A general fictitious domain method with immersed jumps and multilevel nested structured meshes

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(1)A general fictitious domain method with immersed jumps and multilevel nested structured meshes Isabelle Ramière, Philippe Angot, Michel Belliard. To cite this version: Isabelle Ramière, Philippe Angot, Michel Belliard. A general fictitious domain method with immersed jumps and multilevel nested structured meshes. Journal of Computational Physics, Elsevier, 2007, 225 (2), pp 1347-1387. �10.1016/j.jcp.2007.01.026�. �hal-00274475�. HAL Id: hal-00274475 https://hal.archives-ouvertes.fr/hal-00274475 Submitted on 18 Apr 2008. 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..

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(41) hg\^no_Eg>]pbb:aMq$ZvÔ \dk1e{gMtÛ}>b kb carÛ [``ë5b+q+h\dg>kvÍÅw\^rdrVEb(e1b e\^q:a4hb e%hgÅhZ1b(xVk1\dhib f>g>rdm1nobo]pq$Z1b noboe1b+fMgMhb+eUhgsq g>no_1mhib|hZ1\d]Åx1q+h\h\dg>m1]!e1g>n%aM\dkjn%bhZ1g0eÝY[Z1b+k ë5b qh\dg>kvÙ. \^k aMkaMeVa_hb+ecib+x1k1b nob kh(aMrd}>g>ci\hZ1n²VaM]ib+e¿g>kalnlm1rh\dr^b+fMb rVÖVm5Ð{cib+]phici\^q¦®. h\dg>k\d]~\^khcpg5em1q b+ehgÄ\dk1q cpb:aM]pb.hZb©_1cib+q \d]i\^gMklgMtVhZb©]ig>rdmh\dg>kÝ4¹â\dkVaMr^r. \dkë5b qh\dg>kç ]ig>nob|k m1nob ci\dq:aMrcib+]im1rh]Ätg0q m1]pb eËb \dhiZ1b c!gMkje1\dáEm1]i\dg>kjg>c!g>kjq+g>kf>b+q+h\dg>k5®¸e1\dáEm1]i\dg>k _1cigM1r^b+n%][\dr^r^m]phciahb(hZ1bÅ_Pb ctg>cin%aMk1q+b!gMthiZ1bÅn%bhZ1g0eÝ. ø. ù[NOLMDHNXDHNOGPJKúlIGES¯QVN CÀSUûDHüKGÝIRý%NDHüRN SUSUû Fú+û5I8þ:JKSUÿâú. m1cg>0Èpb qh\f>b(\^]hg]ig>rf>b. w\hZ%aÅx1q+h\h\dg>m1]e1g>n%aM\dk|n%bhZ1g0e h Zb(tg>rdr^g?w\^k}Ä_1cigM5® r^b+n g>cp\^}>\dkVaMrdrd|e1bxVk1b+eÃ\^k¿alPg>m1ke1b ese1g>n%aM\^k d w\hZb \ hZ1b+ca|Î`\^cp\^q$Z1rdb+h a b+m1n{ak1k8gMcaoÞg>1\dk¿Eg>mk1eVaMcvq gMk1e1\dhi\^g>kËg>cn%\Ð5b eêÆgMk ± Ü D R xVk1e ¹Vg>c aMke 2 ∞ d×d ∞ d ∞ 1 ]im1q$ZshZVa4h. ˜ ⊂R Ω. ˜ ˜ a ∈ L (Ω). ˜ (P).       . 06αR. \^k. div(−˜ a∇˜ u+v ˜u ˜) + ˜b u ˜ = f˜. g>k. u ˜ = uD.      . wZ1b cpb. ˜ = ∂Ω ˜ ∪ ∂Ω ˜ ∂Ω ˜ ˜ ˜ f ∈ L (Ω) u ˜ ∈ H (Ω). ˜ , ˜b ∈ L (Ω) ˜ , v ˜ ∈ L (Ω). g>k. −(˜ a∇˜ u).n = αR u˜ + gR \^]UhiZ1bTg>mhw.aMcpe. n ˜ R) ∈ L∞ (∂ Ω. ak1e. Y[Z1bhb k]ig>cugMtÝe\dáEm1]i\dg>k. mk1\dh×kg>cin%aMr%fMb qhg>c×g>k . ˜ R) gR ∈ L2 (∂ Ω aMk1ehZ1b(cib ˜ a ≡ (˜ aij )16i,j6d a5bV\^k. . ¤. |.|. ¢^¬ª. Æ ê¨£¢O0iMÃÛ«4¤Â¢X . ∃b0 > 0, ˜b(x) > b0. aÉbMV\dk. g>_Pb kPg>m1k1e1b+e_EgMrd0}>g>kVarPeg>n{a\^k. ˜b f>b. cp\dtXlhiZ1b. OºH. ˜ Ω,. Rd Oº. . . OÍ. ˜ R, ∂Ω. aMq+hi\^g>koq g0b+Ïoq \db kh. ˜ Ω.. Ók¿axVq+hi\dhi\^g>m1]Æe1g>n%aM\^k{aM_1_1cpgaMq$Z. . ˜ R uD ∈ H 1/2 (∂ Ω ˜ D) ∂Ω. q râM]i]p\^q:arÝb rdr^\^_5h\^q+\dhoaM]p]im1no_hi\^g>k1]+±. ∃a0 > 0, ∀ξ ∈ Rd , ˜ a(x)ξ.ξ > a0 |ξ|2. ,H. ˜ Ω, ˜D, ∂Ω. hZ1bÄgMci\^}M\^kVaMrVeg>n{a\^k ]pm1q$ZvhZVa4h. \^].b nlPb e1eb e\^k]i\êb!aMk. ˜ Ω ˜ ∪ Σ ∪ Ωe Ω=Ω. wZ1b cpb. . Ω ⊂ Rd Ωe hiZ1bq g>non%gMks\dkhb ctOaMq bPb+hwÆb b k ˜ aMk1e Σ Ω ]ib+bÄ¹â\d}>m1cibÄak1evï¦ Y[Z1\^]Æ\^khb+cptOaMq+b \^]uq:ar^r^b+e8¢X|{¤¬?i¢XVª¤X¶+0¨$ Ωe Σ ⊂ Rd−1 Y[Z1bÅxVq+hi\dh\dg>m1]©eg>n{a\^k \d]q$Z1g>]ib+k8higoPbÅ}>b gMn%bhci\dq:aMrdrdo]i\dn%_r^bcib qh$aMk}>m1r ac©tg>c Ω bÐaMno_1r^b.\^k ®Î!Y[Z1b[EgMm1k1eVaMclgMt ˜ \^]~eb+xVk1b+e% ∂ Ω ˜ =Γ ˜ ∪ Σ aMk1e|hZb[Eg>mk1e5® Ω aMcpvgMt ˜ ∪ Γe O]ib+b¹â\^}1EH Ω ∂Ω = Γ ôb+hm1]!xVci]hk1gMh\dq blhiZVahÅwci\h\^k}vhZ1boÞg>1\^kjPg>m1keVaMcp´q g>k1e\dh\dg>kÕ Í>!gMk \^kËa Σ xVq+hi\dhi\^g>m1]ue1g>n%aM\dk%nob+hiZ1g5eow\dhiZ ˜ \d].k1gMh.e1\^cpb qhrd_Pg>]p]i\^r^b(m1k1r^b+]i][aq q b+_h\dk1} Ω⊃Ω ÖVm5Ð´g>cÄ]igMr^mhi\^g>k|Èpm1n%_]Äg>k g>cÄPgMhZÝEY[Z1\^]`_1cig>r^b n \^]zhZ m1]zñ,b¦Ð0hb ke1b e5òl\^kÃhiZ1b Σ tg>r^rdg?w\^k1}vn%aMk1kb cÄhg¿hZbwZg>r^b|xVqh\dhi\^g>m]Äe1g>n%aM\^k Ìm1]i\dk1}hZ1btg>cinm1r ahi\^g>k Ω \^][hZbb¦Ð5hib cpkVaMrxVq+hi\dh\dg>m1]e1g>n%aM\dk8ak1e. tg>c[}>b+k1b ciaMrb nlPb ee1b e8Eg>mk1eVaMcvq gMk1e1\dhi\^g>k][\^khcpg5e1mq b es\^kÌÚÙ. Ù. . .

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(43) ¦ 5ÌªO¤pM¬|¢^¬$¬¢"«4¯¨$«4PM¢XªO¢"«4¬«4. oi0i. Pbzb+\dhiZ1b chZ1b(g>mhw.aMcpevm1k1\h.k1gMcin%aMr1f>b+q+hgMc.g>k. Σ. g>chZ1b(m1k1\h.k1gMcin%aMr1f>b+q®. Γe hg>câg>k g>cp\^b+k hib e|tcigMn hg ¹Vg>caÄtm1k1qh\^gMk \^k ˜ ˜ ∪ Ωe ) r^bh ψ − aMke ψ + Σ Ω Ωe ψ H 1 (Ω Σ Σ EbuhZ1buhciaMq b+]âgMt ak1e g>klb:aq$Zl]i\de1b.gMt cib ]p_Eb+q+hi\df>b+rd + − ψ|Σ = (ψΣ +ψΣ )/2 ψ|Ω˜ ψ|Ωe Σ hZ1bÅaci\dhiZ1nob+h\dqn%b aMkgMthciaMq b+]©gMt aMk1e hZ1bÆÈpm1no_vgMthicaMq+b ] ψ [[ψ]]Σ = (ψΣ+ − ψΣ− ) g>k g>cp\^b+k hib es gMt ψ Σ n ¹Vg>cuhiZ1bÄeVa4h$a ak1e }>\dfMb kv\dk w.b!q+g>k1]p\ê1b+cÆhZbÄ]ib+q g>k1e0®g>cpe1b c f ∈ L2 (Ω) g q L2 (Σ) b rdr^\^_5h\^q(_1cpg>1r^b+n tg>c©hiZ1bÅcib aMr®"fMar^m1b+e{tm1k1qh\^gMk e1b+x1k1b es\^k ak1e¿\^k1q+r^m1e1\dk1}v¢X u Ω {¤¬?i|ªO¤pM¬|¢^¬$¬¢"«4´¨$«4PM¢Xª ¢"«4¬gMk wZ1\^q$ZvaM]i]pm1n%bÈpm1n%_]ÆgMtPgMhiZ%hZbÄk1g>cpn{aMr Σ e1\dáEm1]p\df>b[ÖVm0Ð aMkeohiZ1b[hcaq b ]ugMtPhZ1b]pg>r^mhi\^g>k hZ1cpg>m1}>Z|hZb ϕ(u).n ≡ −(a∇u).n u \^khb+cptOaMq+b ± Σ. (P). n.            . \^k. div(−a∇u + vu) + b u = f.           . gMci\^}M\^kVaMrE``. g>k. ]im1\h$aMr^bÅ``. g>k gMk. [[(a∇u).n]]Σ = α u|Σ − q. gMk. (a∇u).n|Σ = β [[u]]Σ − g. Ù . Ω, ˜ Γ,. ç> Oè. Γe , Σ, . Σ,. wZ1b cpb|hZ1bm1k1\tg>cinords_Pg>]i\h\f>ble1bxVk1\hb|hb+k1]ig>c!gt.e1\áÛm]i\^gMk cib aMq+hi\^g>klq+g5bÏoq \^b+kh. b>0. . aMk1elhiZ1bÄªO¤iM¬¶:¤(¨$«? ·¨¢"VªX¬. . é ï hiZ1b. a ≡ (aij )16i,j6d g>k aMcpb.nob:aM]pm1c® α, β>0 Σ. aM1rdb|aMk1ejPg>m1k1eb ejtm1k1qh\^gMk1]Äf>b+ci\dtX0\dk1}vq râM]i]p\^q aMrâb+r^r^\d_h\dq \hsaM]i]pm1n%_5h\^gMk1] ÝÑ8g>cib¦®. g?f>b c. w.bÅq$Z1g0g>]ib . a|Ω˜ = ˜ a,. b|Ω˜ = ˜b,. v|Ω˜ = v ˜,. \^kg>cieb chg(cib q+g?f>b cKhZ1bug>cp\^}>\dkVaMr>_1cpg>1rdb nÇ,+®,ÍÝ\dk. f |Ω˜ = f˜. ˜ HY[Z1bÆ``1OèÝg>k Γe Ω. \^]q$Z1gM]ib k. hgb+k1]im1cpb|hZ1b|]pg>rdfaM1\dr^\hgMthZ1baEg?f>b|xVqh\h\^gMm1]è1gMn{aM\dkńog5e1b+r"êY[Z1\d]!_1cig>r^b n \^](_1cig?fMb eÃhig{PbwÆb rdr®_Pg>]pb es\^k¯Í. ·Z1b. k. α=g=q=0. aMk1e. . Ù . β → ∞. aMk1eshZ1b]igMr^mhi\^g>k. u. Pb r^gMk1}>]hg. ˜ ∪ Ωe ) H 1 (Ω. hZ1b_Pb cptb+q+h(hciaMk1]ino\^]p]i\dg>k_1cigM1r^b+nà\^]cib+q g?f®. ç. .

(44) b cpb e¿w\dhZ. u ∈ H 1 (Ω) . ]ib+bÅY[Z1b g>cpb n. \^k. Y[Z1bjeVaha. Í. \^kË . 4. aMk1e. a v b f Ωe α β g q g>cieb c[hgo}>b+h g>c(ah(rdb:aM]h wZ1b u| ˜ = u ˜ uη |Ω˜ ≈ u ˜ _Eb+kVaMrh_VaMciaMn%Ω bhb+c η→0. ß. . ¤iMªO{Vª©«,¶Åª. ß ß ôb+h. . . . l«4¤¦¢ M¢XPM£ ©ß zß ß«4. Æoi0i«M. . g>k. Σ. w\^rdrzPbje1bxVk1b e·tm1chZ1b+cv\^k. kshZ1bÅnog5e1b+r\^]_Pb kVaMrd\^Ø+b ew\dhiZÃa. Σ. 5oMVË«4 0PM¤¦¥¨$«4PM¢XªO¢"«4¬. ¢XU«4¤. ϕ− ≡ −(a∇u)− .n|Σ. aMk1e. PbohiZ1b%hicaMq+b ]gMt.hiZ1b{e1\áÛm0®. ϕ+ ≡ −(a∇u)+ .n|Σ cib ]p_Eb+q+hi\df>b+rdMuY[ZbÃaMrd}>b Σ. Σt ]i\f>bsk1Σ g>cin%aMr.Ö1m5Ð³g>kÕb:aMq$ZR]i\de1b¿gM q g>ke1\dhi\^g>k1] é. ®,ï>uwci\hb. + ϕ− Σ − ϕΣ =. caM\dq{hciaMk1]ino\^]p]i\dg>k. α + (uΣ + u− Σ ) − q, 2. Oä. 1 − + − − (ϕ+ Σ + ϕΣ ) = β (uΣ − uΣ ) − g. 2 Y[Z1b©tg>m1cm1k1Ô k1g?wkoæ0m1aMkh\dhi\^b+]. ,?ð. + − ϕ− Σ ϕ Σ uΣ. ak1e. u+ Σ. g>k. aMr^}Mb 1ciaM\^q©hicaMk1]pn%\d]i]p\^g>kq gMk1e1\dhi\^g>k]( ä4®?ð>ózb kq b bÐ0hb+ci\dg>cÅæ0m1aMkh\dhi\^b+] b æ m1\fMar^b kh[w.a:5] . b \hZ1b+c. ϕ. + g>c. Σ± cpb p ] _Eb+q+h\f>b+rdj. Σ. aMcibÆhZ1b+k|r^\^kÔMb elw\hZhiZ1b. b rd\^no\^kVahi\^k1}`g>k1bgMtÛhiZ1bhw.g q M a Ë k E | b wci\hihib kË\^kjhZ1bhwÆgstg>rdr^g?w\^k} −. u+ Σ ϕΣ. g>k α − α + q uΣ − β − uΣ + g − , Σ, = β+ 4 4 2 α α 1 − + αβ u + β − ϕ + ϕ− = g − β q , Σ Σ Σ α 4 2 β+ 4. . ϕ− Σ. . . P¤¦¢"«4¤¦¢Pe1b+xVkb. gMhib!hZVah( æê \^] Ü Y Z1b©_Vacph\dq m1râMcq$Z1g>\dq b [ aMk1e·,. u+ Σ. gMc. . β = α/4. eÃ\t. α. ak1e. β. gMk. . Σ.. aMcib!e\dáEb cib+kh(tcpg>n. 0. . ±. . \^k1eb _Pb k1e1b+khrd¯gMt. α − q g>k :Í Σ. uΣ + g − , 2 2 k%hZ1b`Þ(g>\^k{Pg>m1keVaMcpoq+g>k1e1\h\^gMk%\dkj Í>\d]Æe1b+]i\dcib e%tg>cuhiZ1b`gMci\^}M\^kVaMr_1cig>r^b n − ϕ− Σ ≡ −(a∇u) .n|Σ =. ·Z1b. +®Í~\^k. ˜ Ω. hiZ1bztg>rdr^g?w\^k}\dn%nob cp]ib+e{Pg>m1k1eVacpq gMk1e1\dhi\^g>kog>k. vhZ1bÅ]pg>r^mhi\^g>kgtâhZbÄxVq+hi\dh\dg>m1][e1gMn{aM\dk_1cigM1r^b+nÂ\^k. − − ϕ− Σ ≡ −(a∇u) .n|Σ = αR uΣ + gR. Ω. ± gMk. Σ Σ.. nlm1]ph.Eb(]a4h\^]xVb e. Ù . Y[Z1\^]}>\dfMb ]lhZbv]imÏoq+\^b khq+g>k1e1\h\^gMk1]tg>chZ1bve1ah$a´]pm1q$Z×hZVah| æê,?ÍÅ]ahi\^]px1b ] :Ù. . cib æ m1\dcib+]©Û«p¡>ª¤¦¢"«4¤(¨$«4VªO¤«4£?]i\^kq b.PgMhiZl æ ] V,>H. !5\db rde¯hZb{]an%b{Þ(g>\^kUPg>m1keVaMcp¯q gMk1e1\dhi\^g>kËPb r^g?w. ϕ+ Σ. MH. . . ]pb b{ Þz.\^khZ1bÅYaM1rdb|>. è.

(45) Y[Z1b wZ1b cpb. b+m1n%aMk1k°Pg>m1k1e1aMcp¯q g>ke1\dhi\^g>k¯\^]gMh$aM\dk1b e°aM]hZbv_VaMch\^q+m1r acÞg>1\dk°q:a]ib Ü. αR = 0. . Y[Z1\^]Þg>1\dk!g>c \^k¯ÉÙ. b+m1n%aMk1k [``?nob+hZg5eÅ\^]Û_1cig?f>b+ehig(Pbq gMk1]i\d]phb+kh ¢ ßO ß O Ü hZb c[faMci\âMkh](aMcibÅ_1cpg>_Pg>]ib+e8hZb cibÅwZ1\dq$Z8cpb æ m1\^cpbb+\dhiZ1b c g c + M. . ñ,q+g>khcigMr^r^b+e5ò. hZ1bÅ_VacaMnob+hib ci]. H1 aMk1e f. Æoi0iÅ¢X¤¦¢"¨ . ß ß ôb+h. gMc. L2 a b. 0<η1. f>gMr^m1nobl_Pb kVaMrhńob+hiZ1g5e]| \^k. Ωe. . 5. :Í . ÍMè. uΣ. u|Ω˜ = u ˜ higoPb ϕ+ Σ. _Pb ctg>cinob e´w\hZ . £^ªÆ«4 0PM¤¥¨$«4PM¢XªO¢"«4. VPM£ª ¥l¨$«?T¨¢"Vª wZ\^q$Z\^]. Eba!cpb:aMr1_Pb k1aMrdh_VaMcan%bhb c(q:aMrdr^b+e. !. r^\dÔMb rhighib k1eohgÅØ b+cig1 ·Z1b+k%hZb`Î`\dci\dq$Z1r^bhq g>k1e\dh\dg>k. u− Σ = uD. \d]~hgEb(ZVaMk1er^b e. "VPM£ªO¥ . hZ1blÞ(g>\^kÃq+g>k1e1\h\dg>kÃ\dkj æP~ Íg>c|,:Ù q:aMk´Pbl_Pb kVar^\^Ø+b e8 ás¬ 0¤¶+0¨$. uak1e Y[Z1b`_cib+f0\dg>m1]ÆÞ(g>\^k% `` 1 1 αR → + ∞ g R = − uD η→0 η η n%bhZ1g0eq:aksPbÄhZb k¿_Eb+kVaMrd\^Ø b+e¿aM][\^kj ÎH.\dkvhZ1b!YaM1rdbl>Vó(b k1q+b!tg>c aMk1e g=0 hZb ` nob+hiZ1g5e´0\db rê]!k1g¿]ig>rdmh\dg>koÈpm1no_ \^k1eb® β → +∞ (D1) [[uη ]]Σ → 0 _Eb+k1e1b+k hirdÃg>k EY[Z0m] wÆb|q:aMkjm]ibltg>cÄb¦ÐaMn%_r^bhiZ1b|_VaMciaMnob+hb+ci]l± 1 α α=β = η ak1e 1 q = uD g=0 η g>k. Σ. w\dhZ. αR =. º`k1ghZ1b+cÄh5_PbgMtun%bhZ1g0ejq gMk1]i\d]ph]!\dkjm1]i\dk1}sa bÐ0hb+ci\dg>czeg>n{a\^k aMk1eÌhZ m1]. g =0. bÐ0hb+k1]i\dg>k ]ib+bvOÎ . Ωe. ]pm1q$ZjhZVah. lim u+ ηΣ = uD. L2. aM]Å_1cig>_Pg>]pb eË\^kÕ. η→0. ue ∈ H 1 (Ωe ). gMt. .\^khiZ1bÅYâaMr^b|>. uD. g?fMb c. Ωe. H1. f>g>rdm1n%b_Eb+kVaMrdh8\^k´hiZ1b. tg>czhiZ1bnog0e1b rw\dhiZ. η→0. [[uη ]]Σ → 0 . g>c. 4. ÓkÌhZ1ahq:aM]pb. ]im1q$ZshZVah[hiZ1b!hciaMq b. Y[Z1b¿q g>kfMb ci}Mb k1q+b´gMt(hZ1b+]ib¿n%bhZ1g0e1]\^]|_cig?f>b+eÀ\^k·ÚÙ m1r aci\dhÃaM]p]im1no_h\dg>k1]`hiZ1b|tg>r^rdg?w\^k1}%b cicpg>cÄb+]ph\dn{a4hbv±. . . 1 → ∞ η gMk1]i\de1b caMk. β= w.boq. γ0 (ue ) ≡ ue |Σ = uD . aMke³tgMc|m1]imVar©w.b aMÔ×cib+}®. ||uη | ˜ − u ˜||. 1. ˜. √ = O( η). Hcib+ (Ω) \^]ahrdb:aM]hcib aMq$Z1b eÛEÆb+hihib cb ]h\dn{ahib ](aMcpb!_Eg>]p]i\d1r^bÄw\hZ¿]pΩ hicig>k1}Mb c }>m1r aci\dhM. ë5\^kq bhiZ1b\^non%b+ci]ib+eÃPg>m1k1e1aMcpsq gMk1e1\dhi\^g>k]zacibÅhcib ahb+eÃw\dhiZ1\^k¿hZb]an%bm1k1\dx1b e tg>cinm1r ahi\^g>k%w\dhZ8aM_1_cig>_1cp\ ahibÅeVah$as]ib bYâa1r^bH. \dh\^]]hca\^}>ZhitgMcpw.aMciehgoe1b aMr w\dhiZÃno\Ð5b esb nlPb ee1b e´Pg>m1keVaMcp¿q+g>k1e1\h\dg>k1]g>k8hiZ1b\^non%b+ci]ib+e8\dkhb ctOaMq b êº`k. Σ. bÐaMno_1r^b!\d][}>\dfMb ks\^k8ë5b+q+h\dg>k8ç5ÉÙ1. é.

(46) ÔÕ×ÖÕÙØmÕÉ

(47) Ãc¼o¹ È ¾ ÜÈ[ÎºÄ)ÓÜÚ ¥n+aof<a3b3^nbjp}n[]Xfbn[ àTºÂoº¯Çe¹Ì¯Ãc¼jÓ$à Ú §;bg¡r5pea9a3]Xr°fô. #ÀFÂeÀFÉ

(48) Ãc¼}ÃcÂ}»mºÄ. a|Ωe = Id, v|Ωe = 0,. a|Ωe = Id, v|Ωe = 0,. a|Ωe. ' n[°

(49) a9ac]Xr[Òfô ¸ÀFÎÌ¯Ã( ). Ωe. *. ÀFÄ ¾ËÈ[Ä. Σ. Σ. α = 4β = 2αR , q g − = gR 2 2 α = 4β = , η 1 q − g = uD 2 η. b|Ωe = f |Ωe = 0. àTºÂoº¯Çe¹Ì¯Ãc¼jÓà&%5Ú #.ÀFÂeÀFÉ

(50) Ãc¼}ÃcÂ}»ºÄ. #.ÀFÂeÀFÉ

(51) Ãc¼}ÃcÂ}»TÈ[Ä. Ωe. b|Ωe = f |Ωe = 0    Id L2 = 1 , v|Ωe = 0,   Id H 1 η 1 1 b|Ωe = , f |Ωe = ue η η. ÍÃcÄ;ÀFÌ¼<¿ ÍßÃcÄ;ÀFÌ¼K¿. β=. 1 , η. α=q=g=0. Ê¬È[Â9¼o¹ Ã,ÔÕ×ÖÕÙØmÕ É

(52) Ãc¼o¹ È ¾ËÅ9º¼o¹ÁºÉsÉ

(53) ÃcÂ}»eÃ¾JêY½ÉsÍ »Õ. ,+pù.-0/lIKNOú:L>FHû D?N21QD?NOGPC. ù[NOCNXD?û¿WPG OJKSUû. 3 ß . «4ª¸MªO¢"«4¬. Y[Z1b[_Pg>rd0}>g>k1aMr5eg>n{a\^k f>g>rdm1nob ]. Th = {K}. w\hZÁm]imVaMrÄaM]i]pm1n%_5h\^gMk1]. K∈Th. h = sup{diam(K), K ∈ Th } w\dhiZ. . bMÉ}1ÄÉÍ. é4. g>kfMbÐ×_Pg>r5}Mg>k1]. .Y[Zbjn%b+]iZT]i\dØ b´\d]. . EbÄhiZ1b!tOaMn%\drdgMtK]p\ê1b+] aMrd]ig|q:ar^r^b+eb e1}Mb ]\^k. ¹Vg>cak . gMtPe1\^]XÈpg>\^khb+exVk1\hb. 54"¨$«4VªO¤«4£76M«4£ 0{$¬98"ÄwZ1\dq$Z³aMcib%g>_Eb+k×q. K ⊂ Ω Ω = ∪ K. e1b+x1k1b e8 . E. \d]~n%b+]iZ1b+elw\dhiZa`tOaMn%\drd. Oq:ar^r^b+e. ]im1q$ZThiZVah. ôb+h. Ω. b \dhiZ1b cÅhZb. . ®Î!ÆgMthiZ1bÅq g>khcpg>rÛfMg>r^m1nob ]+ ®e\^nob k1]p\^g>kVarôb Pb ]i}Mm1b. (K, L) ∈ Th2 K 6= L (d − 1) n%b aM]imcib¿gMt \^] g>c tg>c|]igMn%b wZ1\dq$ZÕw\dr^rÆhiZ1b kÕPb K∩L 0 K∩L = σ σ ∈ E e1b kgMhb+e° Y[Z1b{]pb+hlgt[k1b \d}>Z EgMci]gMt \d]eb k1gMhib e° hiZVah σ ≡ K|L K N (K) \^] Æ¹êgMc%b aMq$Z rdb+h Pb N (K) = {L ∈ Th ; ∃σ ∈ E, σ = K ∩ L} K ∈ Th EK hZ1b|]pm11]pb+hgMt ]im1q$ZÌhZVa4h Ý¹Vm1chZ1b+cinog>cib E ∂K = K \ K = ∪ σ ∪ EK E = σ∈E. Ke1b kgMh\dk1}R Y[Z1bË]ib+h8gMt\^khb+ci\^gMcËOcib+]i_Ý(Pg>m1k1eVacpv]i\êb ]s\^]¿. hZVa4h\^]. Eint = {σ ∈ E ; σ 6⊂ ∂Ω}. Ocib+]i_Ý. Eext. K∈Th. Ocib+]i_Ý. Eint ¦ = {σ ∈ E ; σ ⊂ ∂Ω}. Eext. . k1bgMt. . hZ1b8n%gM]ph%\^khb+cib ]hvgMt`hZ1b8 [``.nob+hiZ1g0eÀ\^]hg¯m1]ibÃa¯]p\^no_1r^b¿k1g>k0®q gMktg>cino\^k} aMk1eÕ]phicim1qhm1cpb eÁnob ]pZÝY[Z1b cpb+tg>cpb n{a4hq$ZjhZ1b|\dn%nob cp]ib+ej\^khb+cptOaMq+b. Σ. . hZ1b¿]p\ê1b+]%gMtzhZ1b¿nob ]pZ bMÉ}1Ûw\dhiZÌam1k1\dtgMcin. ë5b qh\dg>k½ç>.º`k½aM_1_1cpg:Ð\dn{a4hbÃ\dkhb ctOaMq b. Σh. Th. e1g¯kgMho}>b k1b+caMrdrd. ©aMcphib ]i\âMkjnob ]iZÌrd\^ÔMb\^k. \^]%hZ m1]e1bxVk1b e½ÁhZbjn%b+]iZÊO]pb b. ë5b qh\dg>ksç>0Y[Z1\d]©\^khb+cptOaMq+bÄ]ib+_VaMciahb+]©hZ1b!a_1_1cig:Ð5\^n%ahib`g>cp\^}>\dkVaMrVe1g>n%aM\dk hZ1baM_1_1cpg:Ð\dn{a4hbÅbÐ0hb+cikVare1gMn{aM\dk. Pôb+h. ˜h Ω. tcig>n. PbhiZ1b]pb+h. Ωe,h EΣ = {σ ∈ E; σ ⊂ Σh } a:4"¨$£X£;i¨$Vª¤i<8M¢^¬?¨¤ªO¢=?Mª ¢"«4Ã_Eg>\dkh Σh K ∈ Th \d]q$Z1g>]ib+k%]pm1q$Z%hiZVahtg>cb aMq$Z hZ1b]pb }>nob kh \d] xK ∈ K σ ∈ EK ; σ = K|L [xK , xL ] g>cphiZ1g>}>gMkVaMr5hig ah~hiZ1b[_EgM\^kh ¹Vg>câb aMq$Z r^b+h σ xσ ∈ σ K ∈ Th dK,σ = |xK − xσ | > 0 EbÄhiZ1b!e1\d]ph$ak1q bÄtcpg>n hig ak1e EbÄhiZ1b!m1k1\h[g>mhw.aMcie¿k1gMcin%aMrEfMb q+hig>cg>k xK σ nK,σ ¹êgMc wÆbvhZ1b+k°e1b kgMhbv hZbv m1q rd\ê1b aMk°e1\d]phaMk1q b%Ebhw.b+b k σ σ = K|L dσ xK gMt[]p\ê1b+]rd0\dk1}8gMk. ¹êgMcb aMq$Z. ï.

(54) aMk1e. Zb k1q+bwÆbÅZVa:f>b. xL . ¹Vg>c|b:aMq$Z. . g>c. dσ = dK,σ + dL,σ Z. ak1e. K ∈ Th σ ∈ E m(K) = m(σ) = dx K hiZ1bÄe1\d]iq+cib+hib`faMr^mb n%b aM]imcibÄgMt gMc ¹êgMc.a}M\df>b+kæ mVaMkh\h K σ ψ cib+]i_Pb q+hi\dfMb rdP± e1b+x1k1b e8 {hZ1bÅnob:aMkfaMr^mb ]gMt g?f>b c gMc ψ K σ ψK =. 3 ß. >. 1 Z ψ(x)dx, m(K) K. V 5{¤¦¢"ïM£E¬?¨. . ¹Vg>c]iaMÔMb{gMt[]i\dn%_r^\^q+\dh e1\dáEm1]p\^g>k³hb+k1]ig>c e1b kgMhb+eË. u. uK. . {¶:«4¤Åª. . "

(55) ¦ 0V¬Å{«. hZ1b{k0mn%b+ci\^q aMr]iq$Z1b+n%b{\^]k1g?w. a = a(x)Id. u¹Vg>cb:aq$Z. ahhZb8k1g0eVaMr_Pg>\^kh. ak1e. K ∈ Th. \d]!aMkËaM_1_cig:Ð5\^n%ah\dg>kÃgMthiZ1blfaMr^m1b. ds ]σ ψK. aMk1e. aMcpb. ψσ. 0£. wcp\dhphb kËtg>caMk°\^]pgMhcpg>_1\dq. hZbÃn%aM\^k³e1\d]iq cpb+hib8m1kÔ0kgHwk. u(xK ). gMthiZ1b|bÐaMqhÅ]ig>rdmh\dg>k. gMt`hiZ1b8nob ]pZÝYw.g×am5Ð5\^r^\âMcp¯m1k1Ô k1g?wk1]. xK σ = K|L ∈ Eint u(xσ )|L. eb k1gMhibshiZ1b. 1 Z ψ(s)ds. m(σ) σ. ψσ =. Å¢X|{¤¬?i. aMcib\dk hicig0e1m1q+b eUgMk. u(xσ )|K. aMk1e. Z. (uK,σ , uL,σ ). wZ1\^q$Z¯aMcpb{aM__1cig:Ð5\^n%ah\dg>k1]Ägt©cib+]i_Pb qh\dfMb r. @?. Y[Z1bvq+b r^r®q+b khb+cib e³¹. àk m1nob ci\dq:aMru]iq$Zb nobm1]ib+]lhZ1bv]pq$Z1b nobv_1cpg>_Pg>]ib+e×\^kTÉÍ tg>c e1\dáEm1]p\^g>k5®¸cib aMq+hi\^g>kv_cig>1rdb no] 5Y[Z1bk m1nob cp\^q aMrÛq+g>kf>b qh\f>bÖ1m5Ð¿\^].g>h$a\^k1b+e¿w\hZsa n%gMk1gMhgMk1b`m1_w\dk1e¿]iq$Z1b+n%bVY[Z1\^]©]pq$Z1b nobÅ\^]©gtxVci]hg>cie1b+c.wZ1\dq$Zs\d]©\^k¿a}>cib+b nob kh. A?. w\dhiZÃhZ1bbÐ5_Eb+q+hib e¯aMq q+m1caMqÃgtg>m1cÄnob+hiZ1g5eÛEY[Z1\d]Ä¹. ]iq$Z1b+n%b|ar^r^g?w]Èpm1n%_]`gMt. e1\dáEm1]p\df>b%aMk1eËq gMk fMb qh\dfMb{ÖVm5Ð5b+]laM]Åw.b+r^ruaM]©Èpm1no_1]ÅgMt.hiZ1b{]pg>r^mhi\^g>kÌg>kËb:aq$Z°]i\de1b. σ ∈ Eint X. ak1e8\hcib aMe1]\^khZbÅtgMr^r^g?w\dk1}]5khiZ1b+hi\^qÅtg>cins±. p m(σ) (FK,σ + vK,σ uUK,σ ) + m(K) bK uK = m(K) fK ,. σ∈EK wZ1b cpbhiZ1bÅk m1n%b+ci\dq:aMrÝe\dáEm1]i\f>bÄÖVm5Ð. FK,σ =.  uK,σ − uK   −aK    dK,σ   . FK,σ \dt. uσ − u K  dK,σ    aK    (κσ uK + φσ )  aK + κσ dK,σ −aK. aMk1e×hZbk0mn%b+ci\^q aMr.q+g>kf>b+q+h\f>bÖVm5Ð ]phicib:anà]iq$Z1b+n%bP±. vK,σ =. cib aMe1][tg>caMrdr. K ∈ Th. ?ç>. ∀K ∈ Th ±. \dt. σ = K|L ∈ Eint. \dt. σ ∈ Eext σ ∈ Eext. w\dhZsaoÎ`\^cp\^q$Z1rdb+hz`Vg>k w\dhZsaoÞ(gM1\^k¿``Vg>k. p VK,σ = vK,σ uUK,σ. 1 Z − v .nK,σ ds m(σ) σ K,σ ä. σ. ±. ±. σ u = gD − (a .∇u).n = κ u + φ :è. \^]g>h$a\^k1b+e°w\dhZ³akÕm_5®.

(56) aMk1e. p uUK,σ. =.     uK       uσ.    u   K,σ    u. \t \t \t \t. L. vK,σ >0 aMke. vK,σ < 0. σ ∈ Eext. vK,σ < 0. aMke. σ ∈ EΣ ∩ Eint. aMke. vK,σ < 0. σ 6∈ (EΣ ∪ Eext ). Y[Z1b[ÔMbl_Pg>\^khâgMtêhZ1\d]â]iq$Zb nob[\^]âhigÄm1]ib©hicaMk]ino\^]i]p\^g>k]âq gMk1e1\dhi\^g>k]âg>k|ar^r. σ ∈ Eint. \^kog>cieb chgZVa:f>bhZ1b(]an%b}>b+k1b cp\^qk m1n%b+ci\dq:aMrV]pq$Z1b nob(tgMcÆar^rhiZ1bznob ]pZÝY[Z m1]ÆhiZ1b e1\^]pq cpb+h\dØ:ahi\^g>kgt~ æ ] é. ®,ïutgMcaMr^r. FK,σ + FL,σ = ασ. σ = K|L ∈ Eint. 0\^b rde1][cib+]i_Pb qh\dfMb rE±. uK,σ + uL,σ − qσ 2. . 1 (FL,σ − FK,σ ) = βσ (uL,σ − uK,σ ) − n.nK,σ gσ 2 wZ1b cpb. n. kgHw¾]phaMk1e1][tg>c©hiZ1bÅm1k1\dhkg>cin%aMrÝg>k. ózb+k1q b. tg>c%aMr^r m1k1Ô k1g?wk. (uK )K∈Th. uK,σ. ±. uK,σ. Σh. é. :ï. . ®?ïlb kVa1r^b¿hZ1b8\^khb+cptOaMq+b é hgjPbbÐ5_1cpb ]i]pb eÕw\dhZ×cpb ]p_Eb+q+hhg´hiZ1bn{a\^kUe\^]iq+cibhb¿m1k1Ô k1g?wk1]. σ = K|L ∈ Eint . æ0]+Ä,?èaMk1ey,. !. !. ασ aˆσ dK,σ ασ a ˆσ a ˆσ dL,σ 1+ uK + 1− uL + uK = aL dσ 4 βσ aK dσ 4 βσ d σ βσ ! ! a ˆσ dK,σ qσ dL,σ 1 a ˆσ dK,σ gσ ασ dL,σ + + − n.nK,σ 1+ aK dσ aL 2 βσ a K d σ βσ 2 aL. :ä. wZ1b cpb. aσ , aσ 1 1+ + ασ ( + λσ ) βσ d σ 4 βσ dK,σ dL,σ λσ = . dL,σ aK + dK,σ aL a ˆσ =. Y[Z1b cpb+tg>cpb. aσ =. hiZ1b´\^khib p c tOaq bjm1k1Ô k1g?wk1] K,σ e1\dáEm1]p\df>b!ÖVm0Ð¿bÐ5_1cpb ]ip ] \^g>kÃaM][cib+_EgMcphb+e8\dkËÍ. FK,σ. u. dσ aK aL , dL,σ aK + dK,σ aL. , uL,σ . q:aMk½aMrd]ig°Pbjb rd\^no\^kVa4hb eR\^kÀhiZ1b. aMk1e¿w.bZVa:f>b±. ασ a ˆ σ uK + u L ασ a ˆσ dL,σ uL − u K =−a ˆσ + + uK dσ 2 βσ dσ 2 aL dσ ! ! 1 a ˆ σ gσ ασ dL,σ a ˆσ qσ dL,σ + + n.nK,σ 1+ , − dσ aL 2 βσ d σ βσ 2 aL . . . ?ð. . . ð.

(57) A?. Y[Z1b{q+b rdr®q+b khb+cib+e°¹. ]iq$Z1b+n%b´HçMÄhiZ1b kUcib æ m1\dcib+]hiZ1b{]h$aMk1e1aMcieÁ. CB. e lH¸®_Pg>\^khi] ]phib k1q+\^r g>k1r>HY[Z1\^]Kk m1nob ci\dq:aMr]pq$Z1b nob©\d]a]q$Z1b aM_|aM]hZb.]phaMk1eVacie]pq$Z1b o n bÆw\hZ1gMmh aMk©Èpm1n%_Û4Y[Zb©q g>kfMb ci}Mb k1q+b[_1cig>_Pb ch\db ]KgMt1hZb._1cpb ]ib+kh]iq$Z1b+n%b.aMcib.aMr^]pg`_1cpb ]ib+khb e \^kËÉÍ. . k·hiZ1bÌ]i\de1b ]. w.bÌ]pb+h¿hZ1bÌe1\d]iq+cib+hib´hciaMk1]ptb+c¿q g0b+Ïoq \db kh]¿gMt. σ ∈ Eint ; σ 6∈ EΣ . æ0]+K ®$?ï\^kvgMcie1b+c©hg|cib+]i_Pb qh(hiZ1b!r^g0q:arEq+g>k1]ib+cpfahi\df0\dhvaMk1evhZb!]ig>rdmh\dg>kvq gMk5® é h\dk0m\dhv_cig>_Pb ch\^b+]!± ak1e. ózb+k1q b. FK,σ = −FL,σ ,. uK,σ = uL,σ ,. aMk1e. ασ = q σ = g σ = 0 hZ1b+cpw\^]pb. H . 1 → ∞ ∀σ ∈ Eint \EΣ η. hZ1boe1\^]pq cpb+hbhciaMk1]tb cq g0b+Ïoq \db kh]aMcibob æ mVarhigshiZ1b. ]ib bYaM1r^b|?. DECΣ F$hGAHJILKNM. . σ = K|L ∈ Eint \EΣ. ∀σ ∈ EΣ ∩ Eint gtÝhZ1b(hciaMk1]ptb+c.q+g5bÏ%q+\^b+k hi].gMthiZ1bzx1q+h\h\dg>m1]_1cigM1r^b+n σ. n b aMk%faMrdm1b(g>k % g>k. βσ =. \dt. ª. ∀σ ∈ Eint ; σ 6∈ EΣ O. ¢^¬M£¬?«v¬ 0¤$i. . P6. 6. |£^«?ïM£u¨$«4¬?¤ 4Mª ¢ ?¢XªO¥Ã«,¶|ª. . Q6. (P). r5\dk1}. 6 .RÆ M¡. v¨$«4 M$¨ªO¢ M. VK,σ = −VL,σ. DECF$GAHJITSUM(V¾ª l¶:«4£X£^« ¢XW×V 0{¤¢"ïM£(¤$¬ 5£ªX¬ XM¤¦ª,$¬¢OM¼{$¬ ¢^¬´ ¬?iß V¯ª ¢^¬vï4¬? ¢ ¶|ª %M¢ Y ¬¢"«4¯ª,¬?«4¤|¢^¬M¢OZ0«4PM£ O ª %«[6M \(]À¬?¨ {%ïM× ^!P £¢"il¢XÌi0O ¨ {$¬ M¢X¤$$¨ªO¢"«4Pß `Û«4Q ªO¤¦¢ ?6 ¢OM£ÝO MV¢^¬?«4ªO¤«_P¢"¨Åï4¬?Ä¤a` 0¢X¤$$¬(¬?«4{ ¢XV «4¤¦ª¸MVª{« M¢ §©ïMªO¢"«4¬«,¶!ª \(U ] ¬?¨ { ¢"¨ M¤¥«4Poª Ä¬?¨$«_V«,¶Äª ¢^¬ O ^V¤+ß 3 ß3 ·Z1b. b. c«M ¢Xd©ßzßß. $«4{ªO¤¢"¨|¨$«4¤¦¤$¨ªO¢"«4¶:«4¤. kvhiZ1b`\dkhb ctOaMq b. Σ. \^]ucig>m}>Z1rdoa_1_1cig:Ð5\^n%ahib e{hZbÄn%b+]iZ%\^khg. q g>cpcib+q+h\dg>ko\^]hZ1b+k{cpb æ m1\^cpb eotg>cuaÞg>1\dkÃg>cuakg>k{Z1gMn%g>}Mb k1b+g>m1] Eb+e1e1b+ePg>m1k1e1aMcp`q g>k1e\dh\dg>kÝ ôbh gMthZbzxVk1\hb(f>g>rdm1n%b+]Æq+cig>]p]ib+es. Ü. a]pm1cptOaq b h b+m1n{ak1kêb n|®. Σ. Pb~hZ1bmk1\^g>k. ωh,Σ = {∪K, K ∈ Th , K∩Σ 6= ∅} 5ºzcig>mk1evb:aq$ZvxVk1\dhibzfMg>r^m1nob Σ K ⊂ ωh,Σ . hiZ1b. r^g0q:aMrEq g>k]ib cfah\f5\h{Pb+hwÆb b+k8hiZ1b!b nlPb ee1b e¿ÖVm5Ð¿\dkhb }McaMrÛaMk1ehZ1bÅa_1_1cig:Ð5\^n%ahib ÖVm5Ð¿\dkhb }McaMrÝ\d][b k1]pm1cib+eÃ. Z. ϕ− Σ ds ˆ Σ∩K. =. Z. wZ1b cpb. ˆ Σ h ∩K. 1 − ϕ ds K Σh . M. . ˆ \^].am1k\^g>kvgtq g>khicig>rPf>g>rdm1n%b+]©aMcpg>m1k1e K _1cib+q \^]pb e + gR K = α R u− ϕ− Σ Σ h h Eb+r^g?w¾aMk1e \^]©hZbrdg5q aMrÝq+g>cicpb q+hi\^g>k¿_VacaMnob+hib c©hge1bhb+cino\^k1b K ¹Vg>cËaMr^r w.bRe1b k1ghbÀ K ⊂ ωh,Σ EˆK = {σ ∈ E; σ ∈ EK ∩ EΣ , g>c w\hZsb \hZ1b+c g>c hiZ1b¿]ibh{gMt σ ∈ Eext σ = K|L L ⊂ Ωe,h L 6⊂ ωh,Σ } ]i\de1b ]rd0\dk1}|g>k aM]p]ig0q \âhb+eshig hiZ1b cw\^]ib tg>c Σh K K 6⊂ ωh,Σ EˆK = ∅ >.

(58) ózb+k1q b. tg>car^r e1\^]pq cpb+hbÅeVa4h$a. hiZ1bse1\^]pq cibhbfaMr^mb. σ ⊂ Σh σ ∈ EˆK ϕ− aMk1e g>k acibÄe1\f0\ê1b+esvhiZ1b!q$ZVaMciaMΣq+hhib αR gR σ. ôb+h!m1]`k1g?wÊeb+h$a\^rhZ1blq aMr^q+m1r a4h\^gMk8gMt. K. Σl,K. \dksb aMq$Zsq g>khcpg>rÝf>gMr^m1nob. ak1eÕhiZ m1]ohiZ1b. cp\^]phi\^q!_VacaMnob+hib c. K. . ~¹Vg>c]iaMÔMbgtz]p\^no_1r^\dq \h. K ⊂ ωh,Σ !gt Σ. O]pb b{¹â\^}1. Σl K ⊂ ωh,Σ. ϕ− Σ. ÛY[Z1\d]Äq gMcicib+q+hi\^g>kÃhaMÔMb+]ÅaMq q+g>m1kh!gMthiZ1b. cib+r ahi\df>b%]im1ctOaMq b+]|q g>k]i\êb cib+e³\^k°b aMq$Z³q b rdr _1\^b+q bw\^]ib%r^\dk1b:aMcÅa_1_1cig:Ð5\^n%ahi\^g>k. gMt. a q gMn%_Pg>]ib+eU¯a8]pb }>nob k h. \^][e1bxVk1b eÛ. Σl. M. N. δK|M. δK|L. K. L. ÷ ºÏ½Â}Ã"%£Õ!e7ÈÇÀFÌAÍ;ÀFÂeÀFÉ

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(60) Ã ë5g. a{kVahim1carKw.a:Ãhg{b+faMr^m1ahbÅhZ1bq g>cpcib qh\dg>k´_VaMciaMnob+hb+c w.gMm1rê¿Pb. K =. X. Σh. Σ. K ⊂ ωh,Σ. K. Õ. \^kÃb aMq$Z. K ⊂ ωh,Σ. m(σ). σ∈EˆK. . Í. meas(Σl,K ) Ù ∀σ ⊂ Σh , σ ∈ EˆK , σ = K ózg?wÆb+f>b+c tg>c]igMn%bq+b r^rd] wÆbn{a:ÄZ1a?fMb tgMcbÐaMno_1r^b K ⊂ ωh,Σ K ⊂ ωh,Σ EˆK = ∅ w\dhiZ1g>mhÅak O]pb bo¹â\d}1 zg>c w\dhiZ σ ∈ EΣ K ⊂ (ωh,Σ ∩ Ωe,h ) σ = K|L ∈ EΣ wZ1b cpb ¹Vg>c]imq$Zaq+b rdr ]p\^k1q+b hZ1bzaM_1_cig:Ð5\^n%ahbnob:aM]pm1cibgMtEhZ1b L ⊂ ωh,Σ K = 0 ]ib+}>n%b+kh \^][k1bf>b+c(haMÔMb+kÃ\^khigaMq q+g>m1kh:êY[Z1b+k Σl,K X Z. σ∈EΣ. σ. X m(σ) 1 = σ σ∈EΣ σ. =. X. 1 X m(σ) K ˆ. X. meas(Σl,K ) < meas(Σl ) ≈ meas(Σ),. K⊂ωh,Σ ;EˆK 6=∅. =. K⊂ωh,Σ ;EˆK 6=∅. σ∈EK. aMk1e¿hZ1bÅq+g>k1]ib+cpfahi\df0\dh{gMthiZ1b!ÖVm5Ð5b ]w\^rdrÝk1gMhPbÅb k1]pm1cib+eÝ. .