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On the numerical approximation of first order Hamilton Jacobi equations

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(1)On the numerical approximation of first order Hamilton Jacobi equations Remi Abgrall, Vincent Perrier. To cite this version: Remi Abgrall, Vincent Perrier. On the numerical approximation of first order Hamilton Jacobi equations. [Research Report] RR-6054, INRIA. 2006, pp.11. �inria-00113948v4�. HAL Id: inria-00113948 https://hal.inria.fr/inria-00113948v4 Submitted on 7 Dec 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. On the numerical approximation of first order Hamilton Jacobi equations R. Abgrall. N° 6054 Décembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6054--FR+ENG. Thème NUM.

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(16) Ecv[oeXQP‘’YEXgÀTV\“cv X  ª@ŽŸG.gcpXR]\_qSG[LT]CEG›YEkEk@G[R]GF \ Qg#cvX`T]\jX`YcpYR ’Y«š Rš g LoeX”qžijc‰ŽGW[LR]GF \ Sg#cvXvTV\“XQYEcvYR Ái°š RWš gfGWX`vGijcpk@GWRJce‘ z Ž \mTVC z[VGWR]k”G>gÀTT]c!opi“i<TVCEG‰oe[V\jop|Ei“G>RšBCEGWPoe[VG"qSG#–XGWq|QP oeX”q z (x) = lim inf z(y). z (x) = lim sup z(y) Ecvi“ijc‰Ž \jXE‚ B } CΪEŽŸG"\“X`T][VcSqSYgGLTVCEG"‘’YEXg#T]\jcpX G ∗. ∗. x→y. BCEGgcpF kEYST/o‰T]\jcpXce‘ G oeX”q. G(x, s, p) =. . x→y. H(x, s, p) x ∈ Ω s − g(x) x ∈ ∂Ω.. \jRG>opR]P¢oeXqŽŸGCo€pGL. \“‘ x ∈ Ω \m‘ x ∈ ∂Ω °} \m‘ x ∈ ∂Ω h i“cSgopi“ijP |”cvYEXqSG>qžY«š RWš g&‘’YEXg#T]\jcpX qSG–XEG>q6cvX \_RÙoQ\jRVg#c`R\“TUP RYE0| >ÎR]cpijYST]\jcpX6ce‘ ~ J\m‘oeXqœcvXEijPœ\m‘Uª<‘’cv[ oeXQP φ ∈ C (Ω) ªE\m‘ x ∈ Ω \jRJo.ijcSgopi)F uoelS\“F.YEFce‘ u −Ω φ ªST]CEGWX ∗. 1. G∗.  G∗ (x, s, p) = G∗ (x, s, p) = H(x, s, p)      G∗ (x, s, p) = min(H(x, s, p), s − g(x))      ∗ G (x, s, p) = max(H(x, s, p), s − g(x)). 1 . 0. NQ\jF \“i_oe[Vi“Pvª ª`o"ijcSgopi“ijP.|@cpYXqSGWq«ªviAš RWš geš‘’YEXgÀTV\“cvX¢qSG–XEGWq cvX \jRo"`\_RVg#cvR]\“TUP.R]YEk”GW[>ÆR]cpijYST]\jcpX ce‘ ~ 8\“‘«oeXq cvXEi“P \“‘UªS‘’cp[fopX`P uφ ∈ C (Ω) ªE\“‘ x ∈ Ω \_R o›i“cSgopi)F \“XE\jF&YF*ce‘ u −Ω φ ªST]CEGWX ’x  G (x , u(x ), Dφ(x )) ≥ 0. h Q\jRVg#c`R\“TUPœRcvi“YSTV\“cvX6\_R"R\jF&YEi“TVopXEGcvYR]i“PœoRYE| œopXq™oR]YEk”GW[ >ÆR]cpijYST]\jcpXžcp‘ ~ š BCE\jR"gWoeX6|@G ‚pGWXEG[/oeij\ WGWq TVc ceTVCEG[TUPQk@GWRcp‘|@cpYEX”qEoe[VP¢g#cvXqS\“T]\jcpXR R]Yg/C opR HJGYEFoeXX«ªEG#T/geš 6ÆXT]CEGgWopR]GÙcp‘ËTVCE5 G 4ŸoeYg/CQP!k[]cv|Ei“GWF ª ∂u °w  + H(x, u(x), Du) = 0 x ∈ R , t > 0 ∂t u(x, 0) = u (x) Ž CEGW[]G u |”GWi“cvXE‚vRŸTVcT]CGRGTfce‘8|”cvYEXqSG>q oeXqYEXE\“‘’cp[VF i“Pg#cvXvTV\“XQYEcvYR‘’YEX”gÀT]\jcpX”Rª BU C(R ) š LXEGgopXœopqEopkST GWovR\jijPT]CEGop[]‚vYEF GX`TVRŸ[/oe\_RG>q‘’cv[T]CEGRT]G>opqSP!k[]cv|Ei“GWF šb—fG[VGpª G \_R R\jF kEi“P G∗ (x0 , u(x0 ), Dφ(x0 )) ≤ 0.. 1. 0. ∗. 0. 0. 0. d. 0. 2. 0. R]cœT]CoeT G = G = G šyNQY|Rcvi“YET]\jcpXR ’[VGWR]k«š™R]YEk@G[/Rcvi“YET]\jcpXR oe[VG!GWi“GWF›GWX`TVR.ce‘ ªER]c›T]CoeTf\“XGW^`Yoeij\“TUP 1  ’[VGWR]k«š x  ŸCEcvijqRš T >0 hfiji)TVCE\jR gWoeX |@GG#lQT]GWXqSGWq¢T]c TVCEG54ŸoeY”g/C`P SzJ\j[]\_g/CEijG#TkE[Vcp|EijGF ∗. ∗. G(x, s, p) = pt + H(x, s, px ), x ∈ Rd , s ∈ Rd , p = (pt , px ) ∈ R × Rd. ∂u + H(x, u(x), Du) = 0 x ∈ Ω ⊂ Rd , t > 0 ∂t u(x, 0) = u0 (x) x∈Ω u(x, t) = g(x, t) x ∈ ∂Ω, t > 0. BU C(Rd × [0, T ]). Ž CG[VG Át . IfX”qSG[&RTVopXqEop[Vq™opRVRYEF kSTV\“cvXRcvX6TVCEG!cpk@GX R]YE|R]G#T ª oeXq oeXq ªcpXEGgWoeX kE[Vc‰pG›G#lS\jRT]GWXg#G!opXq EY XE\_^`YEGXGWRVRfce‘TVCEG&Q\_R]gcvR]\mTUP¢R]cpijYSTV\“cvXRfcp‘ ~ #ª w  opXq t Ω#ª@gRGWG B } CΚ H6ÆX6koe[]T]u\_g#YEi_oe[>ªSTVCE\jRL\_R T][VYEG\“‘T]CEG›—JopF›\ji@> T]cvXE\_oeX H \_R g#cvXQpG#l!\jX p ∈ R oeXq\“‘ ∂Ω ij\“k”R]g/CE\“TgcpX`T]\jXQYEcpY”Rš 6ÆX T]C\jR.koek@G[>ª8ŽŸG¢opRVRYF›G!T]CoeT ~ CovR&oœYXE\j^`YEGWXEGWRVR.kE[]\jXg\“kEijGpªTVCo‰T›\jR›oeXQP RYE| QR]cpijYST]\jcpX u oeXq opXQP R]YEk”GW[ SRcvi“YET]\jcpX v ce‘ ~ RVo‰TV\jR‘’P Î  ∀x ∈ Ω, u(x) ≤ v(x) oeX”q  ∀x ∈ R , t > 0, u(x, t) ≤ v(x, t) \jXTVCEGgovRGÙce‘T]C5G 4ŸoeYg/CQPkE[Vcp|EijGF š 0. d. d. Í N ÍÚ¹.

(17) w &

(18)  Ÿ!b*(# ¢#( BŽcœGlSopF kEi“G>Roe[VG!g#cvXR\_qSGW[]G>q)š  \“T]CEGW[ŽŸGi“cQc896‘’cp[TVCEG!RT]GWovqSP6k[]cv|Ei“GWF ~ َ \“T]C oœg#cvXQpG#ly—JoeF \jimTVcpXE\_oeX˪ cp[َŸG.ijcQ8c 9 ‘’cp[ÙTVCEG 4ŸoeYg/CQP kE[Vcp|EijGF w LŽ \mTVC™GW\mTVCEG[og#cpXQvG#l ’cv[ÙgcpXgWo€pG.—LoeF \jimTVcpXE\_oeX Jcp[og#cvX`vG#l ’cv[ g#cvXgo€vG •\“X\mTV\jopiËg#cvXqS\“T]\jcpX«š BCEG"Foe\jXT]cQcpi)\_RŸT]CEG ËG‚vGXqS[VGÙT][/oeXR‘’cp[VF š 6Α f \jR o g#cvXQpG#l‘’YXgÀTV\“cvX R]Yg/CT]C”o‰T  E 

(19)    ! "# "!$%"&'")(* +,-$+  ". -$/0-$. lim. ŽŸG"qEG#–XEG"TVCEG « GW‚pGWXqS[VGJTV[VopXRU‘’cv[]F*ce‘ f |QP. ||x||→+∞. . f (x) = +∞ ||x||. .  f ? (p) = sup p · y − f (y) .. ΑT]CEGR]YEkE[VGF.YEF\_R []G>opg/CEG>q¢oeT y ªEŽŸGÙC”o€pGLT]CEGGi_o‰TV\“cvX y∈Rd. 6. ?. f ? (p) + f (y ? ) = p · y ? .. BCE\_RJR]CEc‰ŽfR TVCo‰T gWoeXœ|@G&RGWGX6ovRT]CG.oe|”R]g\jRVR]o cp‘T]CEG&TVoeX‚pGX`TJcp‘T]CEG.‚p[/oekEC ce‘ o‰T šJBC\jRJ‚p[/oekECE\_g j\ XvTVG[VkE[VG#TVoeT]\jcpX CEGWfi“kRf(p)T]c¢R]GGT]CoeTWª\“‘ f \jRJ[]GW‚pYEi_oe[ GWXEcpY‚pC«ªT]CEG‚v[VopkEC ce‘ f \jRfT]CEGGWXQpGijfcpk@Gcey‘8\“TVRfTVoeX‚pGX`T>ª”R]c T]C”o‰T (f ) = f. J‘8g#cpY[VR]GJTVCE\_R []GWijoeT]\jcpX\jR ‚vGXEGW[Vopi“\ Wop|EijGfT]c!o gcpXQpGl f Ž CEGWX\“TJRVo‰T]\_R–GWR Àš hfiji)TVCE\jR‚vGXEGW[Vopi“\ G>RT]c!o g#cvXgo€vGf‘’YEXg#T]\jcpXR ÁR]\“XgG −f \_R g#cpXQvG#l Àª ?. ∗. ? ?. . f ? (p) = −(−f )? (p) = inf. y∈Rd. . 

(20)     

(21) !"$#.  − y · p − f (y) .. Œ GopRVRYEF GÙT]C”o‰T T]CG—JoeF \jimTVcpXE\_oeX\_R‚p\jpGX|QP H(x, u, p) = sup {−b(x, v) . p + λu − f (x, v)}. Ž EC GW[]GLT]CEGR]kovg#GÙce‘gcpX`T][Vcpi_R V \_R g#cvF›k”opgÀT>ªEoeXq¢ŽG"Co€vGLRTVopXqEoe[/q¢ovR]R]YEF kSTV\“cvXRŸcpX b ª f oeXq λ > 0 ªER]GG B } CΚ Ecv[•TVCEGLzL\j[]\_g/CEijG#TŸgcpXqE\mTV\“cvX«ª`T]CEGÙRcvi“YSTV\“cvX!ce‘ ~ •\jR‚p\jpGWX|QP.TVCEG"qSPQXoeF \_goei”kE[Vcp‚p[/oeF F \jXE‚kE[V\“Xg\“ki“Gvªv‘’cv[ŸopXQP ª T >0 v∈V. u(x) = inf. v(.). BCEG"TV[VoedUGWgÀTVcp[VP. yx (.). RVo‰TV\jR–GWR. Z. min(T,τ ). f (yx (t), v(t))e−λt dt. 0. + 1{T <τ } u(yx (T ))e−λT + 1{T ≥τ } ϕ(yx (τ ))e−λτ yx (0) = x ∈ Ω. opXq. . ’‘ cv[ t > 0. BCEGWP¢oe[VGqSG#–XGWq\m‘ f _\ R [VG‚vYEijop[GWXEcpY‚pC«ªERVo€P «\jkRVg/CE\“Tg#cvXvTV\“XQYEcvYRWšBCGG#lS\mTTV\“F G τ j\ R d yx (t) = b(yx (t), v(t)) dt. zLG#TVop\“i_R gopX|@G"‘’cpYXq\“X B 1 ª } C. Ê ÀÖÀ×/Ð. NON P0Q. τ = inf{t ≥ 0, yx (t) 6∈ Ω}.. U~Wu .

(22) t. . 

(23)  .  . .    .

(24) $#. BCEGopXoeijP`T]\_goei<GlQk[]G>R]R]\“cvX!‘’cv[T]CEGR]cpijYST]\jcpXce‘ w ÀªŽ CEGX H cpXEijP¢qSGWk”GWXqER cvX ~vš Ž CEGX H \jRYEX\m‘’cv[]F ijP «\jkRVg/CE\“T g#cvXvTV\“XQYEcvYRoeXq u g#cpXQvG#l)ª. p ∈ Rd. ªE\jR‚v\“vGX\“X B x CΪ. 0. . u(x, t) = sup x · p −. Ž CEGX u \_R g#cvXgo€vGpª. p∈Rd. 0. u(x, t) = inf. . u?0 (p). −x·p+. − tH(p) ,. u?0 (p). U~>}. . − tH(p) .. }Eš Ž CEGX u \_RYEXE\“‘’cp[VF i“Pg#cvXvTV\“XQYEcvYRWª`ŽŸG"Co€pGّ’cv[fo›gcpXQpGl¢—JopF›\ji“T]cpX\jopX p∈Rd. U~p~ . . 0. oeXq¢‘’cp[fo g#cvXgo€vGٗLoeF \jimTVcpXE\_oeX. ?. . y−x t. #. ,. U~ 1 . u(x, t) = sup u0 (y) + tH ?. . x−y t. #. .. U~x . u(x, t) = inf. p∈Rd. p∈Rd. ". u0 (y) + tH. ". BCEG›‘’cp[VF&Yijo ~v~ "oeX”q ~>} Ù[VG-7G>gÀT"T]CG!—fYEPQ‚pGWXR  R"kE[V\jXg#\jkEijGpª«Ž CE\jijG ~ 1 "oeXq W~ x "oe[VGg#cpX”RG>^vYGXgG.cp‘•T]CG qSPQXopF›\_gopi)kE[Vcp‚p[/oeF \jXE‚&k[]\jXg#\jkEijG ~Wu Àš HfcpT]GÙT]C”o‰Tf\“‘ u \jR ij\jXEGWop[ \“X x ª u (x) = A + p · x ªEŽŸG"Co€pG 0. 0. u(x, t) = u0 (x) − tH(p).. BCEG>RG[VGWR]YEimT/Roe[VGŸcvXEijP€opi“\_q.‘’cp[bR]k”G>g#\_oeiS\jXE\“T]\_oeiEg#cvXqS\“T]\jcpXRcv[koe[]T]\_g#YEi_oe[8—LoeF \“i“T]cvXE\_oeX«š Œ G C”o€pGT]CEG F cp[VG ‚pGWXEG[/oei<[VGWR]YEi“TVR •¤  ¡ˆ oe[/qS\@> ÙR]CEG[ B w C   u = u + u /0  u -$#  u $ ; " -$#  ;"    /0 & /0 $       w  $ "# $ +$ u. .

(25) . . ψ1 (x, t) = inf sup. q∈Rd p∈Rd. ". ψ2 (x, t) = sup inf. d p∈Rd q∈R. LBCEGkE[VcQce‘YRG>RŸT]CEG"‘ÁovgÀTT]C”o‰T.     . conc 0 . ∀(x, t) ∈ Rd × [0, T ],.

(26)  / . conv 0. 0. . . conv 0 . . conc 0 . . . . . ψ2 (x, t) ≤ u(x, t) ≤ ψ1 (x, t). ? ? x · p − uconv (p) − uconc (q) − tH(p − q) 0 0. . ? ? − x · q − uconv (p) − uconc (q) − tH(q − p) 0 0. uconc (x) 0.  . = supp∈Rd. . x·p−. ? uconc (p) 0. ? uconc (x) ≥ vp,0 (x) := x · p − uconc (p). 0 0. . . Rc›T]C”o‰T‘’cp[fopX`P p ª. BCEGWX6ŽŸG›R]cpijpG&T]CEG 4ŸoeYg/CQP kE[Vcp|EijGF ‘’cv[ Ž C\jg/C™\jR"g#cvXQpG#l)ª<YR]G&TVCEG g#cpF kop[]\_R]cpXžkE[V\jXg#\jkEijG  #ª«opXq TVo89pGÙT]CGF oelS\“F.YEF šBC\jR ‚v\“vGWRT]CG"–[VRTf\jvXEGW+^`Yuopi“\“TUPpš8BCEGR]GWgcpXq¢cpXEG\_Rcp|ST/oe\jXEGWq\jX o R]\“F \jijop[ ŽŸo€Pvš •¤  ¡ˆ B t C   H = H + H /!- H +$#  H $ ;" " +$#  ;"    K -"+  ;"K

(27) K!" $ 

(28) /! $     . 

(29) /! (* " ;/  @ w  $ "# $ -$ 0. . . 0. .

(30) . conv. .   conc #!. conv  $. conc. . . .  . "!. . . Φ2 (x, t) ≤ u(x, t) ≤ Φ1 (x, t). Í N ÍÚ¹.

(31) 

(32)  /  

(33)    ! "# "!$%"&'")(* +,-$+  ". -$/0-$. . #. ". .  y−x ? ? Φ1 (x, t) = inf sup u0 (y) + tHconv +q (q) + tHconc d t q∈R y∈Rd "  # x − y ? ? (q) + tHconv Φ2 (x, t) = sup inf u0 (y) + tHconc +q d t p∈Rd y∈R. BCEG"kE[VcQce‘\_RfR]\“F \jijop[ oeXq\_R‚p\jpGX\jX B t CΚ  À() + fW

(34) "#1% &  J 2 #f

(35) 6ÆX¢cv[VqEG[T]cR\jF kEij\m‘’P TVCEGÙT]G#lQT>ªSŽG"opRVRYF›GJ‘’[]cvF XEc‰Ž cvX ªQ|EYSTfoeiji@T]CEG"[VGWR]YEi“TVRŸgoeX|”G"G>opR]\“ijP ‚pGXG[/oeij\WGWq T]c›ceTVCEG[ qS\jF GXR]\jcpXRWªQ\“Xkoe[]T]\_g#YEi_oe[ d = 3 š Œ GgcpXR]\jqEG[do=TV[]\_2oeXE‚vYEi_o‰T]\jcpX¢ce‘ R ªQT]CEG"vG[]T]\_g#GWRŸoe[VG {M } ª T]CG.TV[]\_oeX‚pijGWR"oe[VG {T } š Œ GqSGWXEceTVG.|QP T o¢‚vGXEGW[]\_gTV[]\_oeX‚pijGpš&BCEG vG[]T]\_g#GWRLce‘ T oe[VG M ª M opXq ª‘’cp[›R\jF kEi“\_g#\“TUPyŽGcp‘ÚT]GX qEGXEcpT]GTVCEGF |QP ª ª cp[&|`P 1 ª 2 ª 3 Ž CEGX T]CEGW[]G\_RXEc™oeF&|\“‚vYE\mTUPvšBCG M ‘ÁoeF \ji“Pce‘T][V\jopXE‚pYEi_o‰TV\“cvXRŸŽGgcpXR]\jqEG[ \_R RCopk”GiG‚vYEii_oe[>š i IfkžT]c¢cvYE[ 9`Xc‰Ž i“G>qS‚pGvªET]CG&–[/RTLkopk”GW[JTVcqS\_R]gYRVRf\jX™qSGTVoe\jiT]CEG oekEk[]c€lS\jF oeT]\jcpX ce‘ w J\_R B  CΚÙhLRJ\jXœTVCE\jR [VG#‘’G[VGX”g#Gp ª w Ÿ\_RfoekEk[]c€lS\jF oeT]G>q!|QP U~>w  u = u − ∆tH , i = 1, · · · n , n ∈ N u = u (M ) Ž CEGW[]G ∆t > 0 \jR8T]CEG TV\“F GfRT]GWk!oeXq u \_RbopX opkEkE[Vc€lS\“Fo‰TV\“cvX&cp‘ u(M , n∆t) ª`oeXq.T]CGfXQYEF G[V\jgWoeiE—JopF \“i“T]cvXE\jopX qEGk@GXqER&cpX ªTVCEG!‰oeijYEG>Rce‘ Ž CEG[VG j ∈ V V \jRT]CEGRGT&ce‘ XEGW\“‚vC`|@cp[/Rcp‘ M \jXg#ijYqS\jXE‚ M |`P H g#cvXQpGX`TV\“cvX ÀªSoeX”qu\“‘XEGWgGWRVR]op[]PcvX Muª U~Wt  ). H := H(M , u , {u } 6ÆXyT]CE\_R[VG#‘’GW[]GWXg#G \_R\jXvTV[]cSqSY”g#GWq™TVCEGXEceTV\“cvX ce‘ g#cpX”R\_RUTVGXgPpš¢BCG!XQYEF G[V\jgWoeib—JopF \“i“T]cvXE\jopX H \jR&g#cpX”R\_RUT/oeX`T Ž CEGWX«ªE\“‘ v = A + p · OM~ ªQT]CEGWXœŽ Co‰T GWpG[ M oeX”q s ∈ R ª ~€  H(M , s, {v } ) = H(M, s, p). h ijGWRVR[]G>RUTV[]\_gÀTV\“vGÙqSG–XE\“T]\jcpX«ªŽ CE\jg/C \_RCEGijkS‘’YEiˑ’cp[TVCEGkE[Vc`cp‘UªS\jR‚v\“vGX\“X B  CΪ $ -¤G  † ° ¨  ;  8

(36)   $;   /0  ":0$ 

(37) /!%  5   "K  " H $    ;0$+ $+, ! <-"   x ∈ Ω " φ ∈ C (Ω)  /0 C ~  lim sup H(y, φ(y) + ξ, φ + ξ) ≤ G (x, φ(x), Dφ(x)) 2. i i=1,ns. j j=1,nt. k. i3. 1. n+1 i 0 i. 0. i2. 3. i. ∗. s. i. n i n j. n i. i. n i. 2. i1. i. i. i. i. i. i. i. i. n i. n i j∈Vi. i. i. i. j. j. !. . ∞. i j∈Vi. . !. . . . ∞ b. . . . . ∗. h→0,y→x,ξ→0. ". ~€ . lim inf. $ ; /0  

(38) /! " $ "# $$-$. H(y, φ(y) + ξ, φ + ξ) ≥ G∗ (x, φ(x), Dφ(x)).. $$+"  "! ; 0$+ $+, ! . ρ→0,y→x,ξ→0. $+  "!;0$+ $+, ! $  /!-  $ B C 6. . ~  . ; "! ; "0$+ $+ -K . B CEGRT][VYgÀTVYE[]G cp‘TVCEG!Rcvi“YET]\jcpXyce‘ w َopRٖ”[VB RTYRG>qy\“X w ª ~ Κ ÆX koe[]T]\_g#YEi_oe[>ª)TVCEGPžY”RG>q6TVCEG[VGWR]YEimT/R"ce‘ ~ T]cqSG–XEG o AÙcQqEYEXEc‰ `ij\@9vGRVg/CEGF Gvš 6ÆX t CAªp\mTb\_RRCEc‰Ž X.T]C”o‰TWªp\“X›‚pGWXEG[/oeiAª€‘’cp[8XEcvX.RT][VYgÀTVYE[VGWq.F›G>RCGWRWª T]CG›‚vGXEGW[Vopi“\ WoeT]\jcpXœcp‘•T]CGAÙcSqSYEXc‰ Qi“\ 9pG›R]g/CGF G.cp‘ B w CbijGWopqRJTVcoXEcvX™g#cvXR]\jRTVopXvT—JopF \“i“T]cvXE\jopX«š&—Jc‰ŽGWpG[>ª «GWF F o } kE[Vc‰Q\jqSG>R o Rcvi“YSTV\“cvX«š hJRVR]YEF GfT]C”o‰T } \_:R 9QXEc‰Ž X¢opXq!qSGWXEceTVGJ|QP u TVCEGJk\“G>g#GŽ \_R]GJij\“XEG>oe[\“X`T]GW[]k@cpi_o‰TV\“cvX!ce‘)TVCE\_RqEoeTVoš Ecv[•oeXQPF G>RCk@cp\j{u X`T M ªvŽGfgcpXR]\_qSG[•R]G#T {Ω } cp‘<opXE‚pYEi_oe[bR]GWg#T]cv[VR8oeT ª`RGWG–‚pY[]G ~ š  ovg/CoeXE‚vYEijop[ R]GWgÀTVcp[ Ω gcp[V[]G>Rk@cpXqR T]ccpXEG.ce‘bTVCEGTV[]\_oeX‚pijGWRJT]CoeTÙR]Coe[VG M oeXq6ŽG›qSGXceT]MG.|QP U TVCEG›‚p[/opqS\jGX`TJcp‘ u \jX T]C”o‰T T][V\_oeXE‚vi“GvšBCGL‘’YXgÀTV\“cvXR Φ oeX”q Φ GW€opi“Y”o‰T]G>qoeT x = M oe[VG «GWF F o. . H. n j j=1,··· ,ns. . n h. i. i i=1,··· ,ωi. i. i. i. 1. 2. i. i.  Ui · z − H1∗ (z − q) − H2∗ (q) q∈R 1≤k≤ωi z∈Ωk   Φ2 (Mi , ∆t) = uni − ∆t max2 min inf Ui · z − H1∗ (z − q) − H2∗ (q). Φ1 (Mi , ∆t) = uni − ∆t min2 max sup. q∈R 1≤k≤ωi z∈Ωk. Ê ÀÖÀ×/Ð. NON P0Q. . n h.

(39)     . Duh|Ωi+1/2 = Ui+1/2 Mi Ωi+1 θi ~ni+1/2 Ωi. \“‚vYE[VG›~L 6ÆijijYRUTV[VoeT]\jcpX ce‘T]CGoeXE‚vYEijop[. RG>gÀT]cv[VR Ω ª θ oeX”q¢TVCEGpG>gÀTVcp[/R ~n T]CoeTJoe[VG"XEGG>qSGWq\jXœX`YF›GW[]\_gopi —JopF \“i“T]cvXE\jopXR qSG–XE\“T]\jcpXRWš BCE*G 9vGPL[]GWF op[9J\jR<TVCo‰ToeXQPJcp‘vTVCEG8T]GW[]FRËF&YEi“T]\jkEij\“G>q"|`P ∆t ª€RVo€P min max sup U ·z−H (z−q)−H (q) ª [VGWqSYgGWRTVc Ž CGX U ≡ p ∀i š—fGWXg#Gvª”opX`Pce‘T]CGWR]G"T]G[VFRfqSG–XEGWRLog#cvXR\_RTVoeX`T XQYEF G[V\jgWoei˗LoeF \jimTVcpXE\_oeX«ª ‘’cp[ GlEoeF kEijGpH(p ª  °}eu  H := max min inf U · z − H (z − q) − H (q) , T]CG›qSGWk”GWXqSGWXg#Pœ\jX u ª j ∈ V oekEk@GWop[VRL\jXžT]CG.‚v[VovqS\“GWX`TVR U šBCE\_RL‘’cp[VF&YEi_o!gWoeX6G>opR]\“ijP |”G›G#lQTVGXqSG>qœTVc!T]CG F cp[VG"‚pGXG[/oei)govRG H = H(x, u, Du) oeX”qR]\jF›ki“\“–GWRŽ CGX H \_RfgcpXQpGl<ª`‘’cv[ G#lEoeF kEijG  °}S~  H = max max U · z − H (z) . h RG>g#cvXq¢[VGFop[ 9\_RT]C”o‰TWª|`Pg#cpX”RUTV[]Yg#T]\jcpX«ª H qEG#–XEG>q|`P }pu Ÿcp[ }S~ Ÿ\_RF cpXEcpT]cvXEGpªQTVCo‰Tf\_R ¤  † ° ¨  ÁKœcvXEceTVcpXEGL—JoeF \jimTVcpXE\_oeX”R 

(40)  $ 

(41) /! " H $ ",  /! "  "- M ∈ Σ u ≤ v " i. i. i+1/2. ∗ 1. i. q∈R2 1≤k≤ωk z∈Ωk. ∗ 2. i. i. n j. ∗ 1. i. q∈R2 1≤k≤ωi z∈Ωk. i. ∗ 2. i. i. 1≤k≤ωi z∈Ωi. ∗. i. i. -" "!. . . !. . i. . s ∈ R. j. . j. BCEG›—JoeF \jimTVcpXE\_oeX }eu f\_RJF cvXEceTVcpXEG|QPœg#cvXRT][VYgÀTV\“cvXœ\m‘ ∆t/h max LT]CE\_RÙ\jR ||D H(p)|| ≤ 1/2 o g#cvXRG>^`YEGXgG"ce‘  opXqT]CEG\jXEG>^vY”oeij\mTV\“G>R hfXceT]CG[ 9pGP[VGFop[9›\jRTVCo‰TT]CGL‰oeijYEGÙce‘ qSG–XEG>q¢|QP }eu •cp[ }E~ qScQGWRŸXEceT qSGWk”GWXq¢cvX!TVCEGÙRT][VYg#T]YE[VG ce‘)TVCEGfF GWR]C«ªQ|EYSTŸcpX TVCEGJ\jX`T]G[Vk@cpi_oeX`T u š 6ÆXcpT]HCEGW[•ŽŸcp[/qERWªp\“‘«cpXEGJR]kEij\mT/RopX!oeXE‚vYEi_oe[•R]GWg#T]cp[ Ω \“X!TUŽc”ª

(42)  /0"  ;/0 " 

(43)   

(44) /!  !   ªeTVCEGfXQYEF G[V\jgWoeiE—LoeF \“i“T]cvXE\_oeX \jRbXEceTbF cSqS\“–GWq)š Œ GfR]o€PT]CoeTbT]CEGfRVg/CEGWF›G \_R

(45) K+

(46) !$;. oeX”q¢ŽŸGCo€pGLT]CG"‘’cpiji“c‰Ž U\jXE‚›G[V[]cv[ŸG>RUTV\“FoeT]Gpª B t C   - H : R → R  ;"K

(47) K!" $ " u ∈ BU C(R ) "  K$ /

(48)  ;!#

(49) !!" $  ¤Q¨ ‡ ¤  H(Mi , s, {uj }j∈Vi ) ≥ H(Mi , s, {vj }j∈Vi ).. p. ||Du|T ||∞ ≤L. i. n. . ∞. k. . . . k.

(50) . . . . . . ; +   8@ "# " /0  h $  /0 @ "0 -$+  "

(51)  $   /0 

(52)   @+$  ; ! - M L T     ;"K, 

(53)  

(54)     /0#+  "   +$ /0 

(55)   $ "-+     "$$+  

(56) /! +$/ $/0 ;0i  i= 1, · · · , n M s i 8@ "    

(57) /!

(58) !

(59)    @  α  

(60) /!#+  "  +$ T $  ! -+ "!     ;" ;     ; 8*'   ! ~>w   /0  

(61) /!-  $+$ ;"0$+ "K  -  ;G /0  $ ; $+

(62) ! $        w  " {un} / / u8 0   " " ! " " $+i;/j=1,···  /0 "-,n"5 !  .

(63)  / c α L T >0 H M n 0 ≤ n∆t ≤ T.

(64) 

(65) / ; "!$; "K.  -. 2. 2. 2.

(66). . s. . . i. . .

(67)

(68)

(69) n

(70)

(71) ui − u(Mi , n∆t)

(72) ≤ c ∆t

(73)

(74). Í N ÍÚ¹.

(75) BCEG kE[VcQce‘@\jRbopXopqEopkSTVoeT]\jcpX›ce‘@T]CEG Foe\jX []G>RYimTbcp‘  ”Ž \“T]CR]cpF GT]G>g/CEXE\_gopi“\“T]\jGWR Á\“Xkop[TV\jgYEi_oe[‘’cp[•R]CEc‰Ž \jXE‚ T]C”o‰T T]CG"T]\jF›GRT]GWk cpXEijP¢qSGWk”GWXqcpX u Rk@GWg\m–”gÙTVc YEXRT][VYgÀTVYE[]G>q¢F G>RCEG>Rš Œ GG#‘’G[TVc B t CAš BCEGkE[/opg#T]\_gopiGW‰oeijYo‰TV\“cvXycp‘TVCEG «GW‚pGWXqS[VG.TV[VopXR‘’cp[VF \jR"XceT&oeijŽo€PQR"opXyG>opR]PœTVovR9<ªRc ceTVCEG[&X`YF›GW[]\_gopi —JopF \“i“T]cvXE\jopXRŸG#lS\_RUT/Rš8BCEG"R\jF kEijGWRTcvXEGJ\_RŸT]CG o‰l E[V\jGWqS[V\jg/C”R•cvXEGpªQŽ CE\_g/C¢\_R\“X”RkE\j[VGWq!|QP TVCEG o‰l []\jGWqS[V\_g/CR RVg/CEGF Gّ’cp[fgcpXR]G[V‰o‰T]\jcpXijo€ŽfRW<š 6ÎTfCopR R]GvG[/oei<pGW[VR]\jcpXRWšBCGL–”[VRTfcpXEG"\_R I A}p}eo  ¯) −  [u (M ) − u (M )]dl, H (Du , · · · , Du ) = H(U h Ž CEGW[]G C ’[VGWR]k«š D Ÿ\_Rfo g#\j[/g#ijG °qS\_R 9 Ÿce‘g#GWXvTVG[ M opXq[VovqS\“Y”R h ª  

(76)    ! "# "!$%"&'")(* +,-$+  ". B -C $/0-$. . 0. LF i. h. h|Ω1. h. h|Ωki. h. i. Ch. h. i. b= U. R. Dh. Duh dxdy. ,. oeX”q \_R8i_oe[V‚pGW[ËTVCoeX oeXQP Ë\“kRVg/CE\“T g#cvXRTVoeX`T8ce‘ H qE\“Q\_qSGWq›|QP 2π šBCE\_R8qSG#–XGWRboLF cpXEcpT]cpXGRVg/CEGWF›GkE[Vc‰Q\jqEGWq T]C”o‰T ∆t/h ≤ š h qE\ ?<G[VGX`TLvG[/R\jcpX cp‘8T]CEG Ëo‰l E[V\jGWqS[V\jg/C”RJ—JopF \“i“T]cvXE\jopX«ª@T]CoeTL\_RfF cpXceT]cvXEG&YXqSG[JT]CEG RVoeF G&g#cvXRT][/oe\jXvT>ª \_RŸT]CEG"‘’cvi“ijc‰Ž \“X‚ L ε 2π. HhLF (Duh|Ω1 , · · ·. BCE\_R pGW[VR]\jcpX¢gWoeX |”GGŽ [V\“TT]GWXovR. , Duh|Ωki ) =. HhLF (Duh|Ω1 , · · · , Duh|Ωki ) =. X. Z. πh2. H(Duh ) Dh. πh2.  − h. I. Ch. A}p}‰| . [uh (M ) − uh (Mi )]dl.. X ~nil−1/2 + ~nil+1/2 θli H(Duh|Ωil ) + ε tan θli · Duh|Ωil . 2π 2. B CEGfvGWg#T]cp[ ~n \_R8TVCEGfYEX\mT•vGWg#T]cp[8cp‘<T]CEGfG>qS‚pGTVCo‰TŸRGWkoe[/o‰TVGWRT]CGJoeXE‚vYEi_oe[bR]GWgÀTVcp[/R Ω oeXq Ω ªvT]CEGJopXE‚pijG \_R•TVCEGoeXE‚vi“GLce‘T]CEG"oeXE‚vYEi_oe[RG>gÀT]cv[ŸoeT M ªER]GG \j‚pY[]G ~ šbBCEGÙkoe[/oeF GT]G[ ε \_R•TVCEG"R]opF›G"opRŸ\jX!TVCEG"kE[]GWQ\“cvYR θ pGW[VR]\jcpX«š h T]CE\j[/qvG[/R\jcpX\jR 0≤l≤ki. 0≤l≤ki. l. l+1/2. i l. l+1. i. H. LF. (Duh|Ω1 , · · · , Duh|Ωki ) =. X. T 3Mi. |T |H(Duh|T ) + α X. |T |. X. Mj ∈T. (ui − uj ). A}p}eg. eo X”q α ≥ h max ||D H|| Ž CEG[VG h \_RŸT]CGijop[]‚vGWRTŸG>qS‚pG"ce‘ T š BCEG Foe\jX!qS\@?@GW[]GWXg#GJ|”GTUŽGWGX T]CGWR]GJqS\@?@GW[]GWXvT•‘’cp[VF&YEi_opR\_R8TVCo‰T }p}po 8oeX”q

(77) }p}e| bop[]G\jXvTV[]\jXR]\jgf\“X TVCEGJR]GXR]G ‚p\jpGWX\jX B t Cˎ CE\ji“G }p}pg Ÿ\_R XEceT>š—JGXgGpªS‘’cvi“ijc‰Ž \“X‚.TVCEGRVoeF G[]G‘’G[VGXgGpª }p}eo oeXq }v}‰| op[]GgcpXQpGW[]‚vGX`TopXq T]CGÙGW[][Vcp[ŸGWRT]\jFo‰TVGÙB \_R O(h ) š Ecv[ }p}pg #ªERYg/C oeXG[V[Vcp[ŸGWRT]\jF oeT]GÙ\jRXceTfo€‰oeB \jijop|Ei“G ÁoeT i“G>opRTŽ CGX‘’cpiji“c‰Ž \jXE‚ T]CG"T]GWg/CXE\j^`YEG"cp‘ t CΪS|EYSTf\“Tf\jRJg#cpXQvG[V‚pGX`T LT]CE\_R \_R oR\jF kEi“GopkEkEij\jgWo‰T]\jcpXce‘  CAš BCEG"opqS‰opXvT/oe‚vGJce‘ }v}eg c‰vG[bTVCEG"ceTVCEG[ŸTUŽŸc.vG[/R\jcpXR•\jR\“TVRR]\“F kEij\_g#\“TUP!\“X g#cSqS\jXE‚ g#cvF›k”oe[VGWq TVc }p}eo opXq }v}‰| š    À##Ÿ-2f

(78) BCEGyoekEkE[Vc€lS\“FoeT]\jcpXce‘T]CG™zL\“[V\jg/Ci“GT¢kE[Vcp|EijGF \jRXEcpTopRR\jF kEi“GyopR¢\mTi“cQ8c 9SRWš hfX \jiji“YRT][/o‰TV\“cvX \jR¢T]c –”Xq R]Yg/C¢TVCo‰T u : [0, 1] → R \jX opXq u(1) = 2 |u | − 1 = 0 x ∈ [0, 1], u(0) = 1 Ž CE\_g/C CopR Xc!g#i_opRVR\_gopi«Rcvi“YSTV\“cvX«ªE|EYETJŽ CE\_g/CQ\_R]gcvR]\mTUPRcvi“YSTV\“cvX«ªqSG–XEG>qcvXEijP¢\jX [0, 1[ \_R u(x) = x š Œ GCo€vG š 6ÆXceTVCEG[bgWopR]GWRWªpRVo€P u(0) = u(1) = 0 ªpŽŸG Co€vG u(x) = |x − 1/2| Ž CE\_g/C›FoeTVg/CEG>R8RUTV[]cvXE‚pijP lim u(x) = 1 6= 2 T]CG|”cvYEXqEop[]P¢gcpXqE\mTV\“cvXRš T 3Mi. T. p. p. T. 1/2. 0. x→1−. Ê ÀÖÀ×/Ð. NON P0Q.

(79) W~ u 6ÆXcp[/qSG[TVc qEG#–XEGoRVg/CEGWF›GvªSŽGRTVop[T ‘’[VcpF ~Wu ªopXqgcpXR]\_qSG[fo.T][V\_oeXE‚vYEijoeT]\jcpXce‘ Ω š \j[VRTfŽGovR]R]YEF  G  T]C”o‰T M ∈ ∂Ω š 6ÆX ~>u Àª<TVCEG RGTÙcp‘g#cvXvTV[]cvijRLgopX6|@G›R]kEij\mT]T]GWq™\jXvTVc!TUŽŸc¢kop[T/R LfT]CEG RGT V ‘’cp[َ CE\_g/C T < τ ª oeX”q V ‘’cp[ Ž C\jg/C T ≥ τ š•—fGWXg#Gvª i. 1. 2. u(x) = min( inf [· · · ] , inf [· · · ]).. «GT. @| G T]CEGœ\jX`T]G[V\jcp[ XEcp[VFoeiŸT]c o‰T š NS\“XgG \_R!oe[V|E\“T][/oe[VPpªb\“T¢gopX|@Gžg/CEc`RGWXopRR]FoeijifopR k@cvRVR\j|EijGp~nš6ÆX™T]CEG ij\“F \“T T → 0 ª<TVCEGRGT ΩV ŽŸxcpYE∈i_qžΩ|@G›T]CEGR]G#T"Tce‘g#cpX`TV[]cvijRJ‘’cp["Ž CE\_g/C b(x, v) . ~n > 0 ª)\Aš GpšTVCEG g#cvX`T][Vcpi`‘’cp[8Ž CE\_g/C.T]CEGTV[VoedUGWgÀTVcp[VP‚pcQGWR\jX`T]c Ω šBCGfqSPQXoeF \_goeiQkE[Vcp‚v[VopF \“XE‚LkE[V\“Xg\“ki“G inf [· · · ]−u(x) = 0 g#cv[][VGWR]k@cpXqERTVc.TVCEG—JopF \“i“T]cvXE\jopX v∈V2. v∈V1. 1. v∈V1. Hb (x, t, p) = sup {b(x, v) . p + λt − f (x, v)}.. Œ G oei_Rc›Co€pGÙTVCEG[]GWijoeT]\jcpX H ≤ H š BCEG  cpX gopX |”GopkEkE[Vc€lQ\jFo‰TVGWq)ªS\“‘ T \_RfR]Foeiji°ªE|QP ϕ(y (τ )) šfNQ\jXgG T ≤ τ ª@oeXq\“‘ŽŸGgopXœg/CEcQcvR]G g#cvX`T][Vcpi_R•‘’infcv[ Ž CE\_g/C V T ' τ ª”ŽG"‚pGT |@GWgopYR]G ϕ \jRJg#cpX`TV\“XQYEcvYRš Œ GRGWGLTVCo‰T ~Wu gWϕ(y oeX |@G(τoe))kEkE'[Vc€ϕ(x) lS\“FoeT]GWq«ªQoeTfo.|@cpYXqEoe[VP!k@cp\jX`TWªE|QP v∈V1. b. 2. x. x. 0 = max(Hib , u(x) − ϕ(x)). Ž CEGW[]G H \_R og#cvXR]\jRTVopXvT oekkE[]c€lS\jFo‰T]\jcpXce‘ H š Œ CEGWX ªvT/o9Q\jXE‚ RFoeiji<GXEcvYE‚pC˪`ŽŸG"goeXRGWGJ‘’cv[]Fopi“ijP›T]CoeTŸTVCEG"|”cvYEXqEop[]P kEi_o€PSR•Xc.[VcpijGÙR]cTVCo‰T ŽŸG"gWoeXTVo89pMGoe6∈XQP¢∂ΩgcpXR]\jRTVopX`T T—LoeF \jimTVcpXE\_oeX«ªQ‘’cv[ G#lEoeF kEijGÙT]CEc`RGqSG–XEG>q\jXT]CEGk[]GW`\jcpY”RŸR]GWg#T]\jcpX«š BCEGRVg/CEGF G"\jR T]CEGWX A} 1  S(M , u , {u } )=0 ∀i Ž \“T]C  \“‘ x 6∈ ∂Ω )  H(x, s, {u } °}‰x  S(x, s, {u } )= W G j i ] R p G š  BCEGžRVg/CEGF G } 1  }ex  gop||@GœG#lQTVGXqSG>qmax(H T]c ceTVCEG(x,[TUs,PQk@{uGWR }cp‘Ù|@cp),YEsX”qE−oe[Vϕ(x)) P¡g#cvXqS\“T]\jcpXRWš BCEGW[]G \_R!oeX\“F kEij\_g#\“T qSGWk”GWXqSGX”g#P cp‘ S Ž \mTVC [VGWR]k@GWgÀTT]c h š Œ G G#lQT]GWXq TVCEG qSG#–”XE\mTV\“cvX ce‘ S T]c oeXQP y ∈ Ω |QP R]o€PQ\jXE‚ TVCo‰T \m‘ |”GWi“cvXE‚vRŸTVc›T]CEGqSY”oeiËgcpX`T][Vcpi<pcvi“YF›GovR]R]cSg#\_o‰T]G>qTVc M š S(x, s, {u } Œ GÙC”o€pGÙT]CEG")‘’=cvi“ijS(M c‰Ž \jXE‚., s,[VGW{uR]YEi“T } ) x  ¤Q¨ ‡ ¤  B C  $;$+   /0 " b. b. i. i. i. j j∈Vi. j j∈Vi. j j∈Vi. b. i. j j∈Vj . . . j j∈Vi. j j∈Vj. i.

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(92) . . HM  HH HM. Ê ÀÖÀ×/Ð. NON P0Q. i. j j∈Vi. i. HH i HM i. i. 0. i. −1. . . . . Hb. . . !. H. i. . i. i. i. . H. i. j j∈Vi. k. 0. k. j j∈Vi. .  . . !. "!. i. . . j j∈Vi. i. i. i. . " $+#"  "! ;0$+ $+, !  "'" "' 5  "  ,!  0$. M. . i. i. j j∈Vi.  . Hb. . . . . . . !.

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(101) #Ÿ  (Ë /( 6ÆX ‚pGWXEG[/oeiAªS\mTL\jRfqE \ !g#YimTfTVc!g#cvF›kYST]GopXoeijP`T]\_goeijijPT]CEGR]cpijYSTV\“cvXcp‘o.–[/RTfcp[/qSG[f—LoeF \“i“T]cvX Q˜`opgcp|E\<G>^vY”o‰T]\jcpX˪ oeX”qT]CEG&R]\“T]YoeT]\jcpX \jRJGvGXœŽŸcp[/RG"Ž CEGWX T]CEG&—LoeF \jimTVcpXE\_oeXœ\_R XEcpTLg#cvX`vG#l ÁXEcp[Ùg#cvXgo€vG |”G>gopYRG"TVCEG.opXoeijcp‚vP Ž \“T]C!CQPQk”GW[]|@cpij\_gfRPSRT]GWF R|@GWg#cvF GWRbijcQcvR]G[b\jX!‚vGXEGW[Vopi°š—fGWXg#Gvªv\“T|@GWgcpF GWR•F›cv[]GLqS \ !gYEimTŸT]cfdUYqS‚pGJT]CEGÙ^`Yoeij\mTUP ce‘ŸXQYEF G[V\jgWoei[]G>RYEi“TVRWš&Bc c‰pG[/g#cvF GT]CE\_RqS\ !g#YEi“TUPœ\“X oRk@GWg\jopibgopR]Gpª<ŽŸGg#cpX”R\_qSG[ H(p) = (||p|| − 1) opXq T]CGkE[]cv|EijGF cpX Ω, H(Du) = 0 cvX Γ , 1 ~  u=0 cpX Γ u = 10 Ž CEGW[]G Ω \_RJqSGWkE\jg#T]G>q \“X –‚vYE[VG 1 šfNQ\“X”g#G t 7→ t \_R F cpXEcpT]cpXG\“Xg[]G>opR]\“X‚ª u \_RJR]cpijYSTV\“cvXcp‘ 1 ~ \m‘bopXq cpXEijP¢\“‘ \“Tf\jR oR]cpijYST]\jcpXce‘ cvX Ω, ||Dv|| − 1 = 0 cvX Γ , 1 }  v=0 p c X Γ . BCEGR]cpijYST]\jcpXce‘ 1 ~ oeXq 1 } Ÿ\_RŸT]CEGqE\jRTVopvX=g#G"10TVc Γ š uh. . . −. . Ω. ∗. +. −. +. +. i. i. j j∈Vj. i. j. j. j. i. i. 2. 3. 4. 5. i i=1,··· ,8. 7. 3. 1. 2. 3. 1. 2. 1. Í N ÍÚ¹.

(102) ~1.  

(103)    ! "# "!$%"&'")(* +,-$+  ". -$/0-$. P7 P6 P5. P8. M. P1 P2 P3 P4. \j‚pY[]G} L 4ŸopR]G"ce‘o›XEcvXœg#cpXE‘’cp[VF opi<F›G>RC˚.                                                                                                                         . \“‚vYE[VG 1 L.4cpF kEYETVo‰TV\“cvXoeiEqEcpFoe\jX›‘’cp[bkE[Vcp|i“GWF 1. ~ š Γ \_RT]CEGJ\“XEXG[•g\“[/g#ijGce‘)g#GX`TVG[ (0, 0) opXq›[VovqS\jYR r = 1 ª _ \ Ÿ R ] T  C  G p c S Y V T  G f [ # g j \ / [ # g j i G °. # g W G v X V T  G [ S ª V [ v o S q j \  Y R Àš Γ (0, 0.5) r=3 6ÆX¡cv[VqEG[.T]c qS\jRVg#[VG#TV\WG 1 ~ #ªbŽŸGŽ [V\mTVG Ž \mTVC H (p) = max(|p|| − 1, 0 opXq H (p) = H = H +H  šBCEG>RG.‘’YEXg#T]\jcpXR"oe[VG&[VGWR]k@GWgÀTV\“vGijPœg#cpXQvG#lœoeXq™g#cpX”go€pGvšLBCEG›XQYEF G[V\jgWoei—JopF›\ji“T]cpX\jopX6\_R min(||p|| − 1, 0 T]CG Ëoel []\jGWqE[]\_g/ C  R•oeX”q T]CEGL|”cvYEXqEop[]P —JopF \“i“T]cvXE\jopX!\_RAÙcSqSYEXEc‰  RšBCEGLX`YF›GW[]\_gopi”R]cpijYST]\jcpX¢\jRqS\jR]kEi_o€PpG>q cpX \“‚vYE[VG x > Áo #šJBCEG&R]cpijYSTV\“cvXœce‘ 1 }  Ž \“T]CžT]CE G AÙcQqEYEXEc‰ —JoeF \jimTVcpXE\_oeXœ\_RLkE[]c‰Q\_qSGWqcvX \j‚pY[]G x > Á| ÀšLh g#ijcvR]G g#cvF koe[V\jR]cpXRCc‰Ž T]CoeT T]CEGWPop[]G Áopi“F c`RUT •\_qSGWXvTV\jgWoeiAš 1. 2. 3. 1. 3. Ê ÀÖÀ×/Ð. NON P0Q. 2. 1. 2.

(104) ~x.    . °o. Á| . \“‚vYE[VGx!L °oL NQcpijYST]\jcpX cp‘k[]cv|Ei“GWF 1 ~ #ªF \jX 0 ªFo‰l 1.48 š Á|  L NQcpijYST]\jcpX cp‘ŸkE[Vcp|i“GWF 1 } ÀªF›\jX 0 ª Fo‰l 1.504 hfX ceT]CG[opkEkEij\jgWo‰TV\“cvX ce‘TVCEG6|@cpYEX”qEoe[VPgcpXqE\mTV\“cvXR!\_R¢‚v\“vGX |QPT]CEG oekkE[]c€lS\jFo‰T]\jcpX cp‘ÙTVCEG™‘’cpiji“c‰Ž \jXE‚ kE[Vcp|EijGF ªScvX¢TVCEGR]opF GقvGcvF›GT][VPpª cpX Ω, H(Du) = 0 u(x, y) = 0 u(x, y) = 3 cos(2πx). 181 . (x, y) ∈ Γ1 , (x, y) ∈ Γ2 .. NQ\jXg#G \_R"XEcpX g#cvXQpG#l)ª)\“T\jRqE\ !g#YimTTVc=9`Xc‰Ž '+ "+ Ž Co‰TŽŸcpYijq™|”G TVCEG‰oeijYEG ce‘T]CEG!R]cpijYST]\jcpXycpX™T]CG |@cpYEXqHoe[VPpšœBCEG¢gcpF kEYSTVGWq R]cpijYST]\jcpX \_R‚v\“vGX cpX \j‚pYE[VG w > Áo Àš6ÎT›gWoeX |@G¢RGWGX TVCo‰TTVCEGR]cpijYSTV\“cvX R]oeT]\_RU–G>R RT][VcpXE‚vi“P"TVCEGf|@cpYEXqoe[VP&g#cvXqS\“T]\jcpX›cpX opXq›cpXEijPŽŸGWo9QijPcpX Ág#cvX`T][/oe[V\“ijP"T]c"TVCEG kE[VGQ\“cvYRG#lEopF›ki“G ÀšHfceTVG CEc‰ŽŸGvG[T]CoeTbT]CGP&C”o€pG|”GWGXXQYEF G[V\_ΓgoeijijP ;  \“F k@cvR]GWq cpX Γ Γ oeXq Γ š8BCEGfRcvi“YET]\jcpX \jR•oei_Rc"\jX›vG[VP‚pcQcSq oe‚v[]GWGF GX`TŽ \mTVCT]CEGcvXEG"cp|ST/oe\jXEGWq‘’[VcpF TVCEGqS\_R]g[]GT]\ Wo‰TV\“cvX¢cp‘ cpX Ω, ||Dv|| − 1 = 0 1 x  v(x, y) = 0 (x, y) ∈ Γ , Ž CE\jg/C\_RfqS\_RkEi_o€PpG>q!cvX \j‚pYE[VG w > ’| #šv(x, y) = 3 cos(2πx) (x, y) ∈ Γ . Œ G¢oei_RcžR]CEc‰Ž CEc‰Ž T]CG!CE\j‚pC cv[VqSGW[G#lQTVGXR]\“cvX ce‘JR]GWgÀTV\“cvX t ŽŸcp[ 9QR"Ž CGX TVCEG AÙcSqSYEXEc‰ R]cpijpG[&Ž \mTVC \jXvTVG[Vk”cvijoeT]\jcpX‘’cp[¢T]CEG6CE\“‚vC cp[/qSG[R]g/CEGWF Gpš BCEG Wc`cvF \_R¢qS\_R]kEijo€PvGWqcpX \j‚pYE[VG t š 4ijGWop[]ijPpª o vG[VP¡i_oe[VP‚pG c‰pGW[VR]CEcQceTÙG#lS\_RUT/R"Ž CEG[VG u \jR"XceT C ªËT]CEGW[]G TVCEG!R]cpijYST]\jcpXyce‘ŸT]CEG|i“GWXqSGWq RVg/CEGF G \_R"F›cvXEceTVcpXEGopXq6\_RvG[VP R]\“F \jijop[&TVcžT]CG¢–[/RT›cv[VqSGW[&cvXEGp.š 6ÆX TVCEGR]F cQceT]C¡koe[]T.cp‘fT]CEG R]cpijYST]\jcpX«ªTVCEG RG>g#cpX”q cp[/qSG[›oeXq T]CEG|EijGXqSG>q RVg/CEGF G"oe[VGLvG[VP R]\jF›\ji_oe[ ÚTVCEG"[]G>RYEi“TVRŸ|QP T]CGÙ|i“GWXqSGWqRVg/CEGF GÙ\jR R]i“\j‚pC`TVi“PF cp[VGLqE\jRVR\jkoeT]\jpGLT]CopX¢TVCEcvR]GLcp‘«T]CG R]GWg#cvXqcp[/qSG[YEXEij\jF›\“T]G>qœR]g/CGF G #š BCEGi_opRT \“‚vYE[VGRCEc‰Ž T]CoeTJcpY[f\“F kEijGF GWXvT/o‰TV\“cvXœce‘T]CEG&|”cvYEXqEop[]Pg#cpX”qS\mTV\“cvXR \_R -G ?<GWg#T]\jpGvš 6ΑbŽŸG\“F k@cvR]G RT][VcpXE‚vi“P TVCEGž|@cpYEX”qEoe[VPgcpXqS\“T]\jcpX”RªopR\jX \“‚vYE[VG  °o BCE\_RCovR T]c |@G6gcpF koe[VGWqŽ \“T]C \j‚pYE[VG w š BCEG \“‚vYE[VG  RCEc‰Ž T]C”o‰TfTVCEG[VG\jRLoRUTV[]cvXE‚|@cpYEXqoe[VP!ijo€PvG[ cvXœkoe[]TVR cp‘TVCEGcvYST]GW[f|”cvYEXqEop[]P ’Ž CEGW[]G"TVCEG\_Rcvi“\jXEG>R oe‚v‚pijcpF G[/o‰TVG #šJBCE\_RL\_RLXEceTÙT]B [VYEG‘’cv[JTVCEG \j‚pY[]G  °o #š 6ÆX6‘ÁopgÀT>ª<cpX™R]cpF G.koe[]TVRJce‘bTVCEG.cvYST]GW[L|@cpYXqEoe[VPpª”T]CG g#cvF ko‰TV\“|E\jij\mTUP g#cvXqS\“T]\jcpXœce‘ 1 C\jRfT][VYEGpª<R]cT]CoeTLcvXEG&gopXœ\“F k@cvR]GT]CG&|@cpYEXqoe[VPgcpXqS\“T]\jcpX”RJRT][VcpXE‚vi“Pvª”oeX”q cpX ceTVCEG[k”oe[]TVRLT]CE\_R"\_R"XEceT"T][VYEGoeX”q6ŽŸG Co€pG.T]cœoekkEi“PžT]CEGWF ŽŸGWo 9QijPpš›BCE\jR"k”oe[]T]\“T]\jcpXyce‘•TVCEG |”cvYEXqEop[]P6\jR"XEcpT 9QXEc‰Ž X o.k[]\jcp[V<\ L8cpYE[ \jF kEi“GWF GX`TVoeT]\jcpXTV8o 9pGÙT]CE\_R\jXvTVc!opgWg#cpYXvT oeYET]cpFoeT]\_goeijijPpš  &Ÿ# (Ë#(  Œ G.Co€vG.qSG>R]g[]\j|@G.R]GpGW[Vopi«TVGWg/CEXE\_^`YEG.‘’cp[LT]CEG Rcvi“YET]\jcpX6cp‘b–[/RUTÙcp[/qSG["—JopF \“i“T]cvX6˜vovg#cv|E\G>^`Yo‰TV\“cvXRš Œ G.C”o€pG T][V\jGWqTVcLGlSkEijop\“XT]CEGCE\_qEqSGWX.qSGTVoe\ji_RopXqTVCEGŸcp[V\“‚v\“XRce‘T]CEGR]g/CGF GWRWšNQGWpGW[VopieT]CGcp[VG#TV\jgWoei`[]G>RYimT/Rop[]GkE[Vc‰`\_qSG>q)ª 2. "!. . 1. 1. 2. 1. 2. 2. 1. Í N ÍÚ¹.

(105) ~>w.  

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(122) Unité de recherche INRIA Futurs Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).   

(123).   . ISSN 0249-6399.

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