Domain decomposition method and the Helmholtz prob

ISSN0249-6399apportderechercheINSTITUTNATIONALDERECHERCHEENINFORMATIQUEETENAUTOMATIQUEADomainDecompositionMethodfortheHelmholtzequationandrelatedOptimalControlProblemsJean-DavidBenamou,BrunoDesprŁsN2791Fevrier1996PROGRAMME6ADomainDecompositionMethodfortheHelmholtzequationandrelatedOptimalControlProblemsJean-DavidBenamou,BrunoDespresProgramme6|Calculscientique,modelisationetlogicielnumeriqueProjetIdentRapportderecherchen2791|Fevrier1996|30pagesAbstract:WepresentaniterativedomaindecompositionmethodtosolvetheHelmholtzequationandrelatedoptimalcontrolproblems.Theproofofconvergenceofthismethodreliesonenergytechniques.Thismethodleadstoecientalgorithmsforthenumericalres-olutionofharmonicwavepropagationproblemsinheterogeneousmediaandtheircontrol.Keywords:Domaindecomposition,Helmholtzequation,Harmonicwaves,Optimalcon-trol,Waveguide,AbsorbingboundaryconditionsAMS(MOS)subjectclassication:35J05,49M99,65N55(Resume:tsvp)INRIA,DomainedeVoluceau,B.P.10578153LeChesnaycedex,France,e-mail:[e-mail]’EnergieAtomique,94250VilleneuveSaintGeorgescedex,France,e-mail:[e-mail]éderechercheINRIARocquencourtDomainedeVoluceau,Rocquencourt,BP105,78153LECHESNAYCedex(France)Téléphone:(331)39635511˘Télécopie:(331)39635330Unemethodededecompositiondedomainepourl’equationdeHemlholtzetlesproblemesdecontr^oleoptimalassociesResume:Nouspresontonsunemethodededecompositiondedomaineiterativepourre-soudrel’equationdeHelmholtzetlesproblemesdecontr^olesoptimalquiluisontassocies.Lapreuvedelaconvergencedecettemethodeutilisedesestimationsd’energies.Cettemethodeproduitdesalgorithmesecacespourlaresolutionnumeriquedeproblemesdepropagationsd’ondesenregimeharmoniqueenmilieuinhomogenesetleurcontr^ole.Mots-Cles:DecompositiondeDomaine,EquationdeHemlholtz,OndesHarmoniques,Contr^oleOptimal,Guidesd’Ondes,ConditionsAbsorbantesDDMforHelmholtzandrelatedOptimalControlProblems31IntroductionThenumericalresolutionoftheHelmholtzequationandrelatedoptimalcontrolprobleminheterogeneousmediaathighwavenumberisachallengingproblem.Thismodelndsapplicationsinelectromagneticandacousticwavepropagation.Becauseoftheoscillatorycharacterofthesolution,thenecessarynenessofdiscretization(atleasttenpointsperwavelength,alsoknownas’ruleofthumb’)leadsathighwavenumberstoverylargenonhermitian(possiblynon-symmetric)complexlinearsystemsexceedingthecurrentcomputermemorycapabilities.Itthereforerulesoutdirectmethodstosolvethedirectproblem.Itmakestheresolutionofoptimalcontrolproblemsforwhichgradienttypemethodsneedtosolveasequenceofsuchdirectequationandsimilaradjointequationsevenharder.Integralmethods[13][37][30]orctitiousdomainmethods[1][7]donotapplytohet-erogeneousmedia.IterativemethodssuchaspreconditionedGMRES[36]orbi-conjugategradient[12]havehieraticnumericalconvergenceandthereisactuallynotheoreticalproofofconvergence.Adomaindecompositionfornon-symmetricandindeniteproblemsisalsodiscussedin[9][10].Forveryhighfrequency,asymptoticmethodsisanecientsolution[8]butstillofdelicateuse.ThereisneverthelessaneedfortheresolutionoftheHelmholtzequationundertheirrangeofvalidity.Thepresentdomaindecompositionmethodisageneralandecientsolution.ItmoreoverextendstothenumericalresolutionofoptimalcontrolproblemsforsystemsgovernedbytheHelmholtzequation.SolvingsuchaproblemclassicallyrequiretoiteratetheresolutionofdirectandadjointHelmholtzproblemsinordertocomputedescentdirectionsforagradient-typemethod.Theideaistosplitthedomainintosmallersubdomainsandsolveasequenceofsimilarsubproblemsonthesesubdomains.Theboundaryconditionsareadjustediterativelybyad-hoctransmissionconditionsbetweenadjacentsubdomains.Thenumberandsizeofsubdomainscannowbechosensoastoenabledirectmethodstosolvethesubproblems.Inthecaseofoptimalcontrol,wedecomposethecoupledsystemmadeofthedirectandadjointHelmholtzequationandtheoptimalityconditionwhichvariationallyexpressthatthecontrolisoptimal.Thismethodactuallysolvesatthesametimetheequationsandtheoptimizationproblemwhereasclassicalmethodsrequiretoiteratetheresolutionofdirectandadjointproblemsinordertocomputedescentdirectionsforagradient-typemethod.Themethodiseasilyimplementedandnaturallyadaptedtoparallelcomputers,whichuseisamajortrendinmodernscienticcomputing.Theaimofthispaperistogiveinauniedframeworkaformalpresentationofthealgorithmsandtheenergyestimatesthatleadtotheproofsofconvergence.Comprehensivemathematicalstudiescanbefoundin[17][16][15][3].Wefocusontheseenergyestimatessince:i)Theseestimatesarenotstandardinthecontextofellipticcoerciveproblems.ii)Theyhelptounderstandwhythealgorithmsconverge.iii)Slightmodicationsofthistechniquecanbeusedinvariouscasesofboundarycon-ditionsandequations.RRn27914Jean-DavidBenamou,BrunoDespresWewilluseacontinuousformulationoftheequationsthroughoutthepaper.Thedomaindecompositionmethodcanalsobeappliedtothediscretizedequationswithcorrespondingenergyestimatesandconvergenceproofs(seereferencesinsection6).Wepresentinsection2thedomaindecompositionmethodfortheresolutionoftheHelmholtzequation.Aproo