A Robust A Posteriori Error Estimate for Elliptic Non-Homogeneous Equations
Texte intégral
(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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(6) . . . . . . . . Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : (33 1) 39 63 55 11 – T´el´ecopie : (33 1) 39 63 53 30.
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(206) p \_jUydiZ Qj"me L pcp \ p LdZcS]M©X"ZcLu\ p ^W\b^ I \_MN\b_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d }_ ¨ST^yX"VTV]h L.d I \ Xkm\bMNayX p Sed\_^ yL0Z L0L0^Z I L X"aWa p \n8STM*X"Z[L.di\_VTUWZcST\_^ S]^ Z I L S]UW^W^WSTSÂZcSY\ 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 LL 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]Xb_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 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 STyLdZ I L=L
(208) \bV]UWZ[S]\b^N\"Z I L=L p[p \ p LdZcSTM*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 LdS f UyXV L pcp \ p \ p Z I L=L0nWXMNaWVTL ¡ $p njX"MNaWVTL*} p + ¨ST f /«kmp\_^,diS f L f p d« XI d\"I ZNM©X"ZcL p SeXV ^jL0S] I y\ p S]^WI XMN$\ p L p S]bS f Se 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 Qj"me L pcp \ p LdZcS]M©X"ZcLu\ p ^W\b^ I \_MN\b_L0^WL\_UydÉL¸ UyX"Z[S]\_^,d v 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 ^ ¡ INRIA-MODULEF. INRIA-MODULEF. INRIA-MODULEF .001819. .2006. .399. .598. .797. INRIA-MODULEF. ¨S]S]b^*U pZ L«I L $ nj ^WX"S]M`Z[SeaWX"V]V L=M`}FL¡ d I + O L0VTL0MNL0^FZ¬d`/X^ ff SediZ p STWUWZcST\_^ \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 LXZc\ Xp ¨S]^NST^yZX"I VL X f $ X"nWaWXZ[MNL f aWV]M`L L}8d ¡ I + ~X"O.L0V]L0M`L0^8Z dq/ X"^ f3f SedZ p S]WUWZ[S]\b^ \"lZ I LL pcp \ p . ÈeÊË ÈeÍ.
(211) p \_jUydiZ Qj"me L pcp \ p LdZcS]M©X"ZcLu\ 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 b d\ _XV]UWLsd$\"*Z I L f SedaWVeXbkmL0MNL0^FZ,+ Z X"^WbL0^8Z[SeXV«X"^ f ^W\ p *M XV /N\ p Z I L nWXMNaWV]L }8¡ Ë¨Ë Ô (*),+-,. .
(212) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. v"O INRIA-MODULEF 7.177E-4. 9.648. 19.294. 28.941. 38.588. INRIA-MODULEF INRIA-MODULEF. INRIA-MODULEF. ¨S]bU p L } V]\byX"VX^ f ayX p cZ SYX"V FS]L \" \_^ t$SediLsdÙdZ p Lsdcd £yLV f \ p Z I L $ njX"M`aWV]L }8¡ ÈeÊË ÈeÍ . .
(213) p \_jUydiZ Qj"me L pcp \ p LdZcS]M©X"ZcLu\ 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 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. S]_U p L O x LdS f U,X"V L p[p \ p \ p Z I L=L0njX"MNaWVTL}F¡ HI L f XZ¬X\ p $ nWXMNaWf V]L vÙX p L_f S L0^=f \_^ S] F¡ HI SYd p L0a p LdiL0^FZ¬d;X M`\ p L p LXV]SedZcSYk yLX"M'diLskmZcST\_^ p L0ST^8\ 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 $ jn X"MNajV]LAv8¡ . ÈeÊË ÈeÍ.
(215) p \_jUydiZ Qj"me L pcp \ p LdZcS]M©X"ZcLu\ 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 $ nj X"^jM`S]ZcaWSYX"V]L=V;MNvF¡ Ld I D+ v OLV]L0MNL^8Z dq/X^ 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 LXZc\ ~p ¨S]^NST^yZX"I VL X f $ X"nWaWXZ[MNL f aWV]M`L Lv8d ¡ I + _ O v*L0V]L0M`L0^8Z dq/ X"^ f3f SedZ p S]WUWZ[S]\b^ \"lZ I LL pcp \ p . ÈeÊË ÈeÍ.
(217) p \_jUydiZ Qj"me L pcp \ p LdZcS]M©X"ZcLu\ 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 d\ _XV]UWLsd$\"*Z I L f SedaWVeXbkmL0MNL0^FZ,+ Z X"^WbL0^8Z[SeXV«X"^ f ^W\ p *M XV /N\ p Z I L nWXMNaWV]L v8¡ Ë¨Ë Ô (*),+-,. .
(218) x ¡ p 0X hFXX"^ f o ¡ yL H XV]V]Lsk. OX INRIA-MODULEF 3.234E-5. 24.036. 48.073. 72.109. 96.146. INRIA-MODULEF INRIA-MODULEF. INRIA-MODULEF. ¨S]bU p L V]\byX"VX^ f ayX p cZ SYX"V FS]L \" \_^ t$SediLsdÙdZ p Lsdcd £yLV f \ p Z I L $ njX"M`aWV]L v8¡ ÈeÊË ÈeÍ . .
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