Adaptive Finite Elements and自适应有元

Adaptive Finite Elements and自适应有元

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时间:2019-06-24

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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

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