A finite element method with edge oriented stabilization for the time-dependent Navier-Stokes equations: space discretization and convergence
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A finite element method with edge oriented stabilization for the time-dependent Navier-Stokes equations: space discretization and convergence Erik Burman — Miguel Ángel Fernández. N° 5630 July 2005. ISSN 0249-6399. ISRN INRIA/RR--5630--FR+ENG. Thème BIO. apport de recherche.
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(206) zwne )Ô\ , ² @g{~5wnegj"ji n{\5~g}|svuxzu0¾Y¥j"uxea}|}&}ux{?gunj¿wne3j*w \ j*zgj0Yg}|zws. + ku − Πkh uk0,∞,Ω kvh k0,Ω .. 2. 1. 2 2 kvk20,∂K ≤ C h−1 K kvk0,K + hK k∇vk0,K ,. )Ô ,. ∀v ∈ H 1 (K),. unjij 5870 9g§©{~2*3n{t{~§²l±g~xw z|i3}N¾ytsDi{~h+ygz3z|3*w e3jyR{-®~j5jNuw zh-w j¥
(207) zwne¶*z|Y®\jiuxj5z|3j0Yg~}zws ) 701, ¾3zw
(208) §©{~}|}{-¥uw eg-w )I~1 , kv k ≤ C h kv k , ∀v ∈ V . d5e3j?3gz³§©{\nh )Óz ,¿uwy3z|}|z³wsÀ{~§UwnegjDgnjNuxj0\w¿h?j wneg{t6 ji}|zjNuµ{\¸w e3jD§Ó~w*w eg-w¿wne3j?\ \vzj0Yw ghDauz| ) J1,
(209) 0À {\Yνwn {~}&un{~h?j"zYw ji a{\}|wnz|{~ji n{\ u{~§w e3j+uxwn j0hD°»}z|3j vj0nz|®--w z®\j~¾gvz|®~j0n\jigij ~g gnjNunun3 j2\ \vzj0Yw0²d5egz|u zu §©{~ h?~}z|¬ijN;¾-z|+w e3j§©{~}|}{-¥
(210) z|3¿}j0hDhg¾yts j0uxw ~y3}zune3zg¿un{~h?jji n{\ yR{~gg3u§©{~5w e3jm¿un¥~}| Yg~unz³°»z|\w ji a{\}|~Yw π )Óunjij 5ä [v¾g\C 9 ,²  ÓÆ 0 0% <A x n EO%V 1 # ( " 8 & & > x )- #$A - /1&C
(211) + & & 8 & ") 00 k π 2 h 0,∂K. −1 T K. 2 h 0,K. h. k h. ∗ h. i. i. 1 ni. X. i. ∗ h. def. πh∗ v(xi ) =. ) ,. v|K (xi ),. {K : xi ∈K}. * Z [ 2Â ¼¢ %V 6 A
(212) O%> V
(213) γi > 0 ! I1 - O%V # %?
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(215) 10%6%& . O%. . H 2 (Th ) . ∀v ∈ [H 2 (Th )]d ,. i = 1, 2, 3. A& >G WO%VY # %. 1 kh 2 wh · ∇vh − πh∗ (wh · ∇vh ) k20,Ω ≤ γ1 jwh (vh , vh ), 1 kh 2 ∇ · vh − πh∗ (∇ · vh ) k20,Ω ≤ γ2 j(vh , vh ), 1 kh 2 ∇qh − πh∗ (∇qh ) k20,Ω ≤ γ3 j(qh , qh ),. " $ (v , q , w ) ∈ [V k ]d × V k × [V 1 ]d h h h h h h ! < " gn{t{§ {§ Ô~ ° I [ 50 yaj*§©{\3g zQ5 J3¾^7NC9I² j"zYw n{vvgij*3{-¥"¾t§©{~
(216) jN~e w ∈ [V k ]d \z®\ji&¾tw e3j*wn z|3}ji°Ig{~ h. )I , )IA; , )I [ ,. ) 1, -) 1,. h. h. 2 def |||(vh , qh )|||wh = kvh k2 + J wh ; (vh , qh ), (vh , qh ) ,. )I J , ê &êìë .
(217)
(218) ! #" $%&(') *
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(220) zwne def. 1. 1. 1. 1. kvh k2 = kν 2 ∇vh k20,Ω + kh 2 ∇ · vh k20,Ω + k(γν ν) 2 h− 2 vh k20,∂Ω + kvh · nk20,∂Ω .. @g{~wnegj*i{~Ywnz|t3zws{§9w e3j qYwn{\¦~jNuunsvuw jih¾v¥
(221) zwnezw u e3j*ya{\3g3~ns¶i{~gvzwnz|{~aui¾tzw5zu
(222) }un{Di{~t®~ji° gzj0\w
(223) w {?zYwn {vvgij*D3{\nh.®¯~}z§©{~5§©3awnz|{~au (v, q) ∈ [H (Ω)] × H (Ω) ¾ 3 2 +. 2 def. 1. 1 2 +. d. 1. 1. ||](v, q)[|| = kν 2 ∇vk20,Ω + kh 2 ∇ · vk20,Ω + kh− 2 qk20,Ω + kh. − 21. vk0,Ω + k(νh). 1 2. ∇vk20,∂Ω. +. @g{~5w e3j0unj¿w¥2{?3{\nhu5¥j¿ea¯®~jµwne3j*§©{\}}|{-¥
(224) zg?~33 {¯¹vzh-w z{\¶ j0un3}³wN² 2 ¼¢
(225) 1 # %& I (%& -%& %& * . . kqk20,∂Ω .. ) 1,. 1. 1. |||(u − Πkh u, p − Πkh p)|||0 ≤ C(ν 2 + h 2 )hru −1 kukru ,Ω + Ch. -. rp − 12. 1. kpkrp,Ω ,. 1. ||](u − Πkh u, p − Πkh p)[|| ≤ C(ν 2 + h 2 )hru −1 kukru ,Ω. !. . + Ch. O% r = min(r, k + 1) - u < " @gn{\h 7 J ¥j*eg¯®~j. ) ,. rp. = min(s, k + 1). rp − 21. C>0. kpkrp ,Ω ,. & A& >Y & Y . )Ô~ , )ÔV7 ,. )Ô\1, . γν γ. 1. kν 2 ∇(u − Πkh u)k20,Ω ≤ Cνh2(ru −1) kuk2ru,Ω ,. ~g. j"w njN-w
(226) wnegj ya{\3g3~nswnj0nhuauxz|3?wnegj"wn~ij"z3jNYg}|z³ws Ô) ,5z|¸ {\h+y3z|gwnz|{~¥
(227) z³w e ) 7 J1,
(228) a w e3j Yg~unz°I3gz³§©{\nh?zws{§9w e3j*wn z3\3}|wnz|{~ )!; ,i¾vstzj0}|3z3 1. kh 2 ∇ · (u − Πkh u)k20,Ω ≤ Ch2ru −1 kuk2ru ,Ω .. ku − Πkh uk20,∂Ω ≤ C. X. k 2 h−1 Ke ku − Πh uk0,Ke. e⊂∂Ω. + hKe k∇(u − Πkh u)k20,Ke ≤ Ch2ru −1 kuk2ru ,Ω ,. . ö agõ
(229) b
(230) cô. . )Ô~ ,.
(231) 7¯. 1 #
(232)
(233) - . ¥
(234) egji j 3ji3{~wnj0u+wnegj unzh?3}|j ¹ ungeªw eg-w ²½d5egjz|Ywnji z|{~+Rjia}ws½wnji hu~nj w njN-w j0Kz wnegj un~hDj¿§Ó\uxe3z|{~~uw e3j"yR{~3gg es?⊂wnji ∂K hui² ∩ ∂Ω j"eg¯®\j e. e. j(u − Πkh u, u − Πkh u) =. X. h2K. K∈Th. X. ≤C. h2K k∇(u. −. Z. J∇(u − Πkh u)K2 ds ∂K. Πkh u)k20,∂K. K∈Th. X. ≤C. hK k∇(u − Πkh u)k20,K + h3K k∇2 (u − Πkh u)k20,K. K∈Th. ≤ C hk∇(u − Πkh u)k20,Ω + h3 k∇2 (u − Πkh u)k20,Ω ≤ Ch2ru −1 kuk2ru ,Ω .. . . mµyt®tz|{~gun}s\¾-wnegj3 j0u ux3 j#3h??wnj0nh zu`wn j0wnjNauxz|3"w e3ju h?j ~3h?j0\wN¾¥
(235) e3ze¶ {\hDg}jiwnj0u^wne3j g n{t{§ {§ )Ó&7,² d{ª3 {-®~j )ÓY ,?¥j¸uxz|h?3}s 3{~wnjwnea-wyts¼w \ jÀz|3j0Yg~}zwnz|j0ugPwnegj¨uwy3z|}|z³wsP{§*wne3j L ° gn{~j0w z{\¶w e3ji j"e3{~}3u 2. 1. 1. k(νh) 2 ∇(u − Πkh u)k20,∂Ω ≤ Ckν 2 ∇(u − Ihk u)k20,Ω X 1 kν 2 (u − Ihk u)k22,K + Ch2 K∈Th 2ru −2. d { {\g }|gvj\¾v¥2j"g3}swne3j"z|3jN\a}|z³ws)Ó~ ,2wn{?wnegjµw ji h kp − Π pk ² @ z|g}|}s\¾-¥2juxea}|}Y~}|un{h¦\jUgunjU{~§vwne3j2§©{~}|}{-¥
(236) z|3gn{~j0w z{\ {~Rji-w {~N¾0ya~unj0"{\+µqYw {~¦\j0ux°I}|z¦\j gn{\y3}j0h² @3{\jN~e u ∈ [H (Ω)∩H (Ω)] ∩H (div; Ω) ¾-¥2jvjig{wnjyts S u = (P u, R u) ∈ wnegj"33z\gj ux{\}vw z{\{~§ W ≤ Cνh. kuk2ru ,Ω .. k h. 3 2 +. 1 0. k h. d. 2 0,∂Ω. k h. 0. def. k k k (Ph u, vh ) + ah (Ph u, vh ) + bh (Rh u, vh ) + γj(Phk u, vh ) = (u, vh ) + ah (u, vh ), − bh (qh , Phk u) + j(Rhk u, qh ) = 0,. k h. k h. )Ô ,. §©{\}|} (v , q ) ∈ W ² ks \unun3h?zg¨w eg-w zu}ux{½ux".izj0Ywn}|s¼nj0~3}Dz| w zh?j~¾5un{6w eg-w?w e3j3n{~j0 wnz|{~ h~¦~j0u unjiauxj?-w*j0\eÀwnz|h?j t ¾;¥ujDeg¯®~j+wne3jD§©{~}|}|{-¥
(237) z3 3gn{¯¹vz|h?wnz|{~À j0un3}w0¾;¥
(238) e3{YuxjD3 {Y{~§^z|u*ya~unj0À{\ w e3j" j0un3}³wu5nj0a{\xw j0zZ 587N 9»² ¢ u ∈ [L (0, T ; H (Ω) ∩ H (Ω)] ∩ H (div; Ω) -%&T" >Y
(239)
(240) 2Â ¼ h. k h. h. " %& . %V 8C 2. Phk. r. 1 0. d. . 1. O%. 0. 1. ku − Phk ukL∞ (0,T ;L2 (Ω)) ≤ C(ν 2 + h 2 )hru −1 kukL2 (0,T ;H ru (Ω)) ,. C>0. - & & ". ν. -. )Ô\1,. h. ê &êìë .
(241) 7N.
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(244) z³w enjNuxRj0 w#wn{*wnz|h?j~¾\¥2j5{~y3w z|Dwne3junz|hDz|}` j0un3}³w#§©{\#wne3j
(245) w zh?jvj0nz|®--w z®\j {~§9w e3j"3 {j0 wnz|{~&² Æ ¡ Æ 0 0Ó¢v¡ Ã O% ∂ u ∈ [L (0, T ; H (Ω))] u ∈ [L (0, T ; H (Ω)∩H (Ω)∩H (div; Ω))] 2. %&S%& " >6
(246) $
(247)
(248) ! # %V C r. 1 0. d. 0. 1. t. 2. ru. 1. k∂t (u − Phk u)kL∞ (0,T ;L2 (Ω)) ≤ C(ν 2 + h 2 )hru −1 k∂t ukL2 (0,T ;H ru (Ω)) .. )Ô~ ,. : '"( i l±¶wne3zu2unj0 wnz|{~¶¥jz|t®~j0uxwnz|\wnj
(249) w e3j¥ji}|}|a{YuxjNv3j0u uU~gun{~h?jµuxw ~y3z}|zws3 {~Rjinwnz|j0uU{§;w e3jµvzu j wnj u e3j0hDj ) J ,² . . " ". . . " " !
(250) !
(251) !#". l± w e3j"uxjN\gji}I¾3¥2j"uneg}|};h¦\j¿auxj*{§ wnegj*§©{~}|}{-¥
(252) z|3?vz|u j w j*3njNunun3 j¿~g ®~j0}{v zws¶un3ygung~ij0u<] def. 1 Ch,k =. def. div Vh,k =. . . qh ∈ Qkh : j(qh , qh ) = 0 ,. vh ∈ [Vhk ]d : bh (qh , vh ) = 0,. l±À\3vzwnz|{~&¾ Q \C ¥
(253) z}|}&uwg§©{\5wne3j un333}|jih?j0\w s{§ k h. 1 h,k. 1 ∀qh ∈ Ch,k .. 1 Ch,k. z|. Qkh. ¾3zI² j~²|¾. 1 1 Qkh = (Qkh \Ch,k ) ⊕ Ch,k .. d5e3j*§©{\}}|{-¥
(254) z|3}|jih?h?jiaux3 j0u0¾vz|g~xw z|i3}N¾twnegw V z u
(255) 3{~w5wn z®tz} )©zI² j~² ¢ %V )A
(256) $ V
(257) & β > 0 T V-A " h
(258) 1%?O%V 2Â ¼ div h,k. ! < "!9j w. inf. sup. 1 qh ∈Ch,k vh ∈[Vh ]d. 1 qh ∈ Ch,k. |bh (qh , vh )| ≥ β. kqh k0,Ω kvh k1,Ω. ²T@gn{\h 5 ~v¾-P2{\n{\}} s¶v² 9»¾twne3j0nj*j ¹vzuwu. vq ∈ [H01 (Ω)]d. d5etgu0¾vgunzgDz|Ywnj0~-wnz|{~ytsg~xwu
(259) g) 7N1,¾3¥j*eg¯®~j ∇ · vq = q h ,. kvq k1,Ω ≤ Ckqh k0,Ω .. kqh k20,Ω = (qh , ∇ · vq ) = (qh , ∇ · vq − ∇ · Πkh vq ) + (qh , ∇ · Πkh vq ) = (∇qh , vq − Πkh vq ) − hqh , (Πkh vq ) · ni∂Ω + (qh , ∇ · Πkh vq ) = (∇qh , vq − Πkh vq ) − bh (qh , Πkh vq ).. . ö agõ
(260) b
(261) cô. div Vh,k 6= {0}. uxgewnea-w. ,². )Ô; , )Ô[ ,. d.
(262) 70. 1 #
(263)
(264) - . l±ªanwnz 3}N¾9gunz|3Àwnegj¶{~nwne3{\~{\g}|z³ws¨{§5wne3j a² v¾gg unzgij j(q , q ) = 0 ¾v¥j*~j w h. L2. °»3 {jNwnz|{~9¾\PaeYsY°qvet¥~xw ¬D~g &j0h?h?. h. |(∇qh , vq − Πkh vq )| = |(∇qh − Πkh (∇qh ), vq − Πkh vq )| ≤ k∇qh − Πkh (∇qh )k0,Ω kvq − Πkh vq k0,Ω 1. ≤ γ3 h− 2 j(qh , qh )kvq − Πkh vq k0,Ω. d5etgu0¾t§© {~h Ô) [ , ¾3z§§©{\}}|{-¥u2wnegw. = 0.. l±À\3vzwnz|{~&¾t§© {~h ) <7 [1,5~g Ó) >; ,¾3¥j*eg¯®~j. |bh (qh , Πkh vq )| = kqh k20,Ω .. kΠkh vq k1,Ω ≤ Ckvq k1,Ω ≤ Ckqh k0,Ω ,. ¥
(265) egz|e {\hDg}jiwnj0uw e3j"3 {t{§² j"3{-¥puxw wnj*w e3j*h?~z j0un3}w5{~§w e3zu
(266) g\ ~3e&² ÂtÆ ¡ Â %V - 0 ) J , % u ∈ V V% )1 /1& 1 ! ! < "^Un{\y3}j0h ) J1,5i~ yRj*¥
(267) nzwxw ji&¾gz{~Rji-w {~§©{~ h¾3~u z| [V ] , M ∂ u + A(u )u + B p = M f , z [Q ] , Bu = Jp , div h,k. 0,h. t h. h. T. h. h. ¥
(268) zwne. . M ∈ L [Vhk ]d , [Vhk ]d J ∈ L Qkh , (Qkh )0. . vj «a3j0 yts. 0. ¾. h. uh (0) = u0,h , . )ÔJ ,. k d 0 h k 0 h. h. A ∈ L [Vhk ]d × [Vhk ]d , [Vhk ]d. 0 . ¾. B ∈ L [Vhk ]d , (Qkh )0. . a. def. hM uh , vh i = (uh , vh ), def. hA(wh )uh , vh i = ah (uh , vh ) + ch (wh ; uh , vh ) + jwh (uh , vh ) + γj(uh , vh ), def. hBvh , qh i = b(qh , vh ),. . def. hJph , qh i = j(ph , qh ). 1 B 1 ∈ L [Vhk ]d , (Ch,k )0. j }un{z|Ywn {t3g j¿wnegj"{~Rji-wn{\. def. hB 1 vh , qh i = bh (qn , vh ),. vj «ggj0 yts. 1 ∀(vh , qh ) ∈ [Vhk ]d × Ch,k ,. ê &êìë .
(269) 7¯.
(270) ! #" $%&(') *
(271) +,- .$0/1& ! &. z|{w e3ji
(272) ¥{~3u0¾ def. 9jih?h¶v²87¾3zw§©{~}|}{-¥uw eg-w B zuun3ÓjNw z®\j g (B ) z|uz-jNw z®\j+)Óunjij5äv¾ag\j"[ 9 ,² j*wnegjiÀvj03g j¿wnea-w V = ji (B ) 6= {0} ² 9j wgui{~gunzvjiwne3j¨§©{~}|}{-¥
(273) z|3¼ j0va j0K§©{~ h+3}-w z{\ )Óvj0nz|®~jN §©n{\h ) J ,¶¥
(274) z³w e (v , q ) ∈ ,
(275) ]#«ga (u (t), p˜ (t)) ∈ V × (Q \C ) ungew eg-w V × (Q \C ) z| V , M ∂ u + A(u )u + B p˜ = M f , )Ó\ , z Q \C , Bu = J p˜ , B 1 vh = (Bvh )|C 1 ,. @gn{\h. ∀vh ∈ [Vhk ]d .. h,k. 1. 1 T. div def h,k. 1. h. div h,k. 1 h,k. k h. h. div h,k. h. t h. h. T. h. div 0 h,k. h. h. qvzgij~¾vytsi{~guxwn gw z{\&¾ §© {~h )©\ , ¾v¥j*eg¯®~j C. 0 1 h,k. k h. h. uh (0) = u0,h .. 1 h,k. =. ji (J) ¾3¥j" {~a }|gvj¿wnegw J zu5zt®\jinwnz|y3}j¿z|. 1 Qkh \Ch,k. −1 p˜h = J|Q k \C 1 Buh .. ksª3}|3\~z|3¸w e3z|uDji¹tgnjNununz{\¼z|\w {¨w e3j «guwDj0Yg-w z{\ {~§)Ó\1, ¾U¥j{~y3w z|¼w eg-w un{~}|®~jNu z| V , M ∂ u + A(u )u + B J Bu = M f , h. t h. h. T. h. h. 1 h,k. k h. ²T4jigij~¾ )Ó& 7 ,. h,k. −1 1 |Qk h \Ch,k. div uh (t) ∈ Vh,k. div 0 h,k. h. uh (0) = u0,h ,.
(276) ¥ egz|ePz|u?¨uxw ~g3~ :P~gets½3n{\y3}|jih §©{~ ²ª^¹tzuxwnjia jg 33z\gji3jNunuD{§ §©{~}|}|{-¥u+yts w e3j 9zaune3zwn¬ {~Yw zt3zws{~§ A ² j+h¯swne3j0u¸ j0 {-®\ji p˜ 3gz|Y3j0}s §© {~h )©&7 , ²d5uegji j §©{~ j~¾gwne3j j03g jNgn{\y3}j0h )Ó\ ,
(277) eg~u¶33z\gj+ux{\}3wnz|{~&²mµ wne3j {~wne3j0eg~g;¾g§©n{\hwnegj"«guwjNYg-w z{\ {~§ )Ó\1 , ¾3z§§©{\}}|{-¥uwnegw ji (B ) , M ∂ u + A(u )u + B p˜ − M f ∈ ¥
(278) zwne ji (B ) uxw avzg§©{~*wnegja{\}|~ uxjiw {§ ji (B ) ² @gn{\h &j0h?h?¸3 ² 7~¾zw"§©{~}|}{-¥u¿w eg-w zu*Àzux{\hD{\nge3z|unh §©n{\h C {\\w { ji (B ) )Ôuxj0j 5äv¾[g~~j? [ 9 ,²¿d5etgui¾Rw e3ji j+ji¹tzuxw u B g3z|Y3j p ∈ C uxge w eg-w z| [V ] . )ÓY1 , M ∂ u + A(u )u + B p˜ − M f = (B ) p d5egji j §©{~ j~¾§©n{\h )©t ,`~g )©\ , ¾YgDyts+3{~wnz z|3*wnea-w (B ) p = B p ~g Jp = 0 ¾~z§[§©{~}|}{-¥u w eg-wgn{\y3}j0h ) J ,eg~u
(279) ?3gz|Y3j"un{~}|vwnz|{~&¾v\z®\ji yts (u , p = p˜ − p ) ²
(280) Â ¼¢v¡ G A YV
(281) 1 *C Z%& " ) %& \ u = P u ∈ h. h. h. t h. 1. h. T. h. 0. 1. 1. 1 h,k. 1. 1. 0. 1 h,k. t h. h. h. T. k d 0 h. 1 T 1. h. 1 T 1. T 1. def. h. div Vh,k. . ö agõ
(282) b
(283) cô. 0. 1. h. h. h. 1. 1. 0,h. k h. 0.
(284) 7N. 1 #
(285)
(286) - . . ! . d5egj"§©{~}|}{-¥
(287) z|3 &j0h?h?¶3n{-®tzvj0u
(288) i{~Ywn {~} )©33z§©{~ h z ν ,{§#wnegj+vz|®~ji ~j0g j"i{~guxwnz|Yw
(289) wne3 {~3\e w e3j uxw ygz}|z¬N-w z{\w ji h?u0² ¢ RÂ ¡ RÂ £ Â £ Æ i¡ Æ 0
(290) 1 # 2Â ¼ 0? 1 ! _ " ) J , %V A
(291) &
(292) & C > 0 I V- ? S%& (u, %p A) ∈#W
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