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1、AdaptiveFiniteElementsandCollidingBlackHolesDouglasN.Arnold,ArupMukherjee,andLucPoulyAbstractAccordingtothetheoryofgeneralrelativity,therelativeaccelerationofmassesgeneratesgravitationalradiation.Althoughgravitationalradiationhasnotyetbeendetected,itisbelievedthatextremelyviolentcosmicevents,suchas
2、thecollisionofblackholes,shouldgenerategravitywavesofsucientamplitudetodetectonearth.ThemassiveLaserInterferometerGravitational-waveObservatory,orLIGO,isnowbeingconstructedtodetectgravitywaves.Consequentlythereisgreatinterestinthecomputersimulationofblackholecollisionsandsimilarevents,basedonthenu
3、mericalsolutionoftheEinstein eldequations.Inthisnoteweintroducethescienti c,mathe-matical,andcomputationalproblemsanddiscussthedevelopmentofacomputercodetosolvetheinitialdataproblemforcollidingblackholes,anonlinearellipticbound-aryvalueproblemposedinanunboundedthreedimensionaldomainwhichisakeystepi
4、nsolvingthefull eldequations.Thecodeisbasedon niteelements,adaptivemeshes,andamultigridsolutionprocess.Herewewillparticularlyemphasizethemathematicalandalgorithmicissuesarisinginthegenerationofadaptivetetrahedralmeshes.1IntroductionInEinstein'stheoryofgeneralrelativityspace-timeisrepresentedasafour
5、-dimensionalsemi-Riemannianmanifold.Thegeodesicsofthismanifoldarethepathsoffreelyfallingparticlesandgravityisthemanifestationofthecurvatureofspace-time.Asystemofnonlinearpartialdi erentialequations,theEinstein eldequations,specifytherelationshipbetweenthecurvatureofthespace-timemanifoldandthemass-e
6、nergyitcontains.Aconsequenceofthetheoryisthatwhenamassacceleratesitgivesrisetotinyripplesinthefabricofspacetime,calledgravitywaves.Moreprecisely,ifamassacceleratesindistantspace,gravitywaveswillpropogatefromthemassatthespeedoflight.Theywillbedetectableasslightvariationsinthelengthsofobjectsatthepoi
7、ntofobservation,thesevariationsmodulatingwithtimeanddi eringaccordingtothedirectioninwhichthelengthsaremeasured.Consequentlygravitywavesmayberegardedassignalswhichtransmitinformationaboutthedynamicsofdistan
