Two Level Correction Algorithms for Model Problems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Two Level Correction Algorithms for Model Problems Jichao ZHAO — Jean-Antoine DÉSIDÉRI — Badr ABOU EL MAJD. N° 6246 July 10, 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6246--FR+ENG. Thème NUM.
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(33) )9. f0rtdÃr{_'r vptdc . Enn0. _p. n+1. x{'¬0p u. n0 + 1. vabeupc¾Èlq3dcyzA:3y}}noz}uWvax dy¯q3dyx{z}bdcuCr{p ¬®d _°H±d. E48 = E78 E67 E56 E45 ,. Ëax{b&iL'r{xtz¯ ®ªvvpco¬®d q= ua'¬]rt_r ∂J(Y 0 ) ∂Y 0. ^ _adcxtdy²·x{d . =. ∂Y ∂J(Y 0 ) . ∂Y 0 ∂Y. ∂Y = (Enn0 )T , ∂Y 0. u°. ∂J(Y 0 ) = A(Yk + Enn0 Y 0 − Y¯ ) , ∂Y. É~Ê É'Ê É~Ê É-;Ê. ^ _qvp Æqn@d~|u°pÉHÊÂÁÉscÊ uÉ ;ÊÂa¬®dÃÆor z}u ∂J(Y 0 ) ∂Y 0. ∂Y ∂J(Y 0 ) ∂Y 0 ∂Y = (Enn0 )T A(Yk + Enn0 Y 0 − Y¯ ) = 0 . =. ¨ u/r{_adyx ¬®x apcC¬®dÃ_H±drt_dò·}'¬ z}uaeuady¬ b+'r{xtz¯Qdc|Cvrtz}u Acy Y 0 = bcy ,. ÉJa~Ê. Acy = (Enn0 )T AEnn0 ,. ÉJÊ. _adcxtd rt_d ÌQyzdcur b+'r{xtz¯ u°Qr{_adxtz}_Cr0ptz}od±dcrtx. ÉJÊ. JÉ .5Ê Sdyu pt}±d"r{_adW²·}}'¬ zua¿zrtdcx{rtz}u ur{_adCx psd/}dy±dy ²·x@xtx{dcrtz}uÆCn;zuzµr{z}z}¸yz}ua bcy = (Enn0 )T A(−Yk + Y¯ ) = (Enn0 )T (b − AYk ) .. Y00 = 0. . JÉ CÊ rt_dyuL¬®d«cuWvaIa'r{d«Æqn Y + E Y uWr{_ad :ud}dy±dcJ¾ ¨ rtdcx{rtz}up uWrt_d :uad«dc±dc®ÉscÊhu xtx{dcrtz}upÈu"r{_adx{ptd dc±dySÉ5'Ê ybea}dr{dër¬®³´dc±dcIyx{xtd~Âr{zuo³Jrnq3dhzªodcÁ}xtzrt_b ²·x rt_d+uaua}z}uadcxboodcSax{Æa}dyb/¾h'¬Èdc±dyxÃzrr{.=dcpÃb+uCnzµr{dyx 'rtz}u°p+É·_qvauox{dcpò·xÃr{_az}p boodc ax{Æa}dybQÊrt« _az}dy±dbea}dr{dyuq±dcxtdyuyd¾Á^«pt°dcdc@va+rt_dx 'r{d0²Áyuq±dcxtdyuyd¬®dcu+vptd Y 0j+1 = Y 0j − ρ(Acy Y 0j − bcy ) ,. K. n n0. 0. (*)U+(-,.
(34) .
(35)
(36) "!$#&%'(*)+-,.
(37) 0/12.3
(38) 4(#&%. "Æ°dyrsr{dyxr{dc _auazª|Cvad^ _adcÆqnq _dy±Lzrtdcx{rtz}up¹ '½Sur{_ad:uad+}dy±dcJÁzJ¾ d¾}Ázr _p r{_ax{dyd+pr{dyp²·x dc _"yno}d Y j1 = Y j0 − τ1 (AY j0 − b) ,. ¬ _adcxtd. Y j2 = Y j1 − τ2 (AY j1 − b) , Y j3 = Y j2 − τ3 (AY j2 − b) ,. É. τi i = 1, 2, 3. ÊÈx{dÃz}±dyu"Æqn. . JÉ »Ê ¬ _adcxtd [a, b] zªp rt_d«r x{dyrsrtd~Lz}uCrtdyx{±'8z}ur{_adedyz}dyuq±'}vad λ ² A u r = 0, ±√3/2 @x{qrò rt_dQ^ _adcÆqnq _dy±W3}nCubez}8²®adyx{dyd 5a¾ S _adcu¬Èdeyupt}±deCx psdextx{dcrtz}upd¯art}nWu rt_d«Cx psdÃ}dy±dy5¬®dyu/yb«Æazudrt_ad^ _dyÆqno _ady±Qzrtdcx{rtz}up«ÉJ»Ê ¬ zrt_yx ptdÃyx{xtd~Âr{zup u rt_d x{ptd dc±dyGz}u"rt_adÃb+'r{xtz¯Q²·xtb ¹ H#½ ÉJÊ G = G (I − E ((E ) AE ) (E ) A)G , ¬ _adcxtd G = (I − τ A)(I − τ A)(I − τ A) ¾ ¨ u@rt_adcx ¬Èx{apcrt_adÃbdyrtdÃno}dz}u>rt_ad²·x{b\² b+'r{xtz¯ G z}p z±dyu"ÆCn ÉJCÊ Y =G Y +b , _adcxtd ÉJÊ b = G (b − E ((E ) AE ) (E ) (Ab − b)) + b , u° ÉJ. ;Ê b = ((I − τ A)(I − τ A)τ + (I − τ A)τ + τ I)b , uartd rt_r rt_adxtz}z}u3b+rtx{zµ¯@xtÆadcb 9u :uad}dy±dyGzªp z}±dcu>Æqn AY = b ¾ 1 b+a b−a = + ri τi 2 2. i. y. 3. h. n n0. h. 2. n T n0. h. 3. h. gy. n −1 n0. 2. 1. n T n0. 3. h. 2. h. 3. # &-. $ "$#%
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(40) "+-.. Ëax rt_ad yx{ptd}dy±dyÀyx{xtd~Ârtz}u@² J(Z 0 ) =. ¬ _adcxtd. j y g. n T n0. n n0. h. h. . n T n0. 1. j+1 g. gy. n −1 n0. Q0 = Ωn Pn ΩTn. Z. Z0. Pn. . Pn = . UUWVXZY[\(Y. É5Ê. 1 (Bn (t)T (Yk + Q0 Enn0 Z 0 − Y¯ ))2 dt , 2. u°@r{_adb+'r{xtz¯ γ. bedyrt_aoGa¬®dÃvptdr{_adò·}}'¬ zua+p{ _adybed zªp®rt_ad3dyx{bvar{'r{zu>b+rtx{zµ¯ ¾¾¾ 1. 1. 1. 1 . . (n+1)×(n+1). .
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(42) )9. ^ _ad+zªodc"²Èr{_ad bdyrt_aozªpÃrtWx{dy±dyx psd«rt_ad+°z}xtz}ua/Æ°dyr¬Èdcdyudyz}dcuC±'vad~pudczdyuq±d~Ârtx{p Æqn/bvartz}a}nCz}ua@rt_aZdb+'rtx{z¯ Q ptQrt_°'rªx{dcxhdczdyuq±'}vadcphz}x¬ zrt__az_adyx²·x{dc|Cvadyu°n/uWr{_ad Cx psdÈdc±dcau«xtdc}¯a'rtz}u°pcuxtdcb'±d_az}_«²·x{dc|Cvadyu°n dcxtx{x pÁdyÌQz}dyuCrt}nQ¹¼»H½´¬ _az}d Y bdyrt_ao oqdcp urc¾ IÀdyr É 5~Ê Z = Y − Y¯ + Q E Z dy|¯qrtx 3{zª'rtd~} ax{dcuozrtz}uad~ yx ptd³´dc±dyIxtx{dcrtz}u Ô®n@iLrtx{zµ¯ ®}yva}vpyo¬®Pd =qua'¬ r{_'r É 5CÊ ∂Z ∂J(Z ) ∂J(Z ) = . ∂Z ∂Z ∂Z ^ _adcxtdy²·x{d É 5 5Ê ∂Z = (Q E ) = (E ) Q , ∂Z u° É 5CÊ ∂J(Z ) = A(Y + Q E Y − Y¯ ) . ∂Z ^ _qvp Æqn@d~|u°pÉ 5ÊÂÁÉ 5.5Ê uÉ 5CÊÂa¬®dÃÆor z}u É 5C»Ê ∂Z ∂J(Z ) ∂J(Z ) = 0. 0. 0. n n0. 0. k. 0. 0. 0. 0. 0. 0. 0. n T n0. n T n0. 0. 0. 0. k. 0. ∂Z 0. n n0. 0. 0. ∂Z 0 ∂Z = (Enn0 )T Q0 A(Yk + Q0 Enn0 Z 0 − Y¯ ) = 0 .. ¨ u/psvabeb+x{no¬®dÃ_H±d r{_adò·}'¬ z}uauady¬ b+'r{xtz¯@dc|Cvrtz}u Acz Z 0 = bcz ,. É5Ê. Acz = (Enn0 )T Q0 AQ0 Enn0 = (Enn0 )T A1 Enn0 ,. É5qÊ. _adcxtd rt_dbertx{zµ¯ u°Qr{_ad±d~Ârtx. É 5Ê uartd0r{_'r b+'r{xtz¯ A = Q A Q qu+±dcÂr{x b − A Y zªp® x{dcptz}ov°u+rt_d :ud}dy±dy5¾ ¨ u>d(@IdcÂr~ ¬®d_°H±dxtd~u°ozµr{zuadcrt_dx{dcptzªov b − AY ÆCn Q ¾SÇde²·vaurt_rÃb+rtzªdcp A u A _H±d oz4@Idyx{dyuCr0dyz}dcuC±'vad~pyqÆavar0rt_adp{bedÃdyz}dcuC±dcrtx pc¾ bcz = (Enn0 )T Q0 (b − AYk ) ,. 1. 0. 0. k. k. 0. cy. cz. (*)U+(-,.
(43)
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(46) 0/12.3
(47) 4(#&%. ;. ^ _adcu+¬Èd0cu+ps±d rt_ad0²·}'¬ z}uazrtdyx 'r{zueuert_dhCx psd }dy±dco²·x®xtx{dcrtz}uÆCn«zuzµr{z}z}¸yz}ua Z00 = 0. . É 5 ;Ê urt_d :uad>dc±dcJ¾ Sd>yu©ªpsWv°psd@^ _adcÆCno _adc±. Z 0j+1 = Z 0j − ρ(Acz Z 0j − bcz ) .. Ëzu°}n¬Èd>yu¿v3artd>ÆCn zrtdyx 'r{zup u>rt_d*:uad}dy±dcJ¾ Y + Q E ¨ u"vax dy¯q3dyx{z}bdcuCr{pco¬Èd r{.=d K. 0. n 0 n0 Z. ·É CÊ pt«r{_'rhI²·x{dc|Cvadcun@bood~p0x{dÃax{dcptdyuCr zu"rt_d psvor{zuÀo¬ _azª _>b+.=dcp ybeavor 'r{zuGx{dcptvar{p u°>vax0u°}nqptzªpÈbextdÃdcuadyx 5¾ S _adcu"¬®d u°}nC¸cdvax0uqvabedyx{zªyGdy¯o°dcxtz}bedyuCr{pco¬Èd ad(:uadÃr{_ad Æor{zuad~eÆqn Bsdyx{xtx É·xÈzrtdcx{rtz}±d0dcxtx{xÂÊrt«Æ°drt_doz4@3dcxtdcudÆ°dyr¬Èdcdyu+r{_adÃvax{xtdcuCrdcpsrtz}b+'rtd Y oz4@3dcxtdcuCr0bdyrt_aoap u>rt_adÃr{xtvdÃpt}vor{zu Y¯ ²Árt_d oz}p{x{dr{dÃaxtÆa}dyb/¾ Sdhvptd ^ _adcÆCno _adc±«zrtdcx{rtz}upu@rt_ad :uad dc±dyIu°Qps±du}nCrtzªy}}n«u+rt_adÃCx psdhdc±dy5 z5¾ d¾Cvptd rt_ad G ynqydr{_adyu+¬®d a}rdcxtx{x p±qpSrt_adhuCvbÆ3dyxS²3zrtdcx{'r{zupz}uQËz°¾Èu²·xtd~|Cvadyuyn ±op8rt_d0uqvabÆ3dyxȲIzµr{dyx 'rtz}u°pzuQËz}¾o¾ ¨ uQËz°¾® (a) pt_a'¬0pz}uazrtzªadcxtx{x p8x uoben dyuadcx{rtd~G u pt_a'¬ dyx{xtx{p Æor z}uadcÆqn Y u Z bedr{_aoap'² rtdcx«uadSptz¯u¿dy}dy±dyu G (b) (c) noyd~p«x{dcpt3dcÂ(d) r{z±dy}n¾©Ëxtb r{_adyb/ȬÈd/yu dcpsz}}n©psdcd@r{_'rrt_ad/x{dcptvar{p«²hr{_ad bedr{_ao;xtd ptva°dcxtz}x~¾0f0'¬-}dr vp0}q.=>rt_ad~psd«xtd~psvar{phzuLrt_ad²·xtd~|vdyuyn"ptyd3Ëz}¾ +}dcxt}Zn>dy¯oa}zu°p0¬ _qn bedr{_ao/z}ppsva3dyx{z}x r{_u bedr{_aoG¾0Ä ² r{dyxhuad noyd@ÉJËz°¾ Ê°¬®dcuWpsdcdrt_°'r Z bedr{_aozªp bxtd«dÌQz}dyuCr ²·xÃYx{dcovyzu@_az}_²·xtd~|vdyuyGn"beood~puLr{_ad (b):uaddc±dy5I¬ _az}drt_d YZ bedr{_aozªpÃyydyor Æa}d ²·xÃxtd~ovz}ua@rt_ade_az}_²·xtd~|vdyuyn"beood~puLr{_ad :uaddc±dy8Æavar}ptQr{_ad }'¬º²·x{dc|Cvadyu°n@beood~py¾Ä0² rtdcxptzµ¯@x0dy}dy±dyu G yno}dcpcaÆ°rt_/²·x{dc|Cvadyu°z}dcp x{dÃx{dcrt}nQxtd~ovydc>ÆCn bedr{_aoG¬ _az}}d ua}n>rt_ad _z_/²·xtd~|vdyuyn>uCrtdyuCr0zªp0x{dcov°dc"Æqn bedyrt_aoG¾hh²Svax{ptd¬Èd Z yuªptQvptdzªodcp0²Ëva}8iLvaµr{zµ³´x{zªWiWdr{_ao ÉÛË8i Êhz}upr{dcW²GYvpsr r¬ÈQ}dy±dyªpyIu°WuWr{_ad Cx psd~pr}dy±dy5Á¬®dQyuvptd@fhdy¬ rtuzµr{dyx 'rtz}u°pÃrtLpt}±d+zrz}upr{dcDz®v°psz}uaWr{_adQoz}xtd~Âr«pt}±dcxc¾ Ä0u°>¬®dupszªodcx rt_ad~psdÃartz}up z}uWptvaÆptdc|CvadcuCrhpsd~Ârtz}u/od~}zuae¬ zrt_Wuauz}uadcx axtÆa}dyb/¾ 1 1 1 Y¯ (x) = 2 sin(πx) + 2 sin(πx) + · · · + 2 sin(nπx) n. j. 0. 0. 0. 0. 0. 0. 0. 0. . 0. !+- .. ! $ ""$#%
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