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 AmericanJournalofComputationalMathematics,2021,11,53-63 https://www.scirp.org/journal/ajcm ISSNOnline:2161-1211 ISSNPrint:2161-1203 ConvergenceandSuperconvergenceofFully DiscreteFiniteElementforTimeFractional OptimalControlProblems YuelongTang CollegeofScience,HunanUniversityofScienceandEngineering,Yongzhou,China Email:[email protected] Howtocitethispaper:Tang,Y.L.(2021) ConvergenceandSuperconvergenceofFully DiscreteFiniteElementforTimeFractional OptimalControlProblems.AmericanJour- nalofComputationalMathematics, 11, 53-63. https://doi.org/10.4236/ajcm.2021.111005 Received:February10,2021 Accepted:March22,2021 Published:March25,2021 Copyright c 2021byauthor(s)andScientific Research PublishingInc. This work is licensed under the Creative Com- monsAttributionInternationalLicense(CC BY4.0). http://creativecommons.org/licenses/by/4.0/ Loremipsumdolorsitamet,consectetuer adipiscingelit.Utpuruselit,vestibulumut, placeratac,adipiscingvitae,felis.Curabitur dictumgravidamauris.Namarculibero, nonummyeget,consectetuerid,vulputatea, magna.Donecvehiculaaugueeuneque.Pel- lentesquehabitantmorbitristiquesenectuset netusetmalesuadafamesacturpisegestas. Maurisutleo.Crasviverrametusrhoncus sem.Nullaetlectusvestibulumurnafringilla ultrices.Phaselluseutellussitamettortor gravida placerat.Integer sapienest,iaculis in, pretiumquis,viverraac,nunc.Praesenteget semvelleoultricesbibendum.Aeneanfau- cibus.Morbidolornulla,malesuadaeu,pulv- inarat, mollisac,nulla.Curabitur auctorsem- pernulla.Donecvariusorciegetrisus.Duis nibhmi,congueeu,accumsaneleifend,sagit- tisquis,diam.Duisegetorcisitametorci dignissimrutrum. 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Nuncadipiscingrutrumante.Nuncipsum massa,faucibussitamet,viverravel,elemen- tum semper, orci.Cras eros sem, vulputate et, tinciduntid, ultriceseget, magna.Nullavarius ornareodio.Donecaccumsanmaurissit amet augue.Sedligulalacus,laoreetnon,aliquam sitamet,iaculistempor,lorem.Suspendisse eros.Namporta,leosedconguetempor,fe- lisestultriceseros,idmattisvelitfelisnon metus.Curabiturvitaeelitnonmaurisvar- iuspretium.Aeneanlacussem,tinciduntut, consequatquis,portavitae,turpis.Nullam laoreetfermentumurna.Proiniaculis lectus. Sedmattis,eratsitametgravidamalesuada, elitaugueegestasdiam,tempusscelerisque nuncnislvitaelibero.Sedconsequatfeugiat massa.Nuncporta,erosineleifendvarius, eratleorutrumdui,nonconvallislectusor- ciutnibh.Sedloremmassa,nonummyquis, egestas id,condimentum at,nisl.Maecenas at nibh.Aliquametaugueatnuncpellentesque ullamcorper.Duisnislnibh,laoreetsuscipit, convallisut,rutrumid,enim.Phasellusodio. 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Etiamsitameterat.Nullavarius.Etiamt- inciduntduivitaeturpis.Donecleo.Morbi vulputateconvallisest.Integeraliquet.Pel- lentesque aliquetsodalesurna. Nullameleifendjusto innisl.Inhachabitasse plateadictumst.Morbinonummy.Aliquam utfelis.Invelitleo,dictumvitae,posuereid, vulputatenec,ante.Maecenasvitaepedenec dui dignissim suscipit.Morbimagna.Vestibu- lumidpurusegetvelitlaoreetlaoreet.Prae- sentsedleovelnibhconvallisblandit.Ut rutrum.Donecnibh.Donecinterdum.Fusce sed pede sit amet elitrhoncus ultrices.Nullam atenimvitaepedevehiculaiaculis. Classaptenttacitisociosquadlitoratorquen- tperconubianostra,perinceptoshymenaeos. Aeneannonummyturpisidodio.Integereu- ismodimperdietturpis.Utnecleonecdi- amimperdietlacinia.Etiamegetlacuseget miultriciesposuere.Inplacerattristiquetor- tor.Sedportavestibulummetus.Nullaia- culissollicitudinpede.Fusceluctustellusin dolor.Curabiturauctorvelitasem.Mor- bisapien.Classaptenttacitisociosquadl- itoratorquentperconubianostra,perincep- toshymenaeos.Donecadipiscingurnavehic- ulanunc.Sedornareleoinleo.Inrhoncus leoutdui.Aeneandolorquam,volutpatnec, fringillaid,consectetuervel,pede. Nullamalesuadarisusuturna.Aenean pretiumvelitsitametmetus.Duisiaculis.In hachabitasseplateadictumst.Nullammo- lestieturpisegetnisl.Duisamassaidpede dapibusultricies.Sedeuleo.Inatmaurissit amettortorbibendumvarius.Phasellusjusto risus,posuerein,sagittisac,variusvel,tor- tor.Quisqueidenim.Phasellusconsequat, liberopretiumnonummyfringilla,tortor lacus vestibulumnunc,utrhoncusligulanequeid justo.Nullamaccumsaneuismodnunc.Proin vitae ipsumacmetus dictumtempus.Nam ut wisi.Quisque tortor felis, interdum ac, sodales a,sempera,sem.Curabiturinvelitsitamet duitristiquesodales.Vivamusmaurispede, laciniaeget,pellentesquequis,scelerisqueeu, est.Aliquamrisus.Quisquebibendumpede eudolor. Donec tempusneque vitaeest.Aenean egestas odio sed risus ullamcorper ullamcorper.Sed in nullaatortortinciduntegestas.Namsapien tortor,elementumsitamet,aliquamin,port- titorfaucibus,enim.Nullamconguesuscipit nibh.Quisqueconvallis.Praesentarcunibh, vehicula eget,accumsaneu,tincidunta,nibh. Suspendissevulputate,tortorquisadipiscing viverra, lacus nibh dignissim tellus, eu suscipit risusantefringilladiam.Quisquealiberov- elpedeimperdietaliquet.Pellentesquenunc nibh,eleifenda,consequatconsequat,hen- dreritnec,diam.Sedurna.Maecenaslaoreet eleifendneque.Vivamuspurusodio,eleifend non,iaculisa,ultricessitamet,urna.Mau- risfaucibusodiovitaerisus.Innisl.Praesent purus.Integeriaculis,semeuegestaslacinia, lacuspedescelerisqueaugue,inullamcorper dolorerosaclacus.Nuncinlibero. Fuscesuscipitcursussem.Vivamusrisusmi, egestasac,imperdietvarius,faucibusquis, leo.Aeneantincidunt.Donecsuscipit.Cras idjustoquisnibhscelerisquedignissim.Ali- quamsagittiselementumdolor.Aeneancon- sectetuerjustoinpede.Curabiturullamcorper ligula nec orci.Aliquam purus turpis,aliquam id, ornarevitae,porttitor non,wisi.Maecenas luctusportalorem.Donecvitaeligulaeu ante pretiumvarius.Prointortormetus,convallis et,hendreritnon,scelerisquein,urna.Cras quisliberoeuligulabibendumtempor.Viva- mustellusquam,malesuadaeu,tempussed, temporsed, velit.Doneclaciniaauctor libero. Praesentsednequeidpedemollisrutrum. Vestibulumiaculisrisus.Pellentesquelacus. Ut quisnunc sedodio malesuada egestas.Duis amagnasitametligulatristiquepretium.Ut pharetra.Vestibulumimperdietmagnanec wisi.Maurisconvallis.Sedaccumsansollic- itudinmassa.Sedidenim.Nuncpedeenim, laciniaut,pulvinarquis,suscipitsemper,elit. Crasaccumsaneratvitaeenim.Crassollici- tudin.Vestibulumrutrumblanditmassa. Sedgravidalectusutpurus.Morbilaoreet magna.Pellentesque eu wisi.Proin turpis.In- tegersollicitudinauguenecdui.Fuscelectus. Vivamusfaucibusnullaneclacus.Integerdi- am.Pellentesquesodales,enimfeugiatcursus volutpat,semmaurisdignissimmauris,quis consequatsemestfermentumligula.Nullam justo lectus, condimentum sit amet, posuere a, fringillamollis,felis.Morbinulla nibh,pellen- tesqueat,nonummyeu,sollicitudinnec,ip- sum.Crasneque.Nuncaugue.Nullamvitae quamidquampulvinar blandit.Nunc sitamet orci.Aliquameratelit,pharetranec,aliquet a,gravidain,mi.Quisqueurnaenim,viverra quis,suscipitquis,tinciduntut,sapien.Cras placeratconsequatsem.Curabituracdiam. Curabiturdiamtortor,molliset,viverraac, tempusvel,metus. Curabituraclorem.Vivamusnonjustoin duimattisposuere.Etiamaccumsanligula idpede.Maecenastinciduntdiamnecvelit. Praesentconvallissapienacest.Aliquamul- lamcorpereuismodnulla.Integermollisenim veltortor.Nullasodalesplaceratnunc.Sed tempusrutrumwisi.Duisaccumsangravi- dapurus.Nuncnunc.Etiamfacilisisduieu sem.Vestibulumsemper.Praesenteueros. Vestibulumtellusnisl,dapibusid,vestibulum sit amet,placerat ac,mauris.Maecenaset elit uteratplaceratdictum.Namfeugiat,turpis etsodalesvolutpat,wisiquamrhoncusneque, vitaealiquamipsumsapienvelenim.Maece- nassuscipitcursusmi. Quisque consectetuer.In suscipit mauris a do- lor pellentesque consectetuer.Mauris convallis nequenonerat.Inlacinia.Pellentesqueleo eros,sagittisquis,fermentumquis,tincidun- tut,sapien.Maecenassem.Curabitureros odio,interdumeu,feugiateu,portaac,nis- l.Curabiturnunc.Etiam fermentum convallis velit.Pellentesquelaoreetlacus.Quisquesed elit.Namquistellus.Aliquamtellusarcu, adipiscingnon,tincidunteleifend,adipiscing quis,augue.Vivamuselementumplaceraten- im.Suspendisseuttortor.Integerfaucibus adipiscingfelis.Aeneanconsectetuermattis lectus.Morbimalesuadafaucibusdolor.Nam lacus.Etiamarcu libero,malesuadavitae,ali- quamvitae,blandittristique,nisl. Maecenasaccumsandapibussapien.Duis pretiumiaculisarcu.Curabiturutlacus.Ali- quamvulputate.Suspendisse utpurussed sem tempor rhoncus.Utquam dui, fringillaat, dic- tum eget,ultricies quis,quam.Etiam sem est, pharetra non, vulputate in, pretium at, ipsum. Nunc semper sagittis orci.Sedscelerisque sus- cipitdiam.Utvolutpat,doloratullamcorp- ertristique,erospurusmollisquam,sitamet ornareantenuncetenim. Phasellusfringilla,metusidfeugiatcon- sectetuer,lacuswisiultrices tellus,quis lobor- tisnibhloremquistortor.Donecegestas ornarenulla.Maurismitellus,portafau- cibus,dictumvel,nonummyin,est.Ali- quameratvolutpat.Intellusmagna,port- titorlacinia,molestievitae,pellentesqueeu, justo.Classaptenttacitisociosquadlitora torquentperconubianostra,perinceptoshy- menaeos.Sedorcinibh,scelerisquesitamet, suscipitsed,placeratvel,diam.Vestibulum nonummy vulputate orci.Donec et velit ac ar- cuinterdumsemper.Morbipedeorci,cursus ac,elementumnon,vehiculaut,lacus.Cras volutpat.Namvelwisiquisliberovenenatis placerat.Aeneansedodio.Quisqueposuere purusacorci.Vivamusodio.Vivamusvar- ius,nullasitametsemperviverra,odiomau- risconsequatlacus,atvestibulumnequearcu eutortor.Doneciaculistincidunttellus.Ali- quameratvolutpat.Curabiturmagnalorem, dignissimvolutpat,viverraet,adipiscingnec, dolor.Praesentlacusmauris,dapibusvitae, sollicitudinsitamet,nonummyeget,ligula. Crasegestasipsuma nisl.Vivamusvariusdo- lorutdolor.Fuscevelenim.Pellentesque accumsanligulaeteros.Crasidlacusnon tortorfacilisisfacilisis.Etiamnislelit,cur- sussed,fringillain,conguenec,urna.Cum sociisnatoquepenatibusetmagnisdispar- turientmontes,nasceturridiculusmus.In- tegeratturpis.Cumsociisnatoquepenati- busetmagnisdisparturientmontes,nascetur ridiculusmus.Duisfringilla,ligulasedporta fringilla,ligulawisicommodofelis,utadipisc- ingfelisduiinenim.Suspendissemalesuada ultricesante.Pellentesquescelerisqueaugue sitameturna.Nullavolutpataliquettortor. Crasaliquam,tellusataliquetpellentesque, justosapiencommodoleo,idrhoncussapien quamaterat.Nullacommodo,wisiegetsol- licitudinpretium,orciorcialiquamorci,ut cursusturpisjustoetlacus.Nullaveltor- tor.Quisque eratelit, viverrasit amet, sagittis eget,portasitamet,lacus. In hac habitasse platea dictumst.Proin at est. Curabitur tempusvulputateelit.Pellentesque sem.Praesent eusapien.Duiselitmagna,ali- quet at,tempus sed, vehicula non, enim.Mor- biviverraarcunecpurus.Vivamusfringilla, enimetcommodomalesuada,tortormetusel- ementumligula,necaliquetestsapienutlec- tus.Aliquammi.Utnecelit.Fusceeuismod luctustellus.Curabiturscelerisque.Nullam purus.Namultriciesaccumsanmagna.Morbi pulvinarloremsitametipsum.Donecutjus- tovitaenibhmolliscongue.Fuscequisdiam. Praesent tempuserosutquam. Donecinnisl.Fuscevitaeest.Vivamusante ante,mattislaoreet,posuereeget,conguevel, nunc.Fuscesem.Namvelorcieuerosviver- raluctus.Pellentesquesitametaugue.Nunc sitametipsumetlacusvariusnonummy.In- teger rutrum sem egetwisi.Aenean eusapien. Quisqueornaredignissimmi.Duisaurnavel risus pharetraimperdiet.Suspendissepotenti. Morbijusto.Aeneannecdolor.Inhac habitasseplateadictumst.Proinnonummy porttitorvelit.Sedsitametleonecmetus rhoncusvarius.Crasante.Vestibulumcom- modosemtinciduntmassa.Namjusto.Ae- neanluctus,felisetcondimentumlacinia,lec- tusenimpulvinarpurus,nonportavelitnisl sed eros.Suspendisseconsequat.Mauris adui et tortormattispretium.Sed nulla metus,vo- lutpatid, aliquameget, ullamcorperut, ipsum. Morbieununc.Praesentpretium.Duisali- quampulvinarligula.Ut blanditegestasjusto. Quisqueposueremetusviverrapede. Vivamussodaleselementumneque.Viva- musdignissimaccumsanneque.Sedatenim. Vestibulumnonummyinterdumpurus.Mau- ris ornarevelitid nibh pretiumultricies.Fusce temporpellentesqueodio.Vivamusauguepu- rus,laoreetin,scelerisquevel,commodoid, wisi.Duisenim.Nullainterdum,nunceusem- pereleifend,enimdolorpretiumelit,utcom- modoligulanislaest.Vivamusante.Nulla leomassa,posuerenec,volutpatvitae,rhon- cuseu,magna. Quisquefacilisisauctorsapien.Pellentesque gravidahendreritlectus.Maurisrutrumso- dalessapien.Fuscehendreritsemvellorem. Integerpellentesquemassavelaugue.Integer elit tortor, feugiat quis, sagittis et, ornare non, lacus.Vestibulumposuerepellentesqueeros. Quisquevenenatisipsumdictumnulla.Ali- quam quis quam nonmetus eleifend interdum. Nam eget sapien ac mauris malesuadaadipisc- ing.Etiameleifendnequesedquam.Nulla facilisi.Proinaligula.Sedidduieunibh egestastincidunt.Suspendissearcu. Maecenasdui.Aliquamvolutpatauctorlorem. Crasplaceratestvitaelectus.Curabitur massalectus,rutrumeuismod,dignissimut, dapibus a, odio.Ut eros erat, vulputate ut, in- terdumnon,portaeu,erat.Crasfermentum, felis inportacongue,velit leo facilisis odio,vi- taeconsectetuerloremquamvitaeorci.Sed ultrices,pedeeuplaceratauctor,anteligula rutrum tellus,vel posuere nibhlacus nec nibh. Maecenaslaoreetdoloratenim.Donecmo- lestie dolornec metus.Vestibulum libero.Sed quiserat.Sedtristique.Duispedeleo,fer- mentumquis,consectetuereget,vulputatesit amet,erat. Donecvitaevelit.Suspendisseportafermen- tummauris.Utvelnuncnonmaurisphare- travarius.Duisconsequatliberoquisurna. Maecenasatante.Vivamusvarius,wisised egestastristique,odiowisi luctusnulla,lobor- tisdictumdolorligulainlacus.Vivamusali- quam, urna sed interdum porttitor,metus orci interdum odio,sitameteuismod lectusfeliset leo.Praesentacwisi.Namsuscipitvestibu- lum sem.Praesent eu ipsumvitae pedecursus venenatis.Duis sed odio.Vestibulum eleifend. Nullautmassa.Proinrutrummattissapien. Curabiturdictumgravidaante. Phasellusplaceratvulputatequam.Maece- nasattellus.Pellentesquenequediam,dig- nissim ac, venenatisvitae, consequat ut, lacus. Nam nibh.Vestibulum fringilla arcu mollis ar- cu.Sedetturpis.Donecsemtellus,volut- pat et,variuseu,commodo sed,lectus.Lorem ipsumdolorsitamet,consectetueradipiscing elit.Quisqueenimarcu,suscipitnec,tempus at,imperdietvel,metus.Morbivolutpatpu- rusaterat.Donecdignissim,semidsemper tempus, nibh massa eleifendturpis,sed pellen- tesquewisipurussedlibero.Nullamlobortis tortorvelrisus.Pellentesqueconsequatnulla eutellus.Donecvelit.Aliquamfermentum, wisiacrhoncusiaculis,tellusnuncmalesuada orci,quisvolutpatduimagnaidmi.Nuncvel ante.Duisvitaelacus.Crasnecipsum. Morbinunc.Aliquam consectetuervarius nul- la.Phaselluseros.Crasdapibusporttitorris- us.Maecenasultricesmiseddiam.Praesent gravida velit at elit vehicula porttitor.Phasel- lusnislmi,sagittis ac,pulvinar id,gravidasit amet,erat.Vestibulumest.Loremipsumdo- lorsitamet,consectetueradipiscingelit.Cur- abitur idsem elementum leo rutrumhendrerit. Utatmi.Donectinciduntfaucibusmassa. Sed turpis quam, sollicitudin a, hendrerit eget, pretiumut,nisl.Duishendreritligula.Nunc pulvinarcongueurna. Nuncvelit.Nullamelitsapien,eleifendeu, commodo nec, semper sit amet, elit.Nulla lec- tus risus,condimentum ut,laoreeteget,viver- ra nec, odio.Proin lobortis.Curabitur dictum arcuvelwisi.Crasidnullavenenatistortor congueultrices.Pellentesqueegetpede.Sed eleifendsagittiselit.Namsedtellussitamet lectusullamcorper tristique.Maurisenim sem, tristiqueeu,accumsanat,scelerisquevulpu- tate, neque.Quisquelacus.Donec etipsum sit amet elitnonummy aliquet.Sed viverranisl at sem.Namdiam.Maurisutdolor.Curabitur ornaretortorcursusvelit. Morbitinciduntposuerearcu.Crasvenenatis estvitaedolor.Vivamusscelerisquesemper mi.Donecipsumarcu,consequatscelerisque, viverraid,dictumat,metus.Loremipsum dolor sit amet, consectetuer adipiscing elit.Ut pede sem, tempus ut, porttitor bibendum, mo- lestieeu,elit.Suspendissepotenti.Sedidlec- tus sitamet purusfaucibus vehicula.Praesen- tsedsemnonduipharetrainterdum.Nam viverra ultricesmagna. Aeneanlaoreetaliquamorci.Nuncinterdum elementumurna.Quisqueerat.Nullamtem- porneque.Maecenasvelitnibh,scelerisquea, consequatut,viverrain,enim.Duismagna. Donecodioneque,tristiqueet,tincidunteu, rhoncusac,nunc.Mauris malesuadamalesua- da elit.Etiam lacus mauris, pretium vel, blan- ditin,ultriciesid,libero.Phasellusbibendum eratutdiam.Incongueimperdietlectus. Aeneanscelerisque.Fuscepretiumporttitor lorem.Inhachabitasseplatea dictumst.Nulla sitametnislatsapienegestaspretium.Nunc nontellus.Vivamusaliquet.Namadipiscing euismoddolor.Aliquameratvolutpat.Nul- lautipsum.Quisquetinciduntauctorau- gue.Nuncimperdietipsumegetelit.Ali- quamquamleo,consectetuernon,ornaresit amet,tristiquequis,felis.Vestibulum ante ip- sumprimisinfaucibusorciluctusetultrices posuerecubiliaCurae;Pellentesqueinterdum quamsitametmi.Pellentesquemaurisdui, dictuma,adipiscingac,fermentumsitamet, lorem. Utquiswisi.Praesentquismassa.Vivamus egestasrisusegetlacus.Nunctincidunt,risus quisbibendumfacilisis,lorempurusrutrum neque,necportatortorurnaquisorci.Ae- neanaliquet,liberosempervolutpatluctus, pedeeratlaciniaaugue,quisrutrumsemip- sumsitametpede.Vestibulumaliquet,nibh sediaculissagittis,odiodolorblanditaugue, egetmollisurnatellusidtellus.Aeneanali- quetaliquamnunc.Nullaultriciesjustoeget orci.Phasellustristiquefermentumleo.Sed massametus,sagittisut,semperut,pharetra vel,erat.Aliquamquamturpis,egestasvel, elementumin,egestassitamet,lorem.Duis convallis,wisisitametmollismolestie,libero mauris portadui,vitae aliquamarcuturpis ac sem.Aliquamaliquetdapibusmetus. Vivamuscommodoeroseleifenddui.Vestibu- luminleoeuerattristiquemattis.Crasat elit.Craspellentesque.Nullamidlacussit ametliberoaliquethendrerit.Proinplacerat, minonelementumlaoreet,eroselittincidun- tmagna,arhoncussemarcuidodio.Nul- laegetleoaleoegestasfacilisis.Curabitur quisvelit.Phasellusaliquam,tortornecornare rhoncus,purusurnaposuerevelit,etcommo- dorisustellusquistellus.Vivamusleoturpis, tempussitamet,tristiquevitae,laoreetquis, odio.Proinscelerisquebibendumipsum.Eti- amnisl.Praesentveldolor.Pellentesquevel magna.Curabitururna.Vivamuscongueur- na in velit.Etiam ullamcorper elementum dui. Praesentnon urna.Sed placeratquam non mi. Pellentesquediammagna,ultricieseget, ultri- cesplacerat,adipiscingrutrum,sem. Morbisem.Nullafacilisi.Vestibulumante ipsumprimisinfaucibusorciluctusetultri- cesposuerecubiliaCurae;Nullafacilisi.Mor- bisagittisultriceslibero.Praesenteuligula sedsapienauctorsagittis.Classaptenttaciti sociosquadlitoratorquentperconubianos- tra,perinceptoshymenaeos.Donecvelnunc. Nuncfermentum,lacusidaliquamporta,dui tortoreuismoderos,velmolestieipsumpurus eulacus.Vivamuspede arcu,euismod ac,tem- pusid,pretiumet,lacus.Curabitursodales dapibusurna.Nunceusapien.Doneceget nuncapededictumpretium.Proinmauris. Vivamusluctusliberovelnibh. Fuscetristiquerisusidwisi.Integermolestie massaidsem.Vestibulumveldolor.Pel- lentesquevelurnavelrisusultricieselemen- tum.Quisquesapien urna,blanditnec, iaculis ac,viverrain,odio.Inhachabitasseplatea dictumst.Morbinequelacus,convallisvitae, commodoac,fermentumeu,velit.Sedinor- ci.Infringilla turpisnonarcu.Donecinante. Phasellustemporfeugiatvelit.Aeneanvar- iusmassanonturpis.Vestibulumanteipsum primisinfaucibusorciluctusetultricespo- suerecubiliaCurae; Aliquamtortor.Morbiipsummassa,imperdi- etnon,consectetuervel,feugiatvel,lorem. Quisqueeget loremnec elitmalesuadavestibu- lum.Quisquesollicitudinipsumvel sem.Nul- laenim.Proinnonummyfelis vitaefelis.Nul- lampellentesque.Duisrutrumfeugiatfe- lis.Maurisvelpedesedliberotinciduntmol- lis.Phasellussedurnarhoncusdiameuismod bibendum.Phasellussednisl.Integercondi- mentumjusto id orciiaculis varius.Quisque et lacus.Phaselluselementum, justoat dignissim auctor,wisi odiolobortisarcu,sedsollicitudin felisfeliseuneque.Praesentatlacus. Vivamussitametpede.Duisinterdum,nunc egetrutrumdignissim,nisldiamluctusleo, ettinciduntvelitnislidtellus.Inloremtel- lus,aliquetvitae, portain,aliquetsed,lectus. Phasellussodales.Utvariusscelerisqueerat. Invelnibheuerosimperdietrutrum.Donec ac odio nec neque vulputatesuscipit.Nam nec magna.Pellentesquehabitantmorbitristique senectus et netus et malesuada fames acturpis egestas.Nullamporta,odioetsagittisiaculis, wisi neque fringilla sapien, vel commodo lorem loremidelit.Utsemlectus,scelerisqueeget, placeratet,tinciduntscelerisque,ligula.Pel- lentesque nonorci. Etiamvelipsum.Morbifacilisisvestibulum nisl.Praesentcursuslaoreetfelis.Integer adipiscingpretiumorci.Nullafacilisi.Quisque posuerebibendumpurus.Nullaquammauris, cursuseget,convallisac,molestienon,enim. Aliquamcongue.Quisquesagittisnonummy sapien.Proinmolestiesemvitaeurna.Mae- cenas lorem.Vivamus viverra consequat enim. Nuncsedpede.Praesentvitaelectus.Prae- sentnequejusto,vehiculaeget,interdumid, facilisiset,nibh.Phasellusatpurusetlibero laciniadictum.Fuscealiquet.Nullaeuante placeratleosemperdictum.Maurismetus. Curabiturlobortis.Curabitur sollicitudinhen- dreritnunc.Donecultriceslacusidipsum. Donecanibhutelitvestibulumtristique.In- tegeratpede.Crasvolutpatvariusmagna. Phaselluseuwisi.Praesentrisusjusto,lobor- tis eget, scelerisqueac, aliquet in,dolor.Proin idleo.Nunciaculis,mivitaeaccumsancom- modo, nequesemlacinianulla,quisvestibulum justosemineros.Quisquesedmassa.Morbi lectus ipsum, vulputate a, mollis ut,accumsan placerat,tellus.Nullaminwisi.Vivamuseu ligulaanuncaccumsancongue.Suspendisse aclibero.Aliquameratvolutpat.Donecau- gue.Nuncvenenatisfringillanibh.Fusceac- cumsan pulvinar justo.Nullam semper,dui ut dignissim auctor, orciliberofringilla massa, b- landitpulvinarpedetortoridmagna.Nunc adipiscing justo sed velit tincidunt fermentum. Integerplacerat.Pellentesquehabitantmorbi tristique senectuset netuset malesuadafames acturpisegestas.Sedinmassa.Classaptent tacitisociosquadlitoratorquentperconubi- anostra,perinceptoshymenaeos.Phasellus tempusaliquamrisus.Aliquamrutrumpurus atmetus.Donecposuereodioaterat.Nam nonnibh.Phasellusligula.Quisquevene- natislectusinaugue.Sedvestibulumdapibus neque. Mauris tempus eros at nulla.Sedquis dui dig- nissimmaurispretiumtincidunt.Maurisac purus.Phasellusaclibero.Etiamdapibusia- culisnunc.Inlectus wisi,elementumeu, sollic- itudinnec,imperdietquis,dui.Nullaviverra nequeaclibero.Maurisurnaleo,adipiscing eu,ultricesnon,blanditeu,dui.Maecenas duineque,suscipitsit amet,rutrum a,laoreet in,eros.Uteunibh.Fuscenecerattempus urnafringillatempus.Curabituridenim.Sed ante.Crassodalesenimsitametwisi.Nunc fermentum consequatquam. Utauctor,augueportadignissimvestibulum, arcudiamlobortisvelit,velscelerisqueris- usauguesagittisrisus.Maecenaseujusto. Pellentesquehabitant morbi tristiquesenectus etnetusetmalesuadafamesacturpiseges- tas.Mauriscongueligulaegettortor.Nul- lam laoreeturnasedenim.Donec egeterosut erosvolutpatconvallis.Praesentturpis.Inte- germaurisdiam,elementumquis,egestasac, rutrumvel,orci.Nullafacilisi.Quisqueadip- iscing,nullavitaeelementumporta,semurna volutpatleo,sedportaenimrisussedmassa. Integeracenimquisdiamsodalesluctus.Ut eget eros a ligula commodo ultricies.Donec eu urnaviverradolorhendreritfeugiat.Aliquam acorcivelerosconguepharetra.Quisque rhoncus,justoeu volutpatfaucibus,augueleo posuerelacus,arhoncuspuruspedevelest. Proinultricesenim. Aenean tincidunt laoreet dui.Vestibulumante ipsumprimisinfaucibusorciluctusetultri- cesposuerecubiliaCurae;Integeripsumlec- tus,fermentumac,malesuadain,eleifendut, lorem.Vivamusipsumturpis,elementumvel, hendrerit ut,semperat, metus.Vivamus sapi- entortor,eleifendid,dapibusin,egestaset, pede.Pellentesquefaucibus.Praesentlorem neque, dignissim in,facilisis nec, hendrerit vel, odio.Namatdiamacnequealiquetviver- ra.Morbidapibusligulasagittismagna.In lobortis.Donecaliquetultricieslibero.Nunc dictumvulputatepurus.Morbivarius.Lorem ipsumdolorsitamet,consectetueradipiscing elit.Intempor.Phaselluscommodoporttitor magna.Curabiturvehiculaodioveldolor. Praesentfacilisis,augueaadipiscingvene- natis,liberorisusmolestieodio,pulvinarcon- sectetuerfeliseratacmauris.Namvestibu- lumrhoncusquam.Sedveliturna,pharetra eu,eleifendeu,viverraat,wisi.Maecenasul- tricesnibhatturpis.Aeneanquam.Nulla ipsum.Aliquamposuereluctuserat.Cur- abiturmagnafelis,laciniaet,tristiqueid,ul- tricesut,mauris.Suspendissefeugiat.Cras eleifendwisivitaetortor.Phasellusleopurus, mattissitamet,auctorin,rutrumin,magna. Inhachabitasseplateadictumst.Phasellus imperdietmetusinsem.Vestibulumacenim nonsemultriciessagittis.Sedveldiam. Integervelenimsedturpisadipiscingbiben- dum.Vestibulumpededolor,laoreetnec,po- suerein,nonummyin,sem.Donecimperdiet sapienplaceraterat.Donecviverra.Aliquam eros.Nunc consequat massaid leo.Sed ullam- corper,loreminsodalesdapibus,risusmetus sagittislorem,nonporttitorpurus odioneco- dio.Sedtinciduntposuereelit.Quisqueeu enim.Donecliberorisus,feugiatac,dapibus eget, posuere a,felis.Quisque vellectusutme- tustincidunteleifend.Duis utpede.Duis velit erat,venenatis vitae,vulputate a,pharetrasit amet,est.Etiamfringillafaucibusaugue. Aeneanvelitsem,viverraeu,tempusid, rutrumid,mi.Nullamnecnibh.Proinullam- corper,dolorincursustristique,erosaugue tempornibh,atgravidadiamwisiatpurus. Donec mattis ullamcorper tellus.Phasellus vel nulla.Praesentinterdum,erosinsodalessol- licitudin,nunc nulla pulvinarjusto,aeuismod erossemnecnibh.Nullamsagittisdapibus lectus.Nullamegetipsumeutortorlobortis sodales.Etiampurusleo,pretiumnec,feu- giatnon,ullamcorpervel,nibh.Sedvelelit etquamaccumsanfacilisis.Nuncleo.Sus- pendissefaucibuslacus. Pellentesqueinterdumsapiensednulla.Proin tincidunt.Aliquamvolutpatestvelmassa. Seddolorlacus,imperdietnon,ornarenon, commodoeu,neque.Integerpretiumsem- perjusto.Proinrisus.Nullamidquam. Namneque.Duisvitaewisiullamcorperdi- amcongueultricies.Quisqueligula.Mauris vehicula. Curabiturnuncmagna,posuereeget,vene- natiseu,vehiculaac,velit.Aeneanornare, massaaaccumsanpulvinar,quamlorem laoreetpurus,eusodalesmagnarisusmolestie lorem.Nunceratvelit,hendreritquis,male- suadaut,aliquamvitae,wisi.Sedposuere. Suspendisseipsumarcu,scelerisquenec,ali- quameu,molestietincidunt,justo.Phasellus iaculis.Sedposuereloremnonipsum.Pel- lentesquedapibus.Suspendissequamlibero, laoreeta,tincidunteget,consequatat,est. Nullamutlectusnonenimconsequatfacili- sis.Maurisleo.Quisquepedeligula,auctor vel,pellentesquevel,posuereid,turpis.Cras ipsumsem,cursuset,facilisisut,tempuseu- ismod,quam.Suspendissetristiquedoloreu orci.Maurismattis.Aeneansemper.Viva- mustortormagna,facilisisid,variusmattis, hendreritin,justo.Integerpurus. Vivamusadipiscing.Curabitur imperdiet tem- pusturpis.Vivamussapiendolor,congueve- nenatis,euismodeget,portarhoncus,magna. Proincondimentumpretiumenim.Fusce fringilla,liberoetvenenatisfacilisis,erosen- imcursusarcu,vitaefacilisis odioauguevitae orci.Aliquamvariusnibhutodio.Sedcondi- mentumcondimentumnunc.Pellentesque egetmassa.Pellentesquequismauris.Donec utligulaacpedepulvinarlobortis.Pellen- tesqueeuismod.Classaptenttacitisociosqu adlitoratorquentperconubianostra,perin- ceptoshymenaeos.Praesentelit.Utlaoreet ornareest.Phasellusgravidavulputatenulla. Donecsitametarcuutsemtempormalesua- da.Praesenthendreritaugueinurna.Proin enimante,ornarevel,consequatut,blandit in,justo.Donecfeliselit,dignissimsed,sagit- tisut,ullamcorpera,nulla.Aeneanpharetra vulputateodio. Quisqueenim.Proinvelitneque,tristiqueeu, eleifendeget,vestibulumnec,lacus.Vivamus odio.Duisodiourna,vehiculain,elementum aliquam,aliquetlaoreet,tellus.Sedvelit.Sed velmiacelitaliquetinterdum.Etiamsapi- enneque,convalliset,aliquetvel,auctornon, arcu.Aliquamsuscipitaliquamlectus.Proin tinciduntmagnasedwisi.Integerblanditla- cusutlorem.Sedluctusjustosedenim. Morbimalesuadahendreritdui.Nuncmauris leo, dapibus sitamet, vestibulum et, commodo id,est.Pellentesquepurus.Pellentesquetris- tique,nuncacpulvinaradipiscing,justoeros consequatlectus, sitamet posuerelectusneque velaugue.Cras consectetuerlibero aceros.Ut egetmassa.Fuscesitametenimeleifendsem dictum auctor.In eget risusluctuswisi conva- llis pulvinar.Vivamus sapien risus,temporin, viverra in, aliquetpellentesque, eros.Aliquam euismodliberoasem. Nunc velit augue,scelerisquedignissim,lobor- tiset,aliquamin,risus.Ineueros.Vestibu- lumanteipsumprimisinfaucibusorciluctus etultricesposuerecubiliaCurae;Curabitur vulputateelitviverraaugue.Maurisfringilla, tortorsitametmalesuadamollis,sapienmi dapibusodio,acimperdietligulaenimeget nisl.Quisquevitaepedeapedealiquetsus- cipit.Phasellustelluspede,viverravestibu- lum,gravidaid,laoreetin,justo.Cumsoci- isnatoquepenatibusetmagnisdisparturient montes,nasceturridiculusmus.Integercom- modoluctuslectus.Maurisjusto.Duisvar- iuseros.Sedquam.Craslacuseros,rutrum eget,variusquis,convallisiaculis,velit.Mau- ris imperdiet, metus at tristique venenatis, pu- rusnequepellentesquemauris,aultriceselit lacusnectortor.Classaptenttacitisociosqu adlitoratorquentperconubianostra,perin- ceptoshymenaeos.Praesentmalesuada.Nam lacuslectus,auctorsitamet,malesuadavel, elementumeget,metus.Duisnequepede,fa- cilisiseget,egestaselementum,nonummyid, neque. Proinnonsem.Donecnecerat.Proinlibero. Aliquamviverraarcu.Donecvitaepurus. Donecfelismi,semperid,scelerisqueporta, sollicitudinsed,turpis.Nullainurna.Integer variuswisinonelit.Etiamnecsem.Mau- risconsequat,risusnecconguecondimentum, ligulaligulasuscipiturna,vitaeportaodioer- at quissapien.Proin luctusleoid erat.Etiam massametus,accumsanpellentesque,sagittis sitamet,venenatisnec,mauris.Praesentur- na eros, ornarenec, vulputate eget, cursussed, justo.Phasellus nec lorem.Nullam ligula ligu- la,mollissitamet,faucibusvel,eleifendac, dui.Aliquameratvolutpat. Fuscevehicula,tortor etgravidaporttitor, me- tusnibhcongue lorem,uttempuspurusmau- risapede.Integertinciduntorcisitamet turpis.Aeneanametus.Aliquamvestibulum lobortisfelis.Donecgravida.Sedsedurna. Maurisetorci.Integerultricesfeugiatligu- la.Seddignissimnibhamassa.Donecorci dui,temporsed,tinciduntnonummy,viverra sit amet, turpis.Quisque lobortis.Proin vene- natistortornecwisi.Vestibulumplacerat.In hachabitasseplateadictumst.Aliquampor- tamiquisrisus.Donecsagittisluctusdiam. Nam ipsumelit,imperdietvitae,faucibus nec, fringillaeget,leo.Etiamquisdolorinsapien porttitorimperdiet. Craspretium.Nullamalesuadaipsumut libero.Suspendissegravidahendrerittellus. Maecenas quislacus.Morbifringilla.Vestibu- lumodioturpis,temporvitae,scelerisquea, dictumnon,massa.Praesenteratfelis,por- tasitamet,condimentumsitamet,placer- atet,turpis.Praesentplaceratlacusaen- im.Vestibulumnoneros.Utcongue.Donec tristiquevariustortor.Pellentesquehabitant morbi tristique senectus et netus et malesuada famesacturpisegestas.Namdictumdictum urna. Phasellusvestibulumorcivelmauris.Fusce quamleo,adipiscingac,pulvinareget,mo- lestiesitamet,erat.Seddiam.Suspendisse erosleo,tempuseget,dapibussitamet,tem- puseu,arcu.Vestibulumwisimetus,dapibus vel,luctussitamet,condimentumquis,leo. Suspendissemolestie.Duisinante.Utso- dales sem sit amet mauris.Suspendisseornare pretiumorci.Fuscetristiqueenimegetmi. Vestibulumeroselit,gravida ac,pharetrased, lobortisin,massa.Proinatdolor.Duisac- cumsanaccumsanpede.Nullamblanditelit inmagnalaciniahendrerit.Utnonummyluc- tuseros.Fusceegettortor. Utsitametmagna.Crasaligulaeuurna dignissimviverra.Nullamtemporleoporta ipsum.Praesentpurus.Nullamconsequat. Maurisdictumsagittisdui.Vestibulumsol- licitudinconsectetuerwisi.Insitametdiam. Nullammalesuadapharetrarisus.Proinlacus arcu, eleifendsed, vehiculaat, conguesit amet, sem.Sedsagittispedeanisl.Sedtincidunt o- dioapede.Seddui.Nameuenim.Aliquam sagittislacusegetlibero.Pellentesquediam sem,sagittismolestie,tristiqueet,fermentum ornare,nibh.Nullaettellusnon felisimperdi- etmattis.Aliquameratvolutpat. Vestibulumsodalesipsumidaugue.Integer ipsumpede,convallissitamet,tristiquevi- tae,temporut,nunc.Namnonligulanon lorem convallis hendrerit.Maecenas hendrerit. Sedmagnaodio,aliquamimperdiet,portaac, aliqueteget,mi.Cumsociisnatoquepenati- busetmagnisdisparturientmontes,nascetur ridiculusmus.Vestibulumnislsem,dignissim vel,euismodquis,egestasut,orci.Nuncvi- taerisusvelmetuseuismodlaoreet.Crassit ametnequeaturpislobortisauctor.Sedali- quamsemacelit.Crasvelitlectus,facilisis id,dictumsed,portarutrum,nisl.Namhen- dreritipsumsedaugue.Nullamscelerisque hendreritwisi.Vivamusegestasarcusedpu- rus.Utornarelectussederos.Suspendisse potenti.Maurissollicitudinpedevelvelit.In hachabitasseplateadictumst. Suspendisseeratmauris,nonummyeget, pretiumeget,consequatvel,justo.Pellen- tesqueconsectetuereratsedlacus.Nullam egestasnullaacdui.Doneccursusrhoncus ipsum.Nuncetsemeumagnaegestasmale- suada.Vivamusdictummassaatdolor.Mor- biestnulla,faucibusac,posuerein,interdum ut,sapien.Proinconsectetuerpretiumurna. Donecsitametnibhnecpurusdignissimmat- tis.Phasellusvehiculaelitatlacus.Nullafa- cilisi.Crasutarcu.Sedconsectetuer.Integer tristiqueelitquisfelisconsectetuereleifend. Crasetlectus. Utcongue malesuadajusto.Curabiturcongue, felisathendreritfaucibus,maurislacusport- titor pede, nec aliquamturpis diam feugiat ar- cu.Nullamrhoncusipsumatrisus.Vestibu- lumadolorseddolorfermentumvulputate. Sednecipsumdapibusurnabibendumlobor- tis.Vestibulumelit.Nam ligulaarcu, volutpat eget,laciniaeu,lobortisac,urna.Nammol- lisultricesnulla.Crasvulputate.Suspendisse atrisusatmetuspulvinarmalesuada.Nullam lacus.Aliquamtempusmagna.Aliquamut purus.Prointellus. Vestibulumanteipsumprimisinfaucibusorci luctus et ultrices posuere cubilia Curae; Donec scelerisquemetus.Maecenasnonmiutmetus portahendrerit.Nuncsemper.Crasquiswisi utloremposueretristique.Nuncvestibulum scelerisquenulla.Suspendissepharetrasollic- itudinante.Praesentatauguesitametante interdum porta.Nunc bibendumaugueluctus diam.Etiam nec sem.Sed eros turpis,facilisis nec,vehiculavitae,aliquamsed,nulla.Cur- abiturjustoleo,vestibulumeget,tristiqueut, tempusat,nisl. Nullavenenatisloremidarcu.Morbicur- susurnaaipsum.Donecporttitor.Integer eleifend,estnonmattismalesuada,minulla convallismi,etauctorlectussapienutpu- rus.Aliquamnullaaugue,pharetrasitamet, faucibussemper,molestievel,nibh.Pellen- tesquevestibulummagnaetmi.Sedfringilla dolorveltellus.Nuncliberonunc,venenatis eget,convallishendrerit,iaculiselementum, mi.Nullamaliquam,felisetaccumsanvehic- ula,magnajustovehiculadiam,eucondimen- tumnislfelisetnunc.Quisquevolutpatmauris avelit.Pellentesquemassa.Integeratlorem. Nammetuserat,laciniaid,convallisut,pul- vinarnon,wisi.Crasiaculismaurisutneque. Crassodales,semvitaeimperdietconsequat, pede purussollicitudin urna,ac aliquammetus orciinleo.Utmolestieultricesmauris.Viva- musvitaesem.Aliquameratvolutpat.Prae- sentcommodo,nislacdapibusaliquet,tortor orci sodales lorem, non ornare nulla lorem quis nisl. Sedatsemvitaepurusultricesvestibulum. Vestibulumtinciduntlacusetligula.Pel- lentesquevitaeelit.Vestibulumanteipsum primisinfaucibusorciluctusetultricespo- suerecubiliaCurae;Duisornare,erateget laoreetvulputate,lacusipsumsuscipitturpis, etbibendumnislorcinonlectus.Vestibu- lumnecrisusnecliberofermentumfringilla. Morbinonvelitinmagnagravidahendrerit. Pellentesquequislectus.Vestibulumeleifend lobortisleo.Vestibulumnonaugue.Vivamus dictumtempordui.Maecenasatligulaidfe- liscongueporttitor.Nullaleomagna,egestas quis,vulputatesitamet,viverraid,velit. Utlectuslectus,ultriciessitamet,semper eget,laoreetnon,ante.Proinatmassaquis nuncrhoncusmattis.Aliquamlorem.Cur- abiturpharetraduiatneque.Aliquameutel- lus.Aeneantempus,felisvitaevulputateia- culis,estdolorfaucibusurna,inviverrawisi nequenonrisus.Fusceveldolornecsapien pretiumnonummy.Integerfaucibusmassaac nulla ornarevenenatis.Nullaquissapien.Sed tortor.Phasellusegetmi.Crasnunc.Crasa enim. Quisque nisl.In dignissim dapibus massa.Ae- neansemmagna,scelerisquenec,ullamcorper quis,porttitorut,lectus.Fuscedignissimfa- cilisistortor.Vivamusgravidafelissitamet nunc.Nampulvinarodiovelenim.Pellen- tesquesitametest.Vivamuspulvinarleonon sapien.Aliquameratvolutpat.Utelementum auctormetus.Maurisvestibulumnequevitae eros.Pellentesquealiquamquam.Donecve- nenatistristiquepurus.Innisl.Nullavelit libero,fermentumat,portaa,feugiatvitae, urna.Etiamaliquetornareipsum.Proinnon dolor.Aeneannuncligula,venenatissuscipit, porttitorsit amet,mattis suscipit,magna.Vi- vamusegestasviverraest.Morbiatrisussed sapiensodalespretium. Morbicongueconguemetus.Aeneansedpu- rus.Nampedemagna,tristiquenec,porta id,sollicitudinquis,sapien.Vestibulumblan- dit.Suspendisse ut augue acnibh ullamcorper posuere.Integereuismod,nequeateleifend fringilla,augueelitornaredolor,veltincidunt purusest idlacus.Vivamusloremdui, commo- doquis,scelerisqueeu,tinciduntnon,magna. Cras sodales.Quisquevestibulum pulvinar di- am.Phasellustincidunt,leovitaetristiquefa- cilisis,ipsumwisi interdum sem,dapibussem- pernulla velit vellectus.Crasdapibusmauris etaugue.Quisquecursusnullainlibero.Sus- pendisseetloremsitametmaurismalesuada mollis.Nullamidjusto.Maecenasvenenatis. Doneclacusarcu,egestasac,fermentumcon- sectetuer,tempuseu,metus.Proinsodales, seminpretiumfermentum,arcusapiencom- modomauris,venenatisconsequataugueur- nainwisi.Quisquesapiennunc,variuseget, condimentumquis,laciniain,est.Fuscefacil- isis.Praesentnecipsum. Suspendisseadolor.Namerateros,congue eget,sagittisa,laciniain,pede.Maecenas inelit.Proinmolestievariusnibh.Vivamus tristiquepurussedaugue.Proinegestassem- pertortor.Vestibulumanteipsumprimisin faucibusorciluctusetultricesposuerecubil- iaCurae;Classaptenttacitisociosquadl- itoratorquentperconubianostra,perincep- toshymenaeos.Vestibulumorcienim,sagittis ornare,eleifendut,mattisat,ligula.Nulla molestieconvallisarcu.Uterostellus,condi- mentum at,sodalesin,ultricesvel,nulla. Duismagnaante,bibendumeget,eleifend eget,suscipitsed,neque.Vestibuluminmi sedmassacursuscursus.Pellentesquepulv- inarmollisneque.Fusceut enimvitaemauris malesuadatincidunt.Vivamusaneque.Mau- rispulvinar,sapienidcondimentumdictum, quamarcurhoncusdui,idtemporlacusjusto etjusto.Proinsitametorcieudiameleifend blandit.Nunceratmassa,luctusac,fermen- tumlacinia,tinciduntultrices,sapien.Prae- sentsed orci vitae dolorsollicitudin adipiscing. Crasaneque.Utrisusdui, interdumat,plac- eratid,tristiqueeu,enim.Vestibulumante ipsum primisin faucibusorciluctus etultrices posuerecubiliaCurae;Etiamadipiscingeros vestibulumdolor.Pellentesquealiquam,diam eget eleifendposuere,augueeros porttitorlec- tus,acdignissimduimetusnecfelis.Quisque lacinia.Vestibulumtellus.Suspendissenec wisi.Aeneanacfelis.Aliquamultricesmetus etnulla. Praesentsedestnonnibhtempusvenenatis. Praesentrhoncus.Curabitursagittisestsit ametneque.Sedcommodomalesuadalectus. Phasellusenimtellus,temporut,tristiqueeu, aliquameu,quam.Aeneanquisquamquis wisigravidavehicula.Pellentesqueamassa aleopretiumrhoncus.Suspendisseultrices. Doneclaciniamalesuadamassa.Classapten- ttacitisociosquadlitoratorquentperconu- bianostra,perinceptoshymenaeos.Donec pretiumornaremauris.Phasellusauctorer- ategetenim.Integerscelerisque,feliseucon- sequatfringilla,loremwisiultriciesvelit,id vehiculapurusnullaegetodio.Nullammat- tis,diamarutrumfermentum,odiosapien tristiquequam,idmollistellusquaminodi- o.Mauriseusapien.Donecaliquamloremsit ametlorempharetralobortis. Donecacvelit.Sedconvallisvestibulumsapi- en.Vivamustemporlacussedlacus.Nunc utlorem.Utettortor.Nullamvariuswisiat diam.Etiamultricies,dolorsitametfermen- tumvulputate,nequeliberovestibulumorci, vitaefringillanequearcualiquetante.Lorem ipsumdolorsitamet,consectetueradipiscing elit.Quisquevenenatislobortisaugue.Sed tempor,tellusiaculispellentesquepharetra, pededuimalesuadamauris,velultricesurna maurisacnibh.Etiamnibhodio,ultriciesve- hicula,vestibulumvitae,feugiateleifend,fe- lis.Vivamuspulvinar.Aliquameratvolutpat. Nullaegestasvenenatismetus.Namfeugiat nuncquiselitegestassagittis.Sedvitaefe- lis.Inliberoarcu,rhoncusin,commodoeget, auctorin,enim.Vivamussuscipitest.Nul- ladapibus,magnavelaliquetegestas,massa massahendreritlacus,acrutrumtellustellus sitametfelis.Crasviverra. Suspendisseeununc.Aliquamdignissimurna sit ametmauris.Cras commodo, urna ut port- titor venenatis, arcu metussodales risus,vitae gravidasapienligulainest.Donecvulputate sollicitudinwisi.Donecvehicula,estidinter- dumornare,nibhtellusconsectetuerjusto,a ultrices felis erat at lectus.In est massa, male- suadanon,suscipitat,ullamcorpereu,elit. Namnullalacus,bibendumsitamet,sagit- tissed,temporeget,libero.Praesentligula. Suspendissenulla.Etiamdiam.Nullaante diam,vestibulumet,aliquetac,imperdietvi- tae,urna.Fuscetinciduntlacusvelelit.Mae- cenasdictum,tortornoneuismodbibendum, pedenibhpretiumtellus,atdignissimleoeros egetpede.Nullavenenatiseleifenderos.Ae- nean ut odio dignissim augue rutrum faucibus. Fusceposuere,tellusegetviverramattis,er- attellusportami,atfacilisissemnibhnon urna.Phasellusquisturpisquismaurissus- cipitvulputate.Sedinterdumlacusnonvelit. Vestibulumanteipsumprimisinfaucibusorci luctusetultricesposuerecubiliaCurae; Vivamus vehicula leoajusto.Quisquenecau- gue.Morbimauriswisi,aliquetvitae,dignis- simeget,sollicitudinmolestie,ligula.Indic- tumenimsitametrisus.Curabitur vitaevelit eudiamrhoncushendrerit.Vivamusutelit. Praesentmattisipsumquisturpis.Curabitur rhoncusnequeeudui.Etiamvitaemagna. Namullamcorper.Praesentinterdumbiben- dummagna.Quisqueauctoraliquamdolor. Morbieuloremetestporttitorfermentum. Nuncegestasarcuattortorvariusviverra. Fusce eunulla ut nulla interdum consectetuer. Vestibulumgravida.Morbimattisliberosed est. Abstract Inthispaper,weconsiderafullydiscretefiniteelementapproxi- mationfortimefractional optimalcontrolproblems.Thestateand adjointstateareapproximatedbytriangularlinearfiniteelements inspaceandL1schemeintime.Thecontrolisobtainedbythe variationaldiscretization technique.The mainpurposeof thiswork istoderivetheconvergenceandsuperconvergence.Anumerical exampleispresentedtovalidateourtheoreticalresults. Keywords TimeFractionalOptimalControlProblems,FiniteElement, ConvergenceandSuperconvergence 1.Introduction Therehasbeenagreatdealofresearchonfiniteelementmethods (FEMs)forpartialdifferentialequations(PDEs)oroptimalcontrol problems(OCPs)governedbyPDEs,mostlyfocusedonthecaseof integer order such as elliptic, parabolic and hyperbolic equations.Sys- tematicintroductionscanbefoundin[1–3],andsoon. Inthepasttwodecades,numericalmethodsoffractionalpartial differentialequations(FPDEs)haveattractedmuchattention,since different FPDEsarise invarious physical phenomena orprocesses with memoryandhereditary.Manyauthorshaveinvestigatedfinitediffer- encemethods[4,5],spectralmethods[6–8],mixedFEMs[9,10]and FEMs[11–13]tosolveFPDEs.However,therearerelativelyfewre- searchesonnumericalmethodsforOCPsgovernedbyfractionalPDEs. TherearesomepublishedpapersonFEMsforfractionalOCPs.A finiteelementapproximationalgorithmofoptimalcontrolproblems governed by time fractional diffusion equation is presented in [14], but theauthorsdo notprovide erroranalysisresults.In[15],afastgradi- ent projection method is investigated for a constrained fractional opti- mal control.Finite element approximation of spacefractional optimal DOI:10.4236/ajcm.2021.111005Mar.25,202153AmericanJournalofComputationalMathematics  Y.L.Tang controlproblemisconsideredin[16].In[17],apriorierrorestimates oftime-steppingdiscontinuousGalerkinfiniteelementapproximation fortimefractionaloptimalcontrolproblemisestablished.Afully discretefiniteelementapproximation ofoptimalcontrol problemgov- ernedby atime-fractionalPDEisinvestigatedin [18],wherethetime fractionalderivativeoperatesonthediffusionterm.However,toour bestofknowledge,superconvergenceoffiniteelementapproximation fortimefractionalOCPsisrare.Themainpurposeofthisworkis to derive theconvergence andsuperconvergenceoffully discreteFEM fortimefractionalOCPs. LetJ= [0,T]andΩ ⊂R 2 beaboundedconvexpolygonaldomain withsmoothboundary∂Ω.Weconsiderthefollowingtimefractional OCPs: min u∈U ad 1 2 Z T 0 ky(t,x)−y d (t,x)k 2 L 2 (Ω) +λku(t,x)k 2 L 2 (Ω) dt, 0 ∂ α t y(t,x)−∆y(t,x) = f(t,x)+u(t,x),t∈J,x∈Ω, y(t,x) = 0,t∈J,x∈∂Ω, y(0,x) = 0,x∈Ω. (1) Here 0 ∂ α t (0<α<1)denotestheα-orderleftCaputoderivative withrespecttothetimevariabletanddefinedby 0 ∂ α t y(t,x) = 1 Γ(1−α) Z t 0 1 (t−s) α ∂y(s,x) ∂s ds.(2) y d (t,x),f(t,x) ∈C(J;L 2 (Ω))aregivenfunctions andtheadmissible setU ad isdefinedby U ad = {v(t,x) ∈L 2 (J;L 2 (Ω)) : v(t,x) ≥0,a.e.inΩ×J}.(3) Throughoutthepaper,L s (J;W m,q (Ω))denotesallL s integrable functionsfromJintoW m,q (Ω)withnormkvk L s (J;W m,q (Ω)) = R T 0 kvk s W m,q (Ω) dt 1 s fors∈[1,∞)andthestandardmodificationfor s= ∞,where W m,q (Ω) isSobolev spaceson Ω.Similarly,one cande- fineH l (J;W m,q (Ω))andC k (J;W m,q (Ω))(seee.g.[19]).Inaddition, corCisagenericpositiveconstant. Therestofthispaperisorganizedasfollows.InSection2,we introduceafullydiscreteapproximationschemeof(1).InSection3, weprovideaconvergenceresultofthecontrolvariable.InSection4, wederivethesuperconvergenceresultofthestateandadjointstate variables.Somenumericalexamplesareprovidedinthelastsection toverifyourtheoreticalresults. 2.FiniteElementApproximationofTime FractionalOPCs Forthe stateand co-state,weintroduce triangularlinear finiteelement forthe spatialdiscretization andL1schemefor thetimediscretization. Thecontrolisobtainedbyvariationaldiscretization(VD)technique [20]. For brevity,wedenoteW m,2 (Ω)by H m (Ω)anddropΩorJwhen- DOI:10.4236/ajcm.2021.11100554AmericanJournalofComputationalMathematics  Y.L.Tang everpossible,i.e., k·k m,2,Ω = k·k m,2 = k·k m ,k·k 0 = k·k, k·k L p (J;W m,2 (Ω)) = k·k L p (H m ) ,k·k L p (J;W m,q (Ω)) = k·k L p (W m,q ) , K= v(x) ∈L 2 (Ω) : v(x) ≥0,a.e.inΩ . Furthermore,wesetH 1 0 (Ω)≡ v∈H 1 (Ω) : v| ∂Ω = 0 ,W= H 1 0 (Ω),U= L 2 (Ω).Inaddition, a(v,w) = Z Ω ∇v·∇w,∀v,w∈W, (f 1 ,f 2 ) = Z Ω f 1 ·f 2 ,∀f 1 ,f 2 ∈U. Byusingthevariationalprinciple[2],werecast(1)asthefollowing weakformulation: min u∈U ad 1 2 Z T 0 ky−y d k 2 +λkuk 2 dt, ( 0 ∂ α t y,w)+a(y,w) = (f,w)+(u,w),t∈J,w∈W, y(0,x) = 0,x∈Ω. (4) Itfollowsfrom(seee.g.,[21])that(4)hasauniquesolution(y,u), and thata pair(y,u) is asolution of (4),then there isa co-statepsuch thatthetriplet(y,p,u)fulfillsthefollowingoptimalityconditions: ( 0 ∂ α t y,w)+a(y,w) =(f,w)+(u,w),∀w∈W,t∈J,(5) y(0,x) =0,x∈Ω,(6) ( t ∂ α T p,q)+a(q,p) =(y−y d ,q),∀q∈W,t∈J,(7) p(x,T) =0,x∈Ω,(8) (λu+p,v−u) ≥0,∀v∈K,t∈J.(9) Here t ∂ α T denotestheα-orderrightCaputoderivativewithrespect totanddefinedby t ∂ α T p(t,x) = − 1 Γ(1−α) Z T t 1 (t−s) α ∂p(s,x) ∂s ds. Likein[22],itiseasy toprovethattheinequality (9)isequivalent to u= max(0,− p λ ).(10) LetT h beafamilyofquasi-uniformtriangulationsofΩ,suchthat ¯ Ω = S e∈T h ¯eand h=max τ∈T h {h e }, where h e is the diameterof the element e.Furthermore,weset W h = v h ∈C( ¯ Ω) : v h | e ∈P 1 ,∀e∈T h ,v h | ∂Ω = 0 , whereP 1 isthespaceofpolynomialswhosedegreeatmost1. Let P h be the elliptic projection,defined as follows.For any v∈W, a(v−P h v,w h ) = 0,∀w h ∈W h . Ithasthefollowingerrorestimates(see,[12]): kv−P h vk+hk∇(v−P h v)k≤Ch 2 kvk 2 .(11) DOI:10.4236/ajcm.2021.11100555AmericanJournalofComputationalMathematics  Y.L.Tang Inthenext,wewillconsiderL1schemeforthetimediscretization. Let0 = t 0 <t 1 <···<t N = Tbe agivenuniform partitionof [0,T] withsizeτ= T N andt n =nτ,n=0,1,···,N.Wesetϕ n =ϕ(t n ,x) anddefinethetime-dependentmaximumnorm kvk l ∞ (W m,q ) =max 0≤n≤N {kv n k W m,q }. Then α-order left Caputo derivative can be approximated as follows 0 ∂ α t y n = 1 τ α Γ(2−α) n X k=0 b n k y k +r n y,τ := F α t y n +r n y,τ ,(12) whereb k = (k+1) 1−α −k 1−α ,b n 0 = (n−1) 1−α −n 1−α ,b n k = b n−k − b n−k−1 ,b n n = 1andr n y,τ isthetruncationerror.Itfollowsfrom[5],if y∈W 2,∞ (L 2 ),wehave r n y,τ = | 0 ∂ α t y n −F α t y n |≤Cτ 2−α .(13) Similarly, theα-orderrightCaputo derivative canbeapproximated asfollows t ∂ α T p n = 1 τ α Γ(2−α) N X k=n ˜ b n k p k + ˜r n p,τ := B α t p n + ˜r n p,τ ,(14) where ˜ b n n = −1, ˜ b n N = b N−n−1 , ˜ b n k = b k−n −b k−n−1 ,b k = (k+1) 1−α − k 1−α .Ifp∈W 2,∞ (L 2 ),then ˜r n p,τ = | t ∂ α T p n −B α t p n |≤Cτ 2−α .(15) Thenafullydiscretefiniteelementapproximationschemeofopti- malityconditions(5)-(9)writesas (F α t y n h ,w h )+a(y n h ,w h ) =(f n ,w h )+(u n h ,w h ),∀w h ∈W h ,(16) n=1,2,···,N,y 0 h = 0,(17) (B α t p n h ,q h )+a q h ,p n−1 h =(y n h −y n d ,q h ),∀q h ∈W h ,(18) n=N−1,···,1,0,p N h = 0,(19) λu n h +p n−1 h ,v−u n h ≥0,∀v∈K,n= 1,2,···,N.(20) Asinreference[20],theinequality(20)isequivalentto u n h = max(0,− 1 λ p n−1 h ),n= 1,2,···,N.(21) 3.ConvergenceAnalysis Inthissection,wederivetheconvergenceofthecontrolfortheap- proximationscheme(16)-(20).Foreaseofexposition,wesetα τ = τ α Γ(2 −α).Weintroducesomeusefulintermediatevariables.Let y n h (u),p n−1 h (u) ∈W h ×W h ,n=1,2,···,Nfulfillthefollowinge- quations: (F α t y n h (u),w h )+a(y n h (u),w h ) =(f n ,w h )+(u n ,w h ),∀w h ∈W h , (22) n=1,2,···,N,y 0 h (u) = 0,(23) (B α t p n h (u),q h )+a q h ,p n−1 h (u) =(y n h (u)−y n d ,q h ),∀q h ∈W h , (24) n=N−1,···,1,0,p N h (u) = 0.(25) Thefollowingconclusionswillbeusedinthefollowingconvergence analysis. DOI:10.4236/ajcm.2021.11100556AmericanJournalofComputationalMathematics  Y.L.Tang Lemma1.[10]Let{ξ n } N n=0 beasequenceoffunctionsonΩ.Then ξ n , n X k=0 b n k ξ k ! = 1 2 kξ n k 2 + n−1 X k=0 b n k kξ k k 2 − n−1 X k=0 b n k kξ k −ξ n k 2 ! . Lemma2.[23]Letε k ≥0,satisfyε n ≤ P n−1 k=1 (b k−1 −b k )ε n−k +γ withγ>0,k= 1,2,···,L.Then ε n ≤Cτ −α γ,n= 1,2,···,L. Lemma3.Let (y,p,u)and (y h ,p h ,u h ) be thesolutions of(5)-(9) and (16)-(20),respectively.Supposethaty,p∈W 2,∞ (L 2 ) ∩W 1,∞ (H 2 ). Then ky n −y n h (u)k+kp n −p n h (u)k≤C(h 2 +τ 2−α ),(26) ky n −y n h (u)k 1 +kp n −p n h (u)k 1 ≤C(h+τ 2−α ).(27) Proof.Sety n −y n h (u) = y n −P h y n +P h y n −y n h (u) := η n +ζ n .Choosing v= w h in(5) andsubtracting(22)from(5),we obtainerrorequation (F α t (y n −y n h (u)),w h )+a(y n −y n h (u),w h )+(r n y,τ ,w h ) = 0.(28) BythedefinitionofP h and(28),wehave (F α t ζ n ,w h )+a(ζ n ,w h ) = −(F α t η n ,w h )−(r n y,τ ,w h ).(29) Letw h =F α t ζ n anduseLemma1,weget a(ζ n ,F α t ζ n ) = 1 2α τ k∇ζ n k 2 + n−1 X k=0 b n k k∇ζ k k 2 − n−1 X k=0 b n k k∇ ζ k −ζ n k 2 ! . (30) Fromy∈W 1,∞ (H 2 )and(11),thereholds kF α t η n k= 1 τ α−1 Γ(2−α) n−1 X k=0 b k ∂ t η n−k ≤ 1 τ α−1 Γ(2−α) n−1 X k=0 b k k∂ t η n−k k ≤ 1 α τ n−1 X k=0 b k Z t n−k t n−k−1 kη t kdt ≤Ch 2 ky t k L ∞ (H 2 ) . (31) Then,byapplyingH¨older’sinequalitykf·gk L 1 (Ω) ≤kfk L 2 (Ω) kgk L 2 (Ω) , Young’sinequalityand(31),wearriveat (F α t η n ,F α t ζ n ) ≤kF α t η n kkF α t ζ n k≤ 1 2 kF α t ζ n k 2 + C 2 h 4 .(32) Likewise r n y,τ ,F α t ζ n ≤kr n y,τ kkF α t ζ n k≤ 1 2 kF α t ζ n k 2 + C 2 τ 4−2α .(33) Combining(29)-(33)andnoticethatb n k <0(0 ≤k<n),weobtain k∇ζ n k 2 ≤− n−1 X k=0 b n k k∇ζ k k 2 +Cτ α (h 2 +τ 2−α ) 2 .(34) DOI:10.4236/ajcm.2021.11100557AmericanJournalofComputationalMathematics  Y.L.Tang Itfollowsfrom(34),Lemma2andPoincar´e’sinequalitythat kζ n k 1 ≤Ck∇ζ n k≤C(h 2 +τ 2−α ).(35) Inthesameway,wecanderive kP h p n −p n h (u)k 1 ≤Ck∇(P h p n −p n h (u))k≤C(h 2 +τ 2−α ).(36) Hence,(26)and(27)followfromembeddingtheorem,trianglein- equality,(11)and(35)-(36). Theorem4.Let(y,p,u)and(y h ,p h ,u h )bethesolutionsof(5)-(9) and (16)-(20),respectively.Assume allthe conditions in Lemma3 are true.Then ku n −u n h k≤C h 2 +τ .(37) Proof.Setv= u n h andv= u n in(9)and(20)respectively,wehave λku n −u n h k 2 = λ(u n −u n h ,u n −u n h ) ≤ p n−1 h −p n−1 h (u),u n −u n h + p n−1 h (u)−p n−1 ,u n −u n h + p n−1 −p n ,u n −u n h := I 1 +I 2 +I 3 . (38) ForthefirsttermI 1 ,itfollowsfrom(16)-(19)and(22)-(25)that I 1 = p n−1 h −p n−1 h (u),u n −u n h = F α t (y n h −y n h (u)),p n−1 h (u)−p n−1 h −(B α t (p n h (u)−p n h ),y n h −y n h (u)) −(y n h −y n h (u),y n h −y n h (u)) ≤−ky n h −y n h (u)k 2 . (39) AccordingtoH¨older’sinequalityandYoung’sinequality,weget I 2 = p n−1 h (u)−p n−1 ,u n −u n h ≤ 1 2 kp n−1 h (u)−p n−1 k 2 + 1 2 ku n −u n h k 2 . (40) Forthelastterm,weget I 3 = p n−1 −p n ,u n −u n h ≤ 1 2 kp n−1 −p n k 2 + 1 2 ku n −u n h k 2 ≤ 1 2 τ 2 kp t k 2 L 2 (L 2 ) + 1 2 ku n −u n h k 2 . (41) From(26)and(38)-(41),weobtain(37). 4.SuperconvergenceAnalysis In thissection, we derivethe superconvergenceof thestate andadjoint statevariables. Theorem5.Let(y,p,u)and(y h ,p h ,u h )bethesolutionsof(5)-(9) and(16)-(20),respectively.AssumealltheconditionsinTheorem4 aretrue.Then kP h y n −y n h k 1 +kP h p n −p n h k 1 ≤C h 2 +τ .(42) DOI:10.4236/ajcm.2021.11100558AmericanJournalofComputationalMathematics  Y.L.Tang Proof.LetP h y n −y n h =θ n .From(5),(16)andthedefinitionofP h , forn= 1,2,···,Nand∀w h ∈W h ,weget (F α t θ n ,w h )+a(θ n ,w h ) = −(F α t η n ,w h )+(u n −u n h ,w h )− r n y,τ ,w h . (43) AccordingLemma1,wehave (F α t θ n ,θ n ) = 1 2α τ kθ n k 2 + n−1 X k=0 b n k kθ k k 2 − n−1 X k=0 b n k kθ k −θ n k 2 ! .(44) Notethat a(θ n ,θ n ) ≥ckθ n k 2 1 .(45) Setw h =θ n in(43),thenfrom(44)-(45),H¨older’sinequalityand Young’sinequality,weobtain kθ n k 2 +2α τ ckθ n k 2 1 ≤− n−1 X k=0 b n k kθ k k 2 −2α τ F α t η n −(u n −u n h )+r n y,τ ,θ n ≤− n−1 X k=0 b n k kθ k k 2 +2α τ kF α t η n k+ku n −u n h k+kr n y,τ k kθ n k ≤− n−1 X k=0 b n k kθ k k 2 +α τ kF α t η n k 2 +ku n −u n h k 2 +kr n y,τ k 2 +kθ n k 2 . (46) From (13),(31),Lemma2,Theorem4andembeddingtheorem,we derive kP h y n −y n h k 1 ≤C h 2 +τ .(47) Similarly,wecanget kP h p n −p n h k 1 ≤C h 2 +τ .(48) Then(42)followsfrom(47)-(48). 5.NumericalExperiments Theoptimalcontrolproblemwasdealtnumericallywithcodesde- velopedbasedonAFEPack.Thepackageisfreelyavailableandthe detailscanbefoundat[24].Wesolvethefollowingtimefractional optimalcontrolproblem: min u∈U ad 1 2 Z T 0 Z Ω (y(t,x)−y d (t,x)) 2 +λ Z Ω (u(t,x)) 2 dt, 0 ∂ α t y(t,x)−∆y(t,x) = f(t,x)+u(t,x),inΩ×(0,T], y(t,x) = 0,on∂Ω×(0,T], y(x,0) = y 0 (x),inΩ. Thediscretizationwasdescribed inSection2.For easeofexposition, wedenoteku−h h k l ∞ (L 2 ) ,kP h y−y h k l ∞ (H 1 ) andkP h p−p h k l ∞ (H 1 ) DOI:10.4236/ajcm.2021.11100559AmericanJournalofComputationalMathematics  Y.L.Tang by|||e u |||,|||e y ||| 1 and|||e p ||| 1 ,respectively.Thespatialandtime convergenceratesarecomputedbythefollowingformula: R h = ln(e i+1 )−ln(e i ) ln(h i+1 )−ln(h i ) andR τ = ln(e i+1 )−ln(e i ) ln(τ i+1 )−ln(τ i ) , wheree i (e i+1 )denotestheerrorwhenthespatialpartitionsizeis h i (h i+1 )orthetimestepsizeisτ i (τ i+1 ). Example1LetΩ=[0,1] ×[0,1],T=1,λ=1.Thedataareas follows: y(t,x) = sin(2πx 1 )sin(2πx 2 )t, p(t,x) = sin(2πx 1 )sin(2πx 2 )(1−t), u(t,x) = max(0,−p(t,x)), f(t,x) = 0 ∂ α t y(t,x)−∆y(t,x)−u(t,x), y d (t,x) = y(t,x)− t ∂ α T p(t,x)−∆p(t,x). Fordifferentα,theerrors|||e u |||,|||e y ||| 1 and|||e p ||| 1 areshownin Tables1-3,whereh i+1 = h i /2andτ i+1 = τ i /4. Table1.TheconvergencerateforExample1withα=0.05. hτ|||e u |||R h R τ |||e y ||| 1 R h R τ |||e p ||| 1 R h R τ 1 10 1 10 4.6742e−2--8.2685e−3--7.4163e−3-- 1 20 1 40 1.3123e−21.830.922.3021e−31.840.922.0014e−31.900.94 1 40 1 160 3.3262e−31.980.995.9132e−41.960.985.1089e−41.970.99 1 80 1 640 8.4136e−41.980.991.4876e−41.991.001.2846e−41.991.00 Table2.TheconvergencerateforExample1withα=0.5. hτ|||e u |||R h R τ |||e y ||| 1 R h R τ |||e p ||| 1 R h R τ 1 10 1 10 2.1405e−2--6.0628e−3--6.3384e−3-- 1 20 1 40 5.3908e−31.990.991.5228e−31.991.001.5903e−31.991.00 1 40 1 160 1.3485e−32.001.003.8158e−42.001.003.9828e−42.001.00 1 80 1 640 3.3385e−42.011.019.5292e−52.001.009.7726e−52.031.01 Table3.TheconvergencerateforExample1withα=0.95. hτ|||e u |||R h R τ |||e y ||| 1 R h R τ |||e p ||| 1 R h R τ 1 10 1 10 4.5959e−2--7.2163e−3--9.5103e−3-- 1 20 1 40 1.2301e−21.900.951.7037e−32.081.042.2462e−32.081.04 1 40 1 160 3.1189e−31.970.994.2247e−42.011.015.5526e−42.011.01 1 80 1 640 7.8113e−41.991.001.0548e−42.001.001.3847e−42.001.00 DOI:10.4236/ajcm.2021.11100560AmericanJournalofComputationalMathematics  Y.L.Tang FromnumericalresultsinTables1-3,itisclearthat|||u−u h |||= O(h 2 +τ), |||P h y−y h ||| 1 = O(h 2 +τ) and|||P h p−p h ||| 1 = O(h 2 +τ), whichareconsistentwithourtheoreticalanalysis. 6.Conclusion Inthispaper,weestablishtheconvergenceandsuperconvergencere- sults ofa fullydiscrete finiteelement approximation for timefraction- aloptimalcontrolproblems.Numericalexperimentresultsverifythe correctnessofourtheoreticalresults. 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