R f 2 ()] r 2L ()

"!$#&%' "(*)+-,.0/-,213 '45/$#&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$Zjtvuwl HdP ax_ FpoIy i HhZjHdcHdFpFpoplcUoIy i HhZkZRczLNcUo _ MM.LNopPRHhZkl.LNPjFIHhZ X q cUt _ OHhZGOHgLNFIHdPU|opM@HrGoIZzf _ MvcUopM i HhZ~7rTK ] HhZklP _a FI X HhZX _ OdFIHhZ5ZRHdP _ Mc i M@H q y i LNcUo _ MO ^ LOQ"HhfdcUo _ MwvP q LfdcUo _ MO@LNM@Z+FIHfLZtvuwlxHdP a _ FpoIy i HHdc i M@H q y i LNcUo _ MOHOQo i ZUo _ MwvP q LfdcUo _ MO@LNM@ZgFIHfLZTHdFpFpoplcUoIy i H?S+M i oIZRHOHf _ X lF q X HdMc i [email protected] i Mo q HOHhZ X q cUt _ OHhZ$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×ÙØÚ³Ø²ÛÜQÖKÝhÞ7Û²Ô"Üßzà7áÙÛ7âwØãÕ3ävÔvåjØ3Þ+æ³Ô?ßRçKØ²ÝèNØ$éx×êßáÙÛnØÜßzÛnÕUß×pÔwëKëRìhííGÑßRà7ÝNØ$×îßjï&ß×Ù×Ùð²Ø-ÕãØUÚ¯ØãñgòÔó à*ßÝ¯Õ²ØAôkõKö÷@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 [email protected] _ 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 _ [email protected] ^ 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 [email protected]

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 [email protected]

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 [email protected](·, ·)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@[email protected] ¬ _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(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 [email protected] _ 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


?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(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 [email protected] _ 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 [email protected] _ 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 [email protected] 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 $Â 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


_ 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 [email protected] _ 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 [email protected] 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¡[email protected]¡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@fdopFOHgÌºB± [ 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@[email protected] _ 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 [email protected] 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 _ [email protected] 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 [email protected] 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?

» [

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]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@[email protected] 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

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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

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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[ \

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»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?[?[?[

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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

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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

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