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A Robust A Posteriori Error Estimate for Elliptic Non-Homogeneous Equations

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(1)A Robust A Posteriori Error Estimate for Elliptic Non-Homogeneous Equations Rodolfo Araya, Patrick Le Tallec. To cite this version: Rodolfo Araya, Patrick Le Tallec. A Robust A Posteriori Error Estimate for Elliptic Non-Homogeneous Equations. [Research Report] RR-3279, INRIA. 1997. �inria-00073410�. HAL Id: inria-00073410 https://hal.inria.fr/inria-00073410 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Robust A Posteriori Error Estimate for Elliptic Non-Homogeneous Equations R. Araya. P. Le Tallec. N ˚ 3279 October 1997 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(190). 3"B. .. . . 0. '. "5'. .. P. ! !  B  B  ".  ". '. . . 3 .. ". .    B  B  .. . .. I€J"B [. ŸOW¡  / Np ® ‹WŠ HI L p pL*SedX^ L0nFZcL0^,diS  ƒ L ‘ \ ƒ p § p L0ƒ VeXZcST^Wƒ „ ”ž–jƒ —"˜™išõ›mƒ œe— ›œ Lsƒ diZcSTM*X"Z[Ld=X"^ f V]ST^WLX LVeX_d’ZcSYkmS]Z’h ª_Ÿ\ LnWXMNaWV]L  v Dª  }

(191) ðª v ðª O

(192) Dª  

(193) Dª  Dª  ðª } D¡ P. '. . P. .. #. ! 5 " "9$. . P. +. ?. ÈeÊË ÈeÍ.

(194) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d }_v &   A¦%¦ ! =q !  Ž ^žZ I SedÉd’LkmZ[S]\_^ ‘ LQXaWaWVThNZ I L p LdiUWVTZ¬d\ldiLkZcS]\b^*OZc\«Z I L=L0VYX_diZ[SekmSTZ’ha p \b‚WV]L0M f S  B - ! STd^ % 8 P - ! \b^;1 I 8 B + /G  \b;^ 1 B 8 T - X \_, ^ 1 8 ! B + /G  \b;^ 1 8 ‘ S]Z I I p  ->f +€X„8 X„[ ƒ / I " X"^ f I  -<p +DT 8`Tf C/ " ¡ 3LÉZ p h I L p LÉZc\=MN\_^jS]Zc\ p Z I LÙL

(195)  \bV]UWZ[S]\b^ \"Z L Ld’S U,X"VˆP X_dÉZ LM`Ld SYd Lm£,^WL ¡ p Nf® p ‹WŠ L*Ÿ\_VTV]\ I ‘  ƒ v

(196) ÉpST^ \ p pf L p Zcp\“f \b‚WZ¬XS]^ I X\_aWp Z[S]M*‘ XVÉMNLd I I“p Lm£,I ^WL0MNL0^FI Z a \FkL —mU –y™ðL œ ª8SðP” ¡ L C ¡ S¤   SedqZ LAX_kkmU Xbkmh p L¸ UWS L ‚FhZ LU,diL ª f Ld[X0hp Z f X"ZlZ LuMNLd 5ÙSed pL cp \ p  L¸ U,X"V S¤Zc \S]Z¬ dÉ ¡ L0VTL0MNL0^FZ¬dq^FUWM‚yL ÙSedÉMNST^WS]MUWM X^ S]ZÉa \ 8S LduX„_VT\_‚yXV Ž ^=Z I SYdK p X"MNL ‘ \ p §CªmŸ\ p LsX_k I LV]L0MNL^8Z :bAA5*ª ‘ LÉkm\bMNaWUWZ[LX p L£y^WL0MNL^8ZWðX_kZc\ p . 1. . #1. ?. ". -. 3 " 3 ". ‘ÉI L p L 3 " SedZ I LQd’S 0LA\"Z I L=L0VTL0MNL0^FZ?: \ 5*ªyX^ f 3

(197) " Z I Lgd’S 0L\qZ I L=L0V]LMNL0^FZ¬dq\" 5 ‘ STZ I ST^*Z I L9:zX p LXU+ŸS]^.}  k0X_d’L /¬¡ f fS]^ SeŽ k0ˆX^WZcL=\`Z diI Z p X\_ZÉ^jZ „NI L=„ p VTX \Fk0STX"L0Vˆ^8kmZ \bdÉ^8X"Z aWp STay‚WLUWX Zcp ST\_S]^*^*ZcZ \«I LgZ I di\_LVTUWL pcZcp ST\_\ ^ p ªWd IZ I \bL0UW^.V f Xdcak0Xp STV]\L=p V]S ST§ L Lpcp \ p LsdiZcSTM*X"Z[Ld "  - 3 " 3  " " . -. ". ‘ÉI L f p Lp  f L0ayL0^ f dQ\_^Z I L`L0V]L0M`L0^8ZAZihFayL6+ - ƒ Ÿ\ p VTS]^WLX p L0V]L0M`L0^8Z dmª-µ}žŸ\ p ¸ U,X HI X"UyZcdSYk0‘ dÉL L0V]I L0M`X

(198)  L0L^8Z Z I dq/L¡ Ÿ\_VTV]\ ‘ S]^W„M`S]^WSTMNS X"Z[S]\_^«a p \_‚WVTL0M    ƒ  MNS]^  - " " ‘ S]Z I " "B "B - B [ . Ë¨Ë Ô (*),+-,..

(199) x ¡ p X0hFXX"^ f o ¡ yL H XV]V]Lsk }"O Ž ^žZ I Lg}  k0X_d’L ªWZ I SYda p \b‚WV]LMµX f M`S]Z¬dZ I L=L0nFaWVTSekmSTZd’\_VTUWZcST\_^. ". -. I€J. . . ! I" " I€J. . ".  I " €I J"B ! " B"J . [. H I p L^WL ‘ MNLsp d I Sedf Z I L^N\_‚WZ X"p ST^WL f ‚FhNX=MNL0Z p SekÉkp \_^FZ p p \bV]VTL f  L0VeXUW^yX0hM`Ld I „bL0^WL p X  Zc\   ˆkm\b^ydiZ X"ST^WL Zc\«„_L^WL XZcL=V]\FkX"V L¸ UWS]VYX"Z[L XVCZ SeX^W„_VTLd\" diS0L " 3 " ¡ HIWp \_UW„ I Z I LudiLskmZcST\_^ % I Xf ^ f % p L0a p LsdiL0^FZ¨Z ‘ \Sed’\_Z p \_aWSYkM*X"Z[L p SYX"VYd ‘ S]Z I I o8 B - ƒ  X X~X;+ \bUW^W„j¶ÕdMN\ UWVTS / ª  I - X„[ O o8  B - Xo[·v + o \_SYdcd’\_^ ¶·dk\ L*kS]L0^FZ¬d`/¬ª X"^ f % B 8Q% X p LZ ‘ \\ p Z I \_Z p \baWSek=M*X"Z[L p SYX"Ved ‘ S]Z I LVeX_d’ZcSYkkm\ L©kmS]L^8Z d„_S  L0^N‚Fh } X X~X X  Xƒ XX X"^ f  Xƒ } X XX~X XX 8 X X } X X } p Ld’ayLp kmZcS  L0V]h¢¡ I Ž ^*Z‘ÉI I L£ p dif ZuL0njX"MNaWVTL + ˆS]„ ƒ / ª ‘ LQ\b^WV]h*k\_^yd’S f L p XUW^WSe¸ UWL=Sed’\_Z p \_ajSek M*X"Z[L SYX"V;\_^$Z L \_VTL \_M*XS]^ ¡ . . . . . P. Γ2. Γ3. Γ1. 1. Γ4. ¨S]„bU p L ƒ  $ njX"MNaWVTL ƒ ¡ HI I SYdža p \_‚WVTL0M p Sed f Sedck p L0Z[S 0L f f Uyd’S]p ^W„ fI p L0V]LMNp L0^FZ¬dW‘ +ðv ^WI \ f Lsd Z p SeX^Wf „_VTILdq/ ª‚jUWZ Z L©dcX"M`LZihFayL\ LdiUWVTZ¬d=Sed bX"V]S Ÿ\ ¸ UyX STVeXZcL X"Ved + L X

(200)  LZ[LdiZ[L Z L`dcX"M`L .

(201) . ÈeÊË ÈeÍ.

(202) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d }  L0X njf X"XMNaWZcaWL VTf LdMNUyLd’d S]I^WL„dlX"^ B f ¸ UyZ I X Lf p f S]VYSYX"diZcZ Lp S]p ‚WX"UjVYdqZc/ S]¡\b^*Ž ^$\"Z I Z IL L=¨LST„d’ZcdS]M©}&’X"v Zc\ ‘ p Lk0S]X^`^3Z diI L0LLgd’L=Z I MNLgLS]d ^WI STZcLSedmX¡ VˆX"^ f INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .255. 1.392. 2.53. 3.667 4.804. INRIA-MODULEF. ¨S]S]„b^*U pZ L«I L } $ njŽ ^WX"S]M`Z[SeaWX"V]V L M`ƒ L¡ d I + _ƒ ƒ žL0VTL0MNL0^FZ¬d`/X^ ff SediZ p ST‚WUWZcST\_^ \Z I LL p[p \ p LdiZ[S]M*XZc\ p . Ë¨Ë Ô (*),+-,..

(203) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. } INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .1115. .317. .522. .728. .933. INRIA-MODULEF. ¨LdiS]Z[„bS]U M*p LXZc\v p  ¨S]ST^N^yX"Z VuI LX f $ X"nWaWXZ[MNL f aWV]MNL Lƒ d ¡ I +  ~ ƒ 0L VTL0MNL0^FZ¬d`/=X"^ ff Sed’Z p ST‚WUWZ[S]\_^\QZ I L$L pcp \ p . ÈeÊË ÈeÍ.

(204) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d } ~ Ž ^ ¨S]„gO ‘ Lua p Ld’L0^FZZ I LSed’\ bXV]UWLdq\qZ I L=di\bV]UWZ[S]\b^N\" $ njX"MNaWVTL ƒ X"^ f ST^ ¨S]„  ‘ Lgd I \ ‘ ^.Z I L\_^.t$SYdiLdÉd’Z p Lsdcd£yL0V f Ÿ\ p Z I LgdcXMNLL0njX"M`aWV]L ¡ INRIA-MODULEF -22.79. -11.393. .00415. 11.401. 22.799. INRIA-MODULEF INRIA-MODULEF -127.834 -42.59 -170.456 -85.212 .0324. INRIA-MODULEF. ¨$ S]nW„bXU MNp LaWV]OL  ƒ ¡ Ž di\ bX"V]UjLd“\" Z I L f SYdiaWVYX_kmL0M`L0^8Z Ÿ+ Z¬X^W„_L0^FZcSYX"V*X^ f ^W\ p *M X"Vf/žŸ\ p Z I L . Ë¨Ë Ô (*),+-,..

(205) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. } INRIA-MODULEF 1.513E-4. 1.399. 2.798. 4.198. 5.597. INRIA-MODULEF INRIA-MODULEF. ¨S]„bU p L . . INRIA-MODULEF.  . VT\_‚yXV X^ f ayX p cZ SeXV 8STL ‘ \ \b^$t$SYdiLdldiZ p Ld[d £yL0V f Ÿ\ p Z I L $ njX"MNaWVTL ƒ ¡ ÈeÊË ÈeÍ.

(206) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d }_ ¨ST^yX"VTV]h ‘ L.d I \ ‘ X“km\bMNayX p Sed’\_^ ‚yL0Z ‘ L0L0^Z I L X"aWa p \n8STM*X"Z[L.di\_VTUWZcST\_^ S]^ Z I L S]UW^W^WSTSÂZcŸSY\ X"p VuM`MNV]h Ld p I Lmª£y^jZ I L fL3MNX f LX"d ajI Zc/¬L ¡ f MNLd I X"^ f \_U p*p LŸL p L0^,kmL.di\_VTUWZcST\_^ +Dk0XVekmUjVeX"Z[L f ST^ X initial solution adapted solution reference solution. 9.7 8.95 8.2 7.45 6.7 5.95 5.2 4.45 3.7 2.95 2.2 1.45 0.7 -0.05 -0.8 -1.55 -2.3 -3.05 -3.8 -4.55 -5.3 -6.05 -6.8 -7.55 -8.3 -9.05 -9.8 -10.55 -11.3 -12.05 -12.8 0. 38.2. 76.4. 114.6. 152.8. 76.4. 114.6. 152.8. initial solution adapted solution reference solution. 28.2 18.2 8.2 -1.8 -11.8 -21.8 -31.8 -41.8 -51.8 -61.8 -71.8 -81.8 -91.8 -101.8 -111.8 -121.8 -131.8 -141.8 -151.8 -161.8 -171.8 -181.8 -191.8 -201.8 0. 38.2. ¨f SeS]X„b„_U \bp ^yL X" V kmUW Z\_¡ MNayX p SYdi\_^.‚yL0Z ‘ L0L0^.Z I L f S L p 0L ^FZ¬dudi\_VTUWZcST\_^ydu\"lZ I L $ nWXMNaWV]L ƒ ST^3X . Ë¨Ë Ô (*),+-,..

(207) vX. x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk HI L=VYX_diZ£,„_U p L f Ld[k p ST‚yLdZ I L=L

(208)  \bV]UWZ[S]\b^N\"Z I L=L p[p \ p Ld’ZcSTM*X"Z[L ¡ 1.3 ’ex1_log’ 1.2. 1.1. log( error estimate). 1. 0.9. 0.8. 0.7. 0.6. 0.5. ¨S]„bU p L~ x Ld’S f UyXV L pcp \ p Ÿ\ p Z I L=L0nWXMNaWVTL ƒ ¡ $p njX"MNaWVTL*} p + ¨ST„ f /«kmp\_^,diS f L f p d«‘ XI d’\"ŸI ZNM©X"ZcL p SeXV ƒ ^jL0S]„ I ‚y\ p S]^WI „ XMN$\ p L p S]„bS f Seƒ d’¡ \_Z \_aWSYkÙM*XZcL SeX"VyOjª SYdck L0ZcS0L S]Z Z L=dcXMNL £,^WS]Z[LL0V]LMNL0^FZ¨Z X"^*S]^ njX"M`aWV]L 0.4. 2. . 2.5. 3. 3.5. 4. 4.5. log( elements). P. Γ2 1 Γ1. Γ. 3. 4 Γ4. ¨S]„bU p L  $ njX"MNaWVTL}F¡ . ÈeÊË ÈeÍ.

(209) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d v ƒ f vSedCZ I LldcXMNLX_dCST^ $ njX"M`aWV]L  H I I p  L 0 L 8 n j a e V " X , ^ " X c Z T S _ \ = ^ " \ ,  Z ¨ L , £ _ „ U  L Ì d T S  ^ 0 L j n " X N M j a ] V  L y d É } " X ^ ƒ¡ INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .001819. .2006. .399. .598. .797. INRIA-MODULEF. ¨S]S]„b^*U pZ L«I L  $ njŽ ^WX"S]M`Z[SeaWX"V]V L=M`}FL¡ d I + ƒ O žL0VTL0MNL0^FZ¬d`/X^ ff SediZ p ST‚WUWZcST\_^ \Z I LL p[p \ p LdiZ[S]M*XZc\ p . Ë¨Ë Ô (*),+-,..

(210) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. v_} INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .00466. .0381. .0716. .1051. .1385. INRIA-MODULEF. ¨LdiS]Z[„bS]U M*p LXZcƒ\ Xp  ¨S]^NST^yZX"I VL X f $ X"nWaWXZ[MNL f aWV]M`L L}8d ¡ I + ƒ ~X"O.L0V]L0M`L0^8Z dq/ X"^ f3f Sed’Z p S]‚WUWZ[S]\b^ \"lZ I LL pcp \ p . ÈeÊË ÈeÍ.

(211) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d. v_v. INRIA-MODULEF -.303. -.2079. -.113. -.01819. .0766. INRIA-MODULEF INRIA-MODULEF -1.558. -1.169. -.779. -.39. 9.465E-5. INRIA-MODULEF. ¨$ S]„bU p L ƒbƒ  Ž d’\ _XV]UWLsd$\"*Z I L f Sed’aWVeXbkmL0MNL0^FZ,+ Z X"^W„bL0^8Z[SeXV«X"^ f ^W\ p *M XV /NŸ\ p Z I L nWXMNaWV]L }8¡ Ë¨Ë Ô (*),+-,. .

(212) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. v"O INRIA-MODULEF 7.177E-4. 9.648. 19.294. 28.941. 38.588. INRIA-MODULEF INRIA-MODULEF. INRIA-MODULEF. ¨S]„bU p L ƒ }  V]\b‚yX"VX^ f ayX p cZ SYX"V FS]L ‘ \" \_^ t$SediLsdÙd’Z p Lsdcd £yLV f Ÿ\ p Z I L $ njX"M`aWV]L }8¡ ÈeÊË ÈeÍ . .

(213) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d. v. initial solution adapted solution reference solution. 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 -0.11 -0.12 -0.13 -0.14 -0.15 -0.16 -0.17 -0.18 -0.19 -0.2 0. 38.2. 76.4. 114.6. 152.8. 76.4. 114.6. 152.8. initial solution adapted solution reference solution 0.225 0.15 0.075 0 -0.075 -0.15 -0.225 -0.3 -0.375 -0.45 -0.525 -0.6 -0.675 -0.75 -0.825 -0.9 -0.1 -1.05 -1.125 -1.2 -1.275 -1.35 -1.425 -1.5 -1.575 -1.65 -1.725 -1.8 0. 38.2. ¨f S]„bU p L ƒ v   \_M`ayX p Sedi\b^*‚¥LZ ‘ L0L0^NZ I L f S L p 0L ^FZ¬dldi\bV]UWZ[S]\_^,d\"Z I L $ njX"MNajV]LA}QS]^.X SeX„_\b^yX"VkmUWZ¡ Ë¨Ë Ô (*),+-,. .

(214) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. v 0.5. ’ex2_log’ 0.4. log (error estimate). 0.3. 0.2. 0.1. 0. -0.1. ˆS]„_U p L ƒ O  x Ld’S f U,X"V L p[p \ p Ÿ\ p Z I L=L0njX"MNaWVTL}F¡ HI L f XZ¬XŸ\ p $ nWXMNaWf V]L vÙX p L„_f S  L0^=f \_^ ˆS]„ ƒ F¡ HI SYd p L0a p LdiL0^FZ¬d;X M`\ p L p LXV]Sed’ZcSYk ‚yLX"M'diLskmZcST\_^ p L0ST^8Ÿ\ p kmL ‚Fh*X p \ X"^ X"^ X"^WSYdi\bZ p \baWSek=VeX0h L p ¡ -0.2. 2. . 2.5. 3. 3.5. 4. 4.5. log (elements). P Γ2 2 Γ1 Γ3. 1. 3. Γ4. 4. 3. 2. ¨S]„bU p L ƒ   $ jn X"MNajV]LAv8¡ . ÈeÊË ÈeÍ.

(215) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d. v~. INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .02742. .812. 1.597. 2.382. 3.168. INRIA-MODULEF. ¨S]S]„b^*U pZ LI L ƒ  $  njŽ X"^jM`S]ZcaWSYX"V]L=V;MNvF¡ Ld I D+ v OLV]L0MNL^8Z dq/ˆX^ f$f SYdiZ p S]‚jUWZcST\_^*\"Z I L=L p[p \ p LdiZ[S]M*XZc\ p . Ë¨Ë Ô (*),+-,..

(216) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. v INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .0117. .2579. .504. .75. .997. INRIA-MODULEF. ¨LdiS]Z[„bS]U M*p LXZcƒ\ ~p  ¨S]^NST^yZX"I VL X f $ X"nWaWXZ[MNL f aWV]M`L Lv8d ¡ I + _ƒ ƒ O v*L0V]L0M`L0^8Z dq/ X"^ f3f Sed’Z p S]‚WUWZ[S]\b^ \"lZ I LL pcp \ p . ÈeÊË ÈeÍ.

(217) p \_‚jUydiZ ”Q–j—"˜™’š›mœe— ›œ L pcp \ p Ld’ZcS]M©X"ZcLuŸ\ p ^W\b^ I \_MN\b„_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d. v_. INRIA-MODULEF -16.262. 7.211. 30.683. 54.156. 77.629. INRIA-MODULEF INRIA-MODULEF -321.637. -241. -80.409 -160.819. 0. INRIA-MODULEF. ¨$ S]„bU p L ƒ   Ž d’\ _XV]UWLsd$\"*Z I L f Sed’aWVeXbkmL0MNL0^FZ,+ Z X"^W„bL0^8Z[SeXV«X"^ f ^W\ p *M XV /NŸ\ p Z I L nWXMNaWV]L v8¡ Ë¨Ë Ô (*),+-,. .

(218) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. OX INRIA-MODULEF 3.234E-5. 24.036. 48.073. 72.109. 96.146. INRIA-MODULEF INRIA-MODULEF. INRIA-MODULEF. ¨S]„bU p L ƒ   V]\b‚yX"VX^ f ayX p cZ SYX"V FS]L ‘ \" \_^ t$SediLsdÙd’Z p Lsdcd £yLV f Ÿ\ p Z I L $ njX"M`aWV]L v8¡ ÈeÊË ÈeÍ . .

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