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AFREQUENCY{DOMAINMETHODFORFINITEELEMENTSOLUTIONSOFPARABOLICPROBLEMSCHANG-OCKLEE,JONGWOOLEE,ANDDONGWOOSHEENAbstract.Weintroduceandanalyzeafrequency-domainmethodforpar-abolicpartialdi erentialequations.Themethodisnaturallyparallelizable.AftertakingtheFouriertransformationofgivensourcesinthespace-timedo-mainintothespace-frequencydomain,weproposetosolveaninde nite,com-plexellipticproblemforeachfrequency.Fourierinversionwillthenrecoverthesolutioninthespace-timedomain.Existenceanduniquenessaswellaserrorestimatesaregiven.Fourierinvertibilityisalsoexamined.Numericalexperimentsarepresented.1.IntroductionLet beanopenboundedLipschitzdomaininRN,N=2;3,J=[0;1),and =@ .Weareinterestedinanumericalmethodforthefollowingparabolicproblem:1 ut r ( ru)=f; J;(1.1a)u=0; J;(1.1b)u(x;0)=0;x2 ;(1.1c)where 2L2( )and 2H1( )arepositivefunctionsofxde nedon ,whichsatisfy , ,jr j ,where ; ; ; arepositiveconstants.Inthispaper,insteadofsolvingProblem(1.1)inthespace-timedomain,weanalyzetheFourier-transformedproblemsofProblem(1.1),andproposeanat-urallyparallelizablealgorithmbysolvingtheFourier-transformedproblems.Recall rstthattheFouriertransformbv( ;!)ofafunctionv( ;t)intimeisde nedbybv( ;!)=Z1 1v( ;t)e i!tdtDate:September1,1997.1991MathematicsSubjectClassi cation.Primary65N30;Secondary35K20.Keywordsandphrases.parabolicproblems, niteelementmethods,parallelalgorithm,Four-iertransform.TheresearchwassupportedinpartbyKOSEF961-0106-039-2,GARCandBSRI-MOE-97.12CHANG-OCKLEE,JONGWOOLEE,ANDDONGWOOSHEENandtheFourierinversionformulaisgivenbyv( ;t)=12 Z1 1bv( ;!)ei!td!:Weextendfandubyzerotot0andtransformthespace-timeformulationoftheequations(1.1)toaspace-frequencyformulationbytakingtheFouriertransformof(1.1)intime.Wethenobtainasetofthefollowingellipticproblemsdependingon!:i!1 bu r ( rbu)=bf;x2 ;(1.2a)bu=0;x2 :(1.2b)Notethatifv(x;t)isarealfunction,itsFouriertransformsatis estheconjugaterelation:bv(x; !)=bv(x;!);!2R:Then,theFourierinversionformulatakestheformv(x;t)=1 ReZ10bv(x;!)ei!td!:(1.3)Here,weexplainbrie ythereasonwhyweconsiderProblem(1.2)insteadofProblem(1.1).Ourprimaryinterestliesinproposingandanalyzinganatur-allyparallelizablealgorithmwithwhichonemayuseparallelmachinestosolveProblem(1.1)ase cientlyaspossible.Themostpopularstrategytogetnumericalsolutionsof(1.1)istosolvetheprobleminthespace-timedomainbyusingamarchingalgorithmsuchasbackward-EulerorCrank-Nicolsonmethods.Suchmethodshaveproventosolvee ectivelymanypracticalproblems.Inordertoadvancetonexttimestepswhenoneusesamarchingalgorithm,oneneedstosolveellipticproblemsusinginformationsonspacegridsatthecurrentand/orprevioustimesteps.Therecouldbetwoapproachestoparallelizationinsolvingparabolicproblemsinawidesense.Onedirectionmaybetousetheideaofdomaindecomposi-tionmethodstodecomposethespacedomainintosubdomainsinsolvingeachellipticproblemcorrespondingtoeach xedtimestep.Forsuchadirection,werefer[4,5,14,15,18,19,25]andrecentpublicationsinmajornumericalana-lysisjournals.However,thesemethodsrequireheavycommunicationcostamongprocessorsinordertopassinformationsbetweenneighboringsubdomains.Theotherdirectionmaybeattemptstodeviseandimplementparallelalgorithmsinthemarchingaxes,thatis,thetimeaxis.However,thenatureofevolutionmakesitdi cultto ndanaturallyparallelizablealgorithmwhichdoesnotspendtoomuchcommunicationtimeamongprocessors.Inthissense,bothapproachesfor-mulatedinthespace-timedomainarenotnaturallyparallelizable.Inthispaper,weproposeanalternativemethodbytakingtheFouriertrans-formationofProblem(1.1)toobtainasetofellipticproblems(1.2)fordiscreteFREQUENCY{DOMAINMETHODFORPARABOLICPROBLEMS3numberoffrequency!’sofinterest.Thisformulationisbasedontheobservationthatthesetofellipticproblems(1.2)dependingontheparameter!iscompletelyindependent.Wearethusabletosolvethesetofellipticproblems(1.2),byassigningeachsuchproblemtoeachprocessor.Independenceofeachproblemguaranteesnocommunicationcostamongprocessors.Thenournumericalsolu-tionsineachtimeisrecoveredbyadiscreteinverseFouriertransform.Theaboveprocedurehasbeenproventobeverye cientforsolvingwavepropagationswithabsorbingboundaryconditionsinaparallelmachine[9,10].WaveequationsbecomesHelmholtz-typeequationsinthespace-frequencydo-main,whichhaveeigensolutionswithDirichletorNeumannboundaryconditions.Thisisnotthecasewithabsorbingboundaryconditions;withabsorbingboundaryconditionstheHelmholtz-typeequationsareuniquelysolvable,andthusanaturalparallelizationispossiblebysimultaneouslysolvingFourier-transformedproblemswithdi erent!’sindi erentprocessors;fordetails,see[9,10,12,20].Indeed,itturnsoutthatsolvingwavepropagationismoresubtlethansolvingparabolicproblems.ButitisworthytostressthatparabolicproblemswithDirichletandNeumannboundaryconditionscanbesolvedinanaturalparallelizableway.Seealso[21]forananalysisofalinearizedNavier-Stokesequations,whereasimilartreatmentfortheDirichletboundaryconditionhasbeendoneto(1.1)or(1.2).Thispaperisorganizedasfollows.Inx2,weshowthattheequation(1.2)hastheuniquesolutionbu( ;!)for!0,andregularityandstabilityresultsareprovedforsuchsolutions.I
本文标题:A frequency-domain method for finite element solut
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