A Posteriori Error Estimator for Mixed Approximation of the Navier-Stokes Equations with

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1 S tratoal oral o a omtr S a lato Vol 4 No3 3 otror rror tmator or M romato o t Nar-Sto ato t t a oary oto Mao M a lal Maal r laoratory Falty o a t- ot mozzr F laa Dartmt o matmat oal tr or roo o ato a ra F 43 Sro Moroo trat t ar tro t Nar-Sto ato t a oary oto t ott o t t a o t olto o t a ormlato aoat t t roo rolm o ol t lattr t rtzato y m t lmt mto ato to ty o a otror rror ator ar tro a ar o to loal rror tmat tat ar alt to t tr rror orr to alat t rorma o t mto t mral rlt ar omar t om roly l or a t otr om rom ommral o l DN ytm yor Nar-Sto ato; a oary oto; M Ft lmt mto; al rror tmator; NODUON ar r a mral olto o Nar-to ato t a oary oto ralz t ll o a oto ally t Drlt a t Nma oto So ro tat t a ormlato o t roo moll a a olto o allat t lattr t rtzato y m t lmt mto Moror roo to ty o a otror rror ator ar o to loal rror tmat tat ar alt to t tr rror o omar or olto t t om roly o a DN ytm om mral rlt ar o mto trtr a a taalo aa or ty rtzato alortm or D a or lor a lo alortm mral lar a olar alra or ol t aoat rt ytm t a alo a a aoal tool or ty t or mor lmtary o a t rort o rylo a trat mto [5] lattr to D ottt t a or omtatoal mol o t lo o a omrl Ntoa l For t ato or a o o to-moal oma o t rolm a o alo t oary oto a otr at o t rolm a a o o t lmt rtzato o a arlatral lmt m Wra t rt Nar-Sto ato rr a mto a t ralz mmm ral mto MS or o ymmtr ytm [5] y or at olto l t o o t roto trat aa or a ra o oto l alra mto a omlt U atorzato a ll a mor otat a tat-o-tart mltr mto to ta aata o t trtr o t rt larz Nar-Sto ato ato tr a o o trat trat ar trato or Nto mto or ol t olar alra ytm ar rom t lattr rolm otror rror aaly rolm rlat to l yam a t tat a r a lot o attto r t lat a t oorm a tr ar ral ay to rror tmator y t ral ato artlar or t Sto rolm M ort O [] art S F [] Day DSltr [3] a Vrrt [4] tro ral rror tmator a ro tat tat ty ar alt to t ry orm o t rror Otr or or t tatoary Nar-Sto rolm a tro [ ] la o t ar a ollo Sto rt t mol rolm t ar a ormlato rt to to V o t t a o t olto rtzato y m t lmt r to V Sto V tro to ty o a otror rror o o t omt olto Nmral rmt arr ot t t ramor o t lato a tr omaro t otr rlt ar o Sto V OVNN QUONS W ll or t mol o o omrl lo a alz o ot oma o 3 W alo am tat a a olyoal oary o tat t al otar-ot ormal aataor 45 a

2 S tratoal oral o a omtr S a lato Vol 4 No3 3 tor l t loty o t lo a t alar aral rrt t rr Or matmatal mol t Nar-to ytm t a oary oto 3 ot r a a otat all t mat oty t rat t r a t ala orator a a ral matr a a y y y or all y 4 y y r ar to trtly ot otat a tat X y X 5 or all y a X S X X Wr a a ar / t to oto o W FOMUON W t ollo a / ; ; 6 y 7 / 8 / Γ 9 V / taar a ormlato o t Nar-Sto lo rolm - -3 t ollo F a tat or all t t lar orm ; t tr-lar orm 3 4 ; D z z 5 D z z 6 t toal 7 rly a ormlato may rtat a tat 8 or all t l a am tat V XSN ND UNQUNSS OF SOUON t to ll ty t t a o t olto o rolm 8 or tat t ollo rlt orm 4 r ar to trtly ot otat a tat t or all 9 roo ma oto S [6] torm t tr t tat or all U 5 t or all t O t otr a or to 555 [] tr t a tat otat U or all t a ma ; Fally or all aataor 46 a

3 S tratoal oral o a omtr S a lato Vol 4 No3 3 rlt allo to ro tat a lrt a ol oto or to ota t t a o t olto orm 4 a ral lrt a roo a ral a a lo a a ar alt orm t a ral lrt a or to orm orm 43 or all - llt or torm or all roo t ay orm 44 a or all lar orm at t - r t a otat tat or all 6 roo t a t a ' ' [6] / a or all U 9 ' t orm 45 r t a otat m tat z m z or all z z z or all z V V V or all V V D 5 m aly V a m mly tat D m m D roo t z a z m' z [6] t m' 3 z z V a z z z z 3 y r ormla a z z z z S z V t z a z ally z z 3 t ay t ta 8 4 t to aly 9 5 am roo o Vralt a aart [6] a 5 or t torm a 4 atr V [6] t rlt 8-3 r t t at lat o ar at 8 W N z z z V 3 33 a ll-o t oto or tat or to mall t tat t N to aly torm 3 a 4 atr V [6] aataor 47 a

4 orm 46 m tat ν a ollo oto S tratoal oral o a omtr S a lato Vol 4 No3 3 aty t or all 34 N For om mr [[ tr t a at 8 a ol N roo om roo o torm 4 atr V [6] V MXD FN MN OXMON t to am tat a olyomal a 35 ar t t ; a amly o rtalato o For ay o rtal ar at lat o t lmt ~ t t o rtal ar at lat o rt t lo or a lmt ot t o o rtal ar l ~ t t o rtal ar at lat o rt t Nt t t o t or o ot y a N t t o t a rt rtly W lt ot t t o all lt to tror a oary Wr W ot y t amtr o a ml y ma amtr o a a o a t t rt a ormlato t moal a X a M rt ro o 5 X a M tat 36 For all a X M W t arorat a or t t lmt a la to o lar ytm o alra ato arzato o t ytm Nto trato t t moal Sytm X a M tat For all X a M r 37 r a r ar t o lar ral aoat t t rt ormlato 36 o t orro lar alra rolm a t o toral a to So tat ; 38 W tro a t o rr a to a t ; 39 Wr a ar t mr o loty a rr a to rtly W tat t rt ormlato 37 a r a a ytm o lar ato N W U 4 ytm rrr to a t rt Nto rolm matr t tor alaa matr a t r matr a ]; a 4 [ [ ]; 4 tor-oto matr N a t Nto rat matr W ar y N [ ]; 43 W [ ]; 44 For ; a rt-a tor 4 ar ]; 45 [ or For ar trato t rt rolm N U V OSO O SMO 46 t to roo to ty o a otror rror ator a ral rror tmator a loal oo rolm aataor 48 a

5 S tratoal oral o a omtr S a lato Vol 4 No3 3 tmator ar o to loal rror tmat tat ar alt to t tr rror al rror tmator l to o t rr lmt ~ ar a ollo ~ 4 y y ~ ~ y ~ ~ y y ~ ~ 4 ~ ~ y 3 y y r ~ t rr lmt l to a ~ 4 ar rr l to For ay ~ t lmt l to ~ F a t lmt l to ~ ~ F r ~ F t a ma orm to For a tror o tat r For a oary rtal tat r t Wt t l to r t al [3] lmma 4] tal t ollo lmma mma 6 t a artrary rtal a For ay a t ollo alt ol Wr a ar to otat oly o t lmt at rato a t olyomal r a r a ar a a a aoat t r otat a ato oly o t alo ot t atral to to t lmt From t alt 5 a 5 tal t ollo lmma mma 6 t a rtal a For ay t ollo alt ol 5 roo S t otr tr o rtal t a t to t ol o Ω y tt t a U t alt 9 5 a 5 D Wt D t amtr o Ω a D W rall om a-trolato tmat t ollo lmma mma 63 lmt trolato tmat lt X t a-trolat o y ara a [4] For ay ~ 53 a or all ~ 54 W lt ot t olto o 8 a lt ot t olto o 36 t a aromato o a rtalar o Or am to tmat t loty a t rr rror a lmt otrto o t ral rror tmator y 55 a t omot 55 ar y / aataor 49 a

6 S tratoal oral o a omtr S a lato Vol 4 No3 3 5 a aataor / ; ; Wt t y otrto om rom t tr m aoat t a ao lmt a S S / / loal ral rror tmator y Or am o X a t rt to t orm or loty Ω X a t ott orm or t rr Ω M For ay a t ollo to to ; o o t rtal o otr t t tt y o to ol a t a Wt t to to a t ollo lmma mma 64 For ay a 56 roo U y aly t r ormla a o ota mma 65 a 57 W a 58 roo am roo o 56 W a W a a o t otr tr o rtal t W t lar orm ; W alo t ollo toal a or all y y y y mma 66 r t a tat a or all 59 roo U 7 3 a 35 a tat or 6 N N N W a lm t tr t tat or all U t rlt a 6 ota t a or all N

7 S tratoal oral o a omtr S a lato Vol 4 No3 3 5 a aataor W a - ; - ; - ; ; all or ; 6 a tat trror W Saty t o-lar ato all or ; 6 a ar W to t tz roto o t mo ral ; all or 63 mma 67 or all X 64 roo t a 6 ota Nt tal t al t t orm o a t orm o t olto o 63 orm 68 t t oto o torm 46 ol r t to ot otat a t o tat roo am roo o torm 3 [9] orm 69 a all For 65 roo a t W alo a W atr 66 a 67 to t orm 6 For ay m t lmt aromato ot arly - tal o rtalar r t ral tmator at a ' ' ' Not tat t otat t loal lor o t o t oma a roo o tal t r o lt a X t lmt trolato o U 63 6 a 6

8 S tratoal oral o a omtr S a lato Vol 4 No3 3 5 a aataor U 5 a 53 t Fally 65 or to torm 68 a tal t r o r to t loal lor o Frt or t lmt ral art a S tat 56 a 57 U 48 t S '' ' ato rom t r alty Nt om t r art 69 Fally to tmat t m trm For a W a 57 a U t S U 5 a 5 to t ' U 68 7 U t r alty 49 a t 7 7 lo to o tat 7 ol or oary W a a For U a 4

9 S tratoal oral o a omtr S a lato Vol 4 No a aataor ' U 5 a 5 U 69 7 U 49 t a 7 73 Fally om a 73 tal t loal lor o oal oo rolm tmator loal oo rolm tmator a 74 t o V V Sat t ol oo rolm 75 ay or V / 76 orm 6 alt to tmator roo For t r o rt lt a lmt tror l From 75 U 48 t 77 ato rom t r alt to t 78 Nt lt a l to a 5 S tat t matr t t a o U a 78 a a ay Fallyor ay From t rlt a t r alt W a alo t 8 om a 8 tal t r o t al rlato For t lor to 65

10 V U 75 a 76 V S tratoal oral o a omtr S a lato Vol 4 No3 3 F ally trt traml lot aoat t a 3 8 ar r Q aromato a 5 t 8 V No zro at t or rt o a al armt a t al tra torm [5 mma 5] o tat at om t to alt t 8 mmatly t lor o t al rlato o 6 For ay m t lmt aromato ot arly - tal o rtalar r t ral tmator at F orm traml lot omt t DN Sytm aoat t a 3 8 ar r a 5 F3 Vloty tor olto y MF t a 3 8 ar r a 5 Not tat t otat t loal lor o t o t oma V NUM SMUON t to om mral rlt o allato t m t lmt Mto a DN ytm ll rt U or olr r t lo or a otal [5] t a mr o rt mol aramtr aml Flo or a otal aotr laal rolm oma a aoat t moll lo a rtalar al t a ar ylral otrto oll rol mo o t lo oary ; y a olo zro loty oto mo o t otrto a t to a ottom all Nma oto al at t otlo oary atomatally t t ma otlo rr to zro a ot rtalar ro 8 - rat y lt t ar 7/4 9/4 -/4 /4 F4 olto omt t DN ytm lot o t loty tor olto t a 3 8 ar r a 5 to olto ar tror tally tal ry oo ato tat my olr mlmt orrtly F5 rr lot or t lo t a 3 8 ar r aataor 54 a

11 S tratoal oral o a omtr S a lato Vol 4 No3 3 Nmral rlt ar rt to t rorma o t mto a m to trt y omar tm t otr rt rlt F6 tmat rror aoat t 3 8 ar r a Q aromato loal oo rolm rror tmator Flo or a otal t yol mr = tmat loty r rror r ral rror tmator or Flo or a otal t yol mr = r V ONUSON W r trt t or t mr olto or to moal artal rtal ato moll or ar rom mol tay omrl l lo t l alortm or rtzato y m t lmt mto a a otror rror tmato o t omt olto Or rlt ar t a ytm NOWDMN ator ol l to r tr r ta or t rr or /r ll to r [] lar r -mémor lémt F Do ar 5 [] aart oma troto l aaly mér éato a éré artll Mao ar 983 [3] r rt S Na otror rror tmato or t Sto rolm otro a otro rtzato M3S Vol [4] lmt romato y t lmt to loal rlarzato O al Nmr Vol [5] OWW a M ort a otror rror tmat or t lmt aromato o t Nar-Sto ato omt Mto l M r Vol [6] V ralt a aart Ft lmt romato o t Nar-Sto ato Srr-Vrla rl rlr N Yor 98 [7] laa lal N o m t lmt mto or Nar-Sto ato l Mat & ormat Vol8 No [8] Vrrt o otror rror tmato a at M-mt Wly-r tr 996 [9] M ort a O otror rror tmato Ft lmt aly Wly N Yor [ ] [] M ort O otror rror tmat or Sto a O ato SM Nmr al Vol [] a Wlrt otror rror tmat or t Sto rolm SM Nmr al Vol [] arts F otror rror otrol lo-orr t lmt rtzato o omrl tatoary lo rolm Mat om Vol [3] D ay D Sltr otror rror tmato or talz m aromato o t Sto ato SM S omt Vol [4] Vrrt otror rror tmator or t Sto ato Nmr Mat Vol [5] lma D Sltr Wat Ft lmt a Fat trat Solr t lato omrl Fl Dyam Oor Urty r Oor 5 [6] V o al a otror rror tmat or to-ll t lmt mto or t Nar-Sto ato Nmr Mat Vol [7] D o lmtary Fl Dyam Oor Urty r Oor 99 aataor 55 a