixisbounded,symmetricandpositivedefiniteateachpointx2Ω:aij(x)=aji(x);(3)nXi;j=1aij(x)ijnXi=12i;(4)nXi;j=1aij(x)jinXi=12i1=2nXj=12j1=2;(5)forallntuplesofrealnumbers(1;:::;n);(1;:::;n):Heretheconstants0and0areindependentofx.Firstofall,weprovethattheellipticproblem(1)(2)