INTEGRAL FORMULATION OF THE SECOND KIND FOR MULTISUBDOMAIN SCATTERING X - PDF document

INTEGRALFORMULATIONOFTHESECONDKINDFORMULTI-SUBDOMAINSCATTERINGX.Claeys ;;Universit´edeToulouse,ISAE,FrancejointprojectwiththeSAM,ETHZ¨urich,SwitzerlandEmail:xavier.claeys@isae.frTalkAbstractWestudythescatteringofanacousticwavebyanob-jectmadeofseveraladjacentsub-domainsassociatedtodifferentmaterialcharacteristics.Forthisproblemwede-riveanintegralformulationofthesecondkind.Thisfor-mulationonlyinvolvesoneDirichletdatumandoneNeu-manndatumateachpointofeachinterfaceofthediffract-ingobject,sothatourformulationcanbeconsideredtobelongtothesamefamilyastheformulationoftherstkindthatwasderivedbyvonPetersdorffin[2]forscalarproblemsandbyBuffain[1]forMaxwell'sequations.IntroductionWhensimulatingwavepropagationinamediumwithacomplexgeometrybymeansofboundaryelementdis-cretizationwithaniterativesolver,acrucialissueistokeepthenumberofiterationsofthesolveraslowaspos-sible.Fromthispointofview,oneinterestingapproachconsistsinusingadirectintegralequationalongwithapreconditioningstrategy.Forexample,onecanresortontheCalderonpreconditionerintroducedin[6].However,therestillremaintransmissionproblemswithsuchgeometricalcongurationthatnoCalderonprecon-ditionerisavailablesofar.Forthistypeofproblem,Steinbachandco-workers(see[4],[7]andreferencestherein)developeddomaindecompositionmethodsthatcanreadilybepreconditioned.Howeverthisapproachre-quirestheinversionoflocalSteklov-Poincareoperators,anditreliesonadoublingofsomeunknowns,asapartoftransmissionconditionsisenforcedbymeansofLa-grangemultipliers.Morerecently,HiptmairandJerez-Hanckesdevelopedanewintegralformulationoftherstkind(see[8]),thatalsoreliesondoublingpartoftheun-knowns,andthathasgoodpropertiesintermsofprecon-ditioningpossibilities.Inthistalkwepresentyetanotheralternative,thatcon-sistsinusinganinherentlywellconditionedboundaryin-tegralequation.Wepresentthederivationofawellcon-ditionedintegralformulationofthesecondkind.Wecallitasingletraceformulation,meaningthatitdoesnotre-quirethedoublingofanyunknown.TheproblemthatweanalyzeFord=2or3,considerapartitionRd=[ni=0 \niwhere[ni=1 \niisboundedandeach\niisaconnectedsub-domainwithpiecewisesmooth,Lipstick'sboundary.Wealsoset=[ni=1@\ni.Notethattheremayexistpointswherethreeormoresub-domainswouldbead-jacent,whichispreciselythesituationthatwewishtotackle,seegurebelow. i \n0=exteriordomain\n1\n2\n3Fig:1Typicalmulti-subdomaingeometryWewishtostudywavepropagationinmediawithsuchageometricalconguration.Asamodelproblemwecon-sidertheequationsFindu2H1loc(
;Rd)suchthat
u+2ju=0in\nj;j=0:::nuuincoutgoingradiating(1)wherej2R+;j=0:::nanduinc2H1loc(Rd)issomeknownfunctionsatisfying
uinc+20uinc=0inRd.Be-sidesH1loc(
;Rd)=fv2H1loc(Rd)j
v2L2loc(Rd)gInProblem(1),transmissionconditionsaretakenintoac-countbymeansoftheconditionu2H1loc(
;Rd).Theconditionu2H1loc(Rd)impliesthatuhasnoDirich-letjumpacrossanyinterface,andthecondition
u2L2loc(Rd)impliesthatudoesnotadmitanyNeumannjumpanywhere.Problem(1)isclassicallywellposed.Inthistalk,wepresentanintegralformulationofthesecondkindforthisproblem. ReformulationoftransmissionconditionsSincewewishtoderiveanintegralformulationwehavetointroducetraceoperatorsandspaces.Anaturalspaceforintegralformulationsof(1)isH()=hnj=0H1 2(@\nj)ihnj=0H1 2(@\nj)iThisspaceisequippedwiththenormkVk=(Pnj=0kvjk21=2;@\nj+kqjk21=2;@\nj)1=2,andthedualitypairingB(U;V)=nXj=0@\nj(ujqjvjpj)dforU;V2H()U=((uj);(pj))andV=((vj);(qj))Letnjrefertothenormalvectoron@\njdirectedtowardtheexteriorof\njandintroducethespacesX+1 2()=f(vj@\nj)j=0:::njv2H1(Rd)gX1 2()=f(nj
qj@\nj)j=0:::njq2H(div;Rd)gX()=X+1 2()X1 2()ThespaceX()H()isclosedforkk,asthecon-straintscharacterizingthissetareexpressedbymeansofcontinuouslinearfunctionals.ThespaceX+1=2()isex-actlythepolarsetofX1=2()withrespecttothenaturaldualitypairing:(qj)2X1=2()ifandonlyifnXj=0@\njqjvjd=08(vj)2X1 2():(2)Let\rjDand\rjNreferrespectivelytotheDirichletandNeu-manntraceoperatorstakenfromtheinteriorof\nji.e.\rjD(v)=vj@\njand\rjN(v)=nj
rvj@\njwheneverv2C1( \nj).Ifuissolutionto(1),thentransmissionconditionscanbeformulatedas(\rjDu)j=0:::n;(\rjNu)j=0:::n
2X():(3)Thus,wewilltakeintoaccountthetransmissioncondi-tionsbylookingforasolutioninthespaceX().ThisapproachwasdevelopedbyvonPetersdorffin[2]andBuffain[1]forthederivationofintegralformulationsoftherstkind.Notethatsuchawaytoimposetransmissioncondi-tionsisanotabledifferencecomparedtotheBETI-likeformulationsthathavebeenstudiedbySteinbachandco-workers(see[4]forexample)whereatleastpartofthetransmissionconditionsareenforcedbymeansofLa-grangemultipliers.ReformulationofCalderonidentitiesWhatisdifferentinourapproach,comparedto[1]and[2],isthewayweimposetheequations
u+2ju=0in\njj=0:::n.DenethesetofCauchydata,denotedC(),asthesetofelements((vj)j=0:::n;(qj)j=0:::n)ofH()suchthatthereexists'j2H1loc( \nj);j=0:::nsatisfying
'j+2j'j=0in\nj'0outgoingradiating\rjD('j)=vjand\rjN('j)=qj;j=0:::nFor2R+andx2Rdn,introducethesingleanddoublelayerpotentialsSLjfqg(x)=@\njG(xy)q(y)d(y)DLjfvg(x)=@\njnj(y)
rG(xy)v(y)d(y)whereG(x)referstotheoutgoingGreenkernelassoci-atedtotheoperator
+2.Letusconsiderthecontinu-ousoperatorA:H()!H()denedbyfUg(x)=nXj=0DLjjfujg(x)+SLjjfpjg(x)A=(\r0D;:::;\rnD;\r0N;:::;\rnN)NotethatfUg(x)makessenseforanyx2Rdn.ThisoperatorcanbeusedtoreformulateProblem(1),atleastinthecaseofsmallcontrastscatterers.Proposition1Forany02R+,thereexists�0suchthat,ifmaxj=0:::njj0j,thenU2C()ifandonlyifAU=UInthesequel,weassumethat0;:::narecloseenoughtoguarantythatAU=UisequivalenttoU2C()Weshallrefertothisassumptionasa”smallcontrasthypothesis”.Thisassumptionisintroduceinordertodiscardanyspuriousmodephenomenon.LetUinc=(\r0Duinc;0;:::;0;\r0Nuinc;0;:::;0).Weprovethefollowingresult.Theorem1Assumethatthesmallcontrasthypothesisissatised.LetYbeanyclosedcomplementofX()inH()i.e.X()Y=H().Thefunctionu2H1loc(
;Rd)issolutiontoproblem(1)ifandonlyifU=((\rjDu)j=0:::n;(\rjNu)j=0:::n)2X()andB((IdA)U;V)=B((IdA)Uinc;V);8V2Y:(4) MoreoverFormulation(4)takestheform”iden-tity+compact”sothatitisindeedaformulationofthesecondkind.Theorem2ThereexistsacompactoperatorAK:H()!H()suchthatIm(AK)X()andAKU=AUwheneverU2X()Itcanbecheckedbyathoroughderivationthat,intheclassicalcasewherethereisonlytwosubdomainsandoneinterface,Formulation(4)yieldsthealreadywellknownsecondkindintegralformulation.NumericalresultsFornumericalexperiments,wesolveProblem(1)withslightlymoregeneraltransmissionconditions.Assumethat0;1:::n2(0;+1)aregiven.Insteadofim-posing(3),weimpose(\rjDu)j=0:::n;(1j\rjNu)j=0:::n
2X():(5)Insteadof(4),inthismoregeneralsituation,weconsiderthefollowingcontinuousformulationFindU2X()suchthat8V2YB((IdA)U;V)=B((IdA)Uinc;V):(6)whereAhasbeenreplacedbytheoperatorAdenedinaccordancewithConditions(5),A=A
TT(uj);(pj)
=(uj);(j
pj)
:NotethatProblem(4)isaparticularcaseofProblem(6)where0=1=2.WesolveFormulation(6)forthe2-Dscatteringofanincidentplanewaveuinc(x;y)=exp(i0x)byaninhomogeneousdiscsplitintwoparts,seeFigure2below. \n0\n1\n2\n1[\n2= D(0;1)0=11=2;2=3:Fig:2GeometryusedfornumericalresultsForthediscretizationofFormulation(6),werelyonaGalerkinapproach.Weconsiderapaneling=[Qq=1qwhereeachjisasegment.DeneV()=nj=0V(@\nj)whereV(@\nj)=fv2C0(@\nj)jvjqh2P1ifq@\nj;q=1:::QgTakeX+=V()\X1=2()asanapproximationspaceforX1=2().FortheapproximationofX1=2(),webaseourconstructiononamimicofrelation(2):wedeneXasthesetof(qj)j=0:::n2V()suchthatnXj=0 @\njqj vjd=08(vj)2X+:wheremeansthatthequadratureisachievedusingmasslumping.ForthediscretecounterpartsofX()andY,wesetX=X+XandY=XX+:ThankstothisparticularchoiceforthespaceY,thetermB(U;V)forU2XandV2Yisassociatedtoasymmetricmatrix.ThediscreteformulationthatwesolvewritesFindU(2)2Xsuchthat8V2YwehaveB((IdA)U(2);V)=B((IdA)Uinc;V):(7)WecomparethesolutionU(2)providedby(7)tothesolutionU(1)obtainedbymeansoftherstkindformu-lationderivedfromRumsey'sprinciple,seeEquation(27)in[2].Inallthenumericalresultsbelowwetook0=1;1=2and2=3 10-3 10-2 10-1 10-5 10-4 10-3 10-2 10-1 m0 = m1 = m2 = 1 m0 = 1, m1 = 1/4, m2 = 4 RelativeerrorFig:3RelativeerrorkU(1)U(2)k=kU(1)kversusstepofthemeshh ThegureaboveshowsthatthatkU(1)U(2)k=O(h)Moreover,sincethemeshweusedismadeofsegments,therstkindformulationprovidesanapproximationnotsharperthanO(h).Thisleadstotheconclusionthat,inthepresentcontext,Formulation(7)convergeswithquasi-optimalrate.Ontheotherhand,thesecondkindformulationismuchbetterconditioned,asitappearsinFig.4below. 10-3 10-2 10-1 101 102 103 104 105 106 First kind formulation Second kind formulation Fig:4Conditioningversusstepofthemeshhforthecase0=1;1=1=2;2=2Whereastheconditionnumberfortherstkindformula-tionblowsupasO(h1),whichisstandard,theconditionnumberforthesecondkindformulationremainsstable. 0 20 40 60 80 100 10-10 10-8 10-6 10-4 10-2 100 First kind formulation Second kind formulation Fig:5NormoftheresidualforGMRES(withnorestart)versusnumberofiterationforthecase0=1;1=1=2;2=2withh=5
103Finally,inFig.5above,weseethatGMRESconvergesmuchfasterwiththesecondkindformulation.AcknowledgmentsTheauthortakesthisopportu-nitytothankR.HiptmairandA.Bendaliforfruitfuldiscussions,andtothanktheSeminarofAppliedMathematicsofETHZfornancialsupport.References[1]A.Buffa,”Remarksonthediscretizationofsomenoncoerciveoperatorwithapplicationstothehet-erogeneousMaxwellequations”,SIAMJ.Numer.Anal.,43(1):1-18,2005.[2]T.vonPetersdorff,”BoundaryIntegralEquationsforMixedDirichlet,NeumannandTransmissionProb-lems”,Math.Met.App.Sc.,11:185-213,1989.[3]J-C.N´ed´elec,AcousticandElectromagneticEqua-tions,Springer,2001.[4]U.LangerandO.Steinbach,”BoundaryElementTearingandInterconnectingMethods”,Computing71:205-228,2003[5]R.Kress,Linearintegralequations,vol.82ofAppliedMathematicalSciences.Springer-Verlag,1999.[6]S.ChristiansenandJ-C.N´ed´elec,”Aprecondition-nerfortheelectriceldintegralequationbasedonCalderonformulas”,SIAMJ.Numer.Anal.40(2002),no.3,1100-1135.[7]O.SteinbachandM.Windisch,”Stableboundaryel-ementdecompositionmethodsfortheHelmholtzequation”,Numer.Math.,publishedonline.[8]R.HiptmairandC.Jerez-Hanckes,”MultipletracesboundaryintegralformulationforHelmholtztrans-missionproblems”,ETH,ResearchReportoftheSeminarofAppliedMathematicsno.2010-35,2010.