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1、APDEAPPROACHTONUMERICALFRACTIONALDIFFUSIONRICARDOH.NOCHETTO,ENRIQUEOTAROLA,ANDABNERJ.SALGADOAbstract.Fractionaldi usionhasbecomeafundamentaltoolforthemodel-ingofmultiscaleandheterogeneousphenomena.However,duetoitsnonlocalnature,itsaccuratenumericalapproximationisdelicate.W
2、esurveyourre-searchprogramonthedesignandanalysisofecientsolutiontechniquesforproblemsinvolvingfractionalpowersofellipticoperators.Startingfromalo-calizationPDEresultfortheseoperators,wedeveloplocaltechniquesfortheirsolution:aprioriandaposteriorierroranalyses,adaptivityandm
3、ultilevelmethods.Weshowthe
exibilityofourapproachbyproposingandanalyzinglocalsolutiontechniquesforaspace-timefractionalparabolicequation.1.IntroductionDi usionisthetendencyofasubstancetoevenlyspreadintosurroundingspace,andisoneofthemostcommonphysicalprocesses.Theclassicalmo
4、delsofdi usionleadtolocalandthoroughlystudiedequations.However,inrecenttimes,ithasbecomeevidentthatmanyoftheassumptionsthatleadtothesemodelsarenotalwayssatisfactoryornotevenrealisticinpractice.Consequently,di erentmodelsofdi usionhavebeenproposed,fractionaldi usionbeingoneo
5、fthem.Capturingtheessentialbehavioroffractionaldi usionwiththesimplestandcrudestmodelsisofparamountimportanceinscienceandengineering.Thisal-lowsforunderstandingofphysicalapplicationswherelongrangeoranomalousdi usionisconsidered[1],complexphenomenainmechanics[5],biophysics[2
6、2],turbulence[31],imageprocessing[39],nonlocalelectrostatics[46], nance[50]andthecontrolofdevices[2,3].Tounderstandthebehavioroffractionaldi usion,computationalscienceisfundamental.Itisoneofthepillars,togetherwiththe-oryandexperiments,ofscienti cinquiry.Acarefullycraftedcom
7、putationalmodelcanreplaceaveryexpensiveorunrealizableexperimentalsetting,anditcangivenewinsightintothetheoreticaldevelopmentsofaspeci cdiscipline.TheanalysisarXiv:1508.04382v1[math.NA]18Aug2015ofsuchcomputationalschemesistherealmofnumericalanalysis,whicho ersarigorousmathem
8、aticaldescriptionoftheextenttowhichthecomputer'soutputapproximatestheprocessofinte
