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∂tU(t, ·) = L[U(t, ·)](·);<5 35 '),.3* %,b3%X'<lE.]@:8L[V] = S(V) −∇ · F(V) − A(V)∇V $

;<5V : Rd → Rp

'+ '%0><'12\@*X!&123% ',.VSIA<!&%12' 52c% 35c',c'))3:h68

% |S SF;^D%'"3)1^ 2'l3c%,d3%Lh

;<5\5>0 ,2' % E.l80')%*K!&123%',.Vh

%43%]%5DE.l80')%* !&1% '),.Wh

Sq9 52'(%,b3%D'W%'"3123A@:8 0M1*F8,*8 B 1.% ' ;<5 3'"^!&123%'

ϕ2a@:8>433 %wJ^35c%0>

Ω ⊂ Rd;<5\5 '

,.)3F3%$43% WE.F3 10T@b+%!d**'(∆j)j=1,...,N

SIA<'')10<5.V, ϕ

)c')0>%%5T!&1%2'%R35T**'

∆j

@12?0 8]@bM'% 1%12'W%wJD357**/43"!#'SuGD6d;[M!&%0>**<8a433@:8 ,.)3'(5 J0 %ZJ(352**'

∆j

;[>43),l(35 %%2'"J3FJ,%1lϕA(V)∇V

'c$0>'12(ϕA(V)∇V)Φ

' lE.R-K|S9 5 1'<;[7FJ ∫

Ω

L[V]ϕdx =∑

j

∫

∆j

S(V)ϕdx +

∑

j

∫

∆j

F(V) · ∇ϕdx −∑

j

∫

∂∆j

ϕF(V) · ndσ +

∑

j

∫

∆j

ϕA(V)∇Vdx +∑

j

∫

∂∆j

ϕA(V)[V]ndσ

;<5[V]n

%'M5(Y)120>,A%!V

-l3%A%!X35>%)0 *+6M**+ "!#'SϕA(V)[V]n'W35TJ*1$%!+35 0h')1

(ϕA(V)∇V)Φ

,d%4(%![')%4 118 352 wJJ*12

ϕA(V),d'$% 5,.5

ΦS9 5a3*%3' hJ @dl;^ 35**^4)!#'nSS

T % ( & % x C 2 % % %

10)h*.1~6K' !&%W5 %')"J63J ,.)Sq9 5'))3$!&123%Wh := Lh[Vh]

'(E.2C')%5. ∫

ΩWhϕh

CB 1.*'X5 54<5.]'(%! !&%<**b3')<!&1l3%' ϕh ∈ Wh

S9 5 %'"3123% %!Lh

,d'a%35 wJ J*1'1') I;<5\5 ,2%@*0,d S _.%(352 %')J3FJ.1~6AhJ3

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!&123% 6nSÿ352 '0>,2*'"h')R352 ,2,%6K0>3% %!W5R0h'1) %'')3'>%!?5 hJ3@b;[ 35(J*1' %!

ϕhA(V)!&%0 @d%5R')'<%!/5 43)!#(%!

∂∆j

S

bdc"!ac$#Fp ypqi!|dv&%('/kl~p*) kfu+ rTtdudv3wxl~g3udyZrX/w|lgjudyk_.% 5'))3:h% 35 ' ;<535 '),.*W%,d3%

L5.'O 0% %0,*76 !&%)0R

J%*Ja!&%(620,*T55(%) FJ63J'(%!V

%(%K8*2h)3'c 352$%K8%')"J63J,)%K21V;[a0,*%'8A 2%0>,b%'F3%A%!

LS =lT1'T'')10>35.6M;[a5.hJa'),.* %,b363%)'

Ls!&%s = 1, . . . , S

0>,,2Vs−1

%Vs := Ws × Vs−1

;<5V0 = V

WS = V

'1\5 35L = ΠS LS · · · L1

12'2]5>,2%6Y)3%C%,d3%ΠS(VS, . . . , V1, V ) := VS

Sÿ!Xh\5A%!^5%,b3%'

Ls

'+?E.)') %2I%,b3%%! 35 !&%)0 '1'')a@d%wJ 5$;[c %2')3)1 '%,b3%'

Lh,i

'<@b!&%) ]%0T@2W5')7%,b3%'X3% lE.Lh = ΠS Lh,S · · · Lh,1

S= [1' 0%'"3363 35' ,,)%4\5 5ch')D%! ')0,* ')h*+2Jl3% -,u1')% CB13% q

L[u] = −∂x

(

f(u) − K(u)∂x(k(u)))

− A(u)∂xa(u) . 9 5' B 1.%-,d)' ;^% ',bl3' !&%0 '. F %43' O'%A%2 )J3FJT0 5a%8%')J3FJ>,%1lT'7%K8*h A352 JJ>30 8D35'M*' 3% ,2%@*0'MlE.35 0h')1$% 5 @d%12.38 K|SqIAT352l!&%$!&%)0 **<8O);<)3T5'(CB 1.3% 5!&%**%w;<2M!&%0 1'1~6K*38>!&12%'

w1 = (w1,1, w1,2)

w1,1 = ∂xk(u) ,

w1,2 = a(u) ,

w2 = −∂x(f(u) − K(u)w1,1) − A(u)∂xw1,2 .

=?lE.M352?%,b3%)'L1[u] = (w1, u)

L2[w1, u] = (w2, w1, u)

;^W)JDX7%0,d%'%%!

L%2''"3 %!%,d3%'<%!V35 'O!&%)0RSÿa')%0>Dh')'X*%')1D*%'+FJ !&% 620,*c@:8 **,23X%,b3%)'^5.hJ?3%7@dc\A4%

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Q(A−1(U)). 5

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A'%0$*

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A−1h

Sÿ!^5')>hJ*@* 358 h-@b3*8 *1 35 )$!&30;[%A ')@bA@b%ZJSÿ35'W;X8 5 ,,)%4\5-h-@b,2,*O3% %'"312W2'l3M%,d3%'<!&%xB 17%0>,2*3R'38')30>'D%!IlJ%*123%RCB 1.3% ';[ ;<**d0%'"3363 LEK% KS

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∂tU(t, ·) = L[U(t, ·)](·);<5\5 *h2'a3% '8')0 %!TU(=WHI' ddtUh(t) = Lh[Uh(t)]

!&% 35%K:> 'OE.2Uh(t)

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d

dtU(t, ·) = Lexpl[U(t, ·)](·) + Limpl[U(t, ·)](·) K

;<5Limpl[U(t, ·)](·)

%0T@'W**V35 ')3@*18])')3a30'Sq9 52%)'),d%22'l3%,b3%'[c%2')3)15?'0><0>^'[%1K3*'%7!# >!&% 5c0l8')l3:h%T')0F80>,2* <198 <(12)\M05%K'X1') 6K,*F ,2,%4\5O!&%

Lexpl,hOO0,*F

!&%Limpl,h

S;?];^$%*17%1?2'1'')%R;<35 '0>,* 620,*S^%2'2D5TlJ%*123% CB13%;<5(352(',.*d%,b363%X' JO@:8 S!9 5(E.)')<%)X)0>' *ha%>M0l8')3,')3)l3%5%(%!^5 ':

h;<5)h' 352>-,u12'% 3)0 43)%K21'7'"\@*F18 )')3%%!

5 %h2 S9 5)!&%)M$',2*)3T%!V35(!&%)0

Lexpl[u] = −∂xf(u) − A(u)∂xa(u) , Limpl[u] = ∂x(K(u)∂xk(u)) .

' ,,)%,)SCIAI1')[X')0*d2%0>,b%'F3%(' @b!&%E.2Lexpl,1[u] = (w1, u),Lexpl,2[w1, u] =

(w2, w1, u)2

Limpl,1[u] = (w1, u),Limpl,2[w1, u] = (w2, w1, u)@:8

w1 = a(u) ,

w2 = −∂xf(u) − A(u)∂xw1 ,

w1 = ∂xk(u) ,

w2 = ∂x(K(u)w1) .

9 5'<*h'X3%d

dtU(t, ·) = Π2[Lexpl,2[Lexpl,1[U(t, ·)]]](·) + Π2[Limpl,2[Limpl,1[U(t, ·)]]](·) .

9 5c'0>,2*'"I30>98'):h63%T'IFJ>@:8$352c!&%";X@A4;X)aHI1*+0>l35% SI F35 0l8'"3,

∆t352' *'X%

U(tn+1, ·) − U(tn, ·)

∆t= Π2[Lexpl,2[Lexpl,1[U(tn, ·)]]](·) + Π2[Limpl,2[Limpl,1[U(tn+1, ·)]]](·) .

IA %*12 5. 5 '')43*'),'<!&%X352 2'l3:3%'12'' 5 ) 5(!&%**%w;< Sc);<)3(5(',.63* %,b363%<'c$'120 %!/;[%%,d3%'

Lexpl,LimplS

Sc2%0>,b%' @b%35O'),.3* %,b3%' 4% E')c%)<%,b3%' %!V35(!&%)0 SKSc12' 35 =?'%43 1%1' ? *A O,,)%4\5O3% %2')3)1c' %,d3%'SK2Sc12'7O6K,*F0>,* <198 <(123U =?H '%*FJ<%2J 5 '%*123%0S

Gc%3W5.<'),'c% ;^%>(,2%@*0 ,b24D;<52'<35(!&%**%w;<7'))3:h% ')3,'57]!&%1 h]@b 0>,2*04)h**<8>!V')13@*( 10)* .1~6K'?) hJ*@*S

T % ( & % x C 2 % % %

N [3ùø(þýþL[Bø(þ÷ ùvú S ýmø W!/c3bDc32u iip y l~gj|Dv e vjuauaok9 5 .%w; %![;^"8'04 0> 631)' )$%! 3A0%K2* ')3)a!&)%0 5 %0>,2')'@*M;[%,5'(OG?hJ83E%A'XB 1.63%'^!&%[M%OGc; % .1 ;<5 !&W@d%1.38S59/%$)12?5%0>,2*6KF18%!5 0>%* ')5.**%w; .%w; ,2,%4\5 '<1')];<5\5R*h' 3%a>]0>%* ;<5) 5!&) @d%1.38' 0>,2**<8$lE. @:8O'%*FJ OCB13%]!&%X35 ,235]%!V35 .%w;

hS9 5A0 :1*18 4J%*FJ'O352-2FJ3% %!M')13@* 0>%*'a!&%35 ).* @b

!&)%uS HIJJ8>'0>,*cJ)'%'X2*12cJ38a%0>,*30'^4J%*FJKM@b%5 '%2]%FJ63J'R %K8%')"J63JA,2%1'S>9/%0%'"3335-* ']!&%]12'25,2,%4\5R'@b],)J%12'*<8 ;^ %*8a,)')4W')%0>(,)<%!V352 0%* 2FJ

∂tU + ∇ · F(U) − Sg(U) − Sτ (U,∇U,∇2U) = 0 . 9 57@.* B 1.43F3'D 357,K35 %!35 .%w;

hR5M0%04310

huSSg

'c5M)J2!&%)$3%]wJF18

Sτ

' ]')10 %'')3O%! 3)0>' 0%K2* 5>2,d47)*J4 ')')''SIA %*<8 *12W;^% 30'ca35(!&%**%w;<2

U = (h, hu)T ,

F = (hu, huuT +1

2gzh

2I)T ,

Sg = (0, gz∇bh) ,

Sτ = (0,−µh∂2xu1 + sin(φint)gzsign(∂yu1)h∂yh + . . . , . . . ) .

cgz

' 5W0>F31c%! 3hJ\63%.* *3% b2'@d'^35?@da3%,b%*%8 2

φint, µ) %'"\4'D0%K2*75 43.* !&l3%a%!V35 1* 0 )*nSI 5Mc'0>,*3'"!&%0>3%(;^ hMl;<F3+B 1.% 'Vc'8')0 %!2E')/%)VB 1.%2'!&%

U]')c%!F3%.*+B143'

V1

V1 −∇ux = 0 ,

∂tU + ∇ · F(U) − Sg(U) − S2(U,V1) = 0 ;<5 35 l; ')%1 )0 ' FJ '

S2(U,V1) = (0,−µh∂xV1,1 + gz sin(φint)sign(V1,2)h∂yh + . . . , . . . ) .

Gc%3 5a%K8%')JJa)0>'TS2

;<5\5)a*h A35a2FJ3FJ'MB 1 !&%75':3%O,,)%4\5]'))@bR@d%wJS

!/c"!ac32 u|dk5p e vju gjyXu i!udwk p ogj|ÿ 352'('12@'3% ;^ %'W;[%8,5.')0M1*%0>,b% .%w;` ,d%%1'?0>2S+;?,,2*3%' 0> )75T'0T1*3%R%! ;^(2 %* .%w; 35)%15 5T'%* S Sd@%3.63% %D55 '0M1*% %!/;^c]4'(.%w; !&1* **' ')(!&% 760>,* SD;^a!&%12'T% '0>,2*E-'8')0 !&%7%0,''@*;^%68,5.'&.%w; *1 0M1*F8%0>,b%4W3',b%"?;<5\5-'D5T@.')'?!&%D352 ,,*h63%R 0> S9 5$0%K2*V%')'(4'2];^3 %Z; ] ,d%)%1'<010 ;<5 35274'c,25.' *'%$3527'),d'<58)% d%658u

! "$#%'&)(% +*,.-/!1032;^3 '%0T') 4' 0>*8O%2''"3%! % 5.hJ$3%@b$0>%** S9 5 %wJCB13%' )WFJO'^!&%**%w;<'SK_ )%0 0 '' 20>%0410 @.*c!&%^352?;X63^,5.') i = w

2]5 4' ,5.' i = g

;[ ∂t(φsi) + ∇ · (ui) = qi(si), :

ui = −λi(si)K∇pi,

;<F3535(!&%**%w;<2T*%'1) *3%'

sw + sg = 1, $pw = pg − pc(sw) = pg − pc(1 − sg). I F35a35'X0>%*d352 12!A %w;<]J)@*' ?5 '313%2'X%!35 ,5''

si

235(,2')'1)'c%!5 ,5''

pi

V2C35,5.' J*%F3'ui

Sφ%3'75 ,b%%'18%!X35>0>120

pc(sw)' 5>,**8 ,2')'1)/J 'MO!&1%-%!X5 ;X7'313%uS ; ,.30:h% !&%5' !&1l3%R'c!&%c7620>,2* JR@85 0>%* %! ? 12\5 '> S λi(si)

2%3'c5!&%.* .%w;`3$%![5

i35-,5.' 2,b]% 35 '36313% %! 35 ,5.'''),dJ*<8S

ÿ F3%K

%3'D5M@'%*123 ,b)0>@*18 %!570210 qi

),)'43'('%1'W')!A 'Sd_.%<'"\2 35 ,5.') 32'%O@b;[ 35 ;X63?R4'<,5.')M'c0%**5%125qi

S ;?' 2,b24 J@*'?!&%(%1 %0>,212\%2'?;^\5%%'$5 '313% %![5 4' ,5.')sg

352(,''12 %!35?4' ,5.'pg

S~9 5)@:8;^ ''120>235.6X35(,258')'c)W'12\5]5.X;[*F;X8'W5.hJ

sg > 0SqEK120>01, 5TB 1.63%'W 1' 352T*% ' ;^MD!&%

5 '%h***%@.*dJ*%F18u = ug + uw

]5 *%@.*u')%1)(30q(s) = qg(s) + qw(1 − s)

∇ · u = q(sg). 9 5MB 1.% !&%pg

'Dl30 @:8 1') *3% 3% ,2*pw

@:8pg − pc(1 − sg)

')335]E%' %!

u,ui

43%-B 1.63% S 9 5 1' ;[ ;<35 352R**,23a%,b3%A(pg) := −∇ · ((λw(1 − sg) + λg(sg))K∇pg)

A(pg) = q(sg) −∇ · (λw(1 − sg)K∇pc(1 − sg)). K

_.%a\5 ',b' 35 '%1',5.' ;[ 5.hJ 3% %'OC32',b%)CB 1.3% S^_ %c :=

(cH2 , cO2, cH2o, cR)T;[75wJ

∂t(φsgc) + ∇ · (ugc) −∇ · (φsgDg∇c) = rg(sg, sgc). 9 5?B 1.%!&% 5?'),d'

Ra@b?)%,,b @81') 35D*%

cH2 + cO2 + cH2O + cR = 1S9 5l!&% 5 J3%%! 35 %43363%'1'3%

c := (cH2 , cO2, cH2O)TSÿ 5'a0%*

Dg

2%3'T35',b)'%C3'%$2'@ 5a0>)%'%, -,u1')%C%!<35a'),d'Srg

'M')%1 ')!Aa)0 ;<5\5O!&% 760>,*(,'4'?%2' %!/5 ',b'S

GD76h;[(;<)W35(@d%wJ(2'@dR0%K2*d'<T'8')0 %!VE')<% B 1.%'^%)^%,2,*<8]35 =="? 0>l35% Su=?%2'%u;[$2-12,;<F35 5T!&%**%w;<>' %![CB 1.3%'S9 5$J%%!/1!A %w;<'D'

U := (φsg, φsgc)T

V1,V2

WJ%'c%!V30>,b%38a22RJ*1'S

V1,1 + K∇pc(1 − U1/φ) = 0,V1,2 −∇(U2/U1) = 0,

V2 + K∇A−1(

B(V1,1, U1/φ))

= 0, ∂tU + ∇ · (F(U,V1,V2)) − S(U) = 0.

T % ( & % x C 2 % % %

9 5 E.'"?,''D' !&%0R@:8OB 1.%' .3527')% @:8]B 1.%' . 5pg

'<0,*F3*<8lE. @85

pg = A−1(B(V1,1, V1,1));<5

B(x, y) = q(y) − ∇ · (λw(1 − y)x)'(352 54

5.R')S9 5 *'"<,.')'c%')')'<%! CB13% ;<F35352/1~6!&1l3%F

F(U,V1,V2) :=

(

λg(U1/φ)V2

λg(U1/φ)V2U2/U1 − U1DgV1,2

)

]5 '%12 '!Aa30'S

S(U) :=

(

qg(U1/φ)rg(U/φ)

)

.

!/c10>csp | 7l~gjtdp | t/g3pqi %@l~udp k rL@w|l~g3udyk_.% 5D')0T12*%T%! J%Z;<' 4'I1@<%0M@1')%'435<J?GDwJ"8jE3%A' B 1.Z8%'/'/E.2$ 2 1' S9 52X0 ''/T0%04310 %'))J%7,."%!235[%0,'')@*GDwJ"8jE3%A'cB 1.63%' h]@b ;<))3]'

∂ρ

∂t+ ∇ · (ρu) = 0,

∂ρu

∂t+ ∇ · (ρuuT + pI − τ) = 0. :

ÿC5 0%043120 B 1.% τ%'M5>'"3)''732'% ;<5\5C!&%MGc; 3% .1'75'

5(!&%0

τ = µ

(

∇u + (∇u)T −2

3(∇ · u)I

)

';<5

µ%3'<352 J'%'18a%!/5 .1R

I35 1218 ')%S

ÿ]% 3% %$35 J G?hJ83E%A'DCB 1.3%',."3*u-,u4* CB 1.3%!&%^5\50>h*u',b'D

∂ρwi

∂t+ ∇ · (ρwiu) + ∇ · ji = mi, i ∈ 1, nspecies

;<5ji'W 1~6 )')1* !&)%0 ',b'(2,d1'% S9 5T'%1T3)0

mi

%.63'<!&%0 '),d'%4J)'%'M21$3%]\50*I%'S 9 5',b' ,d1')% ' 0%*1'R_A ' *h;

ji =−ρDi∇wi − ρ

∑nspecies

j=1 Dj∇wj

;<5Di

'( -,u1')% %'"\4(;<5\5 h @bT\5%',b843*<8 !&% h\5R',b'

iS

=?1 3%5 %')3%%![-,u1')% %!^'),d'(;<35-,u435*,:8%! !&%0>%∆Hi

5 ')\.)R)8OB 1.%O0T12')<@b ')1,,*043R;<F35]]F3%.*b3)0R2*hT%

∂ρe

∂t+ ∇ · (u(ρe + p)) = −∇ · (jT + τu +

nspecies∑

i=1

∆Hiji). ÿT5' CB 1.3%

jT'/352X30,b3631)^-,u1')% .1~6T$'/J$@:8$_.%12 '*w;'

jT = −λ∇T;<5λ'X5 5h6D%123%]% :4D

T352W30,b3631)S

ÿ%2?%*%'M5$'8')0 %![CB 1.3%'W2'1'')-5) F3%.*6K,)'')%-*5>5)0>%8.0$J@*' ' S cV5 h* '7*h;

p = ρRT' 1' /;<535

! "$#%'&)(% +*,.-/!1032h*/4'(%')34 %!+35 06312M'(6K,)'')C'

R = R∗∑nspecies

j=1wj

Wj

;<5R∗ := 8.136J/mol@b$35 1FJ'3* 4'<%2')\ S

ÿ!?5O',bE.]5h6>,.18cv

''')10>%A@bO2,b4 %!c352O0,d31T5

'),dE %\* 8ρe

' J '

ρe =1

γ − 1p +

1

2ρu2 +

nspecies∑

i=1

∆Hiρwi HK ;<5 35 lE.F3%

γ := cp/cv

2-5 18R = cp − cv

'(1') V'(;[**I'(35 CB13%%!^')33SU?J3** ;[a)J7]')7%!XCB13%' !&%(35,."3* ,d)43*B 1.43'

U =(ρ, ρu, ρe, ρw)T

*%') 1') HK % lE.a352a,'')1p = p(U)

352a*%T = p

Rρ

SI F35]$'0,*?3')!&%0>3%a;^(E.**8a)FJ c$'8')0 %!VE.'"c% B 1.%2'

V1 − µ(

∇u + (∇u)T −2

3(∇ · u)I

)

= 0,

V2 + λ∇ρR

p(U)= 0,

V3 + ρD∇w + ρ∑nspec

i=1Di∇wi = 0,

∂tU + ∇ · F2(U,V1,V2,V3) = S2(U) ;<5 35 l; .156OJl3%

F2O'%12 30

S2 FJ]'

F2(U,V1,V2,V3) =(

ρu, ρuuT + p(U)I + V1,

(ρe + p(U))u + V2 + V1u +

nspecies∑

i=1

∆HiV3i, ρwuT + V3

)T

,

S2(U) =(

0, 0, 0, m(U))T

.

!/ccgj/wgjo J| udwiGe vju k g l |Dk5p |DydpIA %' 35>8.0'M%!<]%0,)'')@*>* B 1K8J,b%1 .%w; 12)%2 ,5.')>3')%';<5\5$'/%wJ)@:875X%0>,2')'@*^G?hJ83E%A'"8=<D%"3;[ '38K'"30 |SH9 5'/'IT63'%%!u35?G?hJ83E%A'X'8'"30 ;<5aaF3%.*25) %) 3)0 35. \HA'[35D ,d^%! ')1)!#')%R<,5.' @d%1.)'c43% %14S

∂tρ + ∇ · (ρu) = 0,

∂t(ρu) + ∇ · (ρuuT ) + ∇p(ρ) = ∇ · τ + λρ∇∆ρ

;<5(352 12!A %w;<'?W352 2')18ρ(x, t) > 0

2O5WJ*%F18u(x, t)

τ'X35 1')1.* GDwJ"8E%A' 3'% 'M35^'l3%T@b!&%)5X,**)18

λ > 0' A %w;< %')34+T5[!&123%

p = p(ρ)' Ja@:8 JK88 IC*'[CB13% %! ')\63D;<52D35?0,d31c'+E6Ka%$7%K8

'"\4c@d*%w; 35 *b0,d31W% **%w; !&% ,5.')(33'F3%'SKGD%W35.6 35 'X2%%,.30X 35'X0%K*b5. h'[5(,5.''Sqÿ35'X0%K*b5(,5.''<Wl30@85(J*1(%!V352 2')18O%*8 *%w; 2')18 ?J,d%12.552'18 W*<B 1 S9 5<'38K'"30 *')% @dX;<) %')JJ !&%)0 @1K[F 312>%1KI3% @d<@b)3 'M Zb)!&)'c35 3% '):(5(35]% 3)0 3%5 ;<535

∇p(ρ)30 O%%K8

'))JJW!&%0 a%)X%$c123(')1*'X;<5O% FJK*u*%K*bB 1*@)10 %2E.13%2'

T % ( & % x C 2 % % %

) ,'4S∂tρ + ∇ · (ρu) = 0,

∂t(ρu) + ∇ · (ρuuT ) + ρ∇κ = ∇ · τ

;<5κ' 2E. @:8a5 *3%

κ = κ(ρ, ∆ρ) := −λ∆ρ + µ(ρ), µ(ρ) :=

ρ∫

0

p′(s)

sds.

9 5 2J4\ %!W5 %%2'"J3FJ !&%)0 '35.>35]J@*κ

,,bh' 76K,**<8 5CB13% S

κ' CB1* %R')%0>%')34T6M]')33CB 1*@)10 ')33 nSSR')33 ;<5

∂tρ = 02u = 0

S EK%]5')l3:h%%![35%%'))JJ>)!&%)0T1*63% %!^5>CB 1.3%-*h'.631**8>%>$;[** 8@.*')\50S

Vρ −∇ρ = 0,

Vu −∇u = 0, :

κ − µ(ρ) + λ∇ · Vρ = 0, $∂tρ + ∇ · (ρu) = 0,

∂t(ρu) + ∇ · (ρuuT ) + ρ∇κ −∇ · τ(Vu) = 0. 4

hN ö X] S S ÷±ø [3ø(þúG÷ úø QTSVU þW(ýmù S ø S ú]¹ü<ø(þ,ú÷ S ù_[3ø(úù9 5'('3%-h*'?;<5 35 *:h%R%! 35$0>3520>3h*V0>%*/!&%0 '% -LONOPRQTSEK12@'3%aKS '))@b'+!&1.043*.%2,K3'^%! LONOPRQ *J4+!&%I352=="? 2'3:<;<52'')1@'%' 2S 22Sa76K,* 352T;<*8 %1 '):h63% %,23' 5?0>,*l804\63% S

0xc3bDc 5ngjk 7i!pl~p e>wy 7l~g3udyk |dyo m pqi!|l~u i!k9 50,*04\%R%!^O2'l3 0>%* %!^],.)3*2,d43* B 1.%'(@.'A%-5$!&%*F8*%Z;<$* %2,K?%!V!&1l3%O',.' !&12%'<R%,d3%'235.6?<%O!&1l3%'S

! #"%$&(')*',+&-+/.0.21#&#*'+/&34(5"67&# 8.91(&#*',+&#; !&1%>',.

V %1[%,2 '^7'^%! 0>,,'+!&%0`5?2%0

D := KdD

3%735D3R := Kn

R

2S S

V := u : KdD → Kn

R

cKD

%'?5M%0>E.* KR

35737E.* d, n

352M0>'%'c%! 352M%0>2R',bJ*<8S9V%!&1"35 ',b!)8a5(!&1%'),.bF3%.*b,%,d)3' R@b 2 .SS5W!&1l3%)

Cm%<% @d*%2 %$35 EK%@b%*JXE,.

HmS; '))3X!&12% ',

Vh

;<F35$E.F3X0>'%m

'+('1@2'[%!dW!&1l3%'),.<;<F35T5,)%,b)18?35. 5 !&1l3%') 2E.7*%K**8c%(35I*04'

e%!35I12)*<82c%0,12\63%.*

! "$#%'&)(% +*,.-/!1032)

TSgÿ!

e%3'W5$l!&)$*0>4(%!

e

Fe

35T0>,,2Fe : e → e

d;[$E.2T35*%K*u@.' !&1%O')

Ve

%]5 l!& *04e35)%15

Ve := ϕ1, ·, ϕdim(Ve).9 5 ') !&12%O',Vh

'X5]FJ]'

Vh :=

uh ∈ V : uh|e := ue :=∑

ϕ∈Ve

g(ue,ϕ) ϕ F−1e , for all e ∈ T

.

IA]**Ve := spanϕ F−1

e : ϕ ∈ Ve *%K*[!&123% ',.

ue ∈ Ve

*%K*[!&12% 2

DOFe := ue,ϕ, ϕ ∈ Ve35M'W%!+*%h*'(%!I!&)%0]Sÿ %D3%a%),d%37*%@.*

,)%,b)'%!.5X')^!&1l3%M',35 !&1l3%T',. 5.'V3%W,2%wJ D0 ,2,g@b;[

5 *%K* ' %!?!&%0 DOFe

35R*%@* ' %!?!&%0DOF := ui : i =

0, · · · , mSIA '120>0>:5.MR'>!&123%C'),.

Vh

' 02 @:8A]!&123%C'),.VV

)T.357@.' !&1l3% 'l3'

Ve

!&%D** )!&)T*043'e2 3570 ,,2

g!&)%0 *%h* 3%

*%@.*d'D%!/!&)2%0RS; ' !&1l3%uh ∈ Vh

'M%*8E.2'$ ')$%!<*%h*I!&1l3%'ue

;<5)O*%K*d!&123%O,%wJ2'D''<3%$5 *%h*d)'D%!V!&%0 DOFe

S ! #"%$&(')*',+&-+/.0+ 4 " * + ! '& +/& #' 5"% " .21#&( '+/&; ') %,d3%

Lh

'<0>,,T35c'<%O')l3 !&1l3%'S2S

Lh : Vh → Wh.9 5@8;^ ')1,,b%'[35.I?'))3^%,d3%0>8T*;^$8' @bX%0,b%'$43%(D*%@*KU(,b3%Lpre

.')c%!/*%h*d%,b363%)'Le

.R$*%@.*u%,d3%Lpost

.nSS

Lpre : Vh → Ve, e ∈ T ,

Le : Ve → We, for all e ∈ T ,

Lpost : We, e ∈ T → Wh,

Lh = Lpost diagLe, e ∈ T Lpre.c

diagLe, e ∈ T '<%*u0>3)6 %0,b%']@:85 43)'

Le

SGD%3$35.6 ;<F35 352'(lE.%%!^O')l3 %,b3%W;[>@*T3%O%0T@2$%,d3%'

L1h2

L2h

]*%K*d;^8 ,)%ZJ35L2

pre L1post = Id

nSS

L2h L1

h = L2post diagL2

e L1e, e ∈ T L1

pre.

&"6*.95" , "6 .9+ #'5*"% " .21#&(*',+/&( &# + 4 "6* +/; %)3% 357@'"3D')),2%]%! 2'l37!&12%'c2 %,b3%)'c@d%wJ ;^7E.2 5!&%**%w;<2T43)!#(*'')'

S 〈 !"#!#$%'&)(#*!#$%'& " +,&-(#*# + 〉9 52'(*''?%)'),d%'(3%352T!&1% '),. TSÿ 'W,.0>l3: @:8 5$%0> E.*

KD = "#!$% 5> E.*KR =

(#*!#$% '7;[**['7352>0>'%' %!^52%0

d = + O32n =

(#*# +uS

T % ( & % x C 2 % % %

S 〈

〉; _.1%>'[,.30>l3):a@:835c18K,bW%! 5D!&12%a'),. [@d*%'%S!9V%>lJ*1.63 T!&1l3% 2352 !&%**%w;<T0352%KO' ,2%wJq #$ &#5uH J*1.'X5(!&1%]c,b%4a]12'<5(J*1#bS

KS 〈 &!%'& 〉9 52' *')' %)',b%' %?35^')l3^!&12%$'),.Vh

SbÿI'/,.30>l3):T@:8M5^18,bX%!5(!&1l3%]'),.

V = O')1\5O35.

Vh ⊂ V235(18,b %!/5 %0,1233%.*

T =

! % 5M18,dM%!+35T@'M!&12% ')Ve =

bS 9 5$*')',2%wJ'( F33%c!&%c5$')'W%!I5M '

e%!I5M) Saÿ %O352M!&%**%w;<

05%'< ,)%wJK $ $ & #$5 <12' 35 *%@.* 10T@bI%!.35 %! !&%0 ;<5

*%K*u 10M@d $>%O352 4318 S9 5 1'bX%)'),d%'D%$35 0>,,g]%1

@')3l<E.23% S

@ *# *<1'<@.'(!&1l3%]')Ve

%!4318 SK2S 〈 &##$ 〉; ')>!&1l3%C'7,.30:A;<5A518,b %!^35>')l3!&1%A',.

Vh = aF[@b*%'+3%2SÐÿ F3% ['[*')%7,.30:;<F35>5D18,b%![F3'W*%K*/!&12%'

ue

##$ A% 5$43F3'eSq9V%]')' 5$*%h*/!&1l3%'

5(!&%**%w;<T05%]' ,%wJ2q $ #$ & $"!# cl31)'c5 *%h*b!&1l3% $"!a%! 18 S

S $#% 〈 #$ $#!"'& !" & (#*# 〉; ')l3 %,d3%Lh

'R,.30>l3): @:8 35C18,b %!$352C!&1l3%' 3'R%0> 352 18,b %!/5W!&1l3%'<F3' 3 (*# Saÿ]23%a352W18K,b %!5c*%h* UW,b3%)'

Le

'+FJ S:9/%M,,*8T352D'?%,b3%5&'h8%,d3%+'+E'<!&%**%w;<'C *'& %# ';?,,2*' 35+%,b3% 3% *W%!K18,b "7 l31)'V5+'1*F32

2'l3(!&1l3% %#%!V18,d (#*#.S

0xc"!ac u ))( gjygjy 5ngjk87+i!pl~p m pqi| l~u i!k+*-,.( k:l~i|7l0/y l~pqi+| 7+p vj|Dk5k5p kÿ 5D!&%**%w;<352?*'')'^!&%[%0T@ ')?%,b3%)'^>352D!&%)0 )B 1>!&% 5 = %h*=?'%43 1%1'? *A =="? '363: '))@b]35 ,)J%1' '3%'D(,2')43 S9 5 ; *'')'c]LON]P]Q H

S21〈1!$ 3 &41# 1 〉;D' '))@b ')l3% c5 %,b3%

L'R%0>,b%') 43% ,.'')'

Li

'L[U] =

LK[LK−1[· · · L2[L1[U ]] · · · ]]SD9 5'*''')J'O' 35R@.') *'' !&%',b*:J)'%'

%! '1@%,b3%Li

Sgÿ(%43)%*'c35T76K1K3% %!I35M,lJK%1'W,'')' ''0T@*'?5\2|SS(%0M@' 35^'%*123%T%! 352 ,lJK%1'I,.'''

(Vi−1, . . . ,V1);<F35T5 .3

US9 52Tl31.*V%0>,1K\%]%!+,.''D'c0,*0>43] >')1@d*')' ;<5\5 '?3%>%wJ)

5 0>l35% >3%$35'< S9 52 *')' \A'<;^% 30>,*3?,.30>l3)' 1!$ 3Bc'8lE. *'' ;<5\5R,2%wJ'<5 ,%@*0 ,d4D18K,b'S

! "$#%'&)(% +*,.-/!1032

@ 1# 1 9d8K,b %!/5 ,lJ%1'<,.''<0,*0>432Li−1

S1# 1 ' F35M]'1@218,ba%!1R% -h')>352 l31.* ,''7' 35 E.'",'' %!V18K,b 1

〈 * +

〉S

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