r f 2 ()] r 2l ()
Post on 02-Apr-2022
1 Views
Preview:
TRANSCRIPT
"!$#&%' "(*)+-,.0/-,213 '45/$#&1*(*" 6"(78
4#"!-5/$,.9/-,;:< 17,.=+(*8-,;>(*?@8A"(*8+-,.B,9DC-C1*(3. "(*8
EGFIHKJLNM@OQPRHTS5PUM VW?WYX LNPRZ W?[?["\
]$^`_aQb HdceOHgfhHTf _i PRZjHhZUckO ^ LNllP _Nmn_ M@OQopPkF ^`q c i OHOHgF ^ LNllP _ JQo X LNcUo _ M9O ^ S$res$Zjtvuwl HdP ax_ FpoIy i HhZjHdcHdFpFpoplcUoIy i HhZkZRczLNcUo _ MM.LNopPRHhZkl.LNPjFIHhZ X q cUt _ OHhZGOHgLNFIHdPU|opM@HrGoIZzf _ MvcUopM i HhZ~7rTK ] HhZklP _a FI X HhZX _ OdFIHhZ5ZRHdP _ Mc i M@H q y i LNcUo _ MO ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ MO@LNM@Z+FIHfLZtvuwlxHdP a _ FpoIy i HHdc i M@H q y i LNcUo _ MOHOQo i ZUo _ MwvP q LfdcUo _ MO@LNM@ZgFIHfLZTHdFpFpoplcUoIy i H?S+M i oIZRHOHf _ X lF q X HdMc i M@HLNM.LNFpuQZRH i Mo q HOHhZ X q cUt _ OHhZ$OHGrnTFLeM _ cUo _ MOHkZUuwZRcR X HhZ+OHjPUoIHhOQPUoIfRt@Z-ZRHdPzLgHhZRy i oIZRZ q H?S+MQ@My i HdFIy i HhZPRHdl dPRHhZ a o a Fpo _ ?PzLNltoIy i HhZZ _ Mc mn_i PUMoIZgHdM>@MDOHO _ f i X HdMc~FLFpopcUc q PzLNc i PRHZ i PgFIHhZ X q cUt _ OHhZOHYrHhZUcl.LNPUcUoIf i FpoIdPRH X HdMvclP _ Fpo.y i HTOHdl i oIZky i HdFIy i HhZGLNMM q HhZK
¡ j¢+£¤&¥z¦j§¨©xª+«j£¤&¥R¦k§
¬ _ opcΩi M2O _ X LNopM@H ax_ PUM q Hdcf _ MM@HKJQHBOH
RdOH m P _ McUoIdPRH
∂ΩFpopl@ZRfRtopcUhoIHdMM@H? ¬ _ opc
f ∈L2(Ω)
µ ∈ L∞(Ω)
Hdcβ ∈ [W 1,∞(Ω)]d
~F ^ tvuwl _ cUt@hZRHOHP q i FLNPUopc q Z i PFIHg?PzLOQoIHdMcGOHβl H i c
® cUPRH¡f _ M@ZUoIO q PzL a FIH X HdMvcTL¯LNo a FpoIH?@czLNMcGy i H∇·β ∈ L∞(Ω)
K&°GM9Z i ll _ ZzHgy i HglPRHhZzy i Hgl.LNPUc _i cO@LNM@Z
Ω
µ− 12∇·β ≥ µ0 > 0.
~²±³°GM´O q ZRop?M@Hµl.LNP
∂Ω± FIHhZl.LNPUcUoIHhZHdMvcUPzLNMcRH0HdcZ _ PUczLNMvcRHOHFL m P _ MvcUoIdPRHBOHΩ ¶ fhHdFpFIHhZ²·*fdojZ _ McO q @MoIHhZGf _ XX HZ i opcT¸
∂Ω± = x ∈ ∂Ω; ± β(x)·n(x) > 0,~ W
_¹n(x)
O q ZUop?M@HFLM _ P X LNFIHYHKJwc q PUoIH i PRHYºΩL i l _ opMc
x ∈ ∂Ω ] HTlP _a FI X H X _ OdFIH¡f _ M@ZUoIZUcRH
ºfRt@HdPRfRt@HdP i M@H mn_ M@fdcUo _ Mu : Ω→ R
cRHdFpFIHy i H
µu+ β·∇u = fO@LNM@Z
Ω,
u = 0Z i P
∂Ω−.
~n»"
] LgZRHhfdcUo _ M±?p±lP q fdoIZRH¡~¼ZzLNM@ZPRHdMvcUPRHdPO@LNM@Z+c _i Z+FIHhZ5O q czLNopFIZ+cRHhfztMoIy i HhZKFIHfLOQPRH X LNcUt q X LNcUoIy i HO@LNM@ZFIHhy i HdF m¼_ P X i FIHdPFIHGlP _a FI X H X _ OdFIHY~n»"KQs i oIZFL¡ZRHhfdcUo _ M±? W O q fdPUopcF ^ LNllP _ JQo X LNcUo _ M0OHfhHTlP _a FI X HTl.LNP i M@H X q cUt _ OHTOH¡rHdcjHdMHK6Hhfdc i HgF ^ LNM.LNFpuQZRHgOHTf _ M"HdPU"HdM@fhH?
½¾*½ ¿À6Á5Â"Ã2ÄÅÀxÆ?Ç$È.ÄÅÀxÆ?É3ÊË5Ã] LlPRH X oIdPRHTfzt _ ZRH¡º m LNopPRHTHhZUcjOHTlP q fdoIZRHdPjF ^ HhZUl.LfhH m¼_ M@fdcUo _ MM@HdFO@LNM@ZkFIHhy i HdF _ MfRt@HdPRfzt@HTFL
Z _ F i cUo _ MOH~n»"K@s _i PkfhHdFLQ _ M0opMcUP _ O i opcGF ^ HhZUl.LfhHTO i ?PzLNlt@H
W = v ∈ L2(Ω); β·∇v ∈ L2(Ω).~Ì
Í3Î Ï.ÐAÑeÒnÎ6Ó?Ô"Ï@Õ²ÖK×ÙØڳزÛÜQÖKÝhÞ7Û²Ô"Üßzà7áÙÛ7âwØãÕ3ävÔvåjØ3Þ+æ³Ô?ßRçKزÝèNØ$éx×êßáÙÛnØÜßzÛnÕUß×pÔwëKëRìhííGÑßRà7ÝNØ$×îßjï&ß×Ù×Ùð²Ø-ÕãØUÚ¯ØãñgòÔó à*ßݯղØAôkõKö÷@ø¼ùKúNûüzõKöRý?þdüKÿõz÷ üdö
±
]$^ tvuwl _ cUt@hZRHβ·∇v ∈ L2(Ω)
ZUop?Mo.HTy i HgFL mn_ P X HgFpopM q LNopPRH
D(Ω) 3 ϕ 7−→ −
∫
Ωv∇·(βϕ) ∈ R
~"
HhZRcf _ McUopM i HDZ i PL2(Ω) ¶ F ^`_aQb Hdc β·∇v O q ZUop?M@H>LNF _ PRZµFL m¼_ M@fdcUo _ MOH
L2(Ω)y i oGl H i c ® cUPRH
LZzZ _ fdo q HGºTfhHdcUcRH mn_ P X HFpopM q LNopPRHGl.LNP5FIH a oLNoIZ-O i cUt qh_ PR X HGOHjPRHdlP q ZRHdMczLNcUo _ MOHjoIHhZUKP q fzt@Hdc ! #"$z&%?')(
(v, w)W = (v, w)L2 + (β·∇v, β·∇w)L2 * W (+", .-(,"/Q0$+(12(#34%657(,, 98: ;( <(8>=*F m L i cj q PUo.HdPkFLf _ X lF q c i OH? ¬ _ opc
(vn)n∈N
i M@HZ i opcRHTOH?L i fztvu0OHW ] HhZGZ i opcRHhZ
(vn)n∈N
Hdc(β·∇vn)n∈N
Z _ McO _ M@fOH@?L i fztvuO@LNM@ZL2(Ω)
r q ZUop?M _ M@Z+l.LNPvHdcwFIH i PRZFpo X opcRHhZ
PRHhZRlxHhfdcUop"HhZGO@LNM@ZL2(Ω)
.°GM L
−
∫
Ωv∇·(βϕ)←− −
∫
Ωvn∇·(βϕ) =
∫
Ω(β·∇vn)ϕ −→
∫
Ωwϕ,
fhHy i o X _ McUPRHy i Hv ∈W
Lh"Hhfβ·∇v = w
HdcGy i HFLZ i opcRH(vn)n∈N
cRHdM@O0"HdPRZvO@LNM@Z
W
] HOH i JwoI X Hl _ opMvco X l _ PUczLNMvc5HhZUcOHlP q fdoIZRHdPFIHZzHdM@ZAOHFLGf _ M@OQopcUo _ ML i J¡Fpo X opcRHhZ+O@LNM@Z~n»"K°GMopMvcUP _ O i opckF ^ HhZUl.LfhH
L2(∂Ω; |β·n|) = vX HhZ i PzL a FIHZ i P
∂Ω;∫
∂Ω |β·n|v2 < +∞.
~A"
°GM Z i ll _ ZzHgy i H~t±³BF ^ HhZUl.LfhH
C10(R
d)OHhZ mn_ M@fdcUo _ M@ZeOH¡fdFLZRZRH
C1 Z i P
RdºZ i ll _ PUcef _ X l.LfdcHhZRceOHdM@ZRH¡O@LNM@Z
W ¶~t W BFIHhZ m P _ MvcUoIdPRHhZkHdMcUPzLNMvcRHYHdcjZ _ PUczLNMvcRH¡OH
ΩZ _ Mvc a oIHdMZ q l.LNP q HhZfhHTy i oZUop?Mo.HTy i H
OQoIZRc(∂Ω−, ∂Ω+) > 0.
~ \
°GM9l H i c X _ MvcUPRHdPl.LNPecUP _ M@fLNc i PRH¡HdceP q i FLNPUoIZzLNcUo _ My i HTF ^ tvuwl _ cUt@hZRH~t±³jHhZUcGZLNcUoIZ m LNopcRH¡O@LNM@ZFIHfLOQPRHOHhZTtul _ cUt@hZRHhZYfdo·*OHhZRZ i Z ]5^ tvuwl _ cUt@hZzH ~t W gHhZRcTlF i ZTPRHhZUcUPUoIfdcUop"Hl i oIZRy i^ HdFpFIHFpo X opcRHFIHfzt.L X lDO ^ LNllFpoIfLNcUo _ MºOHhZelP _a FI X HhZTOH¡cUPzLNM@ZUl _ PUcgLOQ"HhfdcUo m vP q LfdcUo m O@LNM@ZeOHhZec ia HhZ ] HhZtul _ cUt@hZRHhZY~t±³7x~t W l HdP X HdcUcRHdMvcGOH X _ McUPRHdPkFIHgcUt qh_ PR X H¡OHcUPzLfhHTZ i op?LNMvcCBD ~o FEG +H,²0 9( I2( !²2$7(
γ : C10(R
d) 3 v 7−→ v|∂Ω ∈ L2(∂Ω; |β·n|)
~J"
"HG 9( - K$) - ¼ -HL.M ( - ONW8
~opo QP - R%pTS& MI %U(VWG - 9/X&²0 ¼! - QYZQ& ¼(,"
∀(v, w) ∈W ×W,
∫
Ω[(β·∇v)w + (β·∇w)v + (∇·β)vw] =
∫
∂Ω(β·n)vw.
~["
W
: ;( <(8greHF ^ tul _ cUt@hZRH~t W _ MO q O i opc5F ^ HKJQoIZUcRHdM@fhHjOHjOH i J m¼_ M@fdcUo _ M@Zψ− Hdc
ψ+ O@LNM@ZC
10(R
d)cRHdFpFIHhZGy i Hψ− + ψ+ ≡ 1
O@LNM@ZΩ, ψ−|∂Ω+ = 0, ψ+|∂Ω− = 0.
¬ _ opcv ∈ C
10(R
d)x°GM L
∫
∂Ωv2|β·n| =
∫
∂Ωv2(ψ− + ψ+)|β·n| =
∫
∂Ω−
v2ψ−|β·n|+
∫
∂Ω+
v2ψ+|β·n|
= −
∫
∂Ωv2ψ−(β·n) +
∫
∂Ωv2ψ+(β·n)
= −
∫
Ω∇·(v2ψ−β) +
∫
Ω∇·(v2ψ+β)
≤ c(ψ−, ψ+, β)‖v‖2W .
°GMf _ M@fdF i c5l.LNP5OHdM@ZUopc q ~tul _ cUt@hZRHT~t±³UAl _i P q czL a FpopP5FIHl _ opMvcG~o K ] Hl _ opMvce~opo AOHdoIHdMc-LNF _ PRZo XX q OQoLNcY¸QFL mn_ P X i FIH~["HhZRcPzLNoIHTZ i P
C10(R
d)Hdc _ Mf _ M@fdF i cGºM _i "HL i l.LNPkOHdM@ZUopc q
EG + 0 +" (Z(O" 9&²0 ¼! - 2(+" S, - ¼;(+"@( - !U - 9(( " & 7 - 9( - ( ( Q&" !)($7 - / - )(O > 725,% ' - )& M ( 2( !²2$7(>" - "
L2(∂Ω; |β·n|)8?!$d - $) - !;(;(2( M %U(
$7 - "2,)(,%6(V0 M - ( !K¼ - X %p4;(
Ω = (x, y) ∈ R2; 0 < y < 1; |x| < y
( %U(#$w M 92<(;$ ¼ Sβ = (1, 0)t
8 EOS - $ ¼ -u(x, y) = yα
(+", " - "W
+" (α > −1
M ? "(%'%U( - G 0 M ( O( !U0$7(Iv - "
L2(∂Ω; |β·n|) ( "
α > − 12
8°GM0O q @MopcGZ i P
W ×WFL m¼_ P X H a opFpopM q LNopPRH
a(v, w) =
∫
Ω[µvw + (β·∇v)w] +
∫
∂Ω(β·n)−vw,
~²± [
_¹ l _i P i M0P q HdFx ∈ R
_ M0O q ZUop?M@HTZzHhZl.LNPUcUoIHhZjl _ ZUopcUop"HgHdcjM q vLNcUop"Hgl.LNP
x+ = 12 (|x|+ x), x− = 1
2(|x| − x).~²±?±³
° a ZzHdPU"HdPYy i HFL mn_ P X H a opFpopM q LNopPRHaHhZUc a oIHdMf _ MvcUopM i HµZ i P
W ×WOHl.LNP¡FIHFIH XX H W rH
lF i Z³AfhHdcUcRH mn_ P X H a opFpopM q LNopPRHBHhZRcL2 ·*f _ HdPRfdop"HµZ i P W ? ^ HhZUc i M@Hf _ M@Z q y i HdM@fhHo XX q OQoLNcRHµOH
F ^ tul _ cUt@hZRH~²±³HdckOHFL mn_ P X i FIHTO ^ opMc q ?PzLNcUo _ M l.LNPjl.LNPUcUoIHhZY~["-l i oIZRy i H
a(v, v) =
∫
Ω(µ− 1
2∇·β)v2 + 12
∫
∂Ω(β·n)v2 +
∫
∂Ω(β·n)−v2
≥ µ0‖v‖2L2 + 1
2
∫
∂Ω|β·n|v2.
~²± W
°GM f _ M@ZUoIOdPRHgFIHglP _a FI X HTZ i opLNMc¸
?-t@HdPRfzt@HdPu ∈W
cRHdFy i Ha(u,w) =
∫
Ωfw
∀w ∈W.
~²±»"
] LL2 ·*f _ HdPRfdopopc q OH>FL mn_ P X H a opFpopM q LNopPRH
aM _i ZLZzZ i PRH>OH>F ^ i MoIfdopc q OH>FLZ _ F i cUo _ M'OH>fhH
lP _a FI X H? _i ZjM ^`q czL a FpopP _ M@Zkl.LZoIfdoF ^ HKJwoIZUcRHdM@fhH? X LNoIZkLO X HdcUcUP _ M@ZkFIHP q Z i FpczLNckZ i op?LNMvc "!$# &% ('W E (>5,%) M (0~²±»" (+" 5K/( - <"8+*V(>% " * " -Q- (R"Y% ¼! - (+" @" &% ¼ -(¡~n»" 8
»
½¾ $Â5É*ÄÅÀxÆ"É ]5^`_Nb HdckOHTfhHdcUcRHZzHhfdcUo _ MHhZUcjF ^ LNM.LNFpuQZRHgM i X q PUoIy i HYO ^ i M@HLNllP _ JQo X LNcUo _ MOHgc*uwl H¡rOHgFL
Z _ F i cUo _ M>O i lP _a FI X HB~²±»"KxS5M l.LNPUcUoIf i FpoIHdP _ M>Z ^ LNcUczLfRt@HdPzLµºO q fdPUopPRHYlP q fdoIZ q X HdMcFIHhZ q czLNl HhZ~c _i cYº m LNopcgM.LNc i PRHdFpFIHhZgO@LNM@ZgFL m¼_ P X i FLNcUo _ M<O i lP _a FI X HOQoIZRfdPRHdc9LNoIZ¡L i l.LNPzLh?LNMvcopMcUP _ ·O i oIZ _ M@Zky i HdFIy i HhZM _ czLNcUo _ M@Z i cUopFIHhZ
B ! # #¬ _ opc
Thh>0i M@H m L X opFpFIHP q i FpoIdPRH0~nL i ZzHdM@Z i Z i HdF kOH X LNopFpFLN"HhZTLM@HhZgO i O _ X LNopM@H
Ω
°GM>Z i ll _ ZRH¡l _i PZUo X lFpo.HdPy i HΩHhZUc i M l _ Fpuw _ M@H _i9i M l _ Fpu"hOQPRHYOH
RdL¯@M>y i HYfRt.Ly i H
X LNopFpFLN"HTh
PRHhf _i PRH¡HKJLfdcRH X HdMcΩ ¬ LNM@Zl HdPUcRHTOH q M q PzLNFpopc q _ M Z i ll _ ZzHgy i H
h ≤ 1@s _i P
c _i cHdMcUoIHdPs ≥ 1
Hs(Th)
O q ZUop?M@HF ^ HhZUl.LfhHOH ¬ _a _ FIHd a PUoIZ q ~¼O _ MvcFIHhZ q FIH X HdMvcRZZ _ MvcOHhZmn_ M@fdcUo _ M@ZO _ MvcFLPRHhZUcUPUoIfdcUo _ M´ºfzt.Ly i H X LNopFpFIH
T ∈ ThHhZUcO@LNM@Z
Hs(T )K =3FZRHdPzL f _ XX _ OH
OHOQoIZUl _ ZzHdPTOHF ^`_ l q PzLNcRH i P¡OH?PzLOQoIHdMvcYOQoIZRfdPRHdc∇h : H1(Th) → [L2(Ω)]d
cRHdF-y i Hl _i Pv ∈
H1(Th)
(∇hv)|T = ∇(v|T )l _i Pµc _i c
T ∈ ThLNopM@ZUogy i H9OH>F ^`_ l q PzLNcRH i P0OH>OQop"HdPU"HdM@fhH
OQoIZzfdPRdcRH∇h· : [H1(Th)]d → L2(Ω)
cRHdF5y i Hl _i Pσ ∈ [H1(Th)]d
(∇h·σ)|T = ∇·(σ|T )
l _i Pc _i c
T ∈ Th
°GMDO q ZUop?M@Hl.LNPFh
F ^ HdM@ZRH X a FIHOHhZ m LfhHhZgO i X LNopFpFLN"H? ?HdcTHdM@ZRH X a FIHHhZUcgl.LNPUcUopcUo _ MM q HdMFh = F h ∪F∂
h
_¹F h
O q ZUop?M@HeF ^ HdM@ZRH X a FIHgOHhZ m LfhHhZopMvc q PUoIH i PRHhZjO i X LNopFpFLN"HHdcF∂
h
F ^ HdM@ZRH X a FIHOHhZ m LfhHhZGZUopc i@q HhZjZ i PkFL m P _ McUoIdPRH
∂Ω@s _i P
F ∈ F hopFHKJQoIZUcRHTOH i J X LNopFpFIHhZ
T1HdcT2
O@LNM@ZThcRHdFpFIHhZ¡y i H
F = T1 ∩ T2°GMDl _ ZRH
∆F = T1 ∪ T2Hdc _ MDO q ZUop?M@Hl.LNP
nF
FIH"HhfdcRH i P i MopczLNopPRHM _ P X LNFº
Fl _ opMvczLNMvcjOH
T1"HdPRZ
T2Qs _i P
v ∈ H1(Th) _ MBO q @MopcZ _ MBZzL i cjº¡cUPzLh"HdPRZ
FHdcZzL
X _ u"HdMM@H¡Z i PFl.LNPjFIHhZ mn_ P X i FIHhZ
[[v]]F = v1 − v2, vF = 12(v1 + v2),
~²±dÌ
_¹ l _i PLNFpF q "HdPYFIHhZYM _ czLNcUo _ M@Z _ M´Ll _ Z qvi = v|Ti
i ∈ 1, 2
]5^ opM@OQoIfhHF
HhZRc _ X oIZO@LNM@ZFIHhZZzL i cRZeHdceFIHhZ X _ u"HdMM@HhZF _ PRZRy i^ opFAM ^ u Ll.LZeO ^ L X a op i c q 6s _i P
F ∈ F∂h
_ M l _ ZRH∆F = T_¹
THhZUcGF ^ i MoIy i H X LNopFpFIHOH
ThO _ Mc
FHhZUc i M@H m LfhH?
nF
O q ZUop?M@HFLM _ P X LNFIH i MopczLNopPRH¡Z i PFZ _ PUczLNMvcRHTOH
ΩHdcFIHhZZzL i cjHdc?LNFIH i P X _ u"HdMM@HOH
v ∈ H1(Th)Z _ MvclPUoIZl.LNPjf _ Mv"HdMcUo _ M q vL i J
ºFLYPRHhZUcUPUoIfdcUo _ M OHvZ i P
F
sALNPeFLZ i opcRH?.FLM _ czLNcUo _ MA . B
ZUop?Mo.HYy i^ opFAHKJQoIZUcRH i M@H¡f _ M@ZUczLNMcRHYl _ ZUopcUop"Hc - 2 ( -
v - 9( M 4%'%pX2(dcRHdFpFIHy i HA ≤ cB
6s _i PLNFpF q "HdPTFIHhZgM _ czLNcUo _ M@Z _ MDO q ZUop?M@Hl.LNP(·, ·)E
FIHlP _ O i opcgZRfLNFLNopPRH i Z i HdFOH
L2(E)~ _¹
EHhZUc i M@H X LNopFpFIH? i M@H m LfhHYO i X LNopFpFLN"H _i9i M@H i Mo _ M
OHTcRHdFIZ _aQb HdcRZKHdcl.LNP‖ · ‖E
FLM _ P X H¡LZRZ _ fdo q H?¬ _ opc i MHdMvcUoIHdP
p ≥ 1.°GM0opMvcUP _ O i opckF ^ HhZRl.LfhHTOHgLNFIHdPU|wopM@HTreoIZRf _ MvcUopM i
Gph = vh ∈ L
2(Ω);∀T ∈ Th, vh|T ∈ Pp,~²±"
_¹Pp
O q ZUop?M@HBF ^ HhZUl.LfhH0"Hhfdc _ PUoIHdFGOHhZl _ FpuM X HhZOH0OHd?P q c _ czLNFjopM mnq PUoIH i P _i´q vLNFGºp ] L
OQo X HdM@ZRo _ M OHG
ph
HhZUc q vLNFIH>ºN !#" Cp
p+d
_¹N !#" O q ZRop?M@H FIH M _ X a PRHOH X LNopFpFIHhZBOH
ThHdc
Cpp+d
FIHYf _ H$fdoIHdMvcTO i>a opM X HOHsALZRfLNF5LZRZ _ fdo q L i J opM@OQoIfhHhZ(p + d)
Hdcp~ _ M9PzLNll HdFpFIHy i H
Cpp+d
HhZUc q vLNFºFLOQo X HdM@ZUo _ M9OHF ^ HhZUl.LfhHg"Hhfdc _ PUoIHdFPp
K ]5^ HhZUl.LfhHG
ph
l _ ZRZRhOHgOHhZjlP _ lPUo q c q ZO ^ opMcRHdPUl _ FLNcUo _ M _ lcUo X LNFIHhZAO@LNM@ZFIHhZHhZUl.LfhHhZOH ¬ _ax_ FIHd a PUoIZ q Z³EkopM@ZRo7hl _i Pc _i c
z ∈ Hp+1(Th)
opFHKJwoIZUcRHzh ∈ G
ph
cRHdFy i Hgl _i Pc _i cT ∈ Th
‖z − zh‖T + h1
2
T ‖z − zh‖∂T + hT ‖∇(z − zh)‖T . hp+1T ‖z‖Hp+1(T ).
~²±A"
Ì
S5MQ@Mv?PfhHºgFLP q i FLNPUopc q OHkFL m L X opFpFIHThh>0
_ MOQoIZUl _ ZRHjO@LNM@ZG
ph
OHhZ-opM q vLNFpopc q Z-opM"HdPRZRHHdcGOHcUPzLfhHTZ i op?LNMvcRHhZ¸
∀vh ∈ Gph
∀T ∈ Th
‖∇vh‖T . h−1T ‖vh‖T , ‖vh‖∂T . h
− 1
2
T ‖vh‖T .~²± \
B B # < # # % °GMBl _ ZzH
Wh := Gph
] HTlP _a FI X HOQoIZRfdPRHdcjf _ M@ZUoIZRcRHº
?-t@HdPRfRt@HdPuh ∈Wh
cRHdFy i Hah(uh, wh) =
∫
Ωfwh
∀wh ∈Wh,
~²±J"
Hdc¡opF-Z ^ LN?opc X LNopMvcRHdM.LNMcOHf _ M@ZUcUP i opPRHFL m¼_ P X H a opFpopM q LNopPRHOQoIZzfdPRdcRHah
rH i JDtvuwl _ cUt@hZzHhZOHP q i FLNPUopc q Z i llF q X HdMvczLNopPRHhZZ _ Mc m LNopcRHhZjZ i PFLZ _ F i cUo _ MHKJ@LfdcRH¸
u ∈ Hp+1(Th)
fhHTy i oZUop?Mo.Hy i HFIH X LNopFpFLN"HL q c q f _ M@ZUcUP i opckOHTZ _ PUcRHTy i HFLZ _ F i cUo _ MHKJLfdcRHZ _ opcjZ i ZzL XX HdMckP q i FpoIdPRH¡ºYF ^ opMc q PUoIH i PeOHgfzt.Ly i H X LNopFpFIH ¶;l _i P+c _i c
F ∈ F hβ·nF [[u]] = 0
?fhHy i oQo X lFpoIy i Hky i HFLeZ _ F i cUo _ MHKJ@LfdcRHM@HlxH i c+ZzL i cRHdPºcUPzLh"HdPRZ i M@H m LfhHgy i HgZUo6FIHfRt.L X l LOQ"HhfdcUo m HhZUcjczLNM"HdMcGºfhHdcUcRH m LfhH?
s _i Py i HfhHhZYtvuwl _ cUt@hZRHhZZ _ oIHdMvcZzLNcUoIZ m LNopcRHhZYF _ PRZRy i HFL9Z _ F i cUo _ M2HKJ@LfdcRHlP q ZzHdMvcRHBOHhZZUopM i ·FLNPUopc q Z³+opF m L i cHdM2 q M q PzLNFeLO@LNlcRHdPFIH X LNopFpFLN"Hº>fhHhZZUopM i FLNPUopc q Z³ ] L9@ i PRH±µlP _ l _ ZRH i MHKJH X lFIHkO ^ i M@HjcRHdFpFIHjZUopc i LNcUo _ M ] HkfRt.L X lO ^ LOQ"HhfdcUo _ MHhZUc+PRHdlP q ZRHdMc q ºevL i fRt@H ¶ opF.HhZUc5Z i ll _ Z qi Mo mn_ P X H?°GMZ i ll _ ZRHOHlF i ZTy i HZ i PgFIH a _ PRO>FLNc q PzLNF+vL i fRt@HO i O _ X LNopM@H~oIfdo7 i MfLNPUP q K_ M;o X l _ ZRH i M@Hf _ M@OQopcUo _ M L i JFpo X opcRHhZOHBreopPUoIfRtFIHdcM _ M;t _ X _ "dM@HLh"Hhf i M@HOQoIZRf _ MvcUopM i opc qL i l _ opMvc
S+rLNM@ZfhHhZf _ M@OQopcUo _ M@Z+FL>Z _ F i cUo _ M;HKJ@LfdcRHBHhZUcOQoIZRf _ McUopM i H Z i PFL Fpop?M@H0OHBf _i ·
PzLNMckcUPzL"HdPRZzLNMvcGFIHTO _ X LNopM@HTHdcjPRHdlP q ZRHdMc q HHdMB?PzLZjZ i PjFL¡@ i PRH?&°GMf _ M@ZUczLNcRHy i HF _ PRZRy i HFIHX LNopFpFLN"HHhZUcLO@LNlc q º0fhHdcUcRHZUopM i FLNPUopc q ~¼fLZYº0vL i fRt@H³KFIHhZ¡OH i Jtvuwl _ cUt@hZRHhZ¡OHP q i FLNPUopc qfdo·*OHhZzZ i ZZ _ Mvc-PzLNoIZ _ MM.L a FIHhZ ¶ O@LNM@Z$FL¡ZUopc i LNcUo _ MPRHdlP q ZRHdMvc q HgºOQP _ opcRH?fhHhZ-tvuwl _ cUt@hZzHhZ-c _ X a HdMcfdFLNopPRH X HdMceHdM0O qKm L i c
β
S
± >LNopFpFLN"HLO@LNlc q ºTFLTZUopM i FLNPUopc q OHjFLZ _ F i cUo _ MµHKJLfdcRHeºgvL i fRt@H ¶ X LNopFpFLN"HGopM.LO@LNlc qºOQP _ opcRH?
] HhZeOH i J0lP _ lPUo q c q ZeHhZzZRHdMvcUoIHdFpFIHhZOHTFL m¼_ P X H a opFpopM q LNopPRHah
Z i PkFIHhZRy i HdFpFIHhZkPRHdl _ ZRHc _i cRHTFLf _ M@ZRcUP i fdcUo _ M fdo·*OHhZRZ _i ZGZ _ MvcFIHhZGZ i op?LNMvcRHhZT¸~o
L2 ·*f _ HdPRfdopwopc q Z i P Wh
¸opFZ ^ LN?opckO i lxHdM@O@LNMcGOQoIZRfdPRHdcGOHFLlP _ lPUo q c q ~²± W K.ºZLh _ opP
∀vh ∈Wh, ah(vh, vh) & ‖vh‖2L2 +
∫
∂Ω|β·n|v2
h.~²±["
~opo F? _ M@ZUoIZRczLNM@fhH m¼_ PUcRH?&ºZzLh _ opP
ah(u,wh) =
∫
Ωfwh, ∀wh ∈Wh,
~ W?[
_¹u ∈W
O q ZUop?M@HTFLZ _ F i cUo _ MHKJ@LfdcRH?@f ^ HhZUc²·3º¯·*OQopPRHgFLZ _ F i cUo _ MO i lP _a FI X H~²±»"K
] HBfRt _ oJ FIHµlF i ZZUo X lFIHBf _ M@ZUoIZUcRHBº lPRHdM@OQPRHah := a
HdM2F _ fLNFpoIZLNMvcFIH?PzLOQoIHdMvcZ i PfRt.Ly i Hq F q X HdMc
ah(vh, wh) =
∫
Ω[µvhwh + (β·∇hvh)wh] +
∫
∂Ω(β·n)−vhwh,
~ W ±³
fLNPjfhHdFLLZzZ i PRH12( Sh0$ /FLf _ M@ZUoIZUczLNM@fhH? _i cRH mn_ oIZfhHgfzt _ oJM ^ HhZUcl.LZf _ X l.LNcUo a FIHL"HhfgM _ cUPRHHKJQop"HdM@fhHOH
L2 ·*f _ HdPRfdopwopc q OQoIZRfdPRdcRHTf _ XX HFIH X _ MvcUPRHFIHgfLNFIf i FZ i op?LNMvcT¸l _i Pc _i c vh ∈ Vh
a(vh, vh) =
∫
Ω[µv2
h + (β·∇hvh)vh] +
∫
∂Ω(β·n)−v2
h
=
∫
Ω(µ− 1
2∇·β)v2h + 1
2
∫
∂Ω|β·n|v2
h +∑
F∈F h
∫
F
β·nv2h.
°GP&Lh"HhfOHhZjM _ czLNcUo _ M@Z q woIOHdMvcRHhZ
β·nv2h = 1
2(β1·n1v21 + β2·n2v
22) = β·nF [[vh]]vh.
~7°GM PzLNllxHdFpFIH¡y i Hl.LNPjtvuwl _ cUt@hZRH?@FIH¡fRt.L X l9LOQ"HhfdcUo mβHhZUckf _ McUopM i ºcUPzLh"HdPRZ
FêjsALNPef _ M@Z q ·
y i HdMvcf _ XX HeFL¡Z _ F i cUo _ MBHKJLfdcRHe q PUo.Hel.LNPtvuwl _ cUt@hZzHβ·nF [[u]] = 0
l _i Pc _i cF ∈ F h
i M@HX _ OQo.fLNcUo _ Mf _ M@ZRoIZUczLNMvcRHjO i fzt _ oJB~ W ±³+y i oQP q czL a Fpopc5FL
L2 ·*f _ HdPRfdopwopc q OQoIZRfdPRdcRHf _ M@ZUoIZUcRHjºel _ ZRHdP
ah(vh, wh) =
∫
Ω[µvhwh + (β·∇hvh)wh] +
∫
∂Ω(β·n)−vhwh −
∑
F∈F h
∫
F
β·nF [[vh]]wh.~ W?W
=3FZRHdPzLf _ XX _ OHOHPzLNoIZ _ MM@HdPgZ i PGFL mn_ P X i FIH q y i opLNFIHdMcRH _a cRHdM i HHdM opMc q ?PzLNMvcgl.LNPel.LNPUcUoIHhZFLO q PUop q HLOQ"HhfdcUop"H?
ah(vh, wh) =
∫
Ω[(µ−∇·β)vhwh−vh(β·∇hwh)]+
∫
∂Ω(β·n)+vhwh +
∑
F∈F h
∫
F
β·nFvh[[wh]].
~ W »"?HdcUcRH m¼_ P X i FIHgP q Z i FpcRHTOHF ^ oIOHdMcUopc q
∫
Ω(β·∇hvh)wh = −
∫
Ω[(∇·β)vhwh − vh(β·∇hwh)] +
∑
F∈F h
∫
F
2β·nvhwh
+∑
F∈F∂h
∫
F
β·nvhwh,
HdcGO iµm LNopcky i H2β·nvhwh = β·nF (vh[[wh]] + [[vh]]wh).
] L m¼_ P X H a opFpopM q LNopPRHah
O q @MoIHl.LNP~ W?W _i ~ W »"GHhZUcgO ^ i M@Hl.LNPUcTf _ M@ZUoIZRczLNMvcRHHdcTO ^ L i cUPRHl.LNPUcHdFpFIHZzLNcUoIZ m LNopcFLlP _ lPUo q c q OH
L2 ·*f _ HdPRfdopwopc q OQoIZRfdPRdcRH~²±["KE?LNMvcOH0l _i PRZ i opwPRH0FLDf _ M@ZRcUP i fdcUo _ MÅOH FL mn_ P X H a opFpopM q LNopPRH
ah
~¼y i oGM ^ HhZUcl.LZHdM@f _ PRHcRHdP X opM q HêKxLNPUP ® c _ M@Z²·7M _i Z i MBopM@ZRczLNMvcjl _i Pjf _ M@ZUoIO q PRHdPjFIHlP _a FI X HTOQoIZzfdPRHdc¡~²±J" _a cRHdM i Lh"HhfFIH9fRt _ oJ~ W »"l _i PFL m¼_ P X H a opFpopM q LNopPRH
ah
Z?HlP _a FI X H9LO X Hdc i M@H PRH m¼_ P X i FLNcUo _ MÅF _ fLNFIH_a cRHdM i HHdM;f _ M@ZRoIO q PzLNMvc i M@H m¼_ M@fdcUo _ M´cRHhZRcOHµFL mn_ P X H
wh = q1T_¹
qHhZUc i M;l _ FpuwM X H
A
LNP a opcUPzLNopPRHBO@LNM@ZPp
Hdc _¹1T
O q ZUop?M@HFL mn_ M@fdcUo _ MopM@OQoIfLNcUPUoIfhHBOHFL X LNopFpFIHTEGopM@ZUo7
uh
HhZUcZ _ F i cUo _ MOH~²±J"ZUoHdcjZzH i FIH X HdMvcGZRo6l _i Pc _i c X LNopFpFIH
T ∈ ThHdcl _i Pjc _i cjl _ FpuM X H
q ∈ Pp
∫
T
[(µ−∇·β)uhq − uh(β·∇q)] +∑
F⊂∂T
∫
F
φT,F (uh)q =
∫
T
fq,~ W Ì
_¹ FIH φT,F (uh)
HhZUcjO q @Mol.LNP
φT,F (uh) =
β·nT uh, F ∈ F h,(β·n)+uh, F ∈ F∂
h ,
~ W "
nT
O q ZUop?M.LNMcFL9M _ P X LNFIH i MopczLNopPRH0HKJQc q PUoIH i PRHº FL X LNopFpFIHT ]_ PRZRy i^`_ M2lPRHdM@O l _i P
qi M@H
mn_ M@fdcUo _ M f _ M@ZRczLNMvcRH q vLNFIH º2±? _ M _a cUoIHdMvc i M@H q y i LNcUo _ M OH a opFLNMÅOH X LZRZRH0F _ fLNFIH Z _i ZFLmn_ P X H
∫
T
(µ−∇·β)uh +∑
F⊂∂T
∫
F
φT,F (uh) =
∫
T
f.~ W A"
] H i JφT,F (uh)
HhZRcLNll HdF q i J fhHdMvcUP q O i m LNopcOHYF ^ i cUopFpoIZzLNcUo _ MOHYFL?LNFIH i P X _ u"HdMM@HOHuh
l _i PF ∈ F h
_ c _ M@ZkOH i JµlP _ lPUo q c q Zko X l _ PUczLNMvcRHhZGOHTfhH i J ±?? _ M@ZRHdPU?LNcUopopc q ¸ZUo
F = T1 ∩ T2
φT1,F (uh) + φT2,F (uh) = 0.~ W"\
?HdFLBZUop?Mo.Hy i HfhHy i o Z _ PUc l.LNP i M@H m LfhHF ∈ F h
O ^ i M@H X LNopFpFIH PRHdMvcUPRH O@LNM@ZTFLX LNopFpFIHg _ oIZUopM@H?
W ? _ M@ZUoIZUczLNM@fhHLh"HhfeFIHhZ i JHKJ@LfdcRZ?H i J·*fdo Z _ Mvc _a cRHdM i Z m¼_ P X HdFpFIH X HdMcjHdMopMc q ?PzLNMvcl.LNPl.LNPUcUoIHhZZ i P i M@H X LNopFpFIH
T ∈ ThF ^ S$res
µu + β·∇u = fX i FpcUoplFpo q Hµl.LNP i Ml _ FpuwM X H
q ∈ Pp
fhHTy i oO _ MM@H∫
T
[(µ−∇·β)uq − u(β·∇q)] +∑
F⊂∂T
∫
F
ΦT,F (u)q =
∫
T
fq,~ W J"
Lh"HhfTFIH i JµHKJ@Lfdc
ΦT,F (u) =
(β·nT )u, F ∈ F h,(β·n)u, F ∈ F∂
h .
~ W ["
S+M i cUopFpoIZLNMvcFLgf _ M@OQopcUo _ ML i JFpo X opcRHhZu|∂Ω− = 0
Hdc5FLlP _ lPUo q c qβ·nF [[u]] = 0
l _i P5c _i cF ∈ F h
QopFoIHdMcφT,F (u) = ΦT,F (u).
~n» [ kHdlPRHdM _ M@Z X LNopMcRHdM.LNMvcYFLf _ M@ZRcUP i fdcUo _ MOHFL m¼_ P X H a opFpopM q LNopPRH
ah
] HhZ mn_ M@fdcUo _ M@ZTOHWhZ _ Mvc K!&KOQoIZRf _ McUopM i HhZµOH X LNopFpFIHº X LNopFpFIH ¶ _ PM _i ZLh _ M@Z i y i HµFL>Z _ F i cUo _ M;HKJ@LfdcRH
ZLNcUoIZ m LNopcβ·nF [[u]] = 0
l _i Pc _i cF ∈ F h
?Hhfdo-lFLNoIOHl _i P i M@H X _ OQo.fLNcUo _ M OHFL mn_ P X Ha opFpopM q LNopPRH
ah
HdM2u2L b²_i czLNMc i M@Hµl q M.LNFpoIZzLNcUo _ M L i ZRHdM@ZOHhZ X _ opM@OQPRHhZfLNPUP q ZOHhZZzL i cRZL i JopMcRHdP m LfhHhZ³x°GMf _ M@ZUoIOdPRH
ah(vh, wh) =
∫
Ω[µvhwh + (β·∇hvh)wh] +
∫
∂Ω(β·n)−vhwh −
∑
F∈F h
∫
F
β·nF [[vh]]wh
+∑
F∈F h
∫
F
α
2|β·nF |[[vh]][[wh]],
~n»Q±³
\
_i HdM@f _ PRH?@HdM0opMvc q ?PzLNMcGl.LNPl.LNPUcUoIHhZkFLO q PUop q HLOQ"HhfdcUop"H?
ah(vh, wh) =
∫
Ω[(µ−∇·β)vhwh − vh(β·∇hwh)] +
∫
∂Ω(β·n)+vhwh
+∑
F∈F h
∫
F
β·nFvh[[wh]] +∑
F∈F h
∫
F
α
2|β·nF |[[vh]][[wh]],
~n» W
_¹α > 0
HhZUc i Ml.LNPzL X dcUPRHfRt _ oIZRol.LNPFIHgM i X q PUoIfdoIHdM ~¼fhHdF i o·*fdol H i c q "HdMvc i HdFpFIH X HdMc ® cUPRHTlPUoIZOQo q PRHdMvc¡Z i PgfRt.Ly i H m LfhH³KD=3FAHhZRceo X l _ PUczLNMvc¡O ^`_a ZRHdPU"HdPTy i HfhHdcUcRHM _i "HdFpFIH mn_ P X H a opFpopM q LNopPRHah
ZzLNcUoIZ m LNopc a oIHdMBFIHhZlP _ lPUo q c q ZeOHL2 ·*f _ HdPRfdopwopc q Z i P Wh
HdcjOHTf _ M@ZRoIZUczLNM@fhH?&°GM LHdM0HK6Hdc
ah(vh, vh) ≥ µ0‖vh‖2L2 +
∑
F∈F∂h
∫
F
1
2|β·n|v2
h +∑
F∈F h
∫
F
α
2|β·nF |[[vh]]2, ∀vh ∈Wh,
~n»?»"
ah(u,wh) =
∫
Ωfwh, ∀wh ∈Wh.
~n»NÌ
°GM M _ cRHdPzLy i HTFLlP _ lPUo q c q OH¡f _ HdPRfdopwopc q ~n»?»"kHhZUcjlF i Z m¼_ PUcRH¡y i H~²±["l i oIZRy i^ HdFpFIH¡f _ X l _ PUcRHHdMlF i Z i Mf _ MvcUP FIH¡Z i PFIHhZjZzL i cRZkOH
vh
ºYcUPzLh"HdPRZGFIHhZopMvcRHdP m LfhHhZ]5^ LNM.LNFpuQZRH5OH5f _ M"HdPU"HdM@fhH$lP q ZRHdMc q H-ºFLZzHhfdcUo _ MZ i op?LNMvcRH X _ McUPRHdPzLky i HFL mn_ P X H a opFpopM q LNopPRH
ah
LNopM@ZUo+f _ M@ZUcUP i opcRHf _ M@O i opcºµOHhZgHhZRcUo X LNcUo _ M@ZTO ^ HdPUPRH i P~¼y i LZUo·ã _ lcUo X LNFIHhZ _i cRH m¼_ oIZLLNMcO ^ L a _ PROHdPkfhHl _ opMvc _ u _ M@Zjf _ XX HdMvcGFLYl q M.LNFpoIZLNcUo _ M L i ZRHdM@ZOHhZ X _ opM@OQPRHhZkfLNPUP q ZkOHhZjZzL i cRZL i JYopMvcRHdP m LfhHhZ X _ OQo.H-FIHhZ i J6"°GM q PUo.H m LfdopFIH X HdMvc-y i H-FIHhZ i JLZRZ _ fdo q Z+ºkFLkM _i "HdFpFIH mn_ P X Ha opFpopM q LNopPRH
ah
Z ^`q fdPUop"HdMvc
φT,F (uh) =
β·nT uh+ α2 |β·nT |[[uh]]T , F ∈ F h,
(β·n)+uh, F ∈ F∂h ,
~n»Y"
_¹[[uh]]T
O q ZRop?M@HFIHZL i c¡LZRZ _ fdo q ºBFL X LNopFpFIHTO q @Mo-f _ XX HFLBOQo q PRHdM@fhHHdMvcUPRHFLµ?LNFIH i P
opMc q PUoIH i PRHºTX _ opM@ZFL>?LNFIH i PHKJQc q PUoIH i PRH9º
T ¶ _ MÅL[[uh]]T = nT ·nF [[uh]]F
] HhZ i J´fdo·OHhZzZ i ZZ _ Mvcgf _ M@ZRHdPU?LNcUo m ZHdcf _ M@ZUoIZUczLNMcRZTLh"HhffhH i J9OH¡FLZ _ F i cUo _ M>HKJ@LfdcRH? M>fLZel.LNPUcUoIf i FpoIHdPPRH X LNPRy i L a FIHHhZRcjfhHdF i ol _i PkFIHhy i HdF
α = 1&°GM _a cUoIHdMvc
φT,F (uh) =
β·nTu↑h, F ∈ F h,
(β·n)+uh, F ∈ F∂h ,
~n»YA"
_¹u↑h
O q ZUop?M@HFL?LNFIH i PopMc q PUoIH i PRHºTZUoβ·nT > 0
HdcFL¡LNFIH i PHKJQc q PUoIH i PRHgZUopM _ M?H i JµHhZUcf _ MM i Z _i ZFIHM _ X OH i J i lopM@O fLNPopFf _ M@ZUoIZUcRHTºfzt _ oIZUopPkZ i P i M@HopMcRHdP m LfhHgFLY?LNFIH i PjOHuh
ZUopc i@q HHdMµL X _ McjOHhZFpop?M@HhZOHGf _i PzLNMcGLZRZ _ fdo q HhZºβ M@H mn_ P X i FIH q y i opLNFIHdMcRHgf _ M@ZUoIZUcRHgº
q fdPUopPRH
φT,F (uh) =
(β·nT )+uinth + (β·nT )−uext
h , F ∈ F h,(β·n)+uh, F ∈ F∂
h ,
~n» \
L"HhfOHhZM _ czLNcUo _ M@Z q oIOHdMcRHhZ
J
B # Ej@MOH a oIHdMµf _ X lPRHdM@OQPRHf _ XX HdMvcjZ ^ LNPUcUoIf i FIHGF ^ LNM.LNFpuwZzHGOHGf _ Mv"HdPU"HdM@fhH?QPRHdlPRHdM _ M@Z²·*HdMµFIHhZ
q czLNl HhZHhZRZRHdMvcUoIHdFpFIHhZ-O@LNM@Z i MfLOQPRHj q M q PzLNF7 ] LTZ _ F i cUo _ MOQoIZRfdPRdcRHuh
opc$O@LNM@Z-F ^ HhZUl.LfhHWh
Hdc5FLZ _ F i cUo _ MHKJ@LfdcRH
uO@LNM@Z+F ^ HhZUl.LfhH
W2? _ XX HkF ^ LNllP _ JQo X LNcUo _ MM ^ HhZRc+l.LZ$f _ M mn_ P X H~
Wh 6⊂WK
_ M9HhZUcGL X HdM q ºf _ M@ZUoIO q PRHdPGF ^ HhZUl.LfhHT"Hhfdc _ PUoIHdFZ _ XX HZ := W +Wh
O@LNM@ZkFIHhy i HdFwopcjF ^ HdPUPRH i Pu − uh
] L9lPRH X oIdPRHBlP _ lPUo q c q º q czL a FpopPHhZUc i M@HµlP _ lPUo q c q OHµZUczL a opFpopc q ¶ fhHdFpFIHK·*fdolPRHdM@O2HdM q M q PzLNFFL m¼_ P X HO ^ i M@HTf _ M@OQopcUo _ MopM m ·*Z i l0OQoIZRfdPRdcRH m LNoIZzLNMcjopMvcRHdPU"HdMopP i M@HgM _ P X H
||| · |||¸
∀vh ∈Wh, |||vh||| . supwh∈Wh\0
ah(vh, wh)
|||wh|||.
~n»YJ"
? ^ HhZRcGl.LNPeFIH a oLNoIZOHYfhHdcUcRHM _ P X HYy i H¡ZRHdPzLHhZUcUo X q HF ^ HdPUPRH i Pu − uh ¶
fhHdcUcRH¡M _ P X H¡O _ opcO _ M@f® cUPRHO q @MoIHZ i P
Z S+woIOH XX HdMvcZUo _ MDOQoIZUl _ ZRHO ^ i M@HlP _ lPUo q c q OHf _ HdPRfdopopc q Z i P
Wh
l _i PFL mn_ P X H a opFpopM q LNopPRH
ah
~¼f ^ HhZUc²·3º¯·*OQopPRH?6ZUo _ M9ZLNopcGy i H|||vh|||
2 . ah(vh, vh)l _i Pec _i c
vh ∈ Wh
KfhHdFLZ i cGº q czL a FpopP i M@HTf _ M@OQopcUo _ M opM m ·*Z i l0OQoIZRfdPRdcRH?l i oIZRy i H
|||vh||| .ah(vh, vh)
|||vh|||≤ sup
wh∈Wh\0
ah(vh, wh)
|||wh|||.
_i cRH m¼_ oIZ³-FLM _ P X H||| · |||
O@LNM@ZFLDf _ M@OQopcUo _ M opM m ·*Z i lÅO _ opc ® cUPRH Z i ZzL XX HdMvcµ?PzLNM@OH l _i PLZzZ i PRHdP q vLNFIH X HdMvc i M@HglP _ lPUo q c q OHgf _ McUopM i opc q Z _i ZFL mn_ P X H
∀z ∈ Z, ∀wh ∈Wh, ah(z, wh) . |||z|||∗|||wh|||,~n»Y["
_¹||| · |||∗
HhZUc i M@HM _ P X HjO q @MoIHZ i PZHdc5y i oO _ opc5f _ MvcUP FIHdP5FLeM _ P X H
||| · |||"rgLNM@Z+fhHhZ+f _ M@OQopcUo _ M@Z
_ M9LHdMHK Hdcl _i Pjc _i cyh ∈Wh
|||uh − yh||| . supwh∈Wh\0
ah(uh − yh, wh)
|||wh|||= sup
wh∈Wh\0
ah(u− yh, wh)
|||wh|||,
?PfhH¡ºFLYlP _ lPUo q c q OHTf _ M@ZRoIZUczLNM@fhH?@sLNPkZ i opcRH?@OHl.LNPjFLlP _ lPUo q c q OHTf _ MvcUopM i opc q
|||uh − yh||| . |||u− yh|||∗,
Hdckl.LNPjopM q vLNFpopc q cUPUoLNM i FLNopPRH?&opF6oIHdMc
|||u− uh||| ≤ |||u− yh|||+ |||uh − yh|||
. |||u− yh|||∗ + |||u− yh|||∗
. |||u− yh|||∗.
s5PRHdM.LNMvceF ^ opMQ X i X Z i Pyh ∈Wh
_ ML a _i cUopcGºYF ^ HhZUcUo X LNcUo _ M O ^ HdPUPRH i P _ lcUo X LNFIH
|||u− uh||| . infyh∈Wh
|||u− yh|||∗.~Ì [
s _i PFIHBlP _a FI X H0O ^ LOQ"HhfdcUo _ MQ·7P q LfdcUo _ MFL>M _ P X H0OHBZUczL a opFpopc q y i okf _ M@O i opcº>OHhZHhZUcUo·X LNcUo _ M@ZO ^ HdPUPRH i P _ lcUo X LNFIHhZO@LNM@ZFL>M _ P X HO i ?PzLNlt@H a PUoIZ q H0HdcOHhZHhZUcUo X LNcUo _ M@ZO ^ HdPUPRH i Py i LZUo· _ lcUo X LNFIHhZeHdMBM _ P X H
L2 ~¼Z iax_ lcUo X LNFpopc q O ^`_ PROQPRH 12
HhZUcFLZ i op?LNMvcRH¸
|||v|||2 = ‖v‖2L2 + |v|2M + |v|2J +∑
T∈Th
hT ‖β·∇v‖2T ,
~Ì@±³
[
L"Hhf
|v|2M =∑
F∈F∂h
|v|2M,F , |v|2M,F =
∫
F
|β·n|v2,~Ì W
|v|2J =∑
F∈F h
|v|2J,F , |v|2J,F =
∫
F
α2 |β·nF |[[v]]
2.~Ìv»"
s i oIZRy i HDFL<Z _ F i cUo _ M'HKJLfdcRHHhZRcµO@LNM@ZW ∩ Hp+1(Th)
l.LNPBtvuwl _ cUt@hZRH? _ McUPzLLNopFpFIHdPzL´O@LNM@ZZ := (W ∩Hp+1(Th)) +Wh
=*F6HhZUckfdFLNopPGy i HFLYM _ P X H||| · |||
HhZUc a oIHdMO q @MoIHZ i PkfhHdcjHhZUl.LfhH? ~ ¬ czL a opFpopc q P -
∀vh ∈Wh, |||vh||| . supwh∈Wh\0
ah(vh, wh)
|||wh|||.
~Ì?Ì
: ;( <(8 ¬ _ opcvh ∈Wh
.°GMBl _ ZRHS = supwh∈Wh\0
ah(vh,wh)|||wh|||
~o ] LlP _ lPUo q c q ~n»?»"-o X lFpoIy i Hy i H
‖vh‖2L2 + |vh|
2M + |vh|
2J . ah(vh, vh) . S|||vh|||.
~Ì2"
~opo =3FQPRHhZUcRHºkf _ McUP FIHdP5FLeO q PUop q HLOQ"HhfdcUop"H?w°GMYO q @Mopcβf _ XX HFIHfRt.L X l"Hhfdc _ PUoIHdF@f _ M@ZUczLNMc
l.LNP X _ PRfhHL i JBHdc q vLNF6Z i PfRt.Ly i H X LNopFpFIHgºTFL¡?LNFIH i P X _ u"HdMM@HgOHβZ i PfhHdcUcRH X LNopFpFIH? ]5^ tul _ ·
cUt@hZzHTOHgP q i FLNPUopc q Z i Pβo X lFpoIy i HTy i H
∀T ∈ Th, ‖β − β‖L∞(T ) . hT .
°GMf _ M@ZUoIOdPRHFL mn_ M@fdcUo _ Mπh
cRHdFpFIHy i Hl _i P+c _i cT ∈ Th
πh|T = hTβ·∇vh
?sALNP-f _ M@ZUcUP i fdcUo _ Mπh ∈Wh
rHlF i Z@HdM i cUopFpoIZLNMvcGOHhZjopM q vLNFpopc q ZkopM"HdPRZRHhZkHdcjOHgcUPzLfhH?
|||πh|||2 = ‖πh‖
2L2 + |πh|
2M + |πh|
2J +
∑
T∈Th
hT ‖β·∇πh‖2T
. ‖vh‖2L2 +
∑
F∈Fh
h−1F h2
F ‖β·∇hvh‖2∆F
+∑
T∈Th
hTh−2T h2
T ‖β·∇vh‖2T
. ‖vh‖2L2 +
∑
T∈Th
hT ‖β·∇vh‖2T . |||vh|||
2,
l i oIZzy i H‖β·∇vh‖
2T . ‖β·∇vh‖
2T + ‖(β − β)·∇vh‖
2T . ‖β·∇vh‖
2T + ‖vh‖
2T .
sLNPeLNopFpFIH i PRZ³∑
F∈Fh
‖πh‖2F .
∑
T∈Th
h−1T h2
T ‖β·∇vh‖2T . |||vh|||
2.
± [
rHglF i Z³∑
T∈Th
hT ‖β·∇vh‖2T = ah(vh, πh)− (µvh, πh)Ω −
∑
T∈Th
hT (β·∇vh, (β − β)·∇vh)T
−∑
F∈F∂h
∫
F
(β·n)−vhπh +∑
F∈F h
∫
F
β·nF [[vh]]πh
−∑
F∈F h
∫
F
α
2|β·nF |[[vh]][[πh]]
:= T1 + . . .+ T6.°GM f _ M@ZUczLNcRHTy i H
|T1| ≤ S|||πh||| . S|||vh|||.rHglF i Z³HdM i cUopFpoIZzLNMvcGOHhZjopM q vLNFpopc q ZjopM"HdPRZRHhZkHdcjOHTcUPzLfhH?
|T2|+ |T4|+ |T5|+ |T6| . ‖vh‖Ω‖πh‖Ω + |vh|M
∑
F∈F∂h
∫
F
π2h
1
2
+ |vh|J
∑
F∈F h
∫
F
πh2 + [[πh]]2
1
2
.(
‖vh‖2Ω + |vh|
2M + |vh|
2J
)1
2
‖πh‖2Ω +
∑
F∈Fh
‖πh‖2F
1
2
. S1
2 |||vh|||3
2 .S5MQ@M
|T3| .
∑
T∈Th
hT ‖β·∇vh‖2T
1
2
∑
T∈Th
hTh2Th
−2T ‖vh‖
2T
1
2
. γ∑
T∈Th
hT ‖β·∇vh‖2T + ‖vh‖
2Ω,
_¹γl H i c ® cUPRHfRt _ oIZUoL i ZRZUo5l HdcUopcTy i HM q fhHhZRZzLNopPRH?S+M<f _ FpFIHhfdczLNMvcFIHhZTopM q vLNFpopc q Z¡fdo·*OHhZRZ i Z _ M
L a _i cUopcGºF ^ HhZRcUo X LNcUo _ M∑
T∈Th
hT ‖β·∇vh‖2T . S
1
2 |||vh|||3
2 + S|||vh|||.~Ì2A"
~opopo S+Mf _ X a opM.LNMc~Ì2"Hdc~Ì2A"KopFwoIHdMvc
|||vh|||2 . S
1
2 |||vh|||3
2 + S|||vh|||,O ^`_¹ _ MO q O i opc m LfdopFIH X HdMvcGy i H
|||vh||| . SHdM i cUopFpoIZzLNMvceOH i J m¼_ oIZF ^ opM q vLNFpopc q OH _i M@
_i ZLNFpF _ M@Z X LNopMvcRHdM.LNMvc q czL a FpopP i M@HjlP _ lPUo q c q OHGf _ MvcUopM i opc q l _i P-FL m¼_ P X H a opFpopM q LNopPRHah ¶FLlPRH i "HTHdMZRHdPzLYM@HdcUcRH X HdMcklF i ZkZRo X lFIH?&°GMBopMvcUP _ O i opckFLYM _ P X H
|||v|||2∗ = |||v|||2 +∑
T∈Th
[h−1T ‖v‖
2T + ‖v‖2∂T ].
~Ì \
±?±
~/? _ MvcUopM i opc q P -
∀z ∈ Z, ∀wh ∈Wh, ah(z, wh) . |||z|||∗|||wh|||.~Ì2J"
: ;( <(8 ? _ M@ZUoIO q P _ M@ZFL m¼_ P X i FLNcUo _ M ~n» W ¡Hdc X L bU_ P _ M@ZFIHhZOQo q PRHdMcRZcRHdP X HhZf _ X l _ ZzLNMvcFIHX H X a PRHOHTOQP _ opcRH? =*F6woIHdMvc
∫
Ω(µ−∇·β)zwh . ‖z‖Ω‖wh‖Ω,
∫
Ωz(β·∇hwh) .
∑
T∈Th
h−1T ‖z‖
2T
1
2
∑
T∈Th
hT ‖β·∇wh‖2T
1
2
,
∑
F∈F∂h
∫
F
(β·n)+zwh .
∑
F∈F∂h
‖z‖2F
1
2
|wh|M ,
∑
F∈F h
∫
F
β·nF z[[wh]] .
∑
F∈F h
‖z‖2F
1
2
|wh|J ,
∑
F∈F h
∫
F
α
2|β·nF |[[z]][[wh]] . |z|J |wh|J ,
Hdc _ M0f _ M@fdF i ceLNoIZ q X HdMc _i ZkZ _ XX HhZ X LNopMvcRHdM.LNMceHdM X HhZ i PRHgO ^`q M _ M@fhHdPkFIHgP q Z i FpczLNcjlPUopM@fdopl.LNFOHTfhHdcUcRHTZzHhfdcUo _ M
"!$# &% ~/? _ M"HdPU"HdM@fhH³ P -
|||u− uh||| . hp+ 1
2 ‖u‖Hp+1(Th).~Ì2["
: ;( <(8 cUopFpoIZRHdPFIHfLOQPRHgcUt qh_ PUoIy i Hfdo·*OHhZRZ i Zl _i P _a cRHdMopP
|||u− uh||| . infyh∈Wh
|||u− yh|||∗,
l i oIZ i cUopFpoIZRHdPjFIHP q Z i FpczLNcGO ^ opMvcRHdPUl _ FLNcUo _ M ~²±A"l _i PjHhZUcUo X HdPjFIH X H X a PRHTOHTOQP _ opcRH?S+MBl.LNPUcUoIf i FpoIHdP _ ML
‖u− uh‖Ω . hp+ 1
2 ‖u‖Hp+1(Th),~ [
fhHy i oHhZUcF q "dPRH X HdMceZ iax_ lcUo X LNF-~¼O ^`_ PROQPRH 12
KHdckZUo FIH X LNopFpFLN"HTHhZRcky i LZRo· i Mo m¼_ P X H?
‖β·∇h(u− uh)‖Ω . hp‖u‖Hp+1(Th),~Q±³
fhHy i oHhZUc _ lcUo X LNF7
± W
¥&¥R¦j§g¨e© ª+«j£¤x¥R¦j§«Y©x¦g£ Y¢ ¡¢ £ Y«
¬ _ opcf ∈ L2(Ω)
&°GM0f _ M@ZUoIOdPRHFIHlP _a FI X H X _ OdFIH
−∆u+ u = fO@LNM@Z
Ω,
u = 0Z i P
∂Ω.
~ W
_i Z-L _ M@Z-fzt _ oIZUo.OHcUPzLLNopFpFIHdPZ i P5fhHlP _a FI X HkPRHdFLNcUop"H X HdMvcZUo X lFIHkL¯@MOH X HdcUcUPRHkHdMLLNMcFIHcUPzLNopcRH X HdMc$O i cRHdP X HOHOQo i ZUo _ Ml.LNPFIHhZ X q cUt _ OHhZ+OHjr ] HhZ+O q "HdF _ ll H X HdMvcRZ-fdo·*OHhZRZ _i Zl H i "HdMvc ® cUPRH q cRHdM@O i ZkO@LNM@ZkOHgM _ X a PRH i ZRHhZGOQopPRHhfdcUo _ M@Z¡¸
_ MBlxH i cf _ M@ZUoIO q PRHdPGO ^ L i cUPRHhZjf _ M@OQopcUo _ M@ZGL i JFpo X opcRHhZjy i HfhHdFpFIHhZjOHgreopPUoIfztFIHdcjt _ X _ "dM@H?l.LNPkHKJH X lFIHTOHhZjf _ M@OQopcUo _ M@ZL i JFpo X opcRHhZGOH GH i X LNMM _i OH _a opM ¶ _ MBlxH i c q vLNFIH X HdMcGf _ M@ZUoIO q PRHdP i M@H q y i LNcUo _ MOHgOQo i ZUo _ MwwLOQ"HhfdcUo _ MwvP q LfdcUo _ M9 q M q PzLNFIHZ _i ZFL mn_ P X H
−∇·(K∇u) + β·∇u+ µu = f,~?»"
_¹ FIHcRHdP X HOHP q LfdcUo _ MµHdcgFIHfRt.L X l LOQ"HhfdcUo m
βZzLNcUoIZ mn_ MvcTFIHhZgtul _ cUt@hZRHhZ q M _ M@f q HhZ
º>FLZRHhfdcUo _ M;lP q f q OHdMvcRH9Hdc _¹ FIHBcRHdM@ZRH i POH0OQo i ZRo _ MK ∈ [L∞(Ω)]d,d
HhZUcZ i ll _ Z qZUu X q cUPUoIy i HTHdc i Mo mn_ P X q X HdMvcO q @Mol _ ZUopcUo m ¶ _ M0l H i c _ X HdcUcUPRH? M % F ^ i cUopFpoIZzLNcUo _ M O ^ opM q vLNFpopc q ZGOHc*uwlxHs _ opM@fLNP q .FIHcRHdP X HTO ^`_ PROQPRH q P _ O@LNM@Z¡~ W ¶ _ M lxH i cHdMQ@M q c i OQoIHdPO@LNM@ZB~?»"OQop"HdPRZP q ?o X HhZLZRu X lc _ cUoIy i HhZO@LNM@ZFIHhZf _ H$fdoIHdMcRZlt q M _ X q M _ F _ ?oIy i HhZ
KβHdcµ"f _ XX HGl.LNP-HKJH X lFIHkfhHdF i o _¹ FIHcRHdP X HGLOQ"HhfdcUo m O _ X opM@H
FIHcRHdP X HOQo i ZUo mG_i fhHdF i o _¹ FIHcRHdM@ZRH i PK
lP q ZRHdMvcRHBOHhZ mn_ PUcRHhZt q c q P _ q Mopc q Z _i OHhZm¼_ PUcRHhZGLNMoIZ _ cUP _ loIHhZ
rgLNM@ZµfhHdcUcRH9ZRHhfdcUo _ M$FIHlP _a FI X H2~ W HhZUcPRH m¼_ P X i F q Z _i Z mn_ P X H X oJQcRH?fhH y i oGf _ M@ZUoIZRcRH9ºopMcUP _ O i opPRHeFLTLNPUoL a FIHeL i JwopFpoLNopPRH
σ = −∇u~ _a ZzHdPU"HdPy i H
uHhZUc-ºTLNFIH i PRZO@LNM@Z
RHdc$y i H
σHhZUc
ºTLNFIH i PRZ-O@LNM@ZR
dKw°GMOQopc-y i H
uHhZUc5FLTLNPUoL a FIHklPUo X LNFIHGHdc
σFIH i J ~¼OQo i ZUo m K°GMf _ M@ZUoIOdPRH
O _ M@fTFIHglP _a FI X HZ i op?LNMvcT¸
σ +∇u = 0O@LNM@Z
Ω,
∇·σ + u = fO@LNM@Z
Ω,
u = 0Z i P
∂Ω.
~NÌ
] LgO q X LNPRfRt@HeZ i opwoIHjL¯@MOHf _ M@ZUcUP i opPRH i M@HkLNllP _ Jwo X LNcUo _ M0r OH~NÌAHhZUc$FL X ® X Hjy i Hl _i PF ^`q y i LNcUo _ M O ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ M2¸
±?kL i Mop"HL i f _ McUopM i mn_ P X i FIHdPFIHYlP _a FI X HB~NÌGHdM i cUopFpoIZLNMvcF ^ HhZUl.LfhHO i ?PzLNlt@HYHdc i M@Hmn_ P X H a opFpopM q LNopPRHf _ MvcUopM i HeZ i P-fhHdc-HhZRl.LfhHky i o.o X l _ ZRHjFIHhZf _ M@OQopcUo _ M@ZL i JFpo X opcRHhZL i ZRHdM@Zm LNo a FIH ¶
W kL i Mop"HL i OQoIZzfdPRHdc X _ OQo.HdPkFL m¼_ P X H a opFpopM q LNopPRH¡HdMBuPzL bU_i czLNMvcGc _i ckO ^ L a _ PRO i M0cRHdP X HOHf _ M@ZRoIZUczLNM@fhHy i o5lP q ZRHdPU"H i M@HlP _ lPUo q c q OH
L2 ·*f _ HdPRfdopwopc q OQoIZRfdPRdcRH ¶ l i oIZl q M.LNFpoIZRHdPL i ZRHdM@ZOHhZ X _ opM@OQPRHhZjfLNPUP q ZFIHhZZzL i cRZjL i JopMvcRHdP m LfhHhZjOH~¼fhHdPUczLNopM@HhZjf _ X l _ ZLNMvcRHhZjOH³σHdcjOH
u ¶»QjHdMQ@M$HK6Hhfdc i HdPF ^ LNM.LNFpuQZRH0OH f _ Mv"HdPU"HdM@fhH9HdM q czL a FpoIZRZLNMvcf _ XX HlP q f q OH XX HdMvc i M@H
lP _ lPUo q c q OHTZRczL a opFpopc q ~¼f _ M@OQopcUo _ MopM m ·*Z i l0OQoIZRfdPRdcRH³Hdc i M@HglP _ lPUo q c q OHTf _ McUopM i opc q
±»
+¾*½ ¿À6Á5Â"Ã2ÄÅÀxÆ?Ç$È.ÄÅÀxÆ?É3ÊË5Ã]5^ HhZUl.LfhHTO i ?PzLNlt@HM.LNc i PRHdFpFIH X HdMvcgLZRZ _ fdo q ºB~NÌHhZUc
W = H(OQop
; Ω)×H1(Ω)
= (σ, u) ∈ [L2(Ω)]d+1; ∇·σ ∈ L2(Ω); ∇u ∈ [L2(Ω)]d.
~Y"
]$^ HhZUl.LfhHW
LuvLNMvc i M@HgZUcUP i fdc i PRHgO ^ HhZUl.LfhHglP _ O i opcQopFZRHdPzLYf _ XX _ OHTl.LNPFLZ i opcRHgOHgO q f _ X ·l _ ZRHdP i M q F q X HdMc
z ∈WZ _i ZFL mn_ P X H
(zσ , zu)Lh"Hhf
zσ ∈ H(OQop
; Ω)Hdczu ∈ H1(Ω)
=*F6HhZUcGfdFLNopPky i HF ^ HhZRl.LfhH
WX i MoOHFLM _ P X H~¼HdcGO i lP _ O i opckZzfLNFLNopPRHLZRZ _ fdo q
‖z‖2W = ‖zσ‖2L2 + ‖∇·zσ‖2L2 + ‖zu‖2H1
~YA"
HhZRc i M0HhZUl.LfhHTOHGopF a HdPUc.reHlF i Z³ _ M0OQoIZUl _ ZRHgO i cUt qh_ PR X HTOHgcUPzLfhHZ i opLNMc¸QF ^ LNllFpoIfLNcUo _ M
γ : W 3 z 7−→ (n·zσ, zu) ∈ H− 1
2 (Ω)×H1
2 (Ω)~ \
HhZRckf _ MvcUopM i HZ i PWS+MQ@M _ M0O q @MopcGZ i P
W ×WFL mn_ P X H a opFpopM q LNopPRH
a(z, y) =
∫
Ω(zσ +∇zu)·yσ +
∫
Ω(∇·zσ + zu)yu − 〈n·yσ, zu〉
H−12 ,H
12,
~YJ"
y i okHhZUc a oIHdM f _ MvcUopM i H Z i PW × W
+rHBlF i Z5fhHdcUcRH m¼_ P X H0FpopM q LNopPRHHhZUcL2 ·*f _ HdPRfdop"H0Z i P Wl i oIZzy i^ HdM i cUopFpoIZLNMvckFL mn_ P X i FIH¡OHePRHhHdMopFwoIHdMvc
a(z, z) = ‖zσ‖2L2 + ‖zu‖2L2 .~Y["
°GM f _ M@ZUoIOdPRHgFIHglP _a FI X HTZ i opLNMc¸
?-t@HdPRfRt@HdPz ∈W
cRHdFy i Ha(z, y) =
∫
Ωfyu ∀y ∈W.
~A [
_i ZLh _ M@ZFIHeP q Z i FpczLNcZ i op?LNMvcT~FL¡lPRH i "HeHhZRc-FLNoIZRZ q HGHdMµHKJQHdPRfdoIfhH ¶ _ M i cUopFpoIZRHdPzL¡FL¡Z i P b HhfdcUopwopc qOHhZkcUPzLfhHhZjOHH(
OQop; Ω)
l _i PP q f i l q PRHdPkFLf _ M@OQopcUo _ M@ZGL i JBFpo X opcRHhZK
"!$# &% E (>5,%) M (0~A [ (+" 5K/( - <"8+*V(>% " * " -Q- (R"Y% ¼! - (+" @" &% ¼ -(¡~NÌ 8
+¾ ¿ ¯Æ?ÂvËwÆ?É Á5ÃIÀ $ 5É*ÄÅÀxÆ?É °GMBl _ ZzH
Uh = Gph
Σh = [Gp
h]dHdcWh = Σh × Uh
] HTlP _a FI X HOQoIZRfdPRHdckf _ M@ZRoIZUcRHTº
?-t@HdPRfzt@HdPzh ∈Wh
cRHdFy i Hah(zh, yh) =
∫
Ωfyu
h
∀yh ∈Wh.
~AQ±³
°GM9f _ M@ZUoIOdPRHO _ M@f i M@H¡LNllP _ Jwo X LNcUo _ MOH¡r4OH¡OHd?P qpl _i PjFLLNPUoL a FIHlPUo X LNFIHHdcjl _i P
FIH i J rH i JBtvuwl _ cUt@hZzHhZjOHgP q i FLNPUopc q Z i llF q X HdMczLNopPRHhZZ _ Mvc m LNopcRHhZjZ i PFLZ _ F i cUo _ M HKJ@LfdcRH¸z := (σ, u) ∈ [Hp+1(Th)]d
fhHgy i oZUop?Mo.HTy i HFIH X LNopFpFLN"H¡L q c q f _ M@ZUcUP i opckOHTZ _ PUcRHy i HFLZ _ F i cUo _ MHKJ@LfdcRHgZ _ opckZ i ZzL XX HdMvcGP q i FpoIdPRHºF ^ opMvc q PUoIH i PeOHTfRt.Ly i H X LNopFpFIH ¶;l _i Pkc _i c
F ∈ F hnF ·[[σ]] = 0
Hdc[[u]] = 0
±dÌ
SJ@L X opM _ M@ZFLL2 ·*f _ HdPRfdopwopc q OHeFL mn_ P X H a opFpopM q LNopPRH a O q @MoIHl.LNPT~YJ"-Z i P Wh
HdMlPRHdM.LNMcZ _ opMNL i lP q LNFL a FIH?OH$F _ fLNFpoIZRHdPAFIHhZ _ l q PzLNcRH i PRZ+OH5?PzLOQoIHdMc5HdcOHOQop"HdPU"HdM@fhHZ i Pfzt.Ly i H X LNopFpFIH?°GM9LYl _i Pjc _i c
zh = (zσh , z
uh) ∈Wh
a(zh, zh) = ‖zσh‖
2L2 + ‖zu
h‖2L2 +
∑
F∈F h
∫
F
2n·zσhz
uh
= ‖zσh‖
2L2 + ‖zu
h‖2L2 +
∑
F∈F h
∫
F
nF ·[[zσh ]]zu
h+ nF ·zσh[[z
uh ]].
s i oIZRy i HFLYZ _ F i cUo _ MBHKJ@LfdcRHgZzLNcUoIZ m LNopcnF ·[[σ]] = 0
Hdc[[u]] = 0
l _i Pc _i cF ∈ F h
_ MµP q czL a FpopcFLL2 ·*f _ HdPRfdopwopc q Z i P Wh
c _i ckHdMBlP q ZRHdPU?LNMvckFLf _ M@ZRoIZUczLNM@fhHTHdMf _ M@ZUoIO q PzLNMckFL mn_ P X H a opFpopM q LNopPRH
ah(zh, yh) =
∫
Ω(zσ
h +∇hzuh)·yσ
h +
∫
Ω(∇h·z
σh + zu
h)yuh −
∑
F∈F∂h
∫
F
(n·yσh)zu
h
−∑
F∈F h
∫
F
nF ·[[zσh ]]yu
h+ nF ·yσh[[z
uh ]].
~A W
~7° a ZzHdPU"HdPkFIHhZjfzt _ oJ m LNopcRZl _i Pzh
Hdcyh
O@LNM@ZjFIHhZcRHdP X HhZjPRHdFLNcUo m ZkL i JµopMvcRHdP m LfhHhZêk°GM LO _ M@f
ah(zh, zh) ≥ ‖zσh‖
2L2 + ‖zu
h‖2L2 , ∀zh ∈Wh,
~A?»"
ah(z, yh) =
∫
Ωfyu
h, ∀yh ∈Wh.~ANÌ
=3FjPRHhZUcRH9º>l q M.LNFpoIZRHdPL i ZzHdM@ZOHhZ X _ opM@OQPRHhZfLNPUP q ZFIHhZZzL i cRZL i J2opMcRHdP m LfhHhZOHzuh
HdcOHBFLf _ X l _ ZzLNMcRH¡M _ P X LNFIHYOH
zσh
.rHglF i Z³.f _ XX H¡F ^ opM q vLNFpopc q ~A?»"M@H mn_i PUMopceL i f i M>f _ MvcUP FIHZ i PFIHhZe?LNFIH i PRZeL i9a _ PROOH
zuh
.fhHhZOHdPUMoIdPRHhZZRHdP _ Mvc q vLNFIH X HdMvcl q M.LNFpoIZ q HhZgL i ZzHdM@ZkOHhZ X _ opM@OQPRHhZfLNPUP q Z³S+czLNMvcYO _ MM q OH i Jl.LNPzL X dcUPRHhZ
ηu > 0Hdcησ > 0
fzt _ oIZUoIZTl.LNPgFIHM i X q PUoIfdoIHdMÅ~¼Hdcy i ol H i "HdMvc q "HdMc i HdFpFIH X HdMvc ® cUPRH-lPUoIZOQo q PRHdMvcRZ+Z i PfRt.Ly i H m LfhH³K _ MYf _ M@ZRoIOdPRH$FL m¼_ P X H a opFpopM q LNopPRH
ah(zh, yh) =
∫
Ω(zσ
h +∇hzuh)·yσ
h +
∫
Ω(∇h·z
σh + zu
h)yuh −
∑
F∈F∂h
∫
F
(n·yσh)zu
h
−∑
F∈F h
∫
F
nF ·[[zσh ]]yu
h+ nF ·yσh[[z
uh ]]
+∑
F∈Fh
∫
F
ηu[[zuh ]][[yu
h ]] +∑
F∈F h
∫
F
ησ(nF ·[[zσh ]])(nF ·[[y
σh ]]),
~AY"
_i HdM@f _ PRHTHdM0opMvc q ?PzLNMcGl.LNPl.LNPUcUoIHhZkFIH?PzLOQoIHdMvcHdcFLOQop"HdPU"HdM@fhHTZ i PGfRt.Ly i H X LNopFpFIH?
ah(zh, yh) =
∫
Ω[zσ
h ·yσh − z
uh∇h·y
σh ] +
∫
Ω[zu
hyuh − z
σh ·∇hy
uh] +
∑
F∈F∂h
∫
F
(n·zσh)yu
h
+∑
F∈F h
∫
F
nF ·[[yσh ]]zu
h+ nF ·zσh[[y
uh ]]
+∑
F∈Fh
∫
F
ηu[[zuh ]][[yu
h]] +∑
F∈F h
∫
F
ησ(nF ·[[zσh ]])(nF ·[[y
σh ]]).
~AYA"
±
E?LNMvcYOHl.LZzZRHdPYº0F ^ LNM.LNFpuwZRHOHf _ M"HdPU"HdM@fhH?+HKJL X opM _ M@ZYFL m¼_ P X i FLNcUo _ MF _ fLNFIHO i lP _ ·a FI X H~AQ±³$Hdc-FIHhZ i JLZRZ _ fdo q ZQ°GM q PUo.H m LfdopFIH X HdMvcjHdM i cUopFpoIZzLNMvcg~AYA"-y i H
zh := (zσh , z
uh)
HhZUcZ _ F i cUo _ M0OH~AQ±³-ZUo6HdcZRH i FIH X HdMcjZUo&l _i Pc _i cRH X LNopFpFIH
T ∈ Thwl _i Pc _i cl _ FpuwM X H
q ∈ [Pp]d
Hdckl _i Pc _i cjl _ FpuwM X Hr ∈ Pp
∫
T
[zσh ·q − z
uh∇·q] +
∑
F⊂∂T
∫
F
φσT,F (zh)·q = 0,
∫
T
[zuhr − z
σh ·∇r] +
∑
F⊂∂T
∫
F
φuT,F (zh)r =
∫
T
fr,
~A \
L"HhfTFIHhZ i J
φσT,F (zh) =
nT (zuh+ ησnF ·[[z
σh ]]), F ∈ F h,
0, F ∈ F∂h ,
~AYJ"
Hdc
φuT,F (zh) =
nT ·(zσh+ ηunF [[zu
h ]]), F ∈ F h,n·(zσ
h + ηunzuh), F ∈ F∂
h .
~AY["
?HhZ i JµZ _ Mvc a oIHdMf _ M@ZzHdPULNcUo m Zl i oIZRy i HgZUoF = T1 ∩ T2
φσT1,F (zh) + φσ
T2,F (zh) = 0, φuT1,F (zh) + φu
T2,F (zh) = 0.~ \[
=3FIZ5Z _ Mc$OHlF i Z-f _ M@ZUoIZUczLNMvcRZLh"HhfkFIHhZ i JQZ-HKJLfdcRZ$y i o.Z _ Mvc _a cRHdM i Z mn_ P X HdFpFIH X HdMvcHdMopMvc q ?PzLNMcl.LNPl.LNPUcUoIHhZZ i P i M@H X LNopFpFIH
T ∈ ThFIHhZS5rs5Z
σ + ∇u = 0Hdc∇·σ + u = f
X i FpcUoplFpo q HhZPRHhZRlxHhfdcUop"H X HdMcGl.LNP i M0l _ FpuwM X H
q ∈ [Pp]dHdc i MBl _ FpuM X H
r ∈ Pp
fhHTy i oO _ MM@H∫
T
[zσ·q − zu∇·q] +∑
F⊂∂T
∫
F
ΦσT,F (z)·q = 0,
∫
T
[zur − zσ·∇r] +∑
F⊂∂T
∫
F
ΦuT,F (z)r =
∫
T
fr,
~ \ ±³
L"HhfΦσ
T,F (z) = nT zu, Φu
T,F (z) = nT ·zσ .
~ \W PfhHeL i Jtvuwl _ cUt@hZRHhZOHjP q i FLNPUopc q Z i P$FLZ _ F i cUo _ MHKJ@LfdcRHkHdcL i Jf _ M@OQopcUo _ M@ZL i JFpo X opcRHhZ³ _ ML a oIHdM
φσT,F (z) = Φσ
T,F (z), φuT,F (z) = Φu
T,F (z).~ \ »"
+¾ 5À NÃÁ$Ã Ã@ÂÃ QÃ] LYM _ P X HOHgZRczL a opFpopc q HhZUcFLZ i op?LNMvcRH¸
|||z|||2 = ‖zσ‖2L2 + ‖zu‖2L2 + |zu|2M + |zu|2Ju + |zσ|2Jσ
+∑
T∈Th
[hT ‖∇zu‖2T + hT ‖∇·z
σ‖2T ],~ \ Ì
±A
L"Hhf
|zu|2M =∑
F∈F∂h
|zu|2M,F , |zu|2M,F =
∫
F
ηu(zu)2,~ \ "
|zu|2Ju =∑
F∈F h
|zu|2Ju,F , |zu|2Ju,F =
∫
F
ηu[[zu]]2,~ \ A"
|zσ |2Jσ =∑
F∈F h
|zσ|2Jσ ,F , |zσ |2Jσ,F =
∫
F
ησ(nF ·[[zσ ]])2.
~ \?\
~ ¬ czL a opFpopc q P -
∀zh ∈Wh, |||zh||| . supyh∈Wh\0
ah(zh, yh)
|||yh|||.
~ \ J"
: ;( <(8 ¬ _ opczh ∈Wh
.°GM0l _ ZzHS = supyh∈Wh\0
ah(zh,yh)|||yh|||
~o =*FHhZUcjfdFLNopPGy i H
‖zσh‖
2L2 + ‖zu
h‖2L2 + |zu
h |2M + |zu
h |2Ju + |zσ
h |2Jσ . ah(zh, zh) . S|||zh|||.
~opo =*F+PRHhZRcRHºBf _ McUP FIHdP¡FLµM _ P X HO i ?PzLOQoIHdMvcOHzuh
HdcfhHdFpFIHOHFL0OQop"HdPU"HdM@fhHOHzσh
s _i PfhHdFLQ _ M´lP _ fhhOHOH X LNMoIdPRH LNM.LNF _ i Hº>FL9lPRH i "HOHBFL>f _ M@OQopcUo _ M;opM m ·*Z i l´OQoIZRfdPRdcRHBl _i PFIH9lP _a FI X HO ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ MGs _i P _a cRHdMopPBFIH9f _ MvcUP FIHDZ i PFLOQop"HdPU"HdM@fhHOH
zσh
_ MopMcUP _ O i opcGFIHTfRt.L X l9L i JwopFpoLNopPRH
πh
cRHdFy i HgZ i PjfRt.Ly i H X LNopFpFIHT ∈ Th
πh|T = (πσh , π
uh)|T := (0, hT∇·z
σh)|T .
=3FHhZRckfdFLNopPey i Hπh ∈ Wh
.rHlF i Z.HdM i cUopFpoIZzLNMvcOHhZjopM q vLNFpopc q ZeopMv"HdPRZzH¡HdceOHTcUPzLfhH? _ M q PUo.Hy i H|||πh||| . |||zh|||
@S+MQ@M _ M _a ZRHdPU"HTy i H
∑
T∈Th
hT ‖∇·zσh‖
2T = ah(zh, πh)− (zu
h , πuh)Ω +
∑
F∈F h
∫
F
nF ·[[zσh ]]πu
h
−∑
F∈Fh
∫
F
ηu[[zuh ]][[πu
h ]],
HdceHdM X L bU_ PzLNMvceFIHhZky i LNcUPRHTcRHdP X HhZkO i X H X a PRHYOHTOQP _ opcRH? _ M _a cUoIHdMvc~FIHhZGO q czLNopFIZGZ _ MvcGFLNoIZRZ q ZHdM HKJQHdPRfdoIfhH³K
∑
T∈Th
hT ‖∇·zσh‖
2T . S
1
2 |||zh|||3
2 + S|||zh|||.
sLNPjLNopFpFIH i PRZ _ M _a cUoIHdMvc i Mµf _ MvcUP FIHLNM.LNF _ i HZ i P$FIHG?PzLOQoIHdMvcOHzuh
HdMµf _ M@ZUoIO q PzLNMvcFIHkfzt.L X lL i JQopFpoLNopPRH
πh
cRHdFy i HgZ i PkfRt.Ly i H X LNopFpFIHT ∈ Th
πh|T = (πσh , π
uh)|T := (hT∇z
uh , 0)|T ,
fhHy i of _ M@O i opcº∑
T∈Th
hT ‖∇zuh‖
2T . S
1
2 |||zh|||3
2 + S|||zh|||.
± \
~opopo S+Mf _ FpFIHhfdczLNMvcGFIHhZ X L bU_ PzLNcUo _ M@Zkfdo·*OHhZRZ i ZQopF6oIHdMc
|||zh|||2 . S
1
2 |||zh|||3
2 + S|||zh|||.
Hdc _ M0f _ M@fdF i cGHdM i cUopFpoIZzLNMcGOH i J m¼_ oIZF ^ opM q vLNFpopc q OH _i M@s _i PkFLlP _ lPUo q c q OHTf _ McUopM i opc q _ M i cUopFpoIZRHFIHgM _ P X H
|||z|||2∗ = |||z|||2 +∑
T∈Th
[h−1T ‖z
u‖2T + ‖zu‖2∂T + h−1T ‖z
σ‖2T + ‖zσ‖2∂T ].~ \ ["
°GMPzLNllxHdFpFIHTy i HF ^ HhZUl.LfhHZ_¹ wopcF ^ HdPUPRH i PkHhZUc
Z = [Hp+1(Th)]d+1 +Wh
~/? _ McUopM i opc q P -
∀ζ ∈ Z, ∀yh ∈Wh, ah(ζ, yh) . |||ζ|||∗|||yh|||.~J [
: ;( <(8 q PUo.fLNcUo _ MOQopPRHhfdcRHkHdM X L bU_ PzLNMvc i Mº i Mc _i Z+FIHhZ$cRHdP X HhZ$y i o@f _ X l _ ZRHdMc$FIH X H X a PRHOHOQP _ opcRHTOH~AYA"K ] HhZkO q czLNopFIZkZ _ McjFLNoIZRZ q ZjHdM0HKJHdPRfdoIfhH? _i ZkZ _ XX HhZ X LNopMvcRHdM.LNMceHdM X HhZ i PRHgO ^`q M _ M@fhHdPkFIHgP q Z i FpczLNcjlPUopM@fdopl.LNFOHTfhHdcUcRHTZzHhfdcUo _ M
"!$# &% ~/? _ Mv"HdPU"HdM@fhH³ P -
|||z − zh||| . hp+ 1
2 ‖z‖[Hp+1(Th)]d+1 .~JQ±³
: ;( <(8 cUopFpoIZRHdPFIHfLOQPRHgcUt qh_ PUoIy i Hfdo·*OHhZRZ i Zl _i P _a cRHdMopP
|||z − zh||| . infyh∈Wh
|||z − yh|||∗,
l i oIZ i cUopFpoIZRHdPjFIHP q Z i FpczLNcGO ^ opMvcRHdPUl _ FLNcUo _ M ~²±A"l _i PjHhZUcUo X HdPjFIH X H X a PRHTOHTOQP _ opcRH?] HgczL a FIHL i W Ù»P q Z i X HgFIHhZ _ PROQPRHhZGOHTf _ Mv"HdPU"HdM@fhH _a cRHdM i ZGl _i PFIHhZkHdPUPRH i PRZkHdM i J0HdcjHdM
?LNPUoL a FIHlPUo X LNFIH?°GMDM _ cRHdPzL y i HFIHhZ _ PROQPRHhZYOHf _ M"HdPU"HdM@fhHZ _ MvcF q "dPRH X HdMcZ iax_ lcUo X L i J~¼O ^`_ PROQPRH 1
2
Tl _i PFL M _ P X HL2 Hdcy i^ opFIZZ _ Mc _ lcUo X L i J2L i ZRZRo a oIHdM2l _i PFIH?PzLOQoIHdMvcOHFL
?LNPUoL a FIHglPUo X LNFIH¡y i Hl _i PjFLOQop"HdPU"HdM@fhHO i i J
‖zσ − zσh‖Ω ‖∇h·(z
σ − zσh)‖Ω ‖zu − zu
h‖Ω ‖∇h(zu − zuh)‖Ω
O(hp+ 1
2 ) O(hp) O(hp+ 1
2 ) O(hp)
±°GPROQPRHhZ-OHGf _ Mv"HdPU"HdM@fhHel _i P$F ^ LNllP _ JQo X LNcUo _ Mr'ºOH i JfRt.L X l@Z ¶ l q M.LNFpoIZLNcUo _ MOH[[zu
h ]]Lh"Hhf i Mf _ H$fdoIHdMvc
ηu > 0Hdcgl q M.LNFpoIZLNcUo _ MOH
nF ·[[zσh ]]
L"Hhf i MDf _ H$fdoIHdMvcησ > 0 ¶ FLZ _ F i cUo _ MHKJLfdcRHHhZUcjZ i ll _ Z q H ® cUPRHgO@LNM@Z
[Hp+1(Th)]d ×Hp+1(Th)
±J
¥&¥R¦j§g¨e© ª+«j£¤x¥R¦j§«Y©x¦g£ Y¢ ¡§ £ Y«
]5^ LNllP _ JQo X LNcUo _ MDr4ºOH i Jµfzt.L X l@ZklP q ZzHdMvc q HO@LNM@ZjFLZRHhfdcUo _ M0lP q f q OHdMcRHl _i PFIHglP _ ·a FI X HOHOQo i ZUo _ MwvP q LfdcUo _ M lP q ZRHdMvcRHF ^ opM@f _ Mv q MoIHdMcO ^ ® cUPRH _ M q PRH i ZRH?AS+M<HK6HdcFLBczLNopFpFIHO iZRuwZUcR X HFpopM q LNopPRH¡ºP q Z _i OQPRHTHhZUcOH
(d+ 1)Cdp+dN !#"
sALNPkHKJH X lFIH?Ql _i P i M@HTLNllP _ JQo X LNcUo _ MOHOHd?P qGi M _ M _a cUoIHdMc i M@HeczLNopFpFIHeOH
9N !#" HdMµOQo X HdM@ZUo _ M W HdcOH 16N !#" HdMOQo X HdM@ZRo _ MB»QwfhHy i oHhZUc q M _ P X H?]5^ LNllP _ fRt@H O q "HdF _ ll q H O@LNM@Z9fhHdcUcRH ZRHhfdcUo _ M l HdP X Hdc>OHDl.LNFpFpoIHdPDº´fhHdcUcRH OQo f i Fpc q S+FpFIH
f _ M@ZRoIZUcRHGºP q O i opPRHGf _ M@ZUoIO q PzL a FIH X HdMvcFLczLNopFpFIHGO i ZUuQZUcR X HFpopM q LNopPRHGHdM q Fpo X opM.LNMvcFIH i JOQoIZRfdPRHdcF _ fLNFIH X HdMvcjZ i P-fRt.Ly i H X LNopFpFIH? ] HklPUoJµºTl.Lu"HdPHhZRc-y i HGfhHdcUcRH q Fpo X opM.LNcUo _ M0HhZUc$PRHdM@O i Hel _ ZRZUo a FIH
HdM0M@Hl q M.LNFpoIZzLNMvcGlF i ZFIHTZzL i cjOHFLf _ X l _ ZzLNMcRHTM _ P X LNFIHTO i i J0L i JµopMvcRHdP m LfhHhZ ¶ HdM0l q M.LNFpoIZzLNMvcjFIHZzL i cjOHgFLY?LNPUoL a FIHglPUo X LNFIHgl.LNP i Mf _ H$fdoIHdMvcey i oHKJQlF _ ZRHgHdM 1h
¬ i PFIHelFLNM0OHeF ^ LNM.LNFpuQZRHgO ^ HdPUPRH i PQFLYf _ M@Z q y i HdM@fhHgOHgfhHdcUcRH X _ OQo.fLNcUo _ MO@LNM@ZFLYZUcUPzLNc q ?oIHgOHl q M.LNFpoIZzLNcUo _ MHhZUcy i H$F ^`_ PROQPRHOHf _ Mv"HdPU"HdM@fhHl _i PFIH i JZRHdPzLGO q ?PzLO q F ^`_ PROQPRHOH-f _ M"HdPU"HdM@fhHl _i PTFLµ?LNPUoL a FIHlPUo X LNFIHPRHhZUczLNMvcoIOHdMvcUoIy i HºBfhHdF i o m¼_i PUMo5l.LNPF ^ LNllP _ JQo X LNcUo _ M2r ºµOH i Jfzt.L X l@Z
+¾*½ 3É*Ä É $ÀxÆ?É QÀ 7ÃÁ$ËË Ej@MTOH a oIHdMf _ X lPRHdM@OQPRH$f _ XX HdMcAHhZUc6PRHdM@O i H5l _ ZRZRo a FIH+F ^`q Fpo X opM.LNcUo _ MF _ fLNFIH5O i i JgOQoIZRfdPRHdc
zσh
Nf _ M@ZUoIO q P _ M@Z+FLklPRH X oIdPRH q y i LNcUo _ MO@LNM@ZFL mn_ P X i FLNcUo _ MF _ fLNFIHg~A \ -~¼fhHdFpFIH m LNoIZzLNMvc+opMcRHdPU"HdMopPFIH0l _ FpuwM X H
qK?HdFpFIHK·*fdokl HdP X HdcO ^ HKJwlPUo X HdP
zσh |T
i MoIy i H X HdMcHdM mn_ M@fdcUo _ M OHµFL>?LNPUoL a FIHlPUo X LNFIHZUo+FIH i J
φσT,F
M@HO q lxHdM@Ol.LZOHhZgLNFIH i PRZTlPUoIZRHhZgl.LNPzσh
O@LNM@ZgFIHhZ X LNopFpFIHhZT _ oIZUopM@HhZS5MHKJL X opM.LNMcY~AYJ"K _ M _ opcky i HgfhHdFLHhZUcl _ ZzZUo a FIHgHdM0fRt _ oIZUoIZzZzLNMvc
ησ = 0.~J W
rgLNM@Z+fhHhZ+f _ M@OQopcUo _ M@Z _ Mf _ M@ZRczLNcRHHdM q ?LNF i LNMvc+FIH X H X a PRHjOHOQP _ opcRHOH~AY"l _i Pyh = (yσ
h , 0)
y i^`_ M _a cUoIHdMc∫
Ω(zσ
h +∇hzuh)·yσ
h −∑
F∈F∂h
∫
F
(n·yσh)zu
h −∑
F∈F h
∫
F
nF ·yσh[[z
uh ]] = 0.
~J?»"
¬ _ opcF ∈ Fh
.°GM0opMcUP _ O i opckF ^`_ l q PzLNcRH i PeOHPRHdFId"H X HdMvc
rF : [L2(F )]d 3 v 7−→ rF (v) ∈ Σh,~JNÌ
_¹rF (v)
HhZUccRHdFy i H
∀τh ∈ Σh,
∫
ΩrF (v)·τh =
∫
F
v·τh.~JY"
~7°GMBPzLNll HdFpFIHTy i Hel _i PF ∈ F∂
h
_ MLYl.LNPjf _ M"HdMvcUo _ Mτh = τh
ê°GMBM _ cRHdPzLy i HFIHZ i ll _ PUcOH
rFHhZUc
∆F
f ^ HhZUc²·3º¯·*OQopPRHFIHhZ¡OH i J X LNopFpFIHhZl.LNPUczLN"HLNMvcFl _i P
F ∈ F hHdcFLµZRH i FIH X LNopFpFIH
O _ McFHhZUc i M@H m LfhHl _i P
F ∈ F∂h
s i oIZ _ MBopMcUP _ O i opcGF ^`_ l q PzLNcRH i P
Rh : Uh 3 uh 7−→ Rh(uh) =∑
F∈F h
rF (nF [[uh]]) +∑
F∈F∂h
rF (nuh).~JYA"
°GM'M _ cRHdPzL y i HZUogFIHZ i ll _ PUc0OHuh
HhZRcBopM@fdF i Z0O@LNM@Z i M@H X LNopFpFIHT ∈ Th
FIHZ i ll _ PUc0OHRh(uh)
HhZUcef _ M@ZUcUopc i@q OH¡FLP qdi Mo _ M>OHTHdceOHhZ X LNopFpFIHhZOH
Thl.LNPUczLN"HLNMvc i M@H m LfhHLh"Hhf
T
±[
]$^ opMvc q P ® cO ^ opMcUP _ O i opPRHBF ^`_ l q PzLNcRH i PRh
HhZRcYy i^ opFlxHdP X HdcOHP qhq fdPUopPRHF ^`q y i LNcUo _ M'~J?»"¡Z _i ZYFLmn_ P X HTf _ X l.LfdcRH
zσh = −∇hz
uh +Rh(zu
h).~J \
°GMBf _ M@ZUczLNcRHy i HkFIH i JOQoIZzfdPRHdcHhZRc q vLNFL i ?PzLOQoIHdMvcF _ fLNFxOHeFLLNPUoL a FIHelPUo X LNFIHgf _ PUPUop q l.LNPi MPRHdFId"H X HdMvcOHhZjZzL i cRZjOHFL?LNPUoL a FIHglPUo X LNFIHTZ i PFIHhZ m LfhHhZ³
S+?LNF i@_ M@Z X LNopMvcRHdM.LNMvc¡FIH X H X a PRHOHOQP _ opcRHOHB~AYA"Gl _i Pyh = (0, yu
h)HdMuZ ia ZUcUopc i LNMc
F ^ HKJQlPRHhZRZUo _ M OHzσh
O _ MM q HTl.LNP¡~J \ KH=3F6oIHdMc
ah(zh, (0, yuh)) =
∫
Ω[zu
hyuh − z
σh ·∇hy
uh] +
∑
F∈F∂h
∫
F
(n·zσh )yu
h
+∑
F∈F h
∫
F
nF ·zσh[[y
uh]] +
∑
F∈Fh
∫
F
ηu[[zuh ]][[yu
h ]]
=
∫
Ω[zu
hyuh + (∇hz
uh −Rh(zu
h))·∇hyuh] +
∑
F∈F∂h
∫
F
(n·zσh)yu
h
+∑
F∈F h
∫
F
nF ·zσh[[y
uh]] +
∑
F∈Fh
∫
F
ηu[[zuh ]][[yu
h ]].
°GP
∑
F∈F∂h
∫
F
(n·zσh )yu
h +∑
F∈F h
∫
F
nF ·zσh[[y
uh ]] =
∑
F∈F∂h
∫
ΩrF (nyu
h)·zσh +
∑
F∈F h
∫
ΩrF (nF [[yu
h]])·zσh
=
∫
ΩRF (yu
h)·zσh .
sLNPGZ i opcRH?@ZUo _ M0l _ ZRH
aLDGh (zu
h , yuh) =
∫
Ω[zu
hyuh +(∇hz
uh −Rh(zu
h))·(∇hyuh −Rh(yu
h))] +∑
F∈Fh
∫
F
ηu[[zuh ]][[yu
h ]],~JYJ"
_ M9L X _ MvcUP q FIHTP q Z i FpczLNckZ i op?LNMvc # ## BD P - " 0 <"( (
ησ = 08%6&+" * zh = (zσ
h , zuh)
(+" " &% ¼ - 5,%) M (? " $;( ~AQ±³>"O( " ( %U( M ( - O"
zuh
(+" " &% ¼ - ;25,% M ( $) - "K " 9(N
?-t@HdPRfzt@HdPzuh ∈ Uh
cRHdFy i HaLDG
h (zuh , y
uh) =
∫
Ωfyu
h
∀yu
h ∈ Uh,~JY["
( "Kzσh
(+", K¯&% " (,%6 - ~J \ 8] L X q cUt _ OHkOHr'LNopM@ZUoxf _ M@ZUcUP i opcRHGl _ PUcRHkFIHjM _ X OH ] rT OHkF ^ LfdP _ Mvu X HgLNM?FLNoIZ N]_ fLNF
reoIZRf _ McUopM i@_i ZggLNFIHdPU|wopM ] HgZRuwZUcR X HFpopM q LNopPRH¡LZRZ _ fdo q ºB~JY["HhZRcOHczLNopFpFIHCd
p+dN !#"Z _ opc i M
vLNopM O ^ i M m LfdcRH i P(d+ 1)
l.LNPjPzLNll _ PUckL i lP _a FI X H~AQ±³K
W?[
+¾ 5À NÃÁ$Ã Ã@ÂÃ QÃEj@M>O ^`_a cRHdMopPTOHhZHhZUcUo X LNcUo _ M@ZgO ^ HdPUPRH i PTZzLNcUoIZ m LNoIZzLNMcRHhZGl _i PeFL X q cUt _ OH ] rTxFIH¡f _ H$·
fdoIHdMcGOHl q M.LNFpoIZzLNcUo _ Mηu
O _ opc ® cUPRHTfzt _ oIZUoOHTZ _ PUcRHgy i H
∀F ∈ Fh, ηu =α
hF,
~[ [
_¹α > 0
HhZUc i M f _ H$fdoIHdMcGfRt _ oIZRol.LNPjFIHM i X q PUoIfdoIHdM>HdchF
FIHTOQoL X dcUPRHTOHgFL m LfhHF
s _i PkFL X q cUt _ OH ] rTFIHhZM _ P X HhZjOHZUczL a opFpopc q HdckOHgf _ McUopM i opc q Z _ MvcjFIHhZkZ i opLNMcRHhZ¸
|||z|||2 = ‖zσ‖2L2 + ‖zu‖2L2 + |zu|2M + |zu|2Ju +∑
T∈Th
‖∇zu‖2T ,
|||z|||2∗ = |||z|||2 +∑
T∈Th
[h−2T ‖z
u‖2T + h−1T ‖z
u‖2∂T + hT ‖zσ‖2∂T ],
~[Q±³
L"Hhf
|zu|2M =∑
F∈F∂h
|zu|2M,F , |zu|2M,F =
∫
F
αhF
(zu)2,~[ W
|zu|2Ju =∑
F∈F h
|zu|2Ju,F , |zu|2Ju,F =
∫
F
αhF
[[zu]]2.~[?»"
] LlPRH i "HOHhZµP q Z i FpczLNcRZ0fdo·*OHhZRZ _i ZBHhZRcFLNoIZzZ q H>HdM HKJQHdPRfdoIfhH?S+FpFIHZzH mn_ M@OH>Z i PµFIHhZ X ® X HhZcRHhfztMoIy i HhZy i HTlP q f q OH XX HdMc ] LZRH i FIH¡OQo q PRHdM@fhH¡HhZUcGy i HgF ^ tul _ cUt@hZRHYOHTP q i FLNPUopc q Z i PjFLZ _ F i cUo _ MHKJLfdcRHTlxH i c ® cUPRHTL¯LNo a FpoIH¡º
z ∈ [Hp(Th)]d ×Hp+1(Th)
~ ¬ czL a opFpopc q P -
∀zh ∈Wh, |||zh||| . supyh∈Wh\0
aLDGh (zh, yh)
|||yh|||.
~[NÌ
' ~/? _ McUopM i opc q P -
∀ζ ∈ Z, ∀yh ∈Wh, aLDGh (ζ, yh) . |||ζ|||∗|||yh|||.
~[Y" "!$# &% ~/? _ Mv"HdPU"HdM@fhH³ P -
|||z − zh||| . hp‖z‖[Hp(Th)]d×Hp+1(Th).~[YA"
] HgczL a FIHL i »Q W P q Z i X HgFIHhZ _ PROQPRHhZGOHTf _ Mv"HdPU"HdM@fhH _a cRHdM i ZGl _i PFIHhZkHdPUPRH i PRZkHdM i J0HdcjHdM?LNPUoL a FIHYlPUo X LNFIHB~¼f _ X l.LNPRHdPgL"HhfYFIHczL a FIHL i W Ù»"K6°GM M _ cRHdPzLy i HFIHhZ _ PROQPRHhZOH¡f _ M"HdPU"HdM@fhHZ _ MvcTZ ia _ lcUo X L i J2~¼O ^`_ PROQPRHB±³kl _i PFLM _ P X H
L2 Hdcy i^ opFIZgZ _ Mc _ lcUo X L i J>l _i PgFIH¡?PzLOQoIHdMcOH FLLNPUoL a FIH lPUo X LNFIH? ]_ PRZRy i H
p ≥ 2 i M@HHhZUcUo X LNcUo _ MO ^ HdPUPRH i PBO ^`_ PROQPRH
hp−1 l H i c ® cUPRHq czL a FpoIHGl _i P5FLgOQop"HdPU"HdM@fhHeO i i JHdM i cUopFpoIZzLNMvc i M@HopM q vLNFpopc q opMv"HdPRZRHGHdc5F ^ HhZUcUo X LNcUo _ MOH
‖zσ−zσh‖Ω
S+MQ@M6F _ PRZzy i^`_ MOQoIZUl _ ZRHO ^ i M@HlP _ lPUo q c q OHP q i FLNPUopc q HdFpFpoplcUoIy i H? _ M9l H i cTL X q Fpo _ PRHdPF ^ HhZRcUo X LNcUo _ M<OHF ^ HdPUPRH i P
‖zu − zuh‖Ω
l.LNPOHhZgcRHhfRtMoIy i HhZYOHO i LNFpopc q Hdc _a cRHdMopPTl _i PfhHdFpFIHK·*fdoi M@Hf _ Mv"HdPU"HdM@fhH _ lcUo X LNFIHO ^`_ PROQPRH
(p+ 1)
*¡ - "1%p M .2( E * * 4% (+" <"7"K5,%U(K2(K$ "K' Σh = [Gp−1h ]d * $ G (,", N ?')( - (+0 ? M Y ¼ - &% - M n&%U(R G .- (X&) - S,K( )(# 1%U( Q&1U+0 & #N$7(,%4%U(
( %pBN&n25,%U(>K M &%U(8 EW(,"R(,", ¼ M 0 ¼! - "1WG (,,;( $h 2(+"7" " - (V"G ( - ! <( - <" M Z)(+" 8EG ?¯ - 7X(F(+", ( %)G (+", ¼ M Y ¼! - 2(
‖zσ − zσh‖Ω
(K/( - $) + ¼ M &%U(8 9(4S? " * .- (M 2(#E"* ¼4% "d - (+">(X&)+"Z &% - M ¼ ,;( - 4" %U( ( %¯&K¼25%6(K M &%U(H( !)( M - ""K M %U(N>)XY² MIM (,8
W ±
‖zσ − zσh‖Ω ‖zu − zu
h‖Ω ‖∇h(zu − zuh)‖Ω
O(hp) O(hp) O(hp)
W °GPROQPRHhZYOHf _ M"HdPU"HdM@fhHl _i PTF ^ LNllP _ Jwo X LNcUo _ M;r º i MfRt.L X l ¶ l q M.LNFpoIZLNcUo _ M<OH[[zu
h ]]Lh"Hhf i MDf _ H$fdoIHdMvc
ηu = O( 1h)Hdcgl.LZOHl q M.LNFpoIZzLNcUo _ M<OH
nF ·[[zσh ]] ¶ FLµZ _ F i cUo _ MDHKJ@LfdcRH
HhZRckZ i ll _ Z q H ® cUPRHTO@LNM@Z[Hp(Th)]d ×Hp+1(Th)
+¾ 9È&Á$ËwÆ"É Á-Ë ¯ÆÃ É MO q ZzLLNMczLN"HjOHFL X q cUt _ OH ] rTÅHhZUc+y i HO@LNM@ZFL X LNcUPUoIfhHO i ZUuwZRcR X H$FpopM q LNopPRHjLZRZ _ fdo q
FIHhZOHd?P q ZOHjFpo a HdPUc q Z i P i M@H X LNopFpFIHeZ _ Mcf _i lF q ZºTfhH i JOHhZ X LNopFpFIHhZ _ oIZUopM@HhZ X LNoIZ q vLNFIH X HdMcºfhH i JOHhZ X LNopFpFIHhZY _ oIZUopM@HhZYOHfhHhZYOHdPUMoIdPRHhZ ? _ M@ZUoIO q P _ M@Z¡F ^ HKJwlPRHhZRZRo _ M ~JYJ"gl _i P
aLDGh
HdcHKJ@L X opM _ M@ZFIHeZUcRHdM@fdopFLZRZ _ fdo q ºYfzt.Lf i M0OHhZ-cRHdP X HhZjLNlPRhZjL _ opPjO q "HdF _ ll q FIHecRHdP X Hf _ MvcRHdM.LNMcFIHhZ?PzLOQoIHdMcRZgF _ fL i J HdcFIHhZGPRHdFId"H X HdMcRZ°GM>opFpF i ZUcUPRHfdo·*OHhZRZ i Zel _i Pfzt.Ly i HYcRHdP X HFIHYZUcRHdM@fdopFLZzZ _ fdo q ¶ fhHdF i o·*fdo5Z ^`_a cUoIHdMcTHdMf _ M@ZUoIO q PzLNMvc i M@H mn_ M@fdcUo _ McRHhZUc
yuh
O _ MvcFIHZ i ll _ PUcHhZRcGF _ fLNFpoIZ qZ i PeFL X LNopFpFIHL i fhHdMcUPRHO i X LfdP _ · q F q X HdMc¡HdcTHdMfRt@HdPRfzt.LNMvc¡c _i ZFIHhZcUPUoLNM?FIHhZgl _i PFIHhZRy i HdFIZFIHhZOHd?P q ZOHGFpo a HdPUc q F _ fL i JLNll _ PUcRHdMc i M@HGf _ McUPUo ai cUo _ M0M _ MQ·7M i FpFIH ¶ fhH i Jw·*fdo6Z _ MvcPRHdlP q ZRHdMc q ZHdMcRHdopMvcRHT?PUoIZ q H?
∫
Ω[zu
hyuh +∇hz
uh ·∇hy
uh]
∫
Ω[Rh(zu
h)·∇hyuh +∇hz
uh ·Rh(yu
h)]
+∑
F∈Fh
∫
F
αhF
[[zuh ]][[yu
h ]]
∫
ΩRh(zu
h)·Rh(yuh)
°GM f _ M@ZUczLNcRHTO _ M@fTy i HFIHTZRH i F6cRHdP X HgPRHhZRl _ M@ZzL a FIHOHF ^ HKJwcRHdM@ZUo _ M O i ZUcRHdM@fdopFHhZUcFIHcRHdP X H
ρh(zuh , y
uh) :=
∫
ΩRh(zu
h)·Rh(yuh).
~[ \
?HdF i o·*fdo m LNopckl.LZRZRHdPjFIHgZUcRHdM@fdopFOHg̺B± [ cUPUoLNM?FIHhZGHdMOQo X HdM@ZUo _ M W HdcjOH ºB±»c q cUPzLhOQPRHhZkHdMOQo X HdM@ZRo _ M9»Qr ^`_¹ F ^ oIO q HTO ^`q Fpo X opM@HdPefhHcRHdP X HO@LNM@ZFL mn_ P X H a opFpopM q LNopPRH?x°GMl _ ZRH
aSIPGh (zu
h , yuh) = aLDG
h (zuh , y
uh)− ρh(zu
h , yuh).
~[YJ"S5M _a ZRHdPU?LNMvcGy i H
∫
ΩRh(zu
h)·∇hyuh =
∑
F∈Fh
∫
F
nF ·∇hyuh[[z
uh ]],
W?W
opFoIHdMc
aSIPGh (zu
h , yuh) =
∫
Ω[zu
hyuh +∇hz
uh ·∇hy
uh] +
∑
F∈Fh
∫
F
αhF
[[zuh ]][[yu
h]]
−∑
F∈Fh
∫
F
[nF ·∇hyuh[[z
uh ]] + nF ·∇hz
uh[[y
uh ]]].
~[Y["
°GMPRHhf _ MM.LIcFL mn_ P X H a opFpopM q LNopPRHgLZRZ _ fdo q HeºgFL X q cUt _ OH ¬ =3sOHkF ^ LfdP _ Mvu X HgLNM?FLNoIZ ¬ u X ·X HdcUPUoIf=*McRHdPUo _ PksAHdM.LNFpc*u gLNFIHdPU|wopM
°GM0f _ M@ZUoIOdPRHgFIHglP _a FI X H¡OQoIZRfdPRHdc
?-t@HdPRfRt@HdPzuh ∈ Uh
cRHdFy i HaSIPG
h (zuh , y
uh) =
∫
Ωfyu
h
∀yu
h ∈ Uh.~²± [?[
]$^ LNM.LNFpuwZRHgOHf _ M"HdPU"HdM@fhHTOHfhHeZzfRt q X LYM q fhHhZRZUopcRH?Qf _ XX Hc _ibU_i PRZQO ^`q czL a FpopPjOHhZlP _ lPUo q c q ZOHZRczL a opFpopc q Hdc5OHf _ MvcUopM i opc q l _i P+FL m¼_ P X H a opFpopM q LNopPRH
aSIPGh
?s _i P$fhHdFLQ _ M i cUopFpoIZRHFIHFpoIHdMHdMcUPRHFIHhZ mn_ P X HhZ a opFpopM q LNopPRHhZ
aSIPGh
HdcaLDG
h
~ _ opP~[YJ"UGHdcGFIHhZeP q Z i FpczLNcRZgOHhZFIH XX HhZ±»Hdc±dÌl _i PFL m¼_ P X H a opFpopM q LNopPRH
aLDGh
Q P - 2+"6X - (@&
N∂
%6( - M 5,;(I( Sd2$7(+"VWG .- ( M '%4%U(T ∈ Th
8 P - / uh ∈ Uh *
‖Rh(uh)‖2Ω ≤ N∂
∑
F∈Fh
‖rF (nF [[uh]])‖2Ω.~²± [ ±³
: ;( <(8° a ZRHdPU"HdPky i H
‖Rh(uh)‖2Ω =∑
T∈Th
∫
T
(
∑
F⊂∂T
rF (nF [[uh]])
)2
≤∑
T∈Th
∫
T
N∂
∑
F⊂∂T
|rF (nF [[uh]])|2
= N∂
∑
F∈Fh
‖rF (nF [[uh]])‖2Ω.
D P - # V / uh ∈ Uh *
∑
F∈Fh
‖rF (nF [[uh]])‖2Ω . |uh|2M + |uh|
2Ju .
~²± [?W
: ;( <(8° a ZRHdPU"HdPky i Hl.LNPkO q @MopcUo _ M@l _i Pjc _i cF ∈ Fh
Hdcl _i Pjc _i cv ∈ [L2(F )]d
∫
Ω|rF (v)|2 =
∫
F
vrF (v),
l i oIZ i cUopFpoIZRHdP i M@HopM q vLNFpopc q OHcUPzLfhHl _i Pjf _ M@fdF i PRHgy i H‖rF (nF [[uh]])‖Ω . |uh|Ju,F
ZUoF ∈ F hHdc
‖rF (nF [[uh]])‖Ω . |uh|M,F
ZUoF ∈ F∂
h
=*F6HdM0P q Z i FpcRHgFIHP q Z i FpczLNckOHTf _ M"HdPU"HdM@fhHZ i op?LNMvc
W »
"!$# &% % ( " 9(α0 > 0
29( - " - T(V%B¯&%U( R(V%$) - " 7 - 9( - 9(,K<( - - @" - "%)G - /X"&% 4 91(I !²2$7(~²± \ V 9(,%4%U( (I"
α ≥ α0 *%6(T25%) M (R " $)( j~²± [?[ 1(+", @5K/( - <" ( " -
.- ( "&% ¼ - "dY ¼'"'Sh! %U(+"$) - $ % " - " ;& M (Y±H8: ;( <(8ePfhH L i J´FIH XX HhZ9± \ Hdc9±JQ _ M; q PUo.H y i H Z _i ZF ^ tul _ cUt@hZRH
α ≥ α0+FL mn_ P X H
a opFpopM q LNopPRHaSIPG
h
q PUo.HgFLYlP _ lPUo q c q OHL2 ·*f _ HdPRfdopopc q Z i opLNMcRH¸Ql _i Pkc _i c uh ∈ Uh
aSIPGh (uh, uh) & ‖∇huh‖
2L2 + ‖uh‖
2L2 + |uh|
2M + |uh|
2Ju .
°GM f _ M@fdF i cGLNoIZ q X HdMvc°GM2PRHdcUoIHdM@OQPzLDy i HBFL X q cUt _ OH ¬ =ãs- l HdP X Hdc i M@HBP q O i fdcUo _ M Z ia ZUczLNMvcUoIHdFpFIH O i f _ cOHhZ
fLNFIf i FIZkl.LNPPzLNll _ PUceºYFL X q cUt _ OH ] rg4HdMBP q O i oIZzLNMvcGFLYczLNopFpFIHgO i ZUcRHdM@fdopF6c _i ckHdM0f _ M@O i oIZzLNMcL i J X ® X HhZHhZRcUo X LNcUo _ M@ZO ^ HdPUPRH i P ¶ FIHµlPUoJ´º9l.Lhu"HdPHhZRcy i HµFIHBf _ H$fdoIHdMvcopMvcRHdPU"HdM.LNMvcO@LNM@ZFL9l q M.LNFpoIZzLNcUo _ M;OHhZZL i cRZOHµFL9LNPUoL a FIHµlPUo X LNFIHBO _ opc ® cUPRHBZ i ZL XX HdMvc?PzLNM@O6+ZL?LNFIH i PX opMo X LNFIHO q l HdM@O@LNMvcOHGFLg?LNFIH i POHkFLf _ M@ZUczLNMcRHeopMvcRHdPU"HdM.LNMvcjO@LNM@Z-F ^ opM q vLNFpopc q OHkcUPzLfhH~²± \ K _i cRH m¼_ oIZ³opF5HhZUcgl _ ZRZUo a FIHOHZ ^ L¯ PzLNM@fRtopP¡OHfhHdcUcRHO q lxHdM@O@LNM@fhHHdMDf _ M@ZUoIO q PzLNMvc i M@HLNPUoLNMcRHOHkFL X q cUt _ OH ¬ =ãs-OQopcRH X q cUt _ OHZ 9s ¬ ~¼O i M _ X OHGZRHhZ-L i cRH i PRZv _ opP$FL a o a Fpo _ ?PzLNltoIH³K°GMl _ ZRH
aBRMPSh (zu
h , yuh) =
∫
Ω[zu
hyuh +∇hz
uh ·∇hy
uh] +
∑
F∈Fh
∫
ΩαrF (nF [[zu
h ]])·rF (nF [[yuh]])
−∑
F∈Fh
∫
F
[nF ·∇hyuh[[z
uh ]] + nF ·∇hz
uh[[y
uh ]]],
~²± [ »"
Hdc _ M0f _ M@ZUoIOdPRHTFIHlP _a FI X H¡OQoIZRfdPRHdc
?-t@HdPRfzt@HdPzuh ∈ Uh
cRHdFy i HaBRMPS
h (zuh , y
uh) =
∫
Ωfyu
h
∀yu
h ∈ Uh.~²± [ Ì
S5M i cUopFpoIZzLNMvcGFIHFIH XX H± \ _ M0 q PUo.HZzLNM@Zl HdopM@HFIHgP q Z i FpczLNckZ i op?LNMvc "!$# &% B D "1%;G + Y ," (
α > N∂ *%6(>;25,% M (K'"$)( ~²± [ Ì (,", 5d( - &" K( " -
.- ( "&% ¼ - "dY ¼'"'Sh! %U(+"$) - $ % " - " ;& M (Y±H8? _ M@fdF i@_ M@ZgfhHdcUcRHZRHhfdcUo _ M>l.LNP i M@HYf _ X l.LNPzLNoIZ _ MDOHhZ i J9opMvcRHdPU"HdM.LNMcO@LNM@ZFIHhZlP _a FI X HhZ
F _ fL i JLZRZ _ fdo q ZGL i J X q cUt _ OHhZ ] rT4Hdc ¬ =3s@s _i PkFL X q cUt _ OH ] rT.FIHhZ i J0ZRHTO q O i oIZRHdMcOH~AYJ"Hdc~AY["HdM0cRHdM.LNMckf _ X lcRHO iµm LNopcky i H
ησ = 0Hdcηu = α
h
H=3F oIHdMc
φσT,F (zh) =
nT zuh, F ∈ F h,
0, F ∈ F∂h ,
~²± [ "
Hdc
φuT,F (zh) =
nT ·(zσh+ α
hnF [[zu
h ]]), F ∈ F h,n·(zσ
h + αhnzu
h), F ∈ F∂h .
~²± [ A"
s _i PkFL X q cUt _ OH ¬ =ãs- _ M _a ZRHdPU"Hgl _i P i M@H X LNopFpFIHT ∈ Th
Hdc i MBl _ FpuwM X Hr ∈ Pp
ρh(uh, r1T ) =
∫
ΩRh(uh)·Rh(r1T ) =
∑
F⊂∂T
∫
ΩRh(uh)·rF (nT ) =
∑
F⊂∂T
∫
F
nT ·Rh(uh),
W Ì
O ^`_¹ F ^`_ MO q O i opcGHdM0Z _i ZRcUPzLhuvLNMvcGFLf _ MvcUPUo ai cUo _ M>OHgfhHgcRHdP X HL i J i J0OH ] rT4HdckHdMBcRHdM.LNMcf _ X lcRH¡OHFLPRHdFLNcUo _ M<~J \ y i HFIH i Jµl _i PkFL X q cUt _ OH ¬ =ãs- HhZUcjO _ MM q l.LNP
φuT,F (zh) =
nT ·(−∇hzuh+ α
hnF [[zu
h ]]), F ∈ F h,n·(−∇hz
uh + α
hnzu
h), F ∈ F∂h .
~²± ["\
?HhZ i JµZ _ Mcf _ XX H _ M0Z ^ uµLNcUcRHdM@O@LNopc.f _ M@ZRHdPU?LNcUo m ZkHdcjf _ M@ZUoIZUczLNMvcRZGLh"HhfTfhH i JBOHFLZ _ F i cUo _ MHKJ@LfdcRH?
§¡« .¢¡§Y¥ª+¢ ¡¢ ªA¤ ¡¦gY¢ ¡¢
]5^`_aQb Hdc0OH9fhHdcUcRHZRHhfdcUo _ MHhZUc0O ^ HhZRy i oIZRZRHdP a PUoId"H X HdMvc0FIHhZ0?PzLNM@OHhZBFpop?M@HhZBO ^ i M@HcUt qh_ PUoIHi Mo q HeOHhZ X q cUt _ OHhZOHerH X a PzLZRZzLNMcºTFL mn_ oIZ5F ^ LNllP _ JQo X LNcUo _ MBO ^ S5rs5ZtvuwlxHdP a _ FpoIy i HhZ$HdcHdFpFpoplcUoIy i HhZ ] LM _ cUo _ MfdF q HhZRcefhHdFpFIHOHR" <", M (1( K/(;&K!$2" ¶ opFAZ ^ LN?opceOHYZUuQZUcR X HhZO ^ S5rs5ZO ^`_ PROQPRH i M q y i opl q ZµO ^ i M@H9lP _ lPUo q c q OH" M !K(HdcµO ^ i M@H>lP _ lPUo q c q OH <"! ¼4 99~ _i OH? "+" QY ¼4 9zK
¬ _ opc i MµHdMvcUoIHdPm ≥ 1
~m
O q ZRop?M@HGFIHGM _ X a PRHO ^ opM@f _ MM i HhZjO@LNM@ZFIHeZRuwZUcR X HGOHPUoIHhOQPUoIfRt@ZK¬ _ opc
Ωi M _i "HdPUc a _ PUM q "f _ MM@HKJQHeHdc5Fpopl@ZRfztopcUhoIHdMBOH
Rd MZRuwZUcR X HkOHkPUoIHhOQPUoIfzt@ZZzH m¼_ P X i FIH
ºF ^ LNoIOHeOHjFLTO _ MM q HeOH(d+1)
fRt.L X l@Zºg?LNFIH i PRZO@LNM@ZR
m,mKHdcAk1≤k≤d
Q°GMZ i ll _ ZRHy i HTc _i ZefhHhZkfzt.L X l@ZZ _ MvcO@LNM@Z
[L∞(Ω)]m,mHdcGy i H ∑d
k=1 ∂kAk ∈ [L∞(Ω)]m,m
~∂k
O q ZUop?M@HFL O q PUop q HµL i ZRHdM@Z¡OHhZ¡OQoIZUcUPUo ai cUo _ M@ZO@LNM@ZYFL
k·7oI X HOQopPRHhfdcUo _ MZRl.LNcUoLNFIH³K ] HhZ¡lP _ lPUo q c q ZOH
ZRu X q cUPUoIHTHdcGOHl _ ZUopcUopwopc q ~ _i OHTOQoIZzZUopl.LNcUopopc q Z ^ HKJwlPUo X HdMckZ _i ZFL m¼_ P X HgZ i op?LNMvcRH¸
∀k ∈ 1, . . . , d, Ak = (Ak)Tl l@O@LNM@Z
Ω,~
∃µ0 > 0, K +KT −
d∑
k=1
∂kAk ≥ 2µ0Im
l l@O@LNM@ZΩ,
~
_¹Im
O q ZUop?M@HFL X LNcUPUoIfhHµoIOHdMvcUopc q O@LNM@ZR
m,msLNPFL Z i opcRH?FIHhZYopM q vLNFpopc q ZHdMcUPRH X LNcUPUoIfhHhZ
~¼ZRu X q cUPUoIy i HhZ$O@LNM@ZR
m,mZ _ Mvcº¡HdMvcRHdM@OQPRHgL i ZzHdM@Z-OHhZ mn_ P X HhZy i LOQPzLNcUoIy i HhZLZzZ _ fdo q HhZe¸"l _i P
GHdcH
O@LNM@ZR
m,mG ≥ H
ZUop?Mo.HTy i Hl _i Pc _i cξ ∈ R
m(Gξ, ξ)Rm ≥ (Hξ, ξ)Rm
°GMl _ ZRH
L = [L2(Ω)]my i H5F ^`_ M X i MopcO i lP _ O i opcAZzfLNFLNopPRH i Z i HdFwM _ c q
(·, ·)L
N°GM¡opMvcUP _ O i opcF ^`_ l q PzLNcRH i P
A : [C1(Ω)]m 3 z 7−→ Az =
d∑
k=1
Ak∂kz ∈ L.~²± [ J"
?Hdc _ l q PzLNcRH i PGlxH i c ® cUPRH q cRHdM@O i L i J mn_ M@fdcUo _ M@ZkOHLcRHdFpFIHhZjy i HFL m¼_ P X HFpopM q LNopPRH
[D(Ω)]m 3 φ 7−→ −
∫
Ω
d∑
k=1
zT∂k(Akφ) ∈ R
~²± [ ["
HhZRcf _ McUopM i HZ i PL ¶ F ^`_aQb Hdc Az O q ZUop?M@HLNF _ PRZFL mn_ M@fdcUo _ MOH
Ly i ol H i c ® cUPRHLZRZ _ fdo q HºfhHdc
mn_ P X HFpopM q LNopPRHl.LNPgFIH a oLNoIZgO i cUt qh_ PR X HOHPRHdlP q ZzHdMvczLNcUo _ M<OH1koIHhZUKP q fzt@Hdc°GMopMvcUP _ O i opcF ^ HhZRl.LfhHTO i ?PzLNlt@H
W = z ∈ L; Az ∈ L,~²±?± [
W
y i HDF ^`_ M X i Mopc>O ^ i M@HZUcUP i fdc i PRHDtopF a HdPUcUoIHdMM@Hl.LNPFIH a oLNoIZ9O i lP _ O i opc>ZRfLNFLNopPRH(z, y)L +
(Az,Ay)L
sLNPGf _ M@ZUcUP i fdcUo _ MA ∈ L(W ;L)
&°GMBopMvcUP _ O i opc q vLNFIH X HdMckF ^`_ l q PzLNcRH i P
A : [C1(Ω)]m 3 z 7−→ Az = −
d∑
k=1
∂k(Akz) ∈ L,
~²±?±?±³
O _ Mc _ M q cRHdM@OµFIHO _ X LNopM@HTºW
HdMBlP _ f q O@LNMcjf _ XX Hgfdo·*OHhZRZ i ZQS+MQ@M _ MBopMcUP _ O i opcF ^`_ l q PzL¯·cRH i P
D ∈ L(W ;W ′)HdM0l _ ZzLNMvc
∀(z, y) ∈W ×W, 〈Dz, y〉W ′,W = (Az, y)L − (z, Ay)L.~²±?± W
]$^`_ l q PzLNcRH i PDHhZUcAL i c _ ·3LO bU_ opMvcl.LNPf _ M@ZRcUP i fdcUo _ MNrH$lF i Z
DHhZUccRHdFy i H
[D(Ω)]m ⊂ HdP (D) ¶_ M9OQopcky i HD
HhZUc i M _ l q PzLNcRH i PeOH m P _ MvcUoIdPRH?&sALNPLNopFpFIH i PRZ@F _ PRZRy i HgFIHhZGfRt.L X l@ZAk
Z _ MckZ i ·ZL XX HdMvckP q i FpoIHdPRZ³ _ M0l _ ZRH
D =
d∑
k=1
nkAk : ∂Ω −→ R
m,m,~²±?±»"
_¹nk
HhZUc$FLk·7oI X HGf _ X l _ ZLNMvcRHOHjFLM _ P X LNFIHeHKJQc q PUoIH i PRH
n"Hdc _ M _a ZRHdPU"Hey i HjZUo
zHdcyZ _ Mc
Z i ZzL XX HdMvcGP q i FpoIHdPRZ
〈Dz, y〉W ′,W =
∫
∂ΩyTDz.
~²±?±dÌ
EGopM@ZUo7~²±?± W HhZUc i M@H m¼_ P X i FIHO ^ opMvc q ?PzLNcUo _ Ml.LNPl.LNPUcUoIHhZ _ cRHdPy i Hl.LNP+f _ M@ZUcUP i fdcUo _ MD = DT
s _i P
f ∈ L _ Mfzt@HdPRfRt@H
z ∈WcRHdF.y i H
Kz+Az = f_¹
K : L 3 z 7→ Kz ∈ LPRHdlP q ZRHdMvcRH
FIHcRHdP X HO ^`_ PROQPRH [ O@LNM@ZkFIHZUuwZRcR X HTOHPUoIHhOQPUoIfzt@ZxEj@MO ^`_a cRHdMopP i M0lP _a FI X H a oIHdMl _ Z q opFm L i cgf _ X lF q cRHdPTfhH¡ZRuwZUcR X HYO ^ S5rs ¬ l.LNPgOHhZf _ M@OQopcUo _ M@ZTL i J Fpo X opcRHhZ s _i PgfhHdFLQ _ MZ i ll _ ZRHy i^ opFHKJQoIZUcRH i M _ l q PzLNcRH i P
M ∈ L(W ;W ′)cRHdFy i H
∀z ∈W〈Mz, z〉W ′ ,W ≥ 0,
~
W = HdP (D −M) + HdP (D +M).~
° a ZzHdPU"HdP9y i HM
HhZUc i M _ l q PzLNcRH i P>OH m P _ MvcUoIdPRH<l i oIZRy i H ~ µo X lFpoIy i Hy i H HdP (D) = HdP (M)
rgLNM@ZFLYlF i l.LNPUcjOHhZkLNllFpoIfLNcUo _ M@ZF ^`_ l q PzLNcRH i PM
lxH i c ® cUPRHPRHdlP q ZRHdMc q Z i PFL m P _ MQ·cUoIdPRHel.LNP i MµfRt.L X lºLNFIH i PRZO@LNM@Z
Rm,m
f ^ HhZRc²·3º¯·*OQopPRHey i^ opF6HKJwoIZRcRHM
O@LNM@Z[L∞(∂Ω)]m,m
cRHdFy i H
〈Mz, y〉W ′,W =
∫
∂ΩyTMz.
~²±?±"
] HhZf _ M@OQopcUo _ M@ZjL i JFpo X opcRHhZy i HGF ^`_ MZ _i t.LNopcRHeo X l _ ZRHdPZ ^`q fdPUop"HdMvcZ _i Z-FL mn_ P X Hz ∈ HdP (M −
D)&°GM0O q @MopcGZ i P
W ×WFL mn_ P X H a opFpopM q LNopPRH
a(z, y) = (Kz, y)L + (Az, y)L + 12〈(M −D)z, y〉W ′,W ,
~²±?±A"
Hdc _ M0f _ M@ZUoIOdPRHTFIHlP _a FI X H¡y i of _ M@ZUoIZRcRHº
?-t@HdPRfRt@HdPz ∈W
cRHdFy i Ha(z, y) = (f, y)L
∀y ∈W.
~²±?± \
°GM9LYFIHgP q Z i FpczLNckZ i op?LNMvc
W A
"!$# &% B EW( 25%) M (~²±?± \ F(+", 5d( - <" 8 *V( % " * " - .- (K"&% ¼! - (+", " - " HdP (M −D)
( (,%4%6( "d0 ¼ "4Sd! Kz +Az = f
" - "L8
_ oIfdoOH i JBHKJQH X lFIHhZGOHgZUuwZRcR X HhZjOHgPUoIHhOQPUoIfRt@Z³ ]5^`q y i LNcUo _ M O ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ M´~n»"K.°GM L
m = 1 ¶ _ M0l _ ZRH
K = µHdc
Ak = βk,~²±?±J"
_¹βk
O q ZUop?M@HeFLk·7oI X Hgf _ X l _ ZzLNMcRHeO i fzt.L X l0O ^ LOQ"HhfdcUo _ M
β ] LTlP _ lPUo q c q OHZUu X q ·
cUPUoIH~ GHhZUcgcUPUopwoLNFIH X HdMvcYZzLNcUoIZ m LNopcRHl i oIZRy i HFIHhZgfzt.L X l@ZAk
Z _ McºLNFIH i PRZZRfLNFLNopPRHhZ] LlP _ lPUo q c q OHl _ ZUopcUopwopc q ~ eHhZUc i M@Hf _ M@Z q y i HdM@fhHo XX q OQoLNcRHOH0~²±³K ]5^ HhZUl.LfhHO i?PzLNlt@HHhZUc
W = v ∈ L2(Ω); β·∇v ∈ L2(Ω) ]5^`_ l q PzLNcRH i P
DHhZUccRHdFy i H
〈Dz, y〉W ′,W =
∫
∂Ω(β·n)zy,
~²±?±["
HdcgFLf _ M@OQopcUo _ M<L i J Fpo X opcRHhZu|∂Ω− = 0
l H i c ® cUPRHo X l _ Z q HHdMf _ M@ZUoIO q PzLNMcTF ^`_ l q PzLNcRH i PM
cRHdFy i H
〈Mz, y〉W ′,W =
∫
∂Ω|β·n|zy.
~²± W?[
°GM q PUo.H m LfdopFIH X HdMvcy i HFIHhZYtul _ cUt@hZRHhZB~ gHdcB~ Z _ Mc a oIHdM2ZzLNcUoIZ m LNopcRHhZS+MQ@M~²±?±dÌHdc¡~²±?±"Z _ MvckZzLNcUoIZ m LNopcRHhZkL"Hhf
D = (β·n)HdcM = |β·n|
]5^`q y i LNcUo _ M<OHOQo i ZUo _ MwvP q LfdcUo _ M O@LNM@ZZzL mn_ P X i FLNcUo _ M X oJwcRH~NÌK°GM<L
m = d + 1
r i0m LNopcGOHTF ^ LNllP _ fRt@HYl.LNP m¼_ P X i FLNcUo _ M X oJQcRH?.FIHhZGfRt.L X l@ZKHdcAk _ Mc i M@HYZUcUP i fdc i PRH
a F _ f?x°GMBl _ ZRH
K =
[
Id 0
0 1
] HdcAk =
[
0 ek
(ek)T 0
]
,~²± W ±³
_¹Id
O q ZUop?M@H$FL X LNcUPUoIfhH5oIOHdMcUopc q O@LNM@ZR
d,dHdcekFIHk·7oI X H+"HhfdcRH i POH$FL a LZRH5fLNM _ MoIy i H
OHR
d ] HhZ5lP _ lPUo q c q Zg~ 5HdcG~ 5Z _ MvcfdFLNopPRH X HdMvcZzLNcUoIZ m LNopcRHhZ ]5^ HhZRl.LfhHkO i ?PzLNlt@HeHhZUc
W = H(OQop
; Ω)×H1(Ω) ]5^`_ l q PzLNcRH i P
DHhZUccRHdFy i H
〈Dz, y〉W ′,W = 〈n·yσ, zu〉H
−12 ,H
12
+ 〈n·zσ, yu〉H
−12 ,H
12,
~²± W?W
HdcGFLf _ M@OQopcUo _ MDL i J0Fpo X opcRHhZeOHYreopPUoIfRtFIHdct _ X _ "dM@H¡lxH i c ® cUPRHo X l _ Z q H¡HdM9f _ M@ZUoIO q PzLNMcF ^`_ l q PzLNcRH i P
McRHdFy i H
〈Mz, y〉W ′,W = −〈n·yσ, zu〉H
−12 ,H
12
+ 〈n·zσ, yu〉H
−12 ,H
12.
~²± W »"
°GM q PUo.H m LfdopFIH X HdMvcy i HFIHhZYtul _ cUt@hZRHhZB~ gHdcB~ Z _ Mc a oIHdM2ZzLNcUoIZ m LNopcRHhZS+MQ@M~²±?±dÌHdc¡~²±?±"Z _ MvckZzLNcUoIZ m LNopcRHhZkL"Hhf
D =
[
0 n
nT 0
] HdcM =
[
0 −n
nT 0
]
.~²± W Ì
r ^ L i cUPRHhZAHKJQH X lFIHhZAOH$ZRuwZUcR X HhZOH$PUoIHhOQPUoIfzt@ZAZ _ MvcF ^`q y i LNcUo _ M¡O ^ LOQ"HhfdcUo _ MwOQo i ZUo _ MwvP q LfdcUo _ MFIHhZ q y i LNcUo _ M@ZGOHgFL X q fLNMoIy i HOHhZ X opFpoIH i JBf _ McUopM i ZHdcjFIHhZ q y i LNcUo _ M@ZGOH 9L¯J -HdFpFAHdM0P q ?o X HOQo i ZUo m
W"\
]5^ LNllP _ JQo X LNcUo _ M;r OHhZYZUuwZRcR X HhZOH@PUoIHhOQPUoIfRt@ZZRHf _ M@ZUcUP i opcYl.LNPFIH a oLNoIZ¡OHOH i J m L¯·X opFpFIHhZOHfRt.L X l@Zºµ?LNFIH i PRZ¡O@LNM@Z
Rm,m
] LlPRH X oIdPRH?MF F∂
h
HhZUcopM@OHKJ q Hl.LNP¡FIHhZ m LfhHhZOH m P _ MvcUoIdPRH ¶ Z _ M9P FIH¡HhZUceO ^ o X l _ ZzHdPGFIHhZf _ M@OQopcUo _ M@ZgL i JFpo X opcRHhZgL i ZzHdM@Z m LNo a FIH¡HdceOH¡f _ McUP N·FIHdPµfhHdPUczLNopM@HhZLNFIH i PRZBL i ax_ PROÅOH0FL<Z _ F i cUo _ M OQoIZRfdPRdcRH? ] LOH i JQoI X H m L X opFpFIH?
SF F h
HhZUcopM@OHKJ q Hl.LNPFIHhZ m LfhHhZopMvc q PUoIH i PRHhZ ¶ Z _ M>P FIHHhZUcTOHYl q M.LNFpoIZRHdPL i ZzHdM@ZgOHhZ X _ opM@OQPRHhZfLNPUP q ZFIHZL i cGºYcUPzLh"HdPRZkFIHhZ m LfhHhZjopMvc q PUoIH i PRHhZeOHgfhHdPUczLNopM@HhZGf _ X l _ ZzLNMvcRHhZGOHFLZ _ F i cUo _ M0OQoIZRfdPRdcRH?.S+MQ@MopFGHhZUc i cUopFIH0O ^`q cRHdM@OQPRH0FIHfRt.L X l
DL i J m LfhHhZopMc q PUoIH i PRHhZHdM2l _ ZzLNMcl _i PfRt.Ly i H
T ∈ Th
D|∂T =∑d
k=1 nT,kAk _¹
nT = (nT,1, . . . , nT,d)O q ZUop?M@HµFL M _ P X LNFIH i MopczLNopPRHBHKJQc q PUoIH i PRHº
T ] HfRt.L X l
DLNopM@ZUo-O q @MoHhZUc a o·7LNF i@q Z i P¡fzt.Ly i H
F ∈ F hLh"Hhf
D = 0rHlF i ZFIHhZ
?LNFIH i PRZglPUoIZRHhZTl.LNPD
q czLNMvc¡OHhZ X LNcUPUoIfhHhZ1" M !K (,"O ^`_ PROQPRHm _ Ml H i cTO q @MopPTFIHfzt.L X l
|D| ¶ fhHdF i o·*fdo&HhZUc i Mo·7?LNF i@q Z i P$fRt.Ly i H F ∈ F hv°GMZ i ll _ ZRHy i HFLgZ _ F i cUo _ MHKJLfdcRH
zOH~²±?± \
HhZRckZ i ZzL XX HdMcjP q i FpoIdPRHTl _i Pky i H¸z ∈ [Hp+1(Th)]m ¶Mz = Dz
l l.Z i PjfRt.Ly i HF ∈ F∂
h ¶[[zh]] = 0
l l.Z i PjfRt.Ly i HF ∈ F h
]$^ LNllP _ JQo X LNcUo _ MDgrOH
zZzHdPzLfRt@HdPRfzt q HO@LNM@Z
Wh := [Gph]m
f ^ HhZRc²·3º¯·*OQopPRHy i HF ^`_ Mf _ M@ZUoIOdPRHi M@HLNllP _ JQo X LNcUo _ M2grOHOHd?P q
pl _i PfRt.Ly i Hf _ X l _ ZzLNMvcRHOH
z°GMopMvcUP _ O i opcFL mn_ P X H
a opFpopM q LNopPRH
ah(z, y) =∑
T∈Th
[(Kz, y)L,T + (Az, y)L,T ] +∑
F∈F∂h
12 (MF z −Dz, y)L,F
−∑
F∈F h
2(Dz, y)L,F +∑
F∈F h
(SF [[z]], [[y]])L,F ,
~²± W "
FLBM _ czLNcUo _ M(·, ·)L,E
O q ZUop?M.LNMvc¡FIHlP _ O i opc¡ZRfLNFLNopPRH i Z i HdF$OH[L2(E)]m
_¹E
HhZUc i M@H X LNopFpFIH?i M@H m LfhH _iBi M@H i Mo _ M0OHecRHdFIZ _aQb HdcRZ ] HP FIHgO i cUP _ oIZUoI X HTcRHdP X HgO@LNM@ZFIH X H X a PRHTOHgOQP _ opcRHOH ~²± W "HhZUcTO ^ LZRZ i PRHdP i M@HlP _ lPUo q c q OH
L2 ·*f _ HdPRfdopopc q OQoIZRfdPRdcRHl _i PFL m¼_ P X H a opFpopM q LNopPRH ahc _i cHdMTlP q ZRHdPU?LNMvcAFLjf _ M@ZRoIZUczLNM@fhH$O i ZRfRt q X LQ°GM q PUo.HHdMHK HdcLh"Hhf-FIHhZtul _ cUt@hZRHhZAfdo·*OHhZRZ i Zl _i PjFLZ _ F i cUo _ M HKJLfdcRH
zy i H
ah(yh, yh) ≥ µ0‖yh‖2L +
∑
F∈F∂h
(MF yh, yh)L,F +∑
F∈F h
(SF [[yh]], [[yh]])L,F , ∀yh ∈Wh,~²± W A"
ah(z, yh) = (f, yh)L, ∀yh ∈Wh.~²± W"\
] HglP _a FI X HOQoIZRfdPRHdcGf _ M@ZUoIZUcRHTº
?-t@HdPRfRt@HdPzh ∈Wh
cRHdFy i Hah(zh, yh) = (f, yh)L
∀yh ∈Wh.
~²± W J"
°GM _a cUoIHdMvc¡OHhZHhZUcUo X LNcUo _ M@ZO ^ HdPUPRH i Pµ~¼y i LZUo·ã _ lcUo X LNFIHhZ¡ZUo+FIHhZTfzt.L X l@ZMF F∂
h
HdcSF F
hZLNcUoIZ m¼_ McjFIHhZlP _ lPUo q c q ZeZ i op?LNMvcRHhZT¸
HdP (M−D) ⊂ HdP (MF −D),~
(MF v, v)L,F ≥ 0,~
|(MF v −Dv, w)L,F | . |v|M,F ‖w‖L,F ,~
|(MF v +Dv, w)L,F | . ‖v‖L,F |w|M,F ,~
SF = (SF )THdc|D| . SF . Im,
~
W J
_¹|v|2M,F = (MF v, v)L,F
vS+MlP _ f q O@LNMvcf _ XX HjlP q f q OH XX HdMvcT~¼f _ M@OQopcUo _ MopM m ·*Z i lOQoIZRfdPRdcRH?f _ McUopM i opc q Lh"Hhf¡OH i JM _ P X HhZK _ M0lP _i "HgFIHgP q Z i FpczLNcGZ i opLNMc
"!$# &% B BD P - " 0 <"( (∀k ∈ 1, . . . , k
( ∀T ∈ Th * Ak|T ∈ [C0, 1
2 (T )]m,m8 * - "
$+(+"V$) - 4 ¼ - " *
‖z − zh‖L +
∑
T∈Th
hT ‖A(z − zh)‖2T
1
2
. hp+ 1
2 ‖z‖[Hp+1(Th)]m .~²± W ["
°GM _a cUoIHdMvcLNopM@ZUo i M@HHhZUcUo X LNcUo _ M<O ^ HdPUPRH i PYF q "dPRH X HdMvcZ ia _ lcUo X LNFIH ~¼O ^`_ PROQPRH 12
gO@LNM@ZTFLM _ P X H
‖ · ‖LHdc i M@HTHhZUcUo X LNcUo _ MO ^ HdPUPRH i P _ lcUo X LNFIHTO@LNM@ZjFLYM _ P X HTO i ?PzLNlt@H a PUoIZ q H?
s _i PkF ^`q y i LNcUo _ M9O ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ M _ MBl H i cjfRt _ oIZRopP
MF = β·nHdc
SF = α|β·n|~²±» [
l _i P i MBl.LNPzL X dcUPRHα > 0
~¼O _ MvcGFLYLNFIH i Pl H i c q "HdMvc i HdFpFIH X HdMcgfRt.LNM"HdPGOH m LfhHTº m LfhH³K.°GM q PUo.H m LfdopFIH X HdMcGy i HFIHhZlP _ lPUo q c q Z¡~ 7x~ jZ _ MvckZLNcUoIZ m LNopcRHhZ³
s _i PkF ^`q y i LNcUo _ M9OHgOQo i ZUo _ MwvP q LfdcUo _ M _ MBl H i ckfRt _ oIZUopP
MF =
[
0 −n
nT η1
] HdcSF =
[
η2n⊗n 0
0 η3
] ~²±»Q±³
l _i PjOHhZl.LNPzL X dcUPRHhZη1 > 0
η2 > 0
Hdcη3 > 0
~¼O _ McFLYLNFIH i Pl H i c q "HdMvc i HdFpFIH X HdMcefzt.LNM"HdPOH m LfhHTº m LfhH³K.°GM0 q PUo.H m LfdopFIH X HdMvcey i HFIHhZjlP _ lPUo q c q ZY~ 7x~ Z _ MckZzLNcUoIZ m LNopcRHhZ
] HhZYS5rs5ZHdFpFpoplcUoIy i HhZZ _i Z m¼_ P X H X oJQcRHf _ M@O i oIZRHdMcº0OHhZZUuQZUcR X HhZ¡OHPUoIHhOQPUoIfRt@ZLh"Hhfi M@HZUcUP i fdc i PRH a F _ fl.LNPUcUoIf i FpoIdPRH? °GM0O q f _ X l _ ZzHF ^ opM@f _ MM i H
zHdM
(zσ , zu)_¹
zσHhZUckºY?LNFIH i PRZ
O@LNM@ZR
mσzu
HhZUcGºY?LNFIH i PRZjO@LNM@ZR
muHdcm = mσ +mu
.°GMZ i ll _ ZRHgy i H
K =
[
Kσσ Kσu
Kuσ Kuu
]
, Ak =
[
0 Bk
(Bk)T Ck
]
,~²±» W
FIHfzt.L X lKσσ q czLNMvc i Mo mn_ P X q X HdMvcjO q @Mo l _ ZRopcUo m Q° a ZRHdPU"HdPy i HGFLf _ X l _ ZzLNMcRH
zσlxH i c ® cUPRH
q Fpo X opM q H¡O i ZUuQZUcR X HgOHgPUoIHhOQPUoIfzt@ZKz +Az = (fσ, fu)
l i oIZRy i H
zσ = (Kσσ)−1(fσ −Kσuzu −Bzu)~²±»?»"
_¹B
O q ZUop?M@HF ^`_ l q PzLNcRH i PTOQo q PRHdMvcUoIHdF$cRHdFy i HByu =
∑dk=1 B
k∂kyu6sALNPLNopFpFIH i PRZ6OHYl.LNPFL
O q f _ X l _ ZUopcUo _ M a F _ f~²±» W K.opF6oIHdMc.Lh"Hhf¡OHhZM _ czLNcUo _ M@Z q woIOHdMvcRHhZ
D =
[
0 Dσu
(Dσu)T Duu
]
.~²±»NÌ
°GM Z i ll _ ZRHy i HF ^`_ M<o X l _ ZRH i M@Hf _ M@OQopcUo _ M´L i JDFpo X opcRHhZOHc3ul HreopPUoIfRtFIHdcl.LNPYFIH a oLNoIZYO ifzt.L X l
M =
[
0 −Dσu
(Dσu)T 0
]
.~²±»Y"
W [
s _i POHgcRHdFIZkZUuQZUcR X HhZjOH¡PUoIHhOQPUoIfRt@Z³@opFHhZRcl _ ZRZRo a FIHTOHTf _ M@ZUcUP i opPRH~¼f _ XX HTM _i ZGF ^ Lh _ M@Z m LNopcl _i PYFIHlP _a FI X HOHOQo i ZUo _ MwvP q LfdcUo _ M2O@LNM@ZF ^ LNllP _ fRt@HBº i M<fRt.L X l& i M@HLNllP _ JQo X LNcUo _ Mgr l _i PGFLy i HdFpFIHFLf _ X l _ ZzLNMvcRH
zσh
l H i c ® cUPRH q Fpo X opM q HHdM P q Z _ FpLNMceOHhZelP _a FI X HhZF _ fL i Jl _ Z q ZgZ i PfRt.Ly i H X LNopFpFIH? ] LfdF q l _i Pef _ M@ZUcUP i opPRH i M@HcRHdFpFIHLNllP _ Jwo X LNcUo _ M<L"HhfOHhZgHhZUcUo X L¯·cUo _ M@ZYO ^ HdPUPRH i P¡ZLNcUoIZ m LNoIZLNMvcRHhZHhZRcOHM@HlF i Zl q M.LNFpoIZzHdPTFIHhZZzL i cRZ¡OHFL0f _ X l _ ZzLNMcRH
zσh
HdcOHl q M.LNFpoIZRHdPeFIHhZkZzL i cRZeHdckFIHhZG?LNFIH i PRZeºFL m P _ MvcUoIdPRHYOHTFLf _ X l _ ZzLNMcRH
zuh
OH X LNMoIdPRH¡lF i Z mn_ PUcRH?s5F i ZGlP q fdoIZ q X HdMvc _ M Z i ll _ ZRH¡y i HgFIHhZefRt.L X l@Z
MF F∂h
HdcSF F h
ZzLNcUoIZ m¼_ McGFIHhZklP _ lPUo q c q ZZ i opLNMcRHhZ¸
MF =
[
0 −Dσu
(Dσu)T MuuF
]
, SF =
[
0 0
0 SuuF
]
,~
MuuF = (Muu
F )THdc
h−1F ((Dσu)TDσu)
1
2 + hF |Duu| .Muu
F . h−1F Imu ,
~
SuuF = (Suu
F )THdch−1
F ((Dσu)TDσu)1
2 + hF |Duu| . Suu
F . h−1F Imu .
~ °GM9LYFIHgP q Z i FpczLNckZ i op?LNMvc "!$# &% CB P - " 0 &" ( (
∀k ∈ 1, . . . , k( ∀T ∈ Th * Bk|T ∈ [C0,1(T )]mσ ,mu
8 * - "$+(+"V$) - 4 ¼ - " *
‖z − zh‖L +
∑
T∈Th
‖B(zu − zuh)‖2T
1
2
. hp‖z‖[Hp(Th)]mσ×[Hp+1(Th)]mu .~²±»YA"
°GM _a cUoIHdMc i M@HGHhZUcUo X LNcUo _ MBO ^ HdPUPRH i PZ iax_ lcUo X LNFIH~¼O ^`_ PROQPRH¡±³5l _i P$FLgM _ P X H‖ · ‖L
Hdc i M@HHhZRcUo X LNcUo _ MO ^ HdPUPRH i P _ lcUo X LNFIHgl _i PFLYM _ P X HgO i ?PzLNlt@H a PUoIZ q HTLZRZ _ fdo q HgºYFLf _ X l _ ZzLNMcRH
zu
M@HHhZUcUo X LNcUo _ MO ^ HdPUPRH i P _ lcUo X LNFIHHdM>M _ P X HL2 l _i PeFLf _ X l _ ZzLNMvcRH
zulxH i c ® cUPRHYlP _i q H
Z _i ZFIHhZtvuwl _ cUt@hZRHhZkOHP q i FLNPUopc qi Z i HdFpFIHhZj?PfhHºOHhZjcRHhfRtMoIy i HhZGOHgO i LNFpopc q s _i PkF ^`q y i LNcUo _ M9OHgOQo i ZUo _ MwvP q LfdcUo _ M _ MBl H i ckfRt _ oIZUopP
MF =
[
0 −n
nT η1
hF
]
HdcSF =
[
0 0
0 η3
hF
]
.~²±» \
°GM q PUo.H m LfdopFIH X HdMcGy i HFIHhZjlP _ lPUo q c q ZY~ 7x~ Z _ MvcjZzLNcUoIZ m LNopcRHhZ
¦¤&¢¥ z¥R¦© « ¡¥ Y¢
] HhZ X q cUt _ OHhZjOHr _ Mvc q c q opMcUP _ O i opcRHhZkL i O qdai cGOHhZjLNMM q HhZ \[ l _i PFLZUo X i FLNcUo _ MOHlP _a FI X HhZ¡OHYM@H i cUP _ MoIy i H W » * ]$^ LNM.LNFpuwZRHM i X q PUoIy i HOHhZ X q cUt _ OHhZgOHr l _i PeFIHhZ q y i L¯·cUo _ M@ZktvuwlxHdP a _ FpoIy i HhZjPRH X _ MvcRHL i JµcUPzLh?L i JOH ] HhZzLNopMvceHdcTGLoLNPUcp±[ W?[ O@LNM@ZFIH X opFpoIH i OHhZLNMM q HhZ \[ ¶ F ^ HhZUcUo X LNcUo _ M>O ^ HdPUPRH i PGHdMM _ P X H
L2 O ^`_ PROQPRH p+ 12
L q c q_a cRHdM i HO@LNM@ZjFIHhZeLNMM q HhZJ [ l.LNP _ tM@Z _ MHdcf _ FpFL a _ PzLNcRH i PRZp±J* ] HhZ X q cUt _ OHhZ¡OHgr _ MvcYf _ MM ii MPRHdvLNopM<O ^ opMc q ·P ® cgf _ M@ZUoIO q PzL a FIHl _i PeFLZUo X i FLNcUo _ MDO ^`q y i LNcUo _ M@ZgHdcOHZUuQZUcR X HhZGO ^`q y i LNcUo _ M@Ztvuwl HdP ax_ FpoIy i HhZOHdl i oIZ i M@HOQop³LNopM@HO ^ LNMM q HhZAS+Ml.LNPUcUoIf i FpoIHdPAFIHhZFpoIHdM@Z¡HdMcUPRH X q cUt _ OHhZ¡OH _ F i X HhZ¡@MoIZHdcX q cUt _ OHhZOHGr _ Mvc-lxHdP X oIZOHcUPzLNM@ZUl _ ZRHdP-ºfhHhZ$OHdPUMoIdPRHhZ i MfhHdPUczLNopMM _ X a PRHGOHjcRHhfRtMoIy i HhZa oIHdM>P _ O q HhZgO@LNM@ZeFIH¡fLOQPRHOHhZ X q cUt _ OHhZeOHY _ F i X HhZe@MoIZxf _ XX HFLM _ cUo _ M>OH i JM i X q ·PUoIy i HhZOHZ _ Fp"H i PRZOH1joIH X LNMMLNllP _ fzt q ZHdcOHFpo X opcRH i PRZ¡OHlxHdMcRH ¶ _ opPTl.LNPTHKJH X lFIHJl _i P i M@HgPRHd i H?
» [
] HYO q "HdF _ llxH X HdMcOH X q cUt _ OHhZgOHr l _i PLNllP _ fRt@HdPgFLZ _ F i cUo _ MO ^ S5rs5ZgHdFpFpoplcUoIy i HhZPRH X _ McRHYL i JBcUPzLL i JOH GopcRZRfzt@H W ± l _i Pjo X l _ ZRHdPkOHhZkf _ M@OQopcUo _ M@ZL i JBFpo X opcRHhZGOHrGopPUoIfztFIHdcL i ZRHdM@Z m LNo a FIHHdcL i J>cUPzLh?L i JOH L ai |L Ù»* L ai |?LµHdc&F X LNF Ì * r _i ?FLZHdcTr i l _ Mvcp± [ * LN|"HdP &Hdc-EkPUM _ FIO p± &l _i P5F ^ i cUopFpoIZzLNcUo _ MµOHcRHhfRtMoIy i HhZOHjl q M.LNFpoIZzLNcUo _ MvS+Ml.LNPUcUoIf i FpoIHdPFIHhZTcUPzLh?L i JOH LN|"HdPTl i oIZO ^ EkPUM _ FIOºFLµ@MDOHhZLNMM q HhZ \[ Z _ Mc¡ºF ^`_ PUop?opM@HOHFL X q cUt _ OHf _ MM i H Z _i ZFIHBM _ X OH ¬ =ãs- ] L?LNPUoLNMvcRH~opM@OQoIy i@q H9fdo·*OHhZRZ i ZZ _i ZFIHM _ X OH X q cUt _ OHZ s ¬ _¹ FIHhZcRHdP X HhZOHl q M.LNFpoIZzLNcUo _ MBZ _ Mvc mn_ P X i F q Zjº¡F ^ LNoIOHgOHeF ^`_ l q PzLNcRH i PjOHePRHdFId"H X HdMcL q c q opMvcUP _ O i opcRHl.LNP LZRZUo7 GH a LhuHdcf _ FpFL a _ PzLNcRH i PRZ \ *sALNPLNopFpFIH i PRZ6F ^ LNllP _ Jwo X LNcUo _ M l.LNPOHhZ X q cUt _ OHhZjOHTrOHFL m¼_ P X H X oJQcRHgO ^`q y i LNcUo _ M@ZjHdFpFpoplcUoIy i HhZGL q c q lP _ l _ Z q Hl.LNP LZRZUo HdcGH a LhuA xl i oIZ q cRHdM@O i Hkl.LNPZ? _ fR| ai PUMHdc ¬ t i [ &O _ MM.LNMvcjLNopM@ZUo@M.LNoIZzZzLNM@fhHGºTFL@MOHhZLNMM q HhZ[ [ ºFL X q cUt _ OHf _ MM i HZ _i ZgFIHM _ X OH ] rT6S+MQ@M6l _i PgFIHhZg?LNPUoLNMvcRHhZTM _ M<ZUu X q cUPUoIy i HhZOHhZ X q cUt _ OHhZYOHBr l _i P¡FIHhZ¡lP _a FI X HhZHdFpFpoplcUoIy i HhZ³ _ opPHdMl.LNPUcUoIf i FpoIHdP W?W Hdc W Ì * M@HLNM.LNFpuQZRH i Mo q HOHhZ X q cUt _ OHhZOHr4l _i PFIHlP _a FI X HOHs _ oIZzZ _ MDL q c q lP _ l _ Z q HYHdM W?[?[ ±l.LNPGEkPUM _ FIO6 PRHdhho7? _ fR| ai PUMHdc >LNPUopMo W *
]5^ LNM.LNFpuQZRH i Mo q HGOHhZ X q cUt _ OHhZ-OHGgr H X a PzLZRZLNMvcFIHhZ$S5rs<tvuwlxHdP a _ FpoIy i HhZ-Hdc-HdFpFpoplcUoIy i HhZHhZRcO i H L i J´cUPzLL i J´P q fhHdMcRZµOHS+PUMÅHdcµ i HdP X _ M@OÅZ i PF ^ LNllP _ Jwo X LNcUo _ M OHhZZUuwZRcR X HhZOH@PUoIHhOQPUoIfRt@Zel.LNPkOHhZ X q cUt _ OHhZeOH¡r p± W ±»Q±dÌ * ] LM _ cUo _ M9OHTZUuQZUcR X HhZjO ^ S5rsÅO ^`_ PROQPRH i MZRu X q cUPUoIy i HhZHdcOQoIZzZUopl.LNcUo m ZL q c q opMcUP _ O i opcRHl.LNPPUoIHhOQPUoIfzt@ZjHdM9±[YYJ p±A*wrHhZO q "HdF _ llxH X HdMcRZP q fhHdMcRZZ i PF ^ LNM.LNFpuwZRH X LNcUt q X LNcUoIy i HgOHkcRHdFIZZUuQZUcR X HhZ-lxH i "HdMc ® cUPRHGcUP _i q ZO@LNM@Z p± Hdc p± \ *
S+MQ@M&F ^ LNM.LNFpuwZRH¡OHf _ M"HdPU"HdM@fhHYOHhZ X q cUt _ OHhZGO ^`q F q X HdMcRZe@MoIZ a LZ q H¡Z i P i M@H¡f _ M@OQopcUo _ MopM m ·*Z i l OQoIZRfdPRdcRH? i M@HlP _ lPUo q c q OHf _ McUopM i opc q HdcYF ^ i cUopFpoIZzLNcUo _ M2OHOH i JM _ P X HhZHhZUcYO q czLNopFpF q Hl.LNPGHKJQH X lFIHTO@LNM@Zp±?± _¹ HdFpFIHTHhZUckLNllFpoIy i@q HºOHgM _ X a PRH i JHKJQH X lFIHhZ³
ªhª© ¢+§¡£A¢
p± ¡r ¡wEkPUM _ FIO6AEGMopMcRHdPUo _ PlxHdM.LNFpc3u@MopcRHgHdFIH X HdMc X HdcUt _ O opcUtBOQoIZRf _ MvcUopM i@_i ZkHdFIH X HdMcRZ .8 M (,8 - &%U86±[µ¸ \ Ì W \ A [ ±[YJ W
W ¡r ¡&EkPUM _ FIO6 PRHdhho7 ? _ fR| ai PUM6LNM@O ] `rY>LNPUopMo7 Mo.HhO>LNM.LNFpuwZUoIZ _Nm OQoIZRf _ McUo·M i@_i Z¡TLNFIHdPU|opM X HdcUt _ OZ m¼_ PgHdFpFpoplcUoIflP _a FIH X Z .8 M (,8 - &%U8I6»Y[~"Y¸p± \ Ì2[³± \?\ [Q W?[?[ ± [?W
Ù» =L ai |?LQ t@H@MopcRHGHdFIH X HdMvc X HdcUt _ O opcUtl HdM.LNFpc*u"90 8 M D8 W"\ ¸ W?W ± W?W JQ.±[ \ »Q Ì = L ai |?LeLNM@O &F X LNF7 _ M@f _ M m¼_ P X opMHdFIH X HdMcRZ5opMcUt@H-@MopcRHHdFIH X HdMvc X HdcUt _ O opcUt
lxHdM.LNFpc3u" 8 .M ( 8 - &%68 ± [ ~"¸JYA?»³.J \ Q±[ \ »Q `EY LN|"HdPAopMopcRHHdFIH X HdMvc X HdcUt _ OZ mn_ PTHdFpFpoplcUoIfHhy i LNcUo _ M@Z i ZUopMBM _ M@f _ M mn_ P X opMHdFIHK·
X HdMvcRZ9Y 8 M D8»Q±"~²±» \ g¸ Ì2³.Y[Q±[ \?\ A ¡ LZRZUoLNM@O ¬ kH a Lu" E top?tQ· _ PROHdPLfhf i PzLNcRHµOQoIZRf _ McUopM i@_i Z@MopcRHµHdFIH X HdMc X HdcUt _ O
m¼_ PgcUt@HM i X HdPUoIfLNFZ _ F i cUo _ M _Nm cUt@Hf _ X lPRHhZRZRo a FIH eLoIHdP²· ¬ c _ |"HhZHhy i LNcUo _ M@Z 8 M 98: <" 8 ±»Q±"~ W g¸ W A \ W"\ [Q±[Y[ \
\ ¡ LZRZRo7 ¬ GH a Lhu" >LNPUo _ cUcUo7 ¬ xsAHhOQopM _ cUcUo7LNM@O ¬ LhwopMo7¡E top?tQ· _ PROHdP¡Lfhf i PzLNcRHOQoIZRf _ MvcUopM i@_i Z5@MopcRHGHdFIH X HdMvc X HdcUt _ O m¼_ P+opMoIZRfdoIOLNM@OwoIZRf _i Z+c i P a _ X LfRtopM@HdPUu _ jZ =*M6rHhf i uwlxHdPRHLNM@Oreo a HdFpo i ZHhOQopc _ PRZ³ : ;$7()() - X<" ;S - +H(U - - S (;( - $7(K - 5; M 0$v - (, * % * - M !$"0 - " (, M - M !$"K+l.LN"HhZK[Y[³&± [ JQ$EGMvc -HdPUl HdMHdFp?o i X ±[Y[ \ HhfztM _ F _ ?oIZRfRt =3M@ZUcUopc ii c
»Q±
J Z? _ fR| ai PUMG`Sj LNPUMoLO@LN|woIZjLNM@O ?e ¬ t i *T " $) - ¼ -H " j&%U(,³ - ( .&" "(;& * M 7Y ¼ - - O2%î$RY ¼ - "dA _ F i X HD±?± _Nm E (;$ ;( Y 9(+"µ - M 9(, $d/( - $7(Y - - X - ()(,K - X" ¬ lPUopM"HdP W?[?[?[
[ ? _ fR| ai PUM LNM@O ?e ¬ t i t@HDF _ fLNFTOQoIZRf _ McUopM i@_i ZgLNFIHdPU|wopM X HdcUt _ O mn_ PBcUo X HK·OHdlxHdM@OHdMc-f _ M"HhfdcUo _ MQ·*OQo i ZUo _ MBZUuQZUcRH X Z .8 .M (, 8 - &%U8I"»Y¸ W Ì?Ì [ W Ì2A?»Qx±[Y[YJQ
p± [ @r _i ?FLZ?P.LNM@O gr i l _ Mvc +- 9(,K!& : ( - &% : ;$7(; )(+"ZS& %'% ¼!$Y - : &U25;&% $j&%U(,¯ - ( .<"d _ F i X HFYJ _Nm EW()$ )( Y 9(," -Q: <"$,"d ¬ lPUopM"HdP²· 5HdPUFLN@ HdPUFpopM±[ \ AQ
p±?± E¡S5PUM9LNM@O . · ] i HdP X _ M@O6 "(;& - : ²0$ ¼!$7( S $ - 4 9( %U( M ( - 4"K. _ F i X H±Y[ _NmO0%î/(; 9Y ( M 0 ¼!$RY% $d/( - $7(,"K ¬ lPUopM"HdP²· 5HdPUFLN@ eH _ PU|x W?[?[ Ì@
p± W E¡?S5PUMLNM@O @ · ] i HdP X _ M@O66reoIZRf _ MvcUopM i@_i ZTLNFIHdPU|opM X HdcUt _ OZ m¼_ P5APUoIHhOQPUoIfRt@Z ^ ZUuQZUcRH X Z=@HdM@HdPzLNFcUt@H _ PUu" .8 .M ( 8 - Y%68QÌ?Ì.~ W ¸ \ ?»³ \?\ JQ W?[?[ AQ
p±» E¡?S5PUMLNM@O @ · ] i HdP X _ M@O66reoIZRf _ MvcUopM i@_i ZTLNFIHdPU|opM X HdcUt _ OZ m¼_ P5APUoIHhOQPUoIfRt@Z ^ ZUuQZUcRH X Z=)=K ¬ Hhf _ M@Ow· _ PROHdPes5rS5Z 8 .M (,8 - &%U8I W?[?[ AQ
p±dÌ E¡?S5PUMLNM@O @ · ] i HdP X _ M@O66reoIZRf _ MvcUopM i@_i ZTLNFIHdPU|opM X HdcUt _ OZ m¼_ P5APUoIHhOQPUoIfRt@Z ^ ZUuQZUcRH X Z=)=)=K i FpcUo.HdFIOcUt@H _ PUu opcUtl.LNPUcUoLNF@f _ HdPRfdopwopc*u" 8 .M (, 8 - &%68 W?[?["\ _ LNll HLNP
p± E¡.S+PUM @ · ] i HdP X _ M@O6 LNM@O9?LNlFLNopMEGM0opMvcUPUopM@ZUoIfYfdPUopcRHdPUo _ M m¼_ PGcUt@H a o b HhfdcUopopc3u _NmkopF a HdPUc _ lxHdPzLNc _ PRZPRHdFLNcRHhO9c _ PUoIHhOQPUoIfzt@Z ^ ZUuQZUcRH X Z MIM 8 : & ¼n&%68 *T (,8 8I W?[?["\ _ LNll HLNP
p±A ê°Y@PUoIHhOQPUoIfzt@Z ¬ u XX HdcUPUoIfl _ ZUopcUop"HFpopM@HLNPkOQo6HdPRHdMvcUoLNFHhy i LNcUo _ M@Z MIM 8 : )( 0%U89Y 8 ±?±¸Ù»?»?»³vÌ@±JQ±[YYJQp± \ < "HdM@ZRHdM *T'"$) - ¼ -H " j&%U(,¯ - ( .<" S & K(;Y$0" " 9( M "+4 ,;(/X %p&
&% ¼ - "ds+t@rcUt@HhZUoIZ Mop"HdPRZUopc*u _Nm °kJ mn_ PRO6 W?[?[ Ì@p±J ?e _ tM@Z _ M "HdPUcNLNM@O@s+opcU|NPzLNMvczLQwAopMopcRH$HdFIH X HdMc X HdcUt _ OZ mn_ PFpopM@HLNPtvuwlxHdP a _ FpoIf
Hhy i LNcUo _ M@Z³ M 98 ( .<" 0%U8 (;$ 8 - XY!X28Ì2B¸ W JY³w»Q± W ±[YJNÌ@p±[ s ] HhZzLNopMc F%pQ)+"&% ¼! - (+"F" <" M (+" +(,5;&% (+" ;( M /(, &Y;(R&C2(,"
M .2(,"WG %6 M ( - 4" - '"Ks+t@rcUt@HhZUoIZ Mop"HdPRZUopc3u _Nm sLNPUoIZ #=6±[ \ Q W?[ s ] HhZzLNopMvcLNM@O2sA ·*EY GLhwoLNPUc2°GM;L0@MopcRHHdFIH X HdMvc X HdcUt _ O m¼_ PZ _ FpwopMcUt@HM@H i cUP _ M
cUPzLNM@ZUl _ PUcHhy i LNcUo _ M=3M 90 ( M Y ¼$R&% "9(;$ 4"TS $ - 4 9( %U( M ( - 4" -1: & ¼¼&% *T (;( - ¼¼&% Y ¼ - "Kl.LN"HhZ@JY[³&± W »QQs ia FpoIfLNcUo _ M _ w»?»Q >LNcUt kHhZ ?HdMvcRHdP Mopx _Nm ÅoIZRf _ M@ZUopMQ·9LOQoIZ _ MEfLOH X oIfs+PRHhZRZ³GH _ PU| 6±[ \ Ì@
W ± @ kopcRZRfzt@H? a HdPHdopM $LNPUoLNcUo _ M@ZUlPUopMhoplD i P ] Z i Mµ _ MreopPUoIfRtFIHdc²·*s+P _a FIH X HdM a Hdo 5HdP²·-HdM@O i M _ M HdopFpP i X HdMOQoIHj|"HdopM@HdMKGLNM@O a HhOQopM i M"HdM i MvcRHdP _ P m HdMBZUopM@O6 >5 8 9Y 8 ( M 8 - <8D3 M 5 X?x»YAB¸[³&±Q±[ \ ±?
W?W @ g°eOHdM0=K L ai |?LQ"LNM@O ?e`Sj L i X LNMM EÅOQoIZRf _ MvcUopM i@_i Zhp
@MopcRHkHdFIH X HdMc X HdcUt _ Om¼_ PjOQo i ZUo _ MBlP _a FIH X Z .8 M 98 : <"8I6±dÌ2A~ W ¸ Ì2[Q±.Q±[Q±[Y[YJQ
W » YkHhHhO LNM@O g GopFpF7 PUoLNM i FLNP X HhZRt X HdcUt _ OZ mn_ PjcUt@HgM@H i cUP _ McUPzLNM@ZRl _ PUckHhy i L¯·cUo _ M HhfztMoIfLNFGHdl _ PUc ] Ek· · \ »¯·7Ì \ [Q ]_ Z5EGFL X _ Z ¬ fdoIHdMcUo.f ] L a _ PzLNc _ PUu" ]_ Z+EGFL X _ Z ±[ \ »Q
W Ì kopoIdPRH? `Åt@HhHdFIHdP@LNM@O Y@eopPzL i Fpc = X lP _ "HhO0HdM@HdPU?uHhZRcUo X LNcRHhZ m¼_ PopMvcRHdPUo _ PklxHK·M.LNFpc*u"f _ M@ZUcUPzLNopM@HhO LNM@OBOQoIZRf _ MvcUopM i@_i ZgLNFIHdPU|wopM X HdcUt _ OZ m¼_ PjHdFpFpoplcUoIflP _a FIH X Z = M 98 T(;<"$h8 Jµ¸Ù»?» \ w»YA [ ±[Y[Y[Q
» W
top related