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Ò×ÒÑØÖÐÒÝØ×ÝÑÑØÖØ×ÓÖØ××××ÓÓÖÒØ×× ×ÒÒÓØ×ÙÓÖ×ÝÑÑØÖ×ÐÓÐÐÓÒÑØÖÐÙØ×ÙØÒÓÖØÓ ÖÓØÐ×Ø×ØÒÓÑÒ×ÙÙ×Ò×ÙØÒÍ×ÑÑØÒÑØÖÐÐÐÓÒÓÐÑ ÓÒÐ××ØÒ×ÓØÖÓÓÔÔ××ØÐÝÝÑÑÒÓÖØÓØÖÓÓÔÔ××ÌÖ×ØÐÐÒ 

σij = Cijklǫkl. 

34 81ÓÑÔÓÒÒØØÑÙØØÓ××ÒÒØÝ×ØØÚÒÝÑØÒ×ÓÖÓÚØ 

CijlkÌØÒÚÔØÑØÖÐÔÖÑØÖÓÒÒ 

= 

Cjikl = 

36ÔÔÐØØÂÓ×Ð××ÓÐØØÒØØÑÙÓÓÒÑÙÙØÓ×ÒÖ 

6 × 6 = 

W = 1 

2σijǫij = 1 εÓÒÝÚÒÑÖØÐØÝØÝØÝÝÔØ 

ε: C : 2 

∂W 

x ′ 2 

∂ǫij 

= Cijklǫkl. 

ϕ 

ÃÙÚÒ×ÝÑÑØÖØ×ÓÒØÔÙ× 

x1 

ÑØÖØ×ÓÌÐÐÒÓÒ×ØØÙØÚ×ÒØÒ×ÓÖÒØÙÐÔÝ×ÝÒÚÖÒØØÒÔ ÇÐÓÓÒÃÙÚÒ(x1,x2)Ø×ÓÑØÖÐÒÐ×Ø×ØÒÓÑÒ×ÙÙ×Ò×ÝÑ 

−x3ÌÑÒÙÚÙ×ÒÂÓÒÑØÖ×TÓÒ ÐÙ×××x3 → ÅÖØÒÒÒ×ÒÓÒØÖØÓØaijbij = a: b 

x2 

x ′ 1

ÆÁËÇÌÊÇÇÈÈÁËÌÅÌÊÁÄÁÌ 

ËÚÒÝÑØØÒÒØÝ×ØÒ×ÓÖÐÐÔØÓÓÖÒØ×ØÓÑÙÙÒÒÓ× ÌØÒ××ÔØǫ ′ ij = ǫijσ ÔØǫ ′ ÒÝØÓÒ×ØØÙØÚ×ØÝØÝØØ ×ÑÖ×ÒÒØÝ×Òσ23×Ò 

ÎÖØÑÐÐÐÙ×ØÓØØÑÐÐÙÓÑÓÓÒÓÓÖÒØ×ØÓÑÙÙÒÒÓ 

ÔÙÑØØÓÑÓÑÔÓÒÒØØÓÒ ØÚÒÚØ×××ÐÐÓØÒÝÒ×ÝÑÑØÖØ×ÓÒØÔÙ×××ÖÔ ÇÖØÓØÖÓÓÔÔ×ÐÐÑØÖÐÐÐÓÒÓÐÑØÓ×ÒÚ×ØÒÓØ×ÙÓÖ××ÓÐ ÔÔÐØØ ÖÔÔÙÑØÓÒØÓÑÔÓÒÒØ ×ÒÑÖÒÚÓØÒÒØØÔØC2311 = Î×ØÚ×ØÒÒØÝ×Øσ13ÚÖØÑÐÐ×ÒØÙÐÓ×C1311 = 

ØÒÓÖØÓØÖÓÓÔÔ×ÐÐÑØÖÐÐÐÓÒÝØÒ×Ý×ÒÖÔÔÙÑØÓÒØÓÒ× ØØÙØÚ×ÒØÒ×ÓÖÒÓÑÔÓÒÒØØC1111,C1122,C1133,C2222,C2233,C3333× Ú×ÝÑÑØÖØ×ÓÓØÒ×ÙÓÖØØÑÐÐÚ×ØÚØÖ×ØÐÙÑÝ×x1x2 

ÐÐÐÚØÙÙ×ÖÔÔÙÑØÓÒØÓÑÔÓÒÒØØÚÓÒÖÓØØÑØÖ ÇÖØÓØÖÓÓÔÔ×ÒÐØÒØÔÙ×××ÓÐØÙ×ÒÑÙÒØÔÙÑÙÓ 

×ÙÙÒÒ××ÔÝØÒØÙÐÓ×ÒC1211 = 

ÐÚÓÒE1,E2,µ12,µ21×G12,G23G31ÚÙÐÐÑÙÓØÓÓÒ℄ 

= ÓÒÑÙÙØÓ×ØÐØÒÔ×ÙÙ××ÙÙÒÒ××ÓØÒC1133 

13 = −ǫ13σ 

′ 

⎡ ⎤ 

1 0 0 

⎢ ⎥ 

T = ⎢ 

⎣0 

1 0 

⎥ 

⎦ 

0 0 −1 

. 

ǫ ′ ij = TikTjlǫkl, σ ′ 

ij = TikTjlσkl. ′ 3ÌÓ×ÐØ 

ij = σijÙÒi,j = 1, 2Øi=j = 

13 = −σ13×ǫ ′ 23 = −ǫ23σ ′ −σ23ÃÝØØÑÐÐ 

23 = 

σ ′ 23 = C2311ǫ ′ 11 + 2C2312ǫ ′ 12 + 2C2313ǫ ′ 13 + C2322ǫ ′ 22 + 2C2323ǫ ′ 23 + C2333ǫ ′ 33 

σ23 = C2311ǫ11 + 2C2312ǫ12 + 2C2313ǫ13 + C2322ǫ22 + 2C2323ǫ23 + C2333ǫ33 

C1333 = 0Ò×ÓØÖÓÓÔÔ×ÒÑØÖÐØÒ×ÓÖÒ 

C1212,C1313C2323 

C1111 = 

C2222 = 

E1 

C2312 = C2322 = C2333 = 0 

C1312 = C1322 = 

0Ì 

0 

C1222 = C1233 = C2313 = 

C2233 = C3333 = 

, C1212 = G12, 

1 − µ12µ21 

E2 

, C2323 = G23, 

1 − µ12µ21 

C1122 = µ12E2 

, C3131 = G31. 

1 − µ12µ21

ÅÌÅÌÌÁÆÆÅÄÄÁ ÅØÑØØÒÒÑÐÐ 

×ÒÓÙÙØÒÐ××ÙÓÑÓÑÒØ×ÓÒ×ÙÙÒØ×Ø×ÖØÝÑØÓÐÐÓÒÖØ ×ØÚØØÚØ××××ÓÓ×ØÙÙÝØØÝ×ØÐØØÐÚÝØØÚ×Ø ÃÓÑÔÓ×ØØÐØØÑÐÐÖÓÓÒÒÚÖÖÒÒÓÖÑÐ×ØÐØØÑÐÐ×Ø×ÐÐ 

ÐÒÒÒÑØØ×ÒÓÐØÙ×ÒÄ××ØÙÐÐÒÓ×ÓØØÑÒØØÑÐ× ÐØØØØÐÚÝØØÚÐÐÝØØÒØÙÒÒØØÙÝÚÒÝØØÝØÝÚÑÒØÐ Ì××ØÝ××ÝØØÝ××ÑÐÐ××ÐØØØØÚÔÖÙ×ØÙÙÊ××ÒÖÒÅÒ ÑÓÒÑÝ×ÝØØØÝØØÚÝÚÒÑÖØÐØÝÄ××ÐØØÐÚÝØØ ÚÒ×ÙÔÔÒÑ×ÓÑÒ×ÙÙØÔÖÝØÝÚØÑÝ×ÝØØÝÐÐØØÚÐÐ 

Ú×ØÐØÒÝÝÝØÒÓÒÐÙÓÒÒÓÐÐ×ØÝØØÐÙ×ÒÒØÝ×ØÒ×Ñ ÃÓ×ÓÑÔÓ×ØØÐØÓ××ÐÙ×ÒÒØÝ×ØÚÙØØÚØÚÖ×ÒÙÓÑØØ Ê××ÒÖÒÅÒÐÒÒÐØØÑÐÐ 

ÐÄØØÑÐÐÒÐ×××ØÒÑØØ×ØÓÐØÙ×ØÓÚØ℄ Ñ×Ò×ØÒÔÔÖÓ×ÑØÓÒÙÓÑÓÚÊ××ÒÖÒÅÒÐÒÒÐØØÑÐ 

Ã×ÚÚÒÔ×ØØÐÙÚØÚÒx3×ÙÙÒÒ×× ÈÝ×ØÝ×ÙÙÒØÒÒ×ÖØÝÑÖÔÙx3ÓÓÖÒØ×Ø ÄØÒ×ÚÚÒÒÒÓØ×ÙÓÖ××ÓÐÚØÒØÔÝ×ÝÚØ×ÙÓÖÒ 

ÚÆÓÖÑÐÒÒØÝ×σ33Ú 

∂w 

∂x 

ÃÙÚÌÔÙÑÒÖÚØÒÖØÝÑÒÖÓÚ×ÙÙ×Ê××ÒÖÅÒÐÒÐ 

β 

ÓÓÒ×ÚÒÝÑÒÖØÝÑÒβ×ÔÝ×ØÝ×ÖØÝÑÒwÚÐÐÐÔØÚØ×Ù ÃÖØÝÑÒÑÖØÝ×ØÓÒÚÒÒÓÐÐ×ØØØÙÃÙÚ××ÇÐØÙ×ØÒÐ×ÙÙ×× ØÒÔÓÐÙ××× 

ÖÚØÝØÝØÑ××εÓÒÐÒÖÒÒÒÒØÝ×ØÒ×ÓÖε(u) = 1 

2 (∇u+∇uT x3ÓÐÐÓÒ℄ 

) ÑÖØÒÐ××z:=

ÅÌÅÌÌÁÆÆÅÄÄÁ 

ǫαβ = zǫαβ(β), 

ǫ3α = ∂w 

ÃÙØÒÐÐÑÒØØÒÓÑÔÓ×ØØÐØØÓÒØÔÙ×××ÓÙÙØÒÐÙÓ 

− βα, 

∂xα 

uÓÐÐÓÒÚÒÝÑÐÐ×ÒÐÙ×Ø℄ ÐÚÝØØÚØÌÐÐÒÒÑØØ×ØÓÐØØÑÙ×Ø×ÐÝÚØÑÙÙØÒÒÒÐ ÔÙÑÒÓÐØÙ××ØØ×Ó×ÖØÝÑØÓØØÒÙÓÑÓÓÒÝØÑÐÐÐØØ ÐÒÑÙØØØ×ÓÒ×ÙÙÒØ×××ÚÒÝÑ××ØÝØÝÝÙÓÑÓÐÚÝØÐÒ×ÖØÝÑ 

ǫ33 = 0. 

ǫαβ = ǫαβ(u) + zǫαβ(β), 

ǫ3α = γ(w,β) := ∂w 

ÃÓ×ÐØØÓÒØ×ÓÒÒØÝ×ØÐ××ÚÓÒÚÓÒÐÚÓØÐÒÚÓÑÖ×ÙÐ ÃÓÑÔÓ×ØØÐØÒÓÒ×ØØÙØÚ×ØÝØÝØ 

− βα, 

∂xα 

MαβÑÖØØÒ 

ǫ33 = 0. 

×Ò ØØÒ×ÙØnÖÖÓ×ÓØ×ØÚÓÑÑÓÑÒØØÖ×ÙÐØÒØØËÑÐÐØ ØÖÓÑÐÐÒ×ÒÙÒÒÐÑÒØÒÔ×ÙÙÒÝÐ×ØØÒ×ÙÑÑÑÐÐ ØÒØØN = NαβØÚÙØÙ×ÑÓÑÒØØÖ×ÙÐØÒØØM = ÚÓÒÐ×ØÒÑÝ×ÐÙ×ÚÓÑÒÖ×ÙÐØÒØØS = 

t/2 

n 

zk 

Nαβ = σαβdz = σαβdz, 

−t/2 

zk−1 k=1 

t/2 

n 

zk 

Mαβ = σαβzdz = σαβzdz, 

ØÙÓÑÔÓ×ØØÐØØÑÐÐÒÒÑØØ×ÓÐØÙ××ÒÚÓÑÖ×ÙÐØÒ 

−t/2 

zk−1 

ØØÃÝØØÑÐÐ×ÙÖÚ×ÓÒ×ØØÙØÚ×ØÝØÝØØ ÑÙÓØÓÓÒ 

×ÝÐÐÑÒØ 

k=1 

t/2 

n 

zk 

Sα = σ3αdz = σ3αdz. 

−t/2 

zk−1 k=1 

SαÓÐÐÓÒØÙÐÓ××


Nαβ = 

= 

n 

k=1 

zk 

zk−1 

n 

zk 

Cαβγδ(ǫγδ(u) + zǫγδ(β))dz 

Cαβγδdz 

 

ǫγδ(u) + 

n 

zk 

Cαβγδzdz 

zk−1 

zk−1 

k=1 

k=1 

n 

zk 

Mαβ = Cαβγδ(ǫγδ(u) + zǫγδ(β))zdz 

zk−1 k=1 

n 

 

zk 

n 

zk 

= Cαβγδzdz ǫγδ(u) + Cαβγδz 

zk−1 

zk−1 

k=1 

k=1 

2 

ÖÒÒÚÒÝÑε(β)×ÔÒÒÒÖÚÙÙ×γ(w,β)ÐØÒÒÓÖÑÐÒ ÐÐÐ×Ò×ÙÖÚØÚÓÑÖ×ÙÐØÒØØÓ××××ε(u)ÓÒÐÚÓØÐÒÐÒ ÌØÒÊ××ÒÖÅÒÐÒÐØØÑÐÐÒÔÖÙ×ØÙÚÐÐÓÑÔÓ×ØØÐØØÑÐ 

dz ǫγδ(β) 

ÐÙÙÑ 

ØÐ××ÒÐÒÒÒÖØÐÙÚÙÒØÒ×ÓÖØA,B,D×ØÓ×ÒÖØÐÙÚÙÒØÒ 

N = A: ε(u) + B: ε(β), 

×ÓÖA ∗ÓÒÑÖØÐØÝn 

zk 

n 

Aαβγδ = Cαβγδdz = 

 

ǫγδ(β) 

 

M = B: ε(u) + D: ε(β), 

S = A ∗ 

·γ(w,β). 

(z 2 k − z 2 k−1)C k 

αβγδ, 

(z 3 k − z 3 k−1)C k 

αβγδ, 

(zk − zk−1)C k αβγδ, 

zk−1 k=1 

k=1 

n 

zk 

Bαβγδ = Cαβγδzdz = 

zk−1 k=1 

1 

n 

2 

k=1 

n 

zk 

Dαβγδ = Cαβγδz 

zk−1 k=1 

2 dz = 1 

n 

3 

k=1 

A ∗ n 

zk 

n 

αβ = C3α3βdz = (zk − zk−1)C 

zk−1 k=1 

k=1 

k 3α3β, Ñ××C kÓÒÙÒÒÐÑÒØÒÓÒ×ØØÙØÚÒÒØÒ×ÓÖÐÑ×ØÙÒÐØÒ ÔÓÓÖÒØ××nÐÑÒØØÒÐÙÙÑÖÄ××ÓÒÒC kÓÐØ × 

ØÒz×ÙÙÒÒ××ÚÓ×ÌÝÝÔÐÐÒÒÓÑÔÓ×ØØÐØØÓÒ×ØØØÝÃÙÚ×


Ω 

g f 

z 

zk+1 

z 

y 

zk 

x 

Ê××ÒÖÒÅÒÐÒÒÑÐÐÓÑÔÓ×ØØÐØÐÐ ÃÙÚÃÓÑÔÓ×ØØÐØÒÚÙÚÐÑÒØØÖÒÒ 

zk−1 

ÃÙÒÖ×ÙÐØÒØØÚÓÑ×ÙÙÖØÓÒÒÝØÐÑ×ØÙÚÒÝÑÒÚÙÐÐ×ÒÐ ØÒÓÓÒ×ÒÖÖØÓÑÐÐÓ×ØÚÓÑ×ÙÙÖØØÚ×ØÚÐÐÚÒÝÑ ×ÙÙÖÐÐÒØÖÓÑÐÐÓÓÐØÒÐÙÒÝÐØ×ÌØÒÓÑÔÓ×ØØÐØÒ Ý×Ð××ÓÓÒ×ÒÖ×ΠØÙÐÝÐÐÓÐÚÐÐÑÖÒÒÐÐ 

Π(u,w,β) = 1 

 

N : ε(u)dΩ + 

2 Ω 

1 

 

M : ε(β)dΩ 

2 Ω 

+ 1 Ñ××fÓÒÐÚÓØÐÒÐØØÝÚÙÓÖÑÙÒØÓÐØÒØ×Ó××GÐØÒÑÓ ÖÒÚÙÐÐÐÑ×ØÙÙÒÐÒÖ×ÒÑÐÐÒÒÖÒ℄ ×ÓØØÒÖ×ÙÐØÒØØ ÑÒØØÙÓÖÑØÙ×gÔÝ×ØÝ×ÙÓÖÐØØØÐÒÐØØÝÚÙÓÖÑØÙ×ÃÙÒØÒ ÔÝØÒÐÓÔÙÐØÒÓ×ØÒ×ÖØÝÑ×ÙÙ 

 

 

S·γ(w,β)dΩ − f·udΩ − gwdΩ − G·βdΩ, 

2 Ω 

Ω 

Ω Ω 

Π(u,w,β) = 1 

 

 

ε(u): A: ε(u)dΩ + ε(u): B: ε(β)dΩ 

2 Ω 

Ω 

+ 1 

 

ε(β): D: ε(β)dΩ + 

2 Ω 

1 

 

γ(w,β)·A 

2 Ω 

∗ Ó×ØÔÝ×ØÝØÒØÙÒÒ×ØÑÒØÓ×ÐØÖÚÐØÔÖÒØ×ÒÊ××ÒÖÅÒÐÒ ÐØÒÒÖÝÐÑÑÐØÖÚÐØÐÚÓØÐÚ×ØÚÒÖ×ÒÑ 

·γ(w,β)dΩ 

 

− f·udΩ − gwdΩ − G·βdΩ, 

Ω 

Ω Ω

ØØÚØØÓ×Ò×ÝØÚØÖÑÒÖÒÐÙ××Ø××ÙÓÑØÒØ ÅÌÅÌÌÁÆÆÅÄÄÁ 

ØØÓÑÔÓ×ØØÐØ××ÑÝ×ÔÐÔÝ×ØÝ×ÙÙÒØÒÒÙÓÖÑØÙ×ÙØØ 

ÑÙÓØÓÓÒ ÚÐØØÑ×ØØ×Ó×ÖØÝÑØÓ×ÒÙÒ×ÓØÖÓÓÔÔ×××ÐØ×× ØÙÒÖÓÐÐÓÑÔÓ×ØØÐØÒÚÖÒÐÝÝ××ÒÝ×ÒÖØ×ÑÔÒ ÎÖÒÐÝÝ×ÚÖØÒÑÖØÐÐÒ×ÙÖÚÐØÒÔ×ÙÙÐÐ×ÐØ 

ÅÖØÐÑ ØÙ×Ø×ØÒØØËÐØÒÑÙØØÙØÓÒ×ØÙØÚ×ØØÒ×ÓÖØ×ÙÓÖÑ 

ËÓØØÑÐÐÙÙØÑÙÙØØÙØÝ×Ð×ÒÓÓÒ×ÒÖÒÐÙ××Ò ×ÐÑÐÐÒÖØÖÑÐÐt −3×Ò×ÐØÙÐÐÒÖÐÐÙÙ×Ò ÑÙÙØØÙÒ×ÐØØÙÒØÒ×ÓÖÒÚÙÐÐÐÙ× 

ÐÙ×ØØ ÃÓÓÒ×ÒÖÒÑÒÑÓÒØÚ×ØÚÚÖØÓÑÙÓØÓ×ÒÚÖÓÑÐÐ ÎÖØÓÑÙÓØÓ 

ØÓØØÚ××Ò ÌØÚ ×ÐÚÓØÐØØÐØØÑÙÙØØÙÒ×ÙØÒÓÐÐÓÒÚÖ Ø×(u,w,β) VÔØ⎧⎪ ⎨ 

⎧ 

⎪⎨ 

u → 

⎪⎩ 

1 

β → β, 

w → w 

t u, 

, 

⎧ 

A → 

⎪⎨ 

1 

B → 1 

t A, 

D → 

⎪⎩ 

1 

A∗ → 1 

t 2B, 

t 3D, 

t A∗ 

, 

⎧ 

⎪⎨ 

f → 

⎪⎩ 

1 

g → 1 

G → 1 

t 2f, 

t 3g, 

t3G 

ε(u): B: ε(β)dΩ 

Ω 

Π(u,w,β) = 1 

 

 

ε(u): A: ε(u)dΩ + 

2 Ω 

+ 1 

 

ε(β): D: ε(β)dΩ + 

2 Ω 

1 

2t2 γ(w,β)·A 

Ω 

∗ 

·γ(w,β)dΩ 

− f·udΩ − gwdΩ − G·βdΩ, 

Ω 

Ω 

Ω 

∈ U ×W ×V×ØÒØØ∀(v,ν,η) ∈ U ×W × 

(A: ε(u),ε(v)) + (B: ε(v),ε(β)) = (f,v), 

(B: ε(u),ε(η)) + (D: ε(β),ε(η)) 

⎪⎩ 

+t −2 (A ∗ ·γ(w,β),γ(ν,η)) = (g,ν) + (G,η), 

.

ÅÌÅÌÌÁÆÆÅÄÄÁ Ñ××ÚÖØÓÚÖÙÙØÓÚØU× W × V ⊂ [H1 (Ω)] 2 × H1 (Ω) × [H1 (Ω)] 2 ÓÖÑÙÐØÓ××ÙÒ×ØÐÐÒÐÙ×ÚÓÑ ÎÖÒÐÝÝ×ÚÖØÒÚÓÒÐØØØØÚÑÙÓØÓÐÐÑÝ××ÐÑÒØØ 

ÌØÚ ÖÔÔÙÑØØÓÑÒÑÙÙØØÙÒÌÐÐÒØØÚØÙÐÑÙÓØÓÓÒ Ø×(u,w,β,q) 

q = t −2 A ∗ ·γ(w,β) 

ΓÔØ ∈ U×W ×V ×Γ×ØÒØØÐÐ(v,ν,η,s) ∈ 

U × W × V × 

⎧ 

⎪⎨ 

(A: ε(u),ε(v)) + (B: ε(v),ε(β)) = (f,v), 

(B: ε(u),ε(η)) + (D: ε(β),ε(η)) + (q,γ(ν,η)) = (g,ν) + (G,η), 

⎪⎩ 

t2 (A∗−1 

q,s) + (γ(w,β),s) = 0, Ñ××U× W × V × Γ ⊂ [H1 (Ω)] 2 × H1 (Ω) × [H1 (Ω)] 2 × [L2 (Ω)] 2 

ØØØÚ××ÓÒÓ×ÐÐÖÙÒÐÐÑÖØØÚ××ÖØÝÑÒβÑÓÐÑÔÒ ÇÐÒÒÒÒÓ×ØØÚÓÒÖÙÒØÓÒÑÖØØÑÒÒÒ×ÒÒÒÐØ ÊÙÒØÓÒÑÖØØÑÒÒ 

ØÔÙÑÒÖÚÓÝ×ØÐÑÐÐ×Ò×ÙÖÚØÚ×Ý×Ð××ØÑÖ ÓÑÔÓÒÒØØÒ×ØÔÙÑÒwÖÚÓÌÔÙÑÚÓÒÚÚÐÒØ×Ø 

ØÝ×ÐÐ×ØÖÙÒØÓ ÌÖ×ØÐÐÒÒ×ÒÐØØØØÚÐÐ×ØØØÚÖÙÒØÓÃÖØÝÑÒ 

ÂÝ×ØØÙØÙ××ÒÐÐÑÔÖÙÒÓ××ÒÒØØÒÖÙÒÐÐ 

ÑÝ×ÖÙÒÒÒÓÖÑÐÒØÒÒØÒ×ÙÙÒØ×ÒÓÑÔÓÒÒØØÒβnβτ 

ÈÑ××Ý×ØØÙØÙ××ÒÐ×ÓØÐÑÔÖÙÒÓ××Ò ÚÓØÌÐÐÒÐØÒÐÖÙÒÐÐÓÒ×ØØØÝ ×ØÔÙÑÒwØØÑÓÐÑÔÒÖØÝÑÓÑÔÓÒÒØØÒβnβτÖ 

×ÒÖØ××ØØÙØÙ××ÒÐ×ÑÔÐÝ×ÙÔÔÓÖØÚÒØÔÙÑÒÖ ÑÙÙØÓ×ØÓÐÖÓØØØÙ Ô×ÖØÝÑÒØÙÒÔÐÐÑÙØØØÙÒ×ÙÙÒØ×ØÐÙ×ÑÙÓÓÒ 

ÚÓwÒÒØØÒÖÙÒÐÐÌÑÚ×ØÝ×Ð×ØØÐÒÒØØÓ×× ÐØØÔ×ÚÔ×ØÖØÝÑÒØÙÒÔÐÐØÙÒ×ÙÙÒØÒÒÐ Ù×ÑÙÓÓÒÑÙÙØÓ×ÓÒ×ÐÐØØÙ 

ÒØØÒÒÓ×ØÒwÖØÝÑÓÑÔÓÒÒØØβnÌÐÐÒÐØØ

ÚÃÓÚ××Ý×ÒÖØ××ØØÙØÙ××ÒÐÖ×ÑÔÐÝ×ÙÔÔÓÖØÖÙ ÅÌÅÌÌÁÆÆÅÄÄÁ 

ÒÓ××ÒÒØØÒØÔÙÑÒwÐ××ÖØÝÑβτÓÐÐÓÒÐØØ 

ÚÎÔ××ÖÙÒÓ×××ÖØÝÑÖÓØØÑØÒÒ ÒÒÐÙ×ÑÙÓÓÒÑÙÙØÓ×ÓÒ×ØØØÝ Ô×ÐÐÒÚÔ×ØÖØÝÑÒØÙÒÔÐÐÑÙØØØÙÒ×ÙÙÒØ 

uÒÙÑÔÒÓÑÔÓÒÒØØÖ×ÒØÚÚÐÒØ×ØuØÒÒØÐ ÑÒÒÚ×ØÝÒÒØÝ×ØÓÐÐÓÒÑÙÓÓÒÑÙÙØÓ×ÓÒ×ØØØÝÖÙÒÐÐ ÒÓÖÑÐÓÑÔÓÒÒØØÒuτunÅÓÐÑÔÒÓÑÔÓÒÒØØÒÒÒØØ Ì×ÓÐ×Ø×ÙÙ×ØØÚÒÓ×ÐØÚÓÒÙÐÐÒÖÙÒÒÓ×ÐÐÒÒØØ 

ÈÐÒÒÓÖÑÐÓÑÔÓÒÒØÒÒÒØØÑÒÒÚ×ØÝ×Ð××ØÐÙÙØÙ ØÒÒØÐÓÑÔÓÒÒØÒØ×ÒÖÙÐÐØÙÒØ 

ÓÒÝ××ØØÒÒÖÔÔÙÑØØÐØÒÓÓÒ×Ô×ÙÙ×ØtÑÐÐØØ Ì××ÔÔÐ××ÒÝØØÒØØÝ×ØØÝÐÐÑÐÐÐÐÐÝØÝÝÖØ×ÙÓ ÊØ×ÙÒÓÐÑ××ÓÐÓ 

ÓÒÝ×ØØÙØØÙØ×Ó×ÖØÝÑØ×ØØØÝÖÙÒÐÐÂØÓ××ÓÐØØÒÚÖ ÒÐÝÝ×××ÒÒÑÖÙÒÓØÚÙ×ØÖÚØÒ×ÃÓÖÒÒÔÝ ØÐØØÄÜÅÐÖÑÒÐÑÑÓØÑÙÓØÓÐÐÒ×ÙÖÚ×ÐÝÝ×Ø 

ÖÐÐØÒ×ÓÖÐÐÝØØÒÝÐÐÓÐÚÒÓÖÑÓÑÔÓÒÒØØØÒ 

ËÓÓÐÚÚÖÙÙÒÐÓÐÐf ∈ 

ÄÙ× (Ω)ÒÓÖÑÎ×ØÚ×ØÚØÓ 

VÓÖØ× RÖÓØØØÙ 

Ñ××αÓÒÑÙÐØÒ× · 0ÒÓÖÑÐL ÄÜÅÐÖÑÐÑÑÇÐÓÓÒVÀÐÖØÒÚÖÙÙ×a(·,·) : 

ÌÓ×ØÙ×ÄÙ×ÒØÓ×ØÙ×ÐÝØÝÝ×ÑÖ×ÚØØ×Ø℄ ÔÝØÐÓÖ×Ø×ÒÓØØÙÒÑÖØ×ØØÖÒØÒÒÓÖÑÖÔÔÙÙ ÓÐÐÐ××ØÚÒÖÒØÒ×ÝÑÑØÖ××ØÓ××ØÑÐÝÒÔÔÐÒÐ ÓÒ×ØØØÝÔÝØÐÒØÓ×ØÙ×ÝÐ×××ØÔÙ×××ÓÒÙØÒÒÚÖ×Ò ÌÓÒÒÚÐØØÑØÒÔÙÚÐÒÐ×Ø×ÙÙ×ØØÚÒØÙØÑ×××ÓÒÃÓÖÒÒ 

ÙØ×ÙØÒÙ×ÒÑÝ×ÃÓÖÒÒØÓ××ÔÝØÐ× 

ÒÐ×ÒÔÒÙÙØØ××ØÝ××ËÙÖÚ××ÑÙÓØÓÐØÙÚÖ×ÓØ 

Hm (Ω)ÝØØÒËÓÓÐÚÒÓÖÑ 

f 2 m = 

∂ 

|α|≤m 

α f 2 0, 

2 ÚÖØÓØØÚÒ 

V × V → RØÙÚVÐÐÔØÒÒÐÒÖÑÙÓØÓL:V → ÐÒÖÒÒÙÒØÓÒÐÌÐÐÒÐÝØÝÝÝ××ØØÒÒu ∈ 

a(u,v) = L(v), ∀v ∈ V.

ÄÙ× ÅÌÅÌÌÁÆÆÅÄÄÁ ÃÓÖÒÒÔÝØÐÇÐÓÓÒΩ ∈ R3ÚÓÒÖÓØØØÙÓÙ ÒÓÐÐÑØÐÐÒÒÓÙÓÓÐÐÓÒÑÖØØÝÒÓÐÐÖÙÒÓØÌÐÐÒÐÝØÝÝÚ 

Γ0)×ØÒØØÔØ 

∂ΩÓÒ 

ÄÑÑ ÐØØÑÙÙØØÙÒ×ÙØÒÌØÚÖØÒØÖÚØÒ×ÙÖÚØÙÐÓ× ÆÝØØÒ×ÙÖÚ×ØØÌØÚ ÇÒÓÐÑ××ÔÓ×ØÚ×ØÚÓØC1,C2×ØÒØØÓ×ÐÐ ÓÒÐÐÔØÒÒ×ÐÚÓØØ 

Ñ××A,B,DÓÚØÐÐÑÖØÐÐÝØ×ÐØÙØØÒ×ÓÖØÓÐØØÒØØ 

ÐÓÒÓØØÒÙÓÑÓÓÒBÒ×ÝÑÑØÖ×ÝÝ×ÔØ ÌÓ×ØÙ×ÃÖÓØØÒÒ×ÒÖÚÓØÚÐÙ×ÓÖÑÐÒÑØÖ×ØÙÐÓÒÓÐ 

Ñ×× 

×ØÚÒØØÌÖ×ØÐÐÒÝØÒÐÑÒØØÖÖÓ×ÒÐØØÝÚÑØÖ× ÖÓØØÒ×ÒÑØÖ×ÒØÙÐÓÒÑÖØÐÑÒ ÌÐÐÒ××ÐÑÑÒÚØÔØÑÐÖÖÓÒÑØÖ×ÓÒÖÓØØØÙÔÓ 

 

ÚÙÐÐÑÙÓØÓÓÒ 

ÆÒÑØÖ×ÒØÙÐÓ×ÐÚ×ØÓÑÑÙØÓ×ÐÐØÓÒÒÓÒÐÓÓÓÒÐ ÑØÖ×ÓØÒ×ÝÑÑØÖ×ÝÝ×ÔÓ×ØÚÒØØ×ÝÝ×ÔØØÙÐÓÐÐÑÐ× ÔØÙÑÑÐÐÒÑØÖ×ÐÐÖ×ÒÃÓ×ØÒ×ÓÖCkÓÒÓÐØÙ×ÒÑÙ 

 

Ò×ÝÑÑØÖÒÒÔÓ×ØÚÒØØÖØØØÖ×ØÐÐÓÑÒ×ÖÚÓ×ÝÑ 

2ÑØÖ×ÐÐ 

ÑØÖ×ÐÐ2 × 

ÓÓÒÖÙÒÓÒÔÐÓØØÒ×ÐÇÐØØÒÐ××ØØΓ0 ⊂ ÓC= C(Ω, 

 

ε(v): ε(v)dΩ ≥ Cv 

Ω 

2 1, ∀v ∈ [H 1 Γ0 (Ω)]3 

. 

×ÝÑÑØÖ×ÐÐØÒ×ÓÖÐÐτ,σ ∈ [L2 (Ω)] 4ÔØ 

C1(τ 2 0 + σ 2 0) ≤ (A: τ,τ) + 2(B: τ,σ) + (D: σ,σ) ≤ C2(τ 2 0 + σ 2 0), ÓÒ×ØØÙØÚÒÒØÒ×ÓÖC∈ [L2 (Ω)] 4×4 

 

(A: τ,τ) + 2(B: τ,σ) + (D: σ,σ) = τ σ 

 

A B τ 

, 

B D σ 

Ak Bk 

Bk Dk 

 

= 

 

A B 

= 

B D 

n 

 

k=1 

 

1 

t (zk 

1 

− zk−1) 

1 

Ak Bk 

Bk Dk 

2t2(z2 k − z2 k−1 ) 

2t2(z2 k − z2 k−1 ) 1 

3t3(z3 k − z3 k−1 ) 

 

 

. 

Ck 0 

0 Ck


2ÑØÖ×ÐÐÝØÝ××Ø℄ ÇÑÒ×ÖÚÓØ×Ò2 × 

Ñ××ÒÚÖÒØØÓÚØÑØÖ×ÒØÖÑÒÒØØÐ 

Rk := 

1 

t (zk − zk−1) 

1 

2t2(z2 k − z2 k−1 ) 

1 

2t2(z2 k − z2 k−1 ) 1 

3t3(z3 k − z3 k−1 ) 

 

. 

λ1(Rk) = tr(Rk) 

 

1 − 1 − 

2 

λ2(Rk) = tr(Rk) 

 

2 

1 + 1 − 

4 det(Rk) 

tr(Rk) 2 

 

, 

4 det(Rk) 

tr(Rk) 2 

det(Rk) = 1 

3t4(z3 k − z 3 k−1)(zk − zk−1) − 1 

4t4(z2 k − z 2 k−1) 2 = 1 

12t4(zk − zk−1) 4 , 

tr(Rk) = 1 

t (zk − zk−1) + 1 

3t3(z3 k − z 3 k−1). ÅÖØÒÐÑÒØÒÔ×ÙÙØØhk = 

det(Rk) = h4k > 0 

12t4 Ú×ØÚ×ØÐÐÐtr(Rk) ≥ 1 

t (zk − zk−1) = hk Ñ×Ø×ÙÖØØÓÑÒ×ÖÚÓØÓÚØÖÐ×ØÔÓ×ØÚ×Ø×ÐÐ > 0, 

t 

4 det(Rk) 

0 < 

tr(Rk) 2 ≤ h2 −t/2]ÓØÒÐÐÐÔØÖÚÓ 

k 

< 1. 

3t2 Ä××ÐÐkÔØzk ∈ [t/2, 

tr(Rk) ≤ hk 1 t 

+ 

t 3t3(( 2 )3 − ( −t 

2 )3 ) ≤ 13 

12 . 

×ÝÑÑØÖÒÒÔÓ×ØÚÒØØÓØÒÚØÔØ 

ÒØØÓÒÑÝ×ÒÑ×ÙÑÑÑÐÐ×ØÙÓÓÓÑÔÓ×ØØÐØÒÑØÖ× Ó×ØÐÑÒØØÖÖÓ×ØÚ×ØÚÑØÖ×ÓÒ×ÝÑÑØÖÒÒÔÓ×ØÚ ÇÑÒ×ÖÚÓØÓÚØÑÝ×ÖÓØØÙØÚØÒÖÔÙÐØÒÔ×ÙÙ×ØÃÓ× 

 

zk − zk−1ØÖÑÒÒØÐÐÔØ 

,

ÅÌÅÌÌÁÆÆÅÄÄÁ ËÓØØÑÐÐØÙÐÓ×ÒÑÙÙØØÙuβÚ×ØÚØÚÒÝÑØÒ×ÓÖØ×Ó 

ÚÐØÑÐÐÃÓÖÒÒÔÝØÐ ÑÓÐÑÑÐÐÑÙÙØØÙÐÐ×ÒØØ 

ØØÒÒÝØÖØ×ÙÒÓÐÑ××ÓÐÓÝ××ØØ×ÝÝ×ÝÝÒØÑÐÐ×ØÙÐ Ñ××ÚÓCÖÔÙÐØÒÔ×ÙÙ×ØtÐÐ×ØÒÐÙ×ÒÚÙÐÐÒÝ Ô×ØØØÚÐÐØØØÝÖÞÞÒÙ×ÒØÓÖ℄ÅÖØÐÐÒÚÐ ÚÖÙÙ×H −1 ØÝÐÐÐØÒÔ×ÙÙÐÐtØØÚÐÐ ÄÙ× ÊØ×ÙÒÓÐÑ××ÓÐÓÝ××ØØ×ÝÝ×ÂÓ×ÐÐÒÒØ ÓÒÖØ×Ù(u,w,β,q) ∈ 

ÚÐÐÐÐÔØ×ÝÝ×ØÙÐÓ× C(u 2 1 + β 2 Ω)×ÙÖÚ×Ø 

1) ≤(A : ε(u),ε(u)) + 2(B : ε(u),ε(β)) 

+ (D : ε(β),ε(β)), 

(div, 

H −1 (div, Ω) = {q ∈ [H −1 (Ω)] 2 | div q ∈ H −1 (Ω)}. 

[H1 0(Ω)] 2 × 

H1 0(Ω) × [H1 0(Ω)] 2 × [L2 (Ω)] 2 ÌÓ×ØÙ×ÅÖØÐÐÒÐÒÖÑÙÓØÓΥÐÒÖ×ØÓÔÖØØÓÖØBL 

Υ(u,v;β,η) = (A : ε(u),ε(v)) + (B : ε(v),ε(β)) 

+ (B : ε(u),ε(η)) + (D : ε(β),ε(η)) 

L(v,ν,η) = (f,v) + (g,ν) + (G,η) 

B(w,β;q) = (γ(w,β),q) ÖÓØØÒØØÚÒ×ÓÖÑÙÐØÓ×ÙÖÚÒÑÙÓØÓÓÒØ×(u,w,β,q) ∈ 

[H1 0(Ω)] 2 ×H 1 0(Ω) ×[H 1 0(Ω)] 2 ×[L2 (Ω)] 2×ØÒØØ∀(v,ν,η,s) ∈ [H1 0(Ω)] 2 × 

H1 0(Ω) × [H1 0(Ω)] 2 × [L2 (Ω)] 2ÔØ 

⎧ 

⎨ 

Υ(u,v;β,η) + B(ν,η;q) = L(v,ν,η), 

⎩ 

t2A∗−1 ÐÑÒÓÑÐÐØ×ØÐÙ×ÚÓÑÔ×ØÒ×ÖØÝÑÒ×ÙØÒÖÓØØØÙÙÒ 

(q,s) + B(w,β;s) = 0. ØØÚÒØ×(u,w,β) ∈ [H1 0(Ω)] 2 ×H1 0(Ω)×[H 1 0(Ω)] 2×ØÒØØ∀(v,ν,η) ∈ 

[H1 0(Ω)] 2 × H1 0(Ω) × [H1 0(Ω)] 2ÔØ 

Υ(u,v;β,η) + 1 

t2(A∗γ(w,β),γ(ν,η)) = L(v,ν,η). ÌÒ×ÓÖA ∗ÓÒ×ÝÑÑØÖÒÒÔÓ×ØÚÒØØ×ÐÐÓÑÒ×ÙÙ××ÙÖ×ÙÓ ÖÒÓÒ×ØØÙØÚ×ÒØÒ×ÓÖÒCÔÓ×ØÚÒØØ×ÝÝ×ØØ×ØÑÐÐØÒ×Ó 

= 0ÑÙÙÐÐÓÒτ = 0ÌÒ×ÓÖÒA ÖÐÐτÓÐÐτ3α3β ∗ÔÓ×ØÚÒØØ×ÝÝ ÄÜÅÐÖÑÒÐÑÑÒÔÖÙ×ØÐÐÖØ×ÙÓÒÓÐÑ××Ý××ØØÒÒ 

ÒÔÖÙ×ØÐÐÓÓÚ×ÒÔÙÓÐÓÒ××ÓÖ×ÚÒÒÓ×ÐÐt>0ÓÐÐÓÒ

ÅÌÅÌÌÁÆÆÅÄÄÁ ÅÝ×ÐÙ×ÚÓÑÒÓ×ÐØÖØ×ÙÓÒÓÐÑ××Ý××ØØÒÒÌ 

ÑÚØÒØÖ×ØÐÑÐÐÝØÐÖÝÑÒ ÚÑ×ØÖÚÓ×ØÒ ÒØØÝ×××ÓÒÐÐÔØÒÒØØÚÚÖÙÙ××Γ×ÐÐA ∗Ò×ÝÑÑØ 

ΓÓÒÓÐÑ××ÚÓ 

ÔÙ××××ÓÖÑÙÐØÓ×Ø×ÒÃÖÓÒÐØØÑÐÐÚ×ØÚØ 

0ÐÙ×ÚÓÑÒ×ÒÒÐÐ×ÝÝØØÚÓÐÐÓÒÖØ 

Ö×ÝÝÒÔÓ×ØÚÒØØ×ÝÝÒÔÖÙ×ØÐÐ∀q ∈ 

ÌÔÙ×××t → 

Ù×ÚÓÑÒ×ÙØÒÊØ×ÙÒÓÐÑ××ÓÐÓÓÒÚØÒØÐÐÒ℄ÐÒ ÎÐØØÑ×ØÒÒØØÝ×××ÓÒ×ØÙÐÔ×ØØØÚ×ÖØÝÑÒÐ ÖÑÙÓÓÒΥÐÐÔØ×ÝÝ×ÓÔÖØØÓÖÒBÒÓÐÐÚÖÙÙ×× 

ØÔÙÑÒÒØØÒÓÐÐÒÖÙÒÒÓ×ÐÐΓ0ÓÒÓÐÑ××ÚÓC×ØÒ ÃÓ×∇ν −η ØØÙÒ(ν,η) ∈ 

ÑÑÙÙØØÙÒ×ÙØÒBÒÒÓÐÐÚÖÙÙ××ÐÐÝØÝÝÚÓC×ØÒØØ ÇØØÑÐÐÐ××ÙÓÑÓÓÒÔÝØÐ ÓÒΥÐÐÔØÒÒÒ×ÖØÝ 

ÌÓÒÒÚÐØØÑØÒÖØØÚØÓÓÒÒÒÙØ×ÙØØÙÙ×ÖÞÞ 

ÌÑÒÒÖÚÓÑÐÐÚ×ÒØÔÙÓÐØ×ÙÖÚ×Ø 

0×ØÒØØ C > 

t 2 (A ∗−1 q,q) ≥ Ct 2 q 2 0. 

ØÚØ×(u,w,β,q) ∈ [H1 0(Ω)] 2 × H1 0(Ω) × [H1 0(Ω)] 2 × H−1 Ω)×ØÒ (div, 

ØØ∀(v,ν,η,s) ∈ [H1 0(Ω)] 2 × H1 0(Ω) × [H1 0(Ω)] 2 × H−1 Ω)ÔØ (div, 

⎧ 

⎨ 

Υ(u,v;β,η) + B(ν,η;q) = L(v,ν,η) 

⎩ 

B(w,β;s) = 0. 

Ker B = {(ν,η) ∈ H 1 0(Ω) × [H 1 0(Ω)] 2 |(∇ν − η,s) = 0, ∀s ∈ H −1 (div, Ω)}. 

∈ H−1 BÔØ Ω)Ð××ÈÓÒÖÒÔÝØÐÓÒÚÓÑ××ÙÒ 

(div, 

Ker 

×ØÐ×ÙÙ×ØÓ℄sup (ν,η)∈H1 0 (Ω)×[H1 0 (Ω)]2 

ν1 ≤ |ν|1 = ∇ν0 

≤ η − ∇ν0 + η0 = η0 

≤ Cη1. 

Υ(v,v;η,η) ≥ C(v 2 1 + η 2 1 + ν 2 

1). 

(∇ν − η,s) 

ν1 + η1 

≥ Cs H −1 (div,Ω)


(∇ν − η,s) 

sup 

(ν,η)∈H1 0 (Ω)×[H1 0 (Ω)]2 ν1 + η1 

≥ sup 

(ν,η)∈H1 0 (Ω)×[H1 0 (Ω)]2 

ÃÙÒÒÑÓØÓÚØÚÓÑ××ÑÝ×ÖØ×ÙÒÝ××ØØ×ÝÝ×ÖØÔÙ 

(ν, div s) (η,s) 

0×ÙÖ×ÙÓÖÒ×ØÙÐÔ×ØØÓÖ×Ø 

(C1 + C2 ) 

ν1 η1 

= C1s−1 + C2div s−1 

≥ CsH −1 (div,Ω). 

ØØÚÒ×ÒÒÐÐ×ÝÝÒÑÖØØÑÒÒÑ××ØÖÚØÒÚÙ×ÑÙÙØÑ ÃÓÑÔÓ×ØØÐØØØØÚÒ×ÒÒÐÐ×ÝÝ××ØÑØØÒ×ØÚÓ×ÓÒÐØØ ÊØ×ÙÒ×ÒÒÐÐ×ÝÝ× ×××t 

ÑØÑØØ×ÔÙÚÐÒØÌ×ÓÐ×Ø×ÙÙ×ØØÚÒ×ÒÒÐÐ×ÝÝ×ÓÒÔØÖ 

→ 

××Ø ÚÐ×ÙÖÙ×ÖÑÓÒ×ÒËØÓ×ÒØØÚÒ×ÒÒÐÐ×ÝÝ×ÓÑÒ×ÙÙ ÀÐÑÓÐØÞÒÓØÐÑÐÙ×ÚÓÑÐÐ ÐÙ××ØØÒL 2 (Ω)H −1 ÑÒØØÝ×ØÙÒØÓ×Ø℄ÊÓÓØØÓÖÓÒÒÝØÑÖØÐØÝ×ÐÖÖÚÓ×ÐÐ ÓØÐÑÓÒÚÙÐÐÒÚÓÒÖÓØØÖÒØÒÖÓÓØØÓÖÒ×ÙÑ ×ÙÙÖÐÐp×ØÒØØ Ω)ÙÒØÓÐÐÒÒÙØ×ÙØØÙÀÐÑÓÐØÞÒ 

(div, 

rot p = (∂2p, −∂1p). ÄÙ×ÀÐÑÓÐØÞÒÓØÐÑÇÐÓÓÒq ∈ H−1 Ω)ÌÐÐÒÐÝØÝÝ 

(div, Ý××ØØ×Øψ∈ H1 0(Ω)p ∈ L2 (Ω)/R×ØÒØØÔØ 

⎧ 

⎨ 

q = ∇ψ + rot p 

⎩ 

q2 H−1 (div,Ω) = ψ21 + p2 ÌÓ×ØÙ×ÇØØÑÐÐ ÒÒ×ÑÑ××ØÝØÐ×ØÚÖÒ××ÔÙÓÐØØÒ 

0. ÙÓÑÓÑÐÐØØdiv q ∈ H−1 (Ω)×Ò 

div q = div ∇ψ + div rot p = ∆ψ ⇔ 

(∇ψ, ∇v) = (q, ∇v) ∀v ∈ H 1 0(Ω).


ÓÖØÓÓÒÐ×ØÓÐÐÓÒÒÓÖÑ×ØÑØØÔØ 

/RÃÓ× ÃÓ×q∈ H ×ÙÄ××div (q 

ÑÒÚÙÐÐÚÓÑÑØÓ×ØÐÓÔÙÐØÓÓØØÚÐÐ×ÙÖÚÒ×ÒÒÐÐ×ÝÝ× 

ÖÓØØÖÓÓØØÓÖÒÚÙÐÐÔØq − 

×ØÑØÒ×ÙÖØÒÐØØØØÚÒÓ×ÐØÚØØÒ℄×ØÝ×ØËÓÚÐÐØÒ ××ÀÐÑÓÐØÞÒÓØÐÑÐÙ×ÚÓÑÐÐ 

Î×ØÚ×ØÚØØÒ℄ÔÖÙ×ØÐÐÙÒq∈ [L ÓØÐÑ×ØÒØØ(ψ,p) ∈ 

Ú×ØÚÐÐØ×ØÙØÓÐÐ 

ÌÐÐÒ×ÓØØÑÐÐÐÙ×ØØØÚÒ ÒØØÙÓÖØÓÓÒÐ×ÙÙ× ÙÓÑÓÑÐÐÐÐÑ 

×ÒÚÚÐÒØØØØÚÓ××ÐØÓÐÑÝØØØÝØØÚÒ×Ñ 

s 

ÑÙÓØÓ ×ÑÑ×ÒÝØÐÒÑÙÓÓ×Ø××ËØÓ×ÒØØÚÑÙ×ØÙØØÚÒØØ ÑÒÒÝØÐÓÒØÚÐÐÒÒÈÓ××ÓÒÒØØÚ×ÑÓÒÙÒÚÑÒÒÒ 

ε(β),ε(η))ÓÐÐÓÒÐØØØØÚÐÐ×Ò 

(∇ψ, 

×ØÒØØÔØ 

ÚÒÅÖØÒa(β,η) = (D: ÌØÚØ×(β,w,ψ,p) ∈ 

(∇ϕ, ∇w) = (∇ϕ,β) − t2A∗−1 (∇ψ, ∇ϕ) ∀ϕ ∈ H1 

Ñ××ÚÓÑgÓÒÐØÒÔÓØØ××ÙÙÒØ×ØÙÓÖÑØÙ×ØÚ×ØÚÚÓÑ 

GÓÒÐØÒÑÓÑÒØØÙÓÖÑØÙ× 

0(Ω), 

−1 Ω)ÓÒψØÑÒÈÓ××ÓÒÒØØÚÒÝ××ØØÒÒÖØ 

0ÃÓ×ÚÖÒ××ØÒÙÒØÓÚÓÒÒ 

(div, 

− ∇ψ) = 

∇ψ = rot pÑ××p∈L 2 0ÑÐÚÐØ×ÐÐpÓÚØÓØÐÑÒÓ×Ø××ØÙÐÓÒ×ÙØÒ div rot p = 2 (Ω)] 2ÐÝØÝÝÀÐÑÓÐØÞÒ 

H1 (Ω)×[H 1 (Ω)∩L 2 0(Ω)]ÀÐÑÓÐØÞÒÓØÐ 

 

 

q = ∇ψ + rot p, 

= ∇ϕ + rot q. 

rot p) = 0, 

[H 1 0(Ω)] 2 ×H 1 0(Ω)×H 1 0(Ω)×[H 1 (Ω)∩L 2 0(Ω)] 

⎧ 

(∇ψ, ∇ν) = (g,ν) ∀ν ∈ H 

⎪⎨ 

1 0(Ω), 

a(β,η) − (rot p, η) = (∇ψ,η) + (G,η) ∀η ∈ [H1 0(Ω)] 2 , 

t 

⎪⎩ 

2A∗−1 (rot p, rot q) = (rot q,β) ∀q ∈ [H1 (Ω) ∩ L2 0(Ω)],

ÅÌÅÌÌÁÆÆÅÄÄÁ ËÒÒÐÐ×ÝÝ×ÖÙÒÐÐ××ÐÙ×× 

ÊÐÐt → ÖØ×Ù(w 0 ÌÐÐÒÓÓØØÚÒÖØ×ÙÚÓÒÖÓØØÑÙÓØÓÓÒ 

ÃÓ×w 0ÓÒÃÖÓÒÐØØØØÚÒÖØ×ÙÚÓÒ×ÓÚÐØØÙÒÒØ ØÙ×ØÑØØ℄ÓÐÐÓÒÓÒÚ×××ÐÙ××ΩÔØ 

ËÑÓÒÌØÚÒÒ×ÑÑ×ÐÐÓ×ØØÚÐÐ×ÒÈÓ××ÓÒÒØ ØÚÒH 2×ÒÒÐÐ×ÝÝ×ØÓÒÚ×××ÐÙ×××ØÑØØ 

ÇØØÑÐÐÙÓÑÓÓÒØØ××ÑÒ×Ó××ÖÓÓØØÓÖÖÒØØÓÚØ ØØÚÒ ÙÙÒÑÙÙØØÙÒ˜ηÚÙÐÐ×ØÒÖÒËØÓ×ÒØØÚÒÑÙÓØÓÓÒ℄Ø× 

π/2ÖÓØØÓØÚÐÐ×ÑÓÔÖØØÓÖÚÓÒÑÖØØÑÐÐ˜η = ××ÑÑ×ØÝØÐÖÓØØÖØÔÙ×××t=0 

Ä×××ÐÚ×ØÔØ ˜ ÒÐÐ×ÝÝ×ØÙÐÓ× ØØÚÐÐ ÙÓÑÓÑÐÐ ×ÒÖØØÚÐÐt=0×Ò 

ÃÓ×ÔÖØβ0,p0β,pØÓØÙØØÚØØØÚÒ 0×ÒØÙÐÓ× ØÓ×ÒÓÐÑÒÒÒ ÖÚÒÚ×ØÚ×ØØÔÙ×××t = 0t 

0ÔØB(w,β;s) = 0, ∀s ∈ ΓÓÐÐÓÒÖØØÚÒt=0 

,β0 )ØÓØÙØØÃÖÓÒÐØØØØÚÒÖØ×ÙÐÐÔØ 

β 0 = ∇w 0 . 

w = w 0 + w rβ= β 0 + β r . 

w 0 3 ≤ Cg−1. 

ψs ≤ Cgs−2, s = 1, 2. 

(η2, −η1) 

( ˜ β0,p0) ∈ [H1 0(Ω)] 2 ×[H1 (Ω)∩L2 0(Ω)]×ØÒØØ∀(˜η,q) ∈ [H1 0(Ω)] 2 ×[H1 (Ω)∩ 

L2 0(Ω)]ÔØ⎧ ⎨ 

a( 

⎩ 

˜ β0, ˜η) − (p0, div ˜η) = (∇ψ, ˜η) + (G, ˜η) 

(q, div ˜ βsËÓÚÐØÑÐÐ×ØÒÖ×ØÑØØ℄ 

β0) = 0. 

βs = 

β02 + p01 ≤ C(G0 + ψ1) ≤ C(G0 + g−1). 

= 

a(β0 − β,η) − (rot (p − p0),η) + (β − β0, rot q) 

+ t 2 A ∗−1 (rot (p − p0), rot q) = (β, rot q) + t 2 A ∗−1 (rot (p − p0), rot q) 

= t 2 A ∗−1 (rot p0, rot q),

ÅÌÅÌÌÁÆÆÅÄÄÁ Ó×ØÚÐØ×ÑÐÐØ×ØÙÒØÓ×η= β 

ÀÙÓÑÓÑÐÐÐ××(rot p, rot q)ØÖÑÒH 2×ÒÒÐÐ×ÝÝ××ÒØÙÐÓ× 

ÌÑÒÔÖÙ×ØÐÐ×× p1 ÃÓ×ÌØÚÒØÓÒÒÝØÐÓÒH 2×ÒÒÐÐÒÒÐÐÔØÒÒØØÚÔ Ø×ØÒÖ×ØÑØØÒÝØÐÒ ÔÖÙ×ØÐÐ 

ÄÓÔÙ×ØÖ×ØÐÑÐÐÓÐÑØØÖÚÑÙ×ØÑÐÐØØβ0ÓÒØØÚÒ 

rot q)ØÖÑÒH 2×ÒÒÐÐ×ÝÝÒÒÓÐÐÖÚÓ 

ÃÓÓÑÐÐÝØÒÝØÐÒ ØÒ×ØÑØØÒ ØÙÐÓ×ØÔÝ 

ÌÖÚØÒÚÐ×ØÑØØÔÓØØ××ÖØÝÑÒÓ×ÐÐw r wrØÓØÙØØÌØÚÒÒ×ÑÑ×ÒÖÚÒÔÖÙ×ØÐÐÚÑ×ÒÝØÐÒ ÑÙÓÓ×× 

ÓÐÐÔØ×ØÒÖÈÓ××ÓÒÒØØÚÒ×ØÑØØ 

×ØÑØØ 

(∇w 

ÚÐÐ×ÙÖÚ×ÒÒÐÐ×ÝÝ×ØÙÐÓ× ÄÙ× ÃÓÒÚ×××ÐÙ××ΩÖØØÚÒ×ÐÐÐÙÓÖÑÐÐØØÚÒ Ý×ØÑÐÐ×ÒÐÓÔÙÐØÓÓÐØØØØ 

 

p0×ÙÖ×ØÑØØ 

− β0q= p − 

β − β0 2 1 + t 2 p − p0 2 1 ≤ Ct 2 p01p − p01. 

 

β − β01 + tp − p01 ≤ Ctp01 ≤ Ct(G0 + g−1). 

 

≤ C(G0 + g−1). 

β2 ≤ C(p1 + ψ1 + G0) ≤ C(G0 + g−1). 

t = 0ÖØ×Ù×Ò(rot p, 

p2 ≤ Ct −2 β − β01 ≤ Ct −1 

 

(G0 + g−1). 

ψ1 + β2 + p1 + tp2 ≤ C(G0 + g−1). 

= w − w0ÆÝØ r , ∇ϕ) = (β − β0, ∇ϕ) + t 2 A ∗−1 (g,ϕ), 

w r 2 ≤ C(β − β01 + t 2 g0) ≤ C(tG0 + tg−1 + t 2 

ÖØ×ÙÐÐÔØ 

g0). 

w 0 3 + t −1 w r 2 + β2 + ψ1 + p1 + tp2 

≤ C(g−1 + tg0 + G0).

Ä××××ÐÙ××ÚÓÒÓØ×ÙÖÚÔÖÑÔ×ÒÒÐÐ×ÝÝ××ØÑØ ÅÌÅÌÌÁÆÆÅÄÄÁ 

×ÐÙ××ΩiÔØ Ø℄ ÄÙ× ÐÐÑÒØØÙÙÒÓÒÚ×ÒÐÙ×ÒΩÓÑÔØ×ØÙÔÓØØÙ× 

w 0 s+2,Ωi + t−1w r s+1,Ωi + βs+1,Ωi + ψs,Ωi + ps,Ωi + tps+1,Ωi 

+t 2 ps+2,Ωi ≤ C(gs−2 Ð×Ø×ÙÙ×ØØÚÐÐÖÐØÒÖÙÒÓÐÐÌØÚÒÓÑÙÓØÓÓÒ 

ÌØÚ ÄÓÔÙ×ØÖÚØÒÚÐ×ÒÒÐÐ×ÝÝ××ØÑØØÝØÑØØÑÐÐØ×Ó 

+ tgs−1 + Gs−1). 

ÑØØÒ×ÓØÖÓÓÔÔ×××ØÔÙ×××ØÙÐÓ× ×ÝÝÒÒÓÐÐÎØØ××℄ÓÒÒÝØØØÝÒÓÙØÙÒÖÑÓÒ×ÒÝØÐÒ ÌØÚÓÒÐÐÔØÒÒØÒ×ÓÖÒA×ÝÑÑØÖ×ÝÝÒÔÓ×ØÚÒØØ Ø×u ∈ 

[µ1,µ2]ÅÐÓÒ×ØØÙØÚÒÒØÒ×ÓÖÓÐØØÒÖØØÚÒ×ÒÒÐÐ××ÓÒ ×ÝÝØÓÐØØØØÑÝ×ÌØÚÐÐ ÃÝØØÝÒØØÚÒ×ÒÒÐÐ×ÝÝ× ÔØÓÒÚ×××ÐÙ××ÖÚÓ Ñ××ÖÖÓÒCÖÔÙÄÑÒÚÓ×ØλÙÒØÓÒÒÄÑÒÚÓµ∈ 

ÐÐÒÝØØØÒØØÑÓÐÑÑØÓ×ØØÚØÓÚØÖ×ÒH 2×ÒÒÐÐ× 

ÓØØÑÐÐØÖÑΥ(·,·;·,·)ÙÑÙÓÓÒ ÃÝØØÝÒØØÚÒ×ÒÒÐÐ×ÝÝ×ÓÒÙØÒÒÔØÖÚÐÓÑÒ×ÙÙ×ÔÝ× Ó×ÚÐÐÖÓØÙ×ÐÐÃÝØØÝ××ÑÐÐ××ÌØÚÒØÓÒÒÖÚ×Ö ØÝØÒÒÝØØÑÒØÓ×Ø××ÚÒØØÝÐÐØØÚÒ×ÒÒÐÐ×ÝÝ×ÚÓØ 

ÒÝØÐÖÝÑÑÙÓØÓÓÒ ËÖØÑÐÐÝØÒØØÖÑØÚ×ÑÑÐÐÔÙÓÐÐÐÓ×ØØ×ÒØÖÓÑÐÐ× 

[H1 0(Ω)] 2×ØÒØØ∀v ∈ [H1 0(Ω)] 2ÔØ 

(A: ε(u),ε(v)) = (f,v). 

H4×ÒÒÐÐ×ÝÝ×ØÒ××ØÒÖÒËØÓ×ÒØØÚÒØÙÒÒØØÙÒ×Ø 

u2 ≤ Cf0, 

⎧ 

⎨ 

(A: ε(u),ε(v)) + (B: ε(β),ε(v)) = (f,v), 

⎩ 

(D: ε(β),ε(η)) + (B: ε(u),ε(η)) − (p, div η) = (∇ψ,η) + (G,η). 

⎧ 

⎨ 

(A: ε(u),ε(v)) = (f,v) + (div (B: ε(β)),v), 

⎩ 

(D: ε(β),ε(η)) − (p, div η) = (∇ψ,η) + (G,η) + (div (B: ε(u)),η).

ÅÌÅÌÌÁÆÆÅÄÄÁ ÈØÑÐÐÒ×ÑÑ×××ÝØÐ××βÚÓÒ×ÒÒ×ÑÑ×ÒÖÚÒ 

Î×ØÚ×ØÔØÑÐÐØÓ×××ÝØÐ××uÚÓÒÔØÐØÚÒ×ØÑØ 

×ÒÒÐÐ×ÝÝÒÒÓÐÐÓÐÐÒÚÓÐÐC1,C2 > 

ØÙÔÝØÐÝØÒÔÝØÐÑ ËÓØØÑÐÐÒÝØ×ØÑØØ ÔÝØÐÒ Ð×ÑÐÐÒÒ× 

ØÒÔÖÙ×ØÐÐÚÓÐÐC3,C4 > 

×ÒÖÚÓ 

ÑÐÓÐÐÐ××ØÝØÒØØÖÑÒØÒ×ÓÖÒBÒÓÖÑ×ØÐÙ×ØΩÖÔÔÙ ÀÐÙØØÙ×ÒÒÐÐ×ÝÝ×ØÙÐÓ×ÝØØÝÐÐØØÚÐÐ×××ÒÒÓ×ØÒ 

 

0 

u2 ≤ C1(f0 + div (B: ε(β))0) ≤ C1f0 + C2β2. 

 

0 

β2 + p1 ≤ C3(G0 + g−1 + div (B: ε(u))0) 

 

≤ C3(G0 + g−1) + C4u2. 

ÚÐÐÚÓÐÐC4ÔØ 

(1 − C4(1 + C2))β2 + p1 + u2 ≤ C(G0 + f0 + g−1). 

 

1 

C4(1 + C2) < 1 ⇔ C4 < < 1. 

1 + C2

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ Ì××ÔÔÐ××ÓØÒÚÖÖÚÓØÓÑÔÓ×ØØÐØØØØÚÐÐÎÖÖ ÐÑÒØØÑÒØÐÑÅÁÌÐÑÒØÐÐ ÚÓØÐ×ØÒØÚÐÐ×ÒH 1ÒÓÖÑÒ×ÒÚÖÓÖÔÔÙÚ××ÒÓÖÑ××Ó ÓØØÙÓÑÓÓÒÊ××ÒÖÒÅÒÐÒÒÑÐÐÒÔÙÙØØÐÐ×Ò×ÒÒÐÐ×ÝÝÒ ÙÒt→0ÃÝØØÑÐÐØØÒÓÖÑ×ÒØØÚÒÚÖÒÝØØÝØÝÑ ××ØØÖÑÔØØÓÙÒH 1ÒÓÖÑ××ÐØÒÓÐÐ××ÓÙØÑÓÒÚÖ×Ò Ò×ÒÑÖØØÒÚÖÖÚÓØÖ×ÒÐØØÐÚÓØÐÐÐÓÒÐÒ ÒÝØØÒØØÑÐÝØØØÚ××ÓÒÝÚÒØÓÑÚÐØØÐÑÒØØÓ ÝÐ×ØÓÑÔÓ×ØØÖÒØÒØÔÙ××× 

Ý×ØØÒØÓÑÚÒØ×ÓÐÑÒØØÒ×ÒÐÓÔÙÐØÝÚÒÝØØÝØÝÚ ×ØØÝÒÓÑÔÓ×ØØÐØØÑÐÐÒÒÐÝÝ××ÙÓÖØØÒ××Ó××× 

ÑÒØÐÑÑÝ×ÓÓØØÚÐÐ 

ÔØ×ÙÖÚÊ××ÒÖÒÅÒÐÒÐØØÑÐÐÒÚÖÒÐÝÝ×ØÒÔ ÎØ××ØÝ××ÝØØÒÒÓ×ØÒÅÁÌÐÑÒØØÐØØØØÚÐÐ ÄØØØØÚÒÒÐÝÝ× 

ÒÒÑÙÙØÓ×ÒÝÐ×ÑÑÐÐÐÑÒØØÔÖÐÐ×ØØØkÓÐØØÒØØ× ÖØØÓØÝØØØØØÝÔÖÙ×ÚØÑÙ×Ð×ÝÝÒÐ××ØÙÐÓ××ØØÙ Ð×ÙÓÖÚÚ×ÑÔÒÐÝÝ×××ØÖÚØÔØÖÚÐ×ØÚÓÑ××ÓÐÚ ×ÙÙÒÚÙÓ×Ú××ÒÒÐÐ××ÑÙØØØÙÐÓ×ÔØÑÝ×Ô×ÒÒÐÐ×ÐÐ ×ÖØØÀÐÑÓÐØÞÒÓØÐÑÂØÓ××ÓÐØØÒÚÖÓÝ×ÒÖØ ÚÖÓÐÐ℄ ×ÙÙÒØÚÚÓÒÓÙØØÒÃÖÓØØÒÒ×Ò×ØÙÐÔ×ØØØÚ ÌÖ×ØÐÐÒ××ÒÓÖÑÐÊ××ÒÖÒÅÒÐÒÒÐØØÑÐÐÚØØÒ℄ ÅÁÌÐÑÒØØÔÖ 

ÌØÚ Ó×× ××ÒÝØA = 

Ñ××ÐÒÖÑÙÓØÓAÓÒÑÖØÐØÝ 

Ø×(β,w,q) 

ÖÃÙÒαÓÒÚÔ×ØÚÐØØÚ×ØÐÓÒØÔÖÑØÖÑÖØÐÐÒ×ÖØØ 

ÇÐÓÓÒThÐÙÒΩÚ×ÙÒÓÖÑÓÐÑÓÒØhÚÖÓÒØÝ×ÔÖÑØ 

B = 0ÑÖØÒa(β,η) = (D: ε(β),ε(η))ÑÙÓØÓÓÒ 

∈ [H1 0(Ω)] 2 ×H1 0(Ω)×[L2 (Ω)] 2×ØÒØØ∀(η,v,r) ∈ 

[H1 0(Ω)] 2 × H1 0(Ω) × [L2 (Ω)] 2ÔØ 

A(β,w,q;η,v,r) = (G,η) + (g,v), 

A(β,w,q;η,v,r) := a(β,η) + (∇v − η,q) + (∇w − β,r) − t 2 (A ∗−1 q,r).

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÐÒÖÑÙÓØÓ×ÙÖÚ×Ø 

ÌØÚ 

Ah(β,w,q;η,v,r) :=a(β,η) + (Rh(∇v − η),q) + (Rh(∇w − β),r) ÌÐÐÒÚ×ØÚ×ÖØØØØÚÔÔÖÓ×ÑØÓÚÖÙÙ××Vh × 

ØØÚ××ÌØÚÖØÒ×ÖØØÐÒÖÑÙÓØÓAhÑÖØÐÐÒÖÙØÓ ÅÁÌÐÑÒØØÒÔÖÙ×ØÙ×ÓÒÐÙ×ÚÓÑÒÑÓÓÒØ×ÖØ×× 

Ø×(βh,wh,qh) ∈ 

×ÒÐÙØØÙÑ×ÐØÐÙ×ÚÓÑÒ×ÙØÒÅÁÌÐÑÒØØÒØÔÙ××× 

ËØÔÙÑØØÖØÝÑØ×ÙÔÔÒÚØÓÔØÑÐ××ØÓ×ÐÝØÝÝÚ 

ÓÔÖØØÓÖÒRh : 

ÖÙÙ×Qh ∈ 2ÔÖÓØÓ ÈÂÓ×s ∈ ΓhÐÐÔØrot Wh×ØÒØØs = 

h ,Qh)ÓÒ×ØÐÖØ×ÙÚÖÙÙ×ËØÓ×ÒØØÚÐÐ ÊÓÓØØÓÖÓÒÐÐÑÖØÐØÝÚØÓÖÖÚÓ×ÐÐ×ÙÙÖÐÐq×ØÒØØ 

ÌØÒÚÓÒÑÖØÐÐÚÖÙÙ× 

ÚÖÙÙ×ÒÐÝØÑ×× 

ÆÝØÚÓÒ×ÙÖØÚØØ××℄×ØØØÝÓÒ×ØÖÙØÓØ×ÓÔÚÒÚÖØÓ 

− (αh 2 + t 2 )(A ∗−1 ΓÓÒÑÙÓØÓ 

q,r). 

 

Wh × 

Γh ⊂ V × W × 

Vh × Wh × Γh×ØÒØØ∀(η,w,q) ∈ 

Vh × Wh × ΓhÔØAh(βh,wh,qh;η,v,r) 

ÖÙØÓÓÔÖØØÓÖRhÓÒÑÖØÐØÝ×ØÒØØÔØ 

ΓhÚÙÐÐÓÓØØÒÝØØÒÓØØÚÐØÝØØ 

= (G,η) + (g,v). 

 

Vh → 

Rh∇w = ∇w, ∀w ∈ Wh. 

L2 0×ØÒØØ×ÙÖÚØÓÑÒ×ÙÙØÔØÚØ℄ 

È∇Wh ⊂ Γh 

Èrot Γh ⊂ Qh 

Èrot Rhη = Phrot ηÑ××Ph : L2 → QhÓÒL s = 0ÐÝØÝÝv∈ È(V ⊥ 

rot q = ∂1q2 − ∂2q1 = div q ⊥ . 

H0(rot, Ω) = {q ∈ [L 2 (Ω)] 2 | rot q ∈ L 2 (Ω), q·τ |∂Ω = 0}. 

∇v

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ 

•ÃÓ×QhÓÒÒÒØØØÝØ×ØÒÚÖÙÙ×ΓhÓÔÖØØÓÖRh×ØÒ ØØ×ÙÖÚÚÓÓÑÑÙØÓ 

•ÎÐØÒÔÖ(Vh,Qh) ∈ 

[H1 0(Ω)] 2 rot ✲ L2 0(Ω) 

•ÎÐØÒÚÖÙÙ×Wh×ØÒØØ 

ÑÒØ×××ÙÔÔÒÑÒÒ×ÚÙØØÒÝØØÑÐÐÖØÝÑÐÐÙÔÐÑÙÓØÓ ØÒÐÑÒØØÒÒ××ØÖÚØÒÚÖÓ×ØÐÓÒØÓÖÑÑÒ×ØÒÐ ÖÒÒØØÙÚÐØ×ÑÐÐÐÑÒØØÚÖÙÙØ×ØÒØØÒÓ×ØÒÐÒÖ× Ì××ØÝ××ÝØØØÝÝÒÆÙÑÖÖÒÓÐÑ×ØÓÓÒÅÁÌÐØØÐÑÒØØÓÒ 

ÐÒ×Ò 0ÆÑÓÑÒ×ÙÙØ×ÚÙØØÒÚÐØ×ÑÐ 

ÃÓÐÑÓÐÑÒØÐÐVkÑÖØÐÐÒ 

ÓÐÐÓÒÚÓÒ××ÚÐØα = 

Vh 

Ñ××ÚÖÙÙ×SkÓÒÑÖØÐØÝ 

ÆÐÙÐÑÓÐÐÔÙÓÐ×ØÒÚÐØÒÐÐkÒÖÚÓÐÐ 

ÒÐÙÐÑÓÐÑÒØÐÐ ËØÓ×ÒØØÚÒÔÙÚÖÙÙ×QhÓÒÐÐkÒÖÚÓÐÐ×ÓÐÑÓØØ 

[H 1 0(Ω)] 2 ×L 2 0(Ω)×ØÐ×ËØÓ×ÒØØÚÐÐ 

Γh 

Rh 

Ph 

❄ ❄ 

rot 

✲ Qh. 

∇Wh = {s ∈ Γh | rot s = 0}. 

= {η ∈ [H 1 0(Ω)] 2 | η|T ∈ Vk(T), ∀T ∈ Th}. 

⎧ 

⎨ 

[Pk(T)] 

Vk(T) = 

⎩ 2ÙÒk= 1, 

[Sk(T)] 2ÙÒk= 2, 3, 

Sk(T) = {v ∈ Pk+1(T) | v|E ∈ Pk(E)Ó×ÐÐÖÙÒÐÐE⊂ ∂T }. 

Vk(T) = [Qk(T)] 2 . 

Qh = {p ∈ L 2 0 | p|T ∈ Pk−1(T), ∀T ∈ Th}.

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÄÙ×ÚÓÑÒÚÖÙÙ×ÚÐØÒÒÝØÓÐÑÓÐÑÒØÐÐπ/2ÚÖÖÒ 

ÖÖØØÝÊÚÖØÌÓÑ×ÚÖÙÙ×Ú×ØÚ×ØÒÐÙÐÑÓÐÐÝØØÒÖÞÞ ÓÙÐ×ÓÖØÒÅÖÒÚÖÙÙØØ℄ÓÐÐÓÒ××ÓÐÑÓÐÐ 

Γh = {s ∈ H0(rot) | s|T ∈ [Pk−1(T)] 2 ÒÐÙÐÑÓÐÐ 

+ (x2, −x1)Pk−1(T), ∀T ∈ Th} 

ÒÐÙÐÑÓÐÐ 

ÓÐÑÓÒØÔÙ××× ÊÙØÓÓÔÖØØÓÖÚÓÒÑÖØÐÐÊÚÖØÌÓÑ×ÖÞÞÓÙÐ× ÓÖØÒÅÖÒÐÑÒØØÒÚÔÙ××ØØÚ×ØÚ×Ø×ÒÐÙÐÑÓÒØØ 

ÚÐØÒα=0ÇÒÙÓÑÓÒÖÚÓ×ØØØØÔÙ×××k>1ØÖÚØØ×Ò Ä××ØÙÐÙÓÑØØØÐÒÖ×ÒÐÑÒØÒØÓÑÒÒÒÒÒÐØÚÖ 

ÚÖÓ×ØÐÓÙÐÐÐÑÒØÐÐØÓ×Ò×ØÒÖÚØØÓ××ÐØÚÐ×ØÖ 

1ÚÐØÒαÖ×ÙÙÖ×ÙÒÒÓÐ 

ÑÐÒÖÑÙÓØÓÓÒÑÙØØÒÒÝØÐØÚÐØÝØÒÝØØÑÐÐÖØÝ 

Ó×ØÐÓÒØÓÒÓÐÒÒ×ØÓØÒÙÒk= 

×Ý×ØÚÓÒÒÑØØÒÓ×ÓØØØØÒÑ××ØÐÓÒØØÒÓÚØ ÑÐÐ×ÓÔÚÙÔÐÑÙÓØÓÌÑØÓØÙØÙ×ØÔÚÐØØÒÐÒÒÝØÒÒÒ 

ÐÝÐ×ÖÚÓÔÖÑØÖÐÐÓÚØα 

ÚÚÐÒØØ℄ÆÐÐÚÐÒÒÓÐÐÒÐÙÐÑÓÚÖÙÙ×ÓÒÐÙÔÖÒÒÅÁÌ 

= 

ØÔÖ ÚÖÙÙ×ÓÐÑÓÐÑÒØÐÐÚÖÙÙ×ÓÒÚØØÒ℄Ò×ÑÑÒÒÐÑÒØ ÓÒÚÓÅÖØÐÐÒ×ÙÖÚ×ÒÓÖÑØÓ××ÔÖÓÖÖÚÓØÓØÒ 

ÎÖÖÚÓØÅÁÌÐÑÒØÐÐ ÇÐØØÒØØÝØØØÝÚÖÓÓÒÚ××ÒÒÐÐÒÒÐÐÑÒØÒÓÓh 

Γh = {s ∈ H0(rot) | s|T ∈ (Pk(T)\{ξ k }) × (Pk(T)\{η k ÌÔÙÑÐÐÚÐØÒÓÐÑÓÐÐ 

}), ∀T ∈ Th}. 

Wh = {w ∈ H 1 0(Ω) | w|T ∈ Pk(T), ∀T ∈ Th} 

Wh = {w ∈ H 1 0(Ω) | w|T ∈ Qk(T) ∩ Pk+1(T), ∀T ∈ Th}. 

 

E 

((Rhs − s)·τ)ν = 0, ∀ν ∈ Pk−1(E), 

 

(Rhs − s)·r = 0, ∀r ∈ [Pk−2(T)] 2 . 

T 

0.1,..., 0.25ÌÓ×ÐØÙÒk > 1

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÊÙØÓÓÔÖØØÓÖÒÚÙÓ×ÒÓÖÑØÑÖØÐÐÒÒÓ×ØÒÐÚÖÙÙ××× 

Vh,WhΓhËÖØÝÑ×ÙÙÖÐÐÑÖØÐÐÒ 

ÐÙ×ÚÓÑÐÐÚ×ØÚ×Ø 

Ú××ØÓÚÒ×ÐÚÑÙØØ×ØÑ×ÙÙÖÒÑÒÓÖÑØÓÚØÓÚÐÒØ ÐÐÑÖØÐØÝÒÚÖÓÖÔÔÙÚÒÒÓÖÑÒÝØØÓÐÒØÙØ 

×ØÝ××ÒÓÐÐÔÐÙØÙÙÐÙÔÖÒÒØØÚÃÖÓÒÐØØÑÐÐÚ× ÚÖÒÑØØÑ×ÒÚÓÒÔÖÙ×ØÐÐ×ÙÖÚ×ØÄØÒÔ×ÙÙÒtÐ ØÚ×ØØÚ×ÓÒÖØ×ÙÓÒÚÖÙÙ××H 2 (Ω)ÃÓ××ØÐØÚÒ ØØÚÒÖØ×ÙÐÙÙÓ××ÒÖÔÒH 1 h×Øt×ØÖÔÔÙÚÒØÖÑÒÐ×Ý×ÐÐÐ×ÒÓÖÑØÓØ×ØÙÒÐ ÐÐÖØ×ÙÓÒÃÖÓÒØØÚÒÖØ×Ù 

ÄÑÑ ÚÐÒ××ÓÒÚÙÐÐÔ×ØÒÒÒØØÚÒ×ØÐÙØÒ ÂØØÒØ×ÚÖÖÚÓÒØÓ×ØÑ×ØÒÝØØÑÐÐ×ÙÖÚÒÓÖÑÚ 

ÌÓ×ØÙ×ÇÒÔÙÓÐÒÒÔÝØÐÔØØÖÚÐ×ØÌÓ×ÐØÝØØÑÐÐ ÄÝØÝÝÚÓC > 0×ØÒØØÓ×ÐÐ(η,v) 

ÈÓÒÖÒÔÝØÐ×ÙÓÑÓÑÐÐØØÐÝØÝÝÚÓC1ÓÐÐ1 

×ÓÑÒ×ÙÙØØ Ä××ØÖÚØÒ×ÙÖÚÐÑÑ ÌÐÐÒÑÝ×Ú×ÑÑÒÔÙÓÐÒÒÔÝØÐÓÒØÓ× 

Wh×ØÒØØ 

Ó××ÓÒÝØØØÝÚÙ××ØØØÓÔÖØØÓÖRhÓÒÖÓØØØÙÚÓÐÐC2 

ÄÑÑÃÙÒk=2, 3ÚÓÒÚÐØη ∈ Vhv ÑÐÚÐØ×ÐÐq∈ ΓhÔØRh(∇v − η) = h 2 

q. 

|(η,v)| 2 h := η 2 1 + v 2 1 + 

|r| 2 h := (t 2 + h 2 )r 2 0 

1 

h2 + t2 Rh(∇v − η) 2 

0, 

(Ω)H 2 (Ω)ÚÐÐÐ×Ò 

∈ 

ÔØC|(η,v)| 2 h ≤ η 2 1 

1 + 

h2 + t2 Rh(∇v − η) 2 0 ≤ |(η,v)| 2 0ÒÒØØ 

h 

1 

hT > 0, √2C1 > t > 

v 2 1 ≤ ∇v 2 0 ≤ 2(∇v − Rhη 2 0 + Rhη 2 0) 

≤ C1Rh(∇v − η) 2 0 + C2η 2 0 

1 

≤ C( 

h2 + t2 Rh(∇v − η) 2 0 + η 2 1), 

∈ 

Vh × Wh 

√2C1 >

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÌÓ×ØÙ×ÎÖØÓÚÖÙÙ×ÒÚÐÒÒÒÔÖÙ×ØÐÐÐÐ×ÚÙÐÐEÔØ 

ÙÚÙÙÒØÑ××ÚÖÙÙ×××VhWhÓÐØØÒÒØÙÒØÓÒ Ñ×Ø×ÓÐÑÙ××ØÙÚÌÑÒÐÒÑÖØØÒ×ÚÙÐÐÓÐÚÒÝÐÑ Ö×ØÒÑÓÑÒØØÚÔÙ××ØÒÚÙÐÐÐÓÔÙØÒØÙÒØÓØ×ØÒØØ ÖÚÓØÙÐÑÔ×Ø××ÒÒØØÝ×ÓÐÐÓÒÔÓÐÝÒÓÑÔÔÖÓ×ÑØÓÓÒÚÖ ×ÓÐÑÓÐÐØØÒÐÙÐÑÓÐÐη·τ 

ØÒÒØÐÓÑÔÓÒÒØØÓÒØÙÚÐÑÒØÒÖÙÒÒÝÐØ×ËÐÚ×Ø××Ú ÔØÌÐÐÒ×ÚÙØØÒÚÐØØÑ×ØÑÝ×ØÙÚÙÙ××ÚÙÒÝÐ×ÐÐqÒ 

∈ 

ÚÐÐÐ ØÚÒØÙÒØÓÖÙÒÐÐÓÒØÓ×Ò×ØÒÔÓÐÝÒÓÑÓÚÔ×× ÔÙ××ØÐÐÔØÝ×ÝØÒÚ×ØÚÙÙ×ÐÙ×ÖØÝÑÚÖÙÙ×Ò 

ÓØÒη·τÓÒ×ÝÑÑØÖÒÒÔÖÐÐÒÒÖÙÒÒ×Ô×ØÒ×ÙØÒÅÓ ÑÒØØÚÔÙ××ØØØÚ×ØÚÖÚØØØ×ÒÓÒÐÒÖÒÒÚÚÐÒ ÌÖ×ØÐÐÒÒ×ÒØÔÙ×Øk=2ÌÐÐÒÑÓÑÒØØÚÔÙ××ØØØÚ× 

×Ô×Ø××ÓØÒ∇v·τÓÒÖÙÒÒ×Ô×ØÒ×ÙØÒÔÖØÓÒÌÐÐÒ 

− ÔÖØÓÒds ÔÖÐÐÒÒds 

ÔÓÒÒØÒÒ×ÑÑÒÒØÓÒÒÑÓÑÒØØÖÙÒÒÝÐÚÓÒÖØÝÑ 

− 

ØÔÙÑÚÐØÝ××ØØ××ØÌÔÙ×××k=3ØÙÐÓ×ÚÓÒØÓ×Ø ÃÓ××ÅØØÊÌÐÑÒØØÒÚÔÙ××ØØÓÚØØÒÒØÐÓÑ 

ÔÖÐÐÒÒds ÔÖØÓÒds 

ØÓÒÐÒÖÒÒÖÔÔÙÑØØÓÑÙÙ×ÑÙÓÓ×ØÑÐÐÒ×Ø×ÓÔÚÐÒ ÖÓÑÒØÓØ ×ÑÒÐ×ÐÐÔØØÐÝØÙÐÐÙÓÑÓÑÐÐÐ××ÔÖÓ×ÓØÚÒÒØÙÒ 

×ÚÓC×ØÒØØ ÌÑÒÚÙÐÐÚÓÒØÓ×Ø×ÖØÒØØÚÒ×ØÐÙ× ÄÙ×ÂÓ×ÐÐ(β,w,q) ∈ 

Ah(β,w,q;η,v,r) ≥ C(|(β,w)| 2 h + q 2 

h), 

Ñ×× |(η,v)|h 

 

E 

 

(∇v − η)·τ(a + bs)ds = a 

E 

 

+ b 

 

= b 

E 

Pk(E)∇v·τ ∈ Pk−1(E)ÂØ 

∇v·τ 

 

E 

∇v·τs 

 

a 

 

∇v·τsds − a 

E 

 

b 

E 

η·τ 

 

E 

η·τds. 

η·τs 

 

Vh×Wh×ΓhÐÝØÝÝ(η,v,r) ∈ Vh×Wh×Γh 

+ rh ≤ C(|(β,w)|h + qh).


Ñ××Ó××× ÔØÃÓÖÒÒÔÝØÐÒ ÔÖÙ×ØÐÐ ÌÓ×ØÙ×ÇÐÓÓÒ(β,w,q) ∈ ÎÐØÒÒ×Òr1 = wÌÐÐÒÐÒÖÑÙÓÓÐÐAh 

Vh × Wh × ΓhÒÒØØÙÌÓ×ØØÒÚØÓÐ 

−q,η1 = βv1 = 

Ah(β,w,q;β,w, −q) = a(β,β) + (t 2 + αh 2 )(A ∗−1 q,q) 

≥ C1(β 2 1 + (t 2 + αh 2 )q 2 0). ËÙÖÚ×ÚÐØÒØ×ØÙÒØÓ×r2 = (t2 +αh2 ) −1 0 

Rh(∇w −β),v2 = 

η2 = 0ÓÐÐÓÒ 

1 

Ah(β,w,q;0, 0, 

t2 + αh2Rh(∇w − β)) 

1 

= 

t2 + αh2(Rh(∇w − β),Rh(∇w − β)) − t2 + αh2 t2 + αh2(q,Rh(∇w − β)) 

1 

≥ 

t2 + αh2 Rh(∇w − β) 2 0 − (t 2 + αh 2 ) 1/2 1 

q0 

(t2 + αh2 ) 1/2 Rh(∇w − β)0 

1 

≥ 

t2 + αh2 Rh(∇w − β) 2 1 

0 − 

2(t2 + αh2 ) Rh(∇w − β) 2 0 − t2 + αh2 q 

2 

2 0 

1 

≥ C2{ 

t2 + αh2 Rh(∇w − β) 2 0 − (t 2 + αh 2 )q 2 0ØÙÐÓ× 0)}. ÌÐÐÒÚÐØ×ÑÐÐ×ÓÔÚÓÒÚ×ÓÑÒØÓλ(η1,v1,r1)+(1−λ)(η2,v2,r2) ×Ò×ØÐÓÙ××ØÔÙ×××α > 

Ah(β,w,q;λ(η1,v1,r1) + (1 − λ)(η2,v2,r2)) ≥ λC1(β 2 1 + q 2 h) 

1 

+ (1 − λ)C2( 

t2 + αh2 Rh(∇w − β) 2 0 − q 2 ×ØÑØØÒ×ÐÙ×ÚÓÑÒÒÓÖÑÒh×ØÖÔÔÙÚÓ×ËÒ× 

1]×ÒÐÖÚÖÑ×ØÔÓ 

h), ÓÐÐÓÒ×ÓÔÚÐÐÔÖÑØÖÒÖÚÓÐÐλ∈[0, ×ØÚ××ÐÒÖÑÙÓØÓAhÓÒ×ØÐÁÐÑÒÚÖÓ×ØÐÓÒØα=0 ÒÐÐÒÚÓÒÚÐØr2 = (t2 + h2 ) −1 β)ÓÐÐÓÒ×ØÑØØ 0ÄÑÑÒ 

Rh(∇w − ×Ò| · |hÒÓÖÑ××ØÔÙÑÐÐwÖØÝÑÐÐβ ÎÖÓ×ØÐÓÑØØÓÑ××ØÔÙ×××ÚÐØÒÚÐr3 = ÔÖÙ×ØÐÐÚÓÒÚÐØÔÖ(v3,η3)×ØÒØØ∀q ∈ ΓhÔØRh(∇v3 − 

η3) = h2qÄ×××ÐÙ×ÖÙÑÒØØη3ÒÑÖØÐÑÙÓÑÓÒÔØ η31 ≤ ˜ Ch −1 η30 ≤ Ch −1 h 2 q0 = Chq0.

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÎÐØ×ÑÐÐǫ< 2 CÒÒ 

Ah(β,w,q;η3,v3, 0) = a(β,η3) + (h 2 q,q) = a(β,η3) + h 2 q 2 0 

≥ − 1 

2ǫ β21 − ǫ 

2 η3 2 1 + h 2 q 2 0 

≥ − 1 

2ǫ β21 + (1 − Cǫ 

2 )h2q 2 0 

≥ − 1 

2ǫ β21 + Ch 2 q 2 ×Ò×ØÑØÒ 

0, Ñ××C > 0ÂÐÐÒÚÐØ×ÑÐÐÓÒÚ×ÓÑÒØÓÔÖÑØÖÐÐλ1,λ2 

Ah(β,w,q;λ1(η1,v1,r1) + λ2(η2,v2,r2) + (1 − λ1 − λ2)(η3,v3,r3)) 

≥ λ1C1(β 2 1 + t 2 q 2 1 

0) + λ2C2( 

t2 + h2 Rh(∇w − β) 2 0 + t 2 q 2 0) 

+ (1 − λ1 − λ2)(−β 2 1 + C3h 2 q 2 ÐÖ×ØÚÖÑ×ØÔÓ×ØÚÒÒÄ××ÚÐØÙÐÐØ×ØÙÒØÓÐÐÔØ 

0) 

 

|η1,v1|h + r1h = |β,w|h + qh, 

|η2,v2|h + r2h ≤ |β,w|h 

|η3,v3| 2 h + r3 2 h = η3 2 1 

1 + 

t2 + h2 h2q 2 0 ≤ Cq 2 ÌØÒÚØØÑÓÒ××ØÓ×ØØØÙ×ÚÖÓ×ØÐÓÙ××ØØ×ØÐÓÑØ 

ÐÒÖÑÙÓØÓAhÓÐÓÒ××ØÒØØÐÙÔÖ×ÒÐÒÖÑÙÓÓÒAÒ× ØÓÑ××ØÔÙ××× 

×ÎÖÖÚÓÒÐ×Ñ××ØÖÚØÒ××Ò×Ò×ØÑØØÓÒ××ØÒ××ÚÖ ÃÓ×ÐÙ×ÚÓÑÓÒ×ÖØ××ØÔÙ×××ÑÓÓØÙ×ÖØØ 

h. 

ΓÖÓØØ ÐÐËÓØØÑÐÐÐÒÖÑÙÓØÓÓÒAhØØÚÒØÖÖØ×Ù(β,w,q) ÚÓÒ∀(η,v,r) ∈ V × W × 

Ah(β,w,q;η,v,r) = a(β,η) + (Rh(∇w − β),r) + (Rh(∇v − η),q) 

− (t 2 + αh 2 )(A ∗−1 q,r) 

= a(β,η) + (∇w − β,r) + (∇v − η,q) − t 2 (A ∗−1 q,r) 

+ ((Rh − I)(∇v − η),q) + ((Rh − I)(∇w − β),r) 

− αh 2 (A ∗−1 q,r) 

= (G,η) + (g,v) + E(q;η,v,r).

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÀÙÓÑÓÑÐÐØØØÖÐÐÖØ×ÙÐÐÔØ 

q = A∗ 

t2 · (∇w − β) ⇔ ∇w − β = t2A ∗−1 ×ÒÓÒ××ØÒ××ÚÖEÑÙÓØÓÓÒ 

ÒÔÒÐÐÚÖÓÓ×ÚÐÐÖÓØÙ×ÐÐÖÙØÓÓÔÖØØÓÖÐÐRhÓÒÓÔ ÎÖÒÐÝÝ×ÒÐÔÚÑ××ØÖÚØÒÑÙÙØÑÔÙØÙÐÓ×Ò×ÒÒ 

·q, 

C×ØÒØØÔØ ØÑÐ×ØÒØÖÔÓÐØÓÓÑÒ×ÙÙØ℄ 

Ä××ÚÓÒÓØ×ÙÖÚØÙÐÓ×℄ 

ÄÑÑÂÓ×ÐÐη∈ [H 

η ÄÑÑÂÓ×ÐÐs∈[H m−1 ÚÓC×ØÒØØÔØ 

ÌÓ×ØÙ×ÅÐm=1ØÙÐÓ××ÙÖ×ÙÓÖÒËÛÖÞÒÔÝØÐ×ØÐÑ 

T)ÐÑÒØØØÒ Ñ×ØÃÙÒ2 ≤ 

ÌÐÐÒÓÔÖØØÓÖÒRhÑÖØÐÑÒÔÖÙ×ØÐÐÚÐØÙÐÐÚÖØÓÚÖÙÙ 

Ä××ÑÖØÒPT : 

×ÐÐÔØÐÑÒØÐÐTÑÐÚÐØ×ÐÐˆs ∈ 

ÅÖØÐÐÒ×ÙÖÚ×L 2ÔÖÓØÓÐÑÒØÐÐTÖÖÒ××ÐÑÒØÒÚÙÐ Ð ΠT 

E(q;η,v,r) := ((Rh−I)(∇v−η),q)+t 2 (A ∗−1 (Rh−I)q,r)−αh 2 (A ∗−1 q,r). 

m (Ω)] 2Ñ××1≤m≤kÓÒÓÐÑ××ÚÓ 

− Rhη0,T ≤ Ch m ηm,T. 

(Ω)] 2Ñ××1≤m≤kÓÒÓÐÑ×× 

|(s,η − Rhη)T | ≤ h m sm−1,T η1,T. 

m ≤ kÑÖØÐÐÒÔÙÚÖÙÙ×A( ˆ A( ˆ T) = [Pk−2( ˆ T)] 2 . 

 

(PTˆs,η − Rhη)T = 

 

= 

 

= 

[L2 ( ˆ T)] 2 → [L2 (T)] 2ÈÓÐÒÑÙÒÒÓ×Ø 

PTˆs = |JT |JTˆs, ˆs ∈ [L 2 (T)] 2 . 

T 

A( ˆ T) 

PTˆs· (η − Rhη)) dxdy 

|JT | 

ˆT 

−1 JTˆs· (η − J −T 

T 

ˆT 

ˆs· (J T T η − ˆ RhJ T T η)) dξdη = 0. 

= PTΠˆ T P −1 

T , 

ˆRhJ T T η))|JT | dξdη

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ Ñ××Π ˆ 2ÔÖÓØÓÖÖÒ××ÐÑÒØÐÐÌÐÐÒ 

Ä××ØÙÒÒØÒÒØÖÔÓÐØÓ×ØÑØØ℄ ÆÒÚÙÐÐ×ÒÐÓÔÙÐØÐÙØØÙØÙÐÓ××ÙÖÚ×Ø 

ÔØÑÐÚÐØ×ÐÐs ∈ 

ÄÑÑÃÓÒ××ØÒ××ÚÖÐÐÔØ ÌÐÐÒÓÒ××ØÒ××ÚÖØØÚÓÒÖÚÓ×ÙÖÚ×Ø℄ 

ÑÒÑÖØÐÑ×ÒÓÒ××ØÒ××ÚÖØÖÑÐÐ×ÙÖÚÖÚÓ×ÓÚÐØ ÌÓ×ØÙ×ÃÝØØÑÐÐÝÚ×ÓÑÒ×ÙÙØØ ÑÐÐÒ×ÑÑ×ÒØÖÑÒÄÑÑØÓ×ÒÄÑÑÄ××ÙÒ ×ÚÖÓÖÔÔÙÚÒÒÓÖ 

ØÖÔÒ 

Ä××ÙÓÑÓÑÐÐØØ×ØÐ×ÙÙ××ØÑØÒÔÖÙ×ØÐÐÔØ ×ÒÐÙØØÙÖÚÓ 

IhÂØÓÒÐÝÝ×ÚÖØÒÑÖØÐÐÒØÔÙÑÐÐwÖØÝÒÒÒØÖÔÓÐÒØØ 

 

: ÅÖØÐÑÁÒØÖÔÓÐÒØØIh 

 

TÒÖÔ×Ø××, 

T : [L 2 ( ˆ T)] 2 → A( ˆ T)ÓÒL 

[H m−1 (Ω)] 2 

(ΠTs,η − Rhη)T = 0. 

s − ΠTs0,T ≤ Ch m−1 

T sm−1,T. 

(s,η − Rhη)T = (s − ΠTs,η − Rhη)T ≤ s − ΠTs0,T η − Rhη0,T 

≤ Ch m−1 

T sm−1,T η1,T. 

E(q;η,v,r) ≤ Ch m 2ÚÐØÒα=0×ÐÐÚÖÓ×ØÐÓÒØÓÐÙÔÐÑÙÓØÓÒÒ×× 

(qm−1 + tqm) 

m ≥ 

E(q;η,v,r) = (η − Rhη,q) + (tA ∗−1 (Rh − I)q,tr) − αh 2 (A ∗−1 q,r) 

≤ tC1Rhq − q0tr0 + C2h m qm−1η1 + C3αhq0hr0 

≤ C4(|(η, 0)|h + rh)(th m qm + h m qm−1) 

|(η,v)|h + rh ≤ C5 

H1 WhÑÖØÐÐÒÐÑÒØØ 

0(Ω) → ØÒ×ÙÖÚ×ØÓ×ØÓÔÖØØÓÖÐÐIh|T = IT 

((v − ITv) ◦ FT) = 0××ˆ 

((v − ITv) ◦ FT)ˆr dˆs = 0, ∀ˆr ∈ Pk−2( 

Ê 

Ê)ÐÑÒØÒÖÙÒÓÐÐÊ 

((v − ITv) ◦ FT)ˆsdξdη = 0, ∀ˆs ∈ Pk−3( 

ˆT 

ˆ T)ÐÑÒØ××ˆ T.

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÁÒØÖÔÓÐÒØØÓÒÑÖØÐØÝ×ØÒØØÔØ ÄÑÑ 1ÔØ 

ÑÐˆτ×ÚÙÒØÒÒØØÌÐÐÒÓÑÒ×ÙÙ×Ò T×ÚÙ×ˆn×ÚÙÒÒÓÖ ÔÖÙ× 

ÃÐÐv∈ H 

ÌÓ×ØÙ×ÅÖØÒÐÐÒÊÖÖÒ××ÐÑÒØÒˆ ØÐÐÖÖÒ××ÐÑÒØÐÐÔØ∀ˆr ∈ 

×ÐÐ∂ˆr ∂ˆs Ê)ÎÐØÒ×ØØÒˆs ×ÒÝØØÑÐÐÙÒ Ú 

RTËÓÚÐØÑÐÐÝÐÐÓÐÚØÙÐÓ×ÓÔÖØØÓÖÒRhÑÖØÐÑÒÒÒ ÚÐØØÑ×ØØØÖÖÒ××ÐÑÒØÐÐÔØ 

ÌØÒ××ÐÑÒØÐÐTÔØ 

ÅÖØÒÖÙØÓÓÔÖØØÓÖÒÖÓØØÙÑÐÑÒØÐÐTÐÝÝ×ØRh|T = 

Ä××ÚÓÒÓ×ÓØØØØÓÔÖØØÓÖÐÐIhÓÒÓÔØÑÐ×ØÒØÖÔÓÐØÓ ÓÑÒ×ÙÙØ℄ ÄÑÑ ÃÐÐv∈H m 

Pk−3( ˆ T)ÌÐÐÒÓÑÒ×ÙÙ××Ø 

s (Ω)Ñ××s > 

Rh∇(v − Ihv) = 0. 

Pk−1( Ê) 

 

 

ˆ∇((v 

∂ 

− ITv) ◦ FT)· ˆτ ˆrdˆs = 

Ê 

Ê ∂ˆs ((v − ITv) ◦ FT)ˆrdˆs 

 

= 

∂Ê 

((v − ITv) ◦ FT)ˆr − ((v − ITv) ◦ FT) 

Ê 

∂ˆr 

dˆs = 0, 

∂ˆs 

∈ [Pk−2( ˆ T)] 2ÓÐÐÓÒ××ˆ div ˆs ∈ 

∈ Pk−2( 

ˆT 

 

− 

×ØÒØØÔØv − 

 

ˆ∇((v − ITv) ◦ FT)· ˆs dξdη = 

ˆT 

∂ ˆ T 

((v − ITv) ◦ FT) ˆ div ˆs dξdη = 0. 

RT ∇(v − ITv) = J −T 

T 

ˆRT ˆ ∇((v − ITv) ◦ FT) = 0. 

((v − ITv) ◦ FT)ˆs· ˆn dˆs 

ˆRTJ T T ∇((v − ITv) ◦ FT) 

= J −T ˆRT ˆ ∇((v − ITv) ◦ FT) = 0. 

T 

(T)Ñ××1

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÐÐØÓ×ØØØÙÒÓÑÒ×ÙÙ×ÒÚÙÐÐÚÓÒÐÓÔÙÐØÒÝØØØÓ 

××ÙÖÚÚÖÖÚÓÐÐÖØ×ØÚÐÐ×ÙÙÖÐÐÐÙØÙ××ÒÓÖÑ×× 

ÓÐØØÒÙÓÖÑÖØØÚÒ×ÒÒÐÐ×××ØÒØØØÖÒÖØ×ÙÒ×Ò ÄÙ× Ý×ØØÙØÙÒÐØÒØÔÙ××× 

ÒÐÐ×ÝÝÐÐsÔØ1≤s≤kÑ××kÓÒÔÓÐÝÒÓÑÔÔÖÓ×ÑØÓÒ×Ø ÇÐÓÓÒΩÓÒÚ×ÑÓÒÙÐÑÓÐØØÝ×ØØÙØØÙÄ×× 

ΩhiÚÖÓÔÖÑØÖ××ÐÙ××hbÖÙÒÐÐÌÐÐÒ 

ÐÑÖØØØÝÒØÖÔÓÐÒØØÐÙ×ÚÓÑÐÐØ×℄ÄÙ×ÒÔÖÙ×ØÐÐ ØÐØÝÒØÖÔÓÐÒØØØÔÙÑÐÐw˜qÊÌØÅÚÔÙ××ØÒÚÙÐ 

IhwÐÐÑÖ ÌÓ×ØÙ×ÇÐÓÓÒ˜ βÄÖÒÒÒØÖÔÓÐÒØØÖØ×ÙÐÐβ 

ÓÐÐÔØÙÓÑÓÑÐÐÄÑÑ 

ÓÒÓÐÑ××(η,v,r) ∈ 

ÆÝØÚÖØÖÑÒÒ×ÑÑ×ØÓ×ÚÓÒÖÚÓ×ÙÖÚ×Ø 

ÔØ ⊂⊂ ÇÐÓÓÒΩi β − βh1 + w − wh1 + tq − qh0 + q − qh−1 ≤ 

C{h k i (gs−2,Ωi + tgs−1,Ωi + Gs−1,Ωi ) + hb(g−1 + tg0 + G0)}. 

Γh×ØÒØØ 

Ä××ÚÖÓÓÐØØØÒÚ×ÙÒÓÖÑ×ÓØÒhi = hb = h 

Vh × Wh × 

|(η,v)|h + qh ≤ C, 

|(βh − ˜ β,wh − Ihw)|h + |qh − ˜q|h 

≤ Ah(wh − Ihw,βh − ˜ β,qh − ˜q;η,v,r) 

= Ah(w − Ihw,β − ˜ β,q − ˜q;η,v,r) − E(q;v,η,r) 

= Ah(0,β − ˜ β,q − ˜q;η,v,r) − E(q;η,r). 

Ah(0,β − ˜ β,q − ˜q;η,v,r) = a(β − ˜ β,η) + (Rh( ˜ β − β),r) 

+ (Rh(∇v − η),q − ˜q) − (t 2 + αh 2 )(A ∗−1 (q − ˜q),r) 

≤ a(β − ˜ 1 

β,η) + ( 

t2 + αh2 Rh( ˜ β − β) 2 0) 1/2 ((t 2 + αh 2 )r 2 0) 1/2 

1 

+ ( 

t2 + αh2 Rh(∇v − η) 2 0) 1/2 ((t 2 + αh 2 )q − ˜q 2 0) 1/2 

+ ˜ C((t 2 + αh 2 )q − ˜q 2 0) 1/2 ((t 2 + αh 2 )r 2 0) 1/2 

≤ C{β − ˜ 1 

β1 + ( 

t2 + αh2 Rh( ˜ β − β)0) 1/2 + q − ˜qh},

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ×ÐÐÐÐÒÔØ 

ÇØØÑÐÐÙÓÑÓÓÒÄÑÑÔØ 

rh 

ØÙÐÓ× βÓÐÚÐØØÙÄÖÒÒÒØÖÔÓÐÒØ××Ò×ØÒÖ×ØÑØÐÐ 

×ØÓÓÑÐÐ×Ò××ÄÑÑÒ×˜qÒÓÔØÑÐ×ØÒÒØÖÔÓÐØÓ 

ÃÓ×˜ 

ÓÑÒ×ÙÙ×ÒØ×℄ÚÙÐÐÚÐØÙÐÓ× 

β 

Ñ×× 

ÙÒÓÐØØÒÖØ×ÙÒ×ÒÒÐÐ×ÝÝ×β∈ [H 

ÐÒØØÒÓÔØÑÐ×ØÒØÖÔÓÐØÓÓÑÒ×ÙÙØÔ×ØÒÓÐÑÓÔÝØÐÒ ÇØØÑÐÐÐ××ÙÓÑÓÓÒÒÓÖÑÒÑÖØÐÑØ×ÚÐØØÙÒÒØÖÔÓ 

E 

ÚÓÑÐÐÚÖÖÚÓÙÐÒÓÖÑ×××ÙÖÚ×ØÝØØÑÐÐÒ×ÒÙÐÒÓÖ ÆÝØÝØØÑÐÐÈØÖÒÒÒÎÖÖØÒ×ØÑØØ×ÒÐÙ× 

ÚÙÐÐØÙÐÓ×Òβ 

ÑÒÑÖØÐÑ 

− 

 

q . ÎÐØÒ×ÙÖÚ×ÑÐÚÐØ×ÐÐη×ÒÐÑÒØÒÒØÖÔÓÐÒØØη c ÓÐÐÓÒ 

Vh℄ ∈ 

+ |(η,v)|h ≤ C. ËÙÖÚ×ÖÚÓÒØÖÑ1 t2 + αh2 Rh( ˜ β − β)0. 

1 

t2 + αh2 Rh( ˜ β − β) 2 0 ≤ h −2 (β − ˜ β) − (β − ˜ β) + Rh( ˜ β − β) 2 0 

≤ h −2 β − ˜ β 2 0 + h −2 Rh( ˜ β − β) − (β − ˜ β) 2 0 

≤ h −2 β − ˜ β 2 0 + Cβ − ˜ β 2 1 

1 

s+1 (Ω)] 2ÐÐÓÐÚØØÙÐÓ 

− ˜ β 2 1 + 

t2 + αh2 Rh( ˜ β − β) 2 0 ≤ Ch 2s β 2 s+1, 

|(βh − ˜ β,wh − Ihw,qh − ˜q)| ≤ ˜ 

CE, 2 = h 2s β 2 s+1 + h 2s q 2 s−1 + t 2 h 2s q 2 

s. 

βh1 + w − wh1 + q − qhh ≤ CE. 

− qh−1 = sup 

η∈V 

(q − qh,η) 

η1


Ç×ÓØØÒÒ×ÑÑ×ÐÐØÖÑÐÐÔØÒØÖÔÓÐØÓÓÑÒ×ÙÙ×ÒÔÖÙ× ØÐÐ . 

(q − qh,η − η c ) ≤ h 2 q − qh 2 0h −2 η − η c 2 Î×ØÚ×ØÙÓÑÓÑÐÐØØ×ÓÖÑÙÐÓÙÒÐØØÑÐÐÒØÖÐÐ 

0 ≤ |q − qh|hη1. ×ÖØÐÐÓÖÑÙÐØÓÐÐÔØ∀η c ÎÒØÑÐÐÐÑÔÖÚÝÐÑÑ×Ø×ÒØÓ×ÐÐØÖÑÐÐ×ØÑØØÝØ ØÑÐÐÝÚ×ÄÑÑÒÑÙ×ÒØÖÔÓÐØÓÓÑÒ×ÙÙ× 

(q − qh,η − η 

q − qh−1 = sup 

η∈V 

c ) + (q − qh,η c ) 

η1 

∈ V 

a(β,η c ) + (q,η c ) = 0, 

a(βh,η c ) + (qh,Rhη c ) = 0. 

(q − qh,η c ) = (q,η c ) − (qh,Rhη c ) + (qh,Rhη c ) − (qh,η c ) 

= a(β − βh,η c ) + (qh,Rhη c − η c ) 

= a(β − βh,η c ) + (q,Rhη c − η c ) + (qh − q,Rhη c − η c ) 

≤ C{β − βh1η c 1 + q0Rhη c − η c 0 

+ q − qh0Rhη c − η c ÌÐÐÒ××ÙÐÒÓÖÑÒÚÖÐÐ×ÒÖÚÓ 0} 

ÓÒÚÙÐÐÔ×ØÒØÙÐÓ×Ò 

≤ Cη1{β − βh1 + hq0 + hq − qh0} 

q − qh−1 ≤ C(hq − qh0 + hq0 + β − βh1), 

ÄÓÔÙÐÐ×ÒØÙÐÓ×ÒÔ×ØÒ×ÓÚÐØÑÐÐ×ÒÒÐÐ×ÝÝ××ØÑØØ 

hiÐÐ××ÐÙ××ÔØ×ØÒkÐÑÒØÐÐØÐÐÒ ÅÖØ×ÑÐÐhbÐÐÚÖÓÔÖÑØÖÐÙÒÖÙÒÐÐÚ×ØÚ×Ø 

β − βh1 + w − wh1 + tq − qh0 + q − qh−1 ≤ CE. 

E 2 = (h 2k β 2 k+1 + h 2k q 2 k−1 + t 2 h 2k q 2 k) 1/2 

= C{h k i (gs−2,Ωi 

+ hb(g−1 + tg0 + G0)}. 

+ tgs−1,Ωi + Gs−1,Ωi )

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ ÀÙÓÑÓÒÖÚÓ×ØÓÒØØÐÙ×Ò ÚÙÐÐ×ÒÒØÖÔÓÐÒØÐÐ 

Ihw×ÙÔÖÓÒÚÖÒ××ØÙÐÓ×ÚÖÓÖÔÔÙÚÒÒÓÖÑÒÒ×Ó×ØØ×℄ÌÙÐÓ× 

whÚÒØÐÐ×ØÑØØ 

ÃÙØÒÑÑÒØÓØØÒÓÒØ×ÓÐ×Ø×ÙÙ×ØØÚÖÐØÒÖÙÒÓÐÐ ×ØØÝÒÑÐÐÒÒÐÝÝ× 

ÙØÒÒÔÒÓÖÑ××| · |hÖÓØÙ×ÐÐw 

ÐØØÐÚÝØØÚÒÚÐÐÐÓÒÐÙÓÒØÐØÒÐÐÔØÒÒÄ××ÐÚÝÐØ ÔÖÙ×ØÐÐØØÒØØÝØÒØ 

×ÒÒÓ×ØÒÒÓÖÑ×× · 

ØØØÚØÝØÚÒÐÒÖÑÙÓÓÒ 

×ÙØÒÝØØØÝØØÚÓÒ×ÐÚ×ØÓÒ××ØÒØØÌÐÐØÖÓØØÒØ××Ý ØÝ×××ØØØÐØØ×ØÑØ××ØÖÑÒa(β,η)ÓÖÚÑÒÒÝØØÝÒ 

0ÖÚÓ ØØÚÒØÖÑÐÐΥ(u,v;β,η)ÚÙØÓÒ××ØÒ××ÚÖ×ÒE(q;η,v,r) ÌÐÐÒÔØÓÐÐÒÚÓÐÐC > 

ÐÝØØÓ×ØØÚÒ×ÒÒÐÐ×ÝÝ×ÓÑÒ×ÙÙØ×ÒÓÐÑÓÔÝØÐÒ ØØØÝÒØÖÔÓÐÒØØÖØÝÑÐÐ ÅÐÓÐØØÒØØØ×ÓÐ×Ø×ÙÙ×ÐØØØØÚÒÚÐÒÒÝØÒ×× βÐÐÝ Ñ××ũÓÒÄÖÒÒÒØÖÔÓÐÒØØØ×ÓÐ×Ø×ÙÙ×ØØÚÐÐ˜ 

ØÔÙ×××Ý×ØÑÐÐØÑÐÐÓØØÙÙÒÐØØØØÚÒÚÖÖ ÚÓÓÒÐ×ÑÐÐÐÙ×ØÝØÒ×ÒÐÓÔÙÐØØÙÐÓ× ØØÚÒ×ÒÒÐÐ×ÝÝ×ÚÓØÓ×ÚÐÐÖÓØÙ×ÐÐØ×Ó ËÒÒÐÐ×ÝÝ×ÓÑÒ×ÙÙ×ÒÔÖÝØÝÑÒÒÓ×ØØÚÐÐÔØØØÝÐÐÓ× Ì×× 

ÄÙ× ÓÐØØÒÙÓÖÑÖØØÚÒ×ÒÒÐÐ×××ØÒØØØÖÒÖØ×ÙÒ×Ò ÒÐÐ×ÝÝÐÐsÔØ1≤s≤kÑ××kÓÒÔÓÐÝÒÓÑÔÔÖÓ×ÑØÓÒ×Ø ÇÐÓÓÒΩÓÒÚ×ÑÓÒÙÐÑÓÐØØÝ×ØØÙØØÙÄ×× 

− 

1 

H2×ÒÒÐÐÒÒÇÑÒ×ÙÙÒ 

Υ(u,v;β,η) = (A : ε(u),ε(v)) + (B : ε(v),ε(β)) 

+ (B : ε(u),ε(η)) + (D : ε(β),ε(η)) 

uh − ũ1 + βh − ˜ β1 ≤ C(u − ũ1 + β − ˜ 

Ø×ÓÐ×Ø×ÙÙ×ØØÚÒ×ÒÒÐÐ×ÝÝÒÔÖÙ×ØÐÐÖÚÓ 

β1) 

uh − u1 + βh − β1 ≤ Ch s (βs+1 + us+1). 

ÌÐÐÒÔØ 

β − βh1 + w − wh1 + tq − qh0 + q − qh−1 + u − uh1 ≤ 

C{h k i (gs−2,Ωi + tgs−1,Ωi + F s−1,Ωi ) + hb(g−1 

+ tg0 + F 0)},

ÄÅÆÌÌÁÅÆÌÄÅÅÁÌÄÅÆÌÁÄÄ Ñ××F××ÐØ×ÐÚÝØØÑÓÑÒØØÙÓÖÑØÙ×Ø 

F 2 s = f 2 s + G 2 ÐÐhbÓÒÐÐÒÚÖÓÔÖÑØÖÖÙÒÐÐhi××ÐÙ××ÖØÝ××Ø 

s. Ú××ÒÒÐÐ×ÐÐÚÖÓÐÐhb = 

hi = h

ÅÄÄÁÆËÇÎÄÄÍËÈÈÊÁÊÃÁÆÃÍÈÊÍÁÄÍÍÆ ÅÐÐÒ×ÓÚÐÐÙ×ÔÔÖÖÒÙÔÖÙÐÙÙÒ 

ØÓ×ØØÓ×Ò×ÒÓÒØØÚ××ÓÐÐÚÖ×Ò××ØÒÒÓ×ØÙÒØÔÔÖÖ ÙÔÖÙÐÙÙÒÃÙÔÖÙÐÙÐÐØÖÓØØÒÐÓÐÔÔÖÒÔÒÒÒÑÙÓÓÒÑÙÙ ØÑÒÒÓÒÓÒÐÑÐÐ×ØÐÙØØ××ØÙØÓ×ØÙÒÚÙØÙ×ÔÔÖÒ ÈÔÖÓÒÒ×ÓØÖÓÓÔÔÒÒØÖÓÒÒÒÑØÖÐÓØÒ×ÒÑÐÐÒ 

ÝØØÑÒÚØØ××℄×ØØØÝÑÐÐÓ××ÔÔÖØÒÓÙ×ÒÖ ÒÐÓÐ×ØÑÙÓÓÒÑÙÙØÓ××ØÙØÒÝÖ×ØÝÑ××ØÂØÓ××ØÙÐÐÒ 

ÖÓ×ÒÓ××ÑØÖÐÓÐØØÒÓÑÓÒ×× ÒÒÔÐÓÒÎÔÔÖÒÙØÙ×ÙÙÒÒØ×ØØÚØÓÐÐÐÓÐ×ØÚÖ×Ò ÅÐÐ××ÔÔÖØÒÓÙ×ÒÖÖÓ×ÒÓÒÒÖÖÓ×ÐÐÒÔ ÇÖÒØØÓÒ×ÓØÖÓÔ 

ÖØØÚÒØÅÖØÒΘÐÐÔÙØÙ×ÙÙÒØÓÑÖØÐÐÒÙÐÑÒ ÖÓÓÔÔ×ØÑØÖÐÑÐÐÙÒÒØÒÑÖØØÑ×ÒÝØØØÝÚÖÓÓÒ ×ØØÙÑÒÚÖ×Ó×××ÐÑÒØ××ÚÓÒÙØÒÒ×ÓÚÐØÓÖØÓØ 

ÑÖØÒÖÓ××ÖØÓÒ ξÐÐÒ×ÓØÖÓÔÒ×ØØØÃÓÒÒ×ÙÙÒØÒÓØ×ÙÓÖ××ÓÐÚ×ÙÙÒØ ÓÒÒ×ÙÙÒÒÒÅÅÒÖØÓÒÙØÙÒÔ×ÙÙÒÒÒÚÐÐÐ× 

Θ 

MD 

ÃÙÚÇÖÒØØÓÙÐÑÒΘÒ×ÓØÖÓÔÒξÑÖØØÑÒÒ 

n2 

ØØÓÒÚÓÑÙÙÒÑÖØÐÐÒ×ÙÙÖÑÑÒÙØÙÓÒ×ÒØÖØÓÒÚÓÑ 

ÒÒn1ÅÒÚÐ×ÒÙÐÑÒÒ×ÓØÖÓÔξÓÖØÓÓÙØÙÓÖÒ ÇÖÒØØÓÙÐÑΘÑÖØØÒ×ÙÙÖÑÑÒÙØÙÓÒ×ÒØÖØÓÒ×ÙÙÒ 

b 

n1 

a 

CD

ÅÄÄÁÆËÇÎÄÄÍËÈÈÊÁÊÃÁÆÃÍÈÊÍÁÄÍÍÆ ÙÙÒ×ÙØÒØØÚ×ØÒÓØ×ÙÓÖ××ÓÐÚÒÙØÙÓÒ×ÒØÖØÓÓÒ 

×ÒØÖØÓÒÔ×ÙÙÒÒØÓÒÔÖÖØØÝÐÐÔ×ÒÔ×Ð× 

a/bÌÐÒÒØØÚÓÒÚÒÒÓÐÐ×ØÃÙÚÒÚÙÐÐÓ××ÙØÙÓÒ 

ÈÔÖÖÒÓÑÒ×ÙÙ×ÒÑÖØØÐÑ×××ÓÐØØÒÝØØÝÒÚØØ× ÅØÖÐÔÖÑØÖÒÑÖØØÑÒÒ 

ξ 

×℄×ØØÝÒÐØ×ØÑÒØÐÑÓ××Ò×ÑÑ×ÒÔÔÖ×ØÑØØÒ 

= 

×ØÖÓÓØÓÑØÖ××ØÓ×ØÙÒÙØØÑØÓÐÐÒÒÙÔÖÙÐÙÓÒÐÒ 

ÐÝÝ×ÒÓ××ÒÝØØØ×ÒÒØÒÒÓÒ ÙØÙÓÖÒØØÓÑØØÒ ÖØÒ×Ò×ÒÖÙÙØÙÙÒØØÙÙÒÖÖÓ×ÒÓ×× 

ÚÚÐÓÚ×ØÒÌÐÐÒÙØÙÒÖÙÒØÖÓØØÙÚØÚÓÑÒÓÒØÖ×Ø ÖÒØØÒØ×ØÐ×ØÒÙÐÑ×ØÓÖÑÑÙÒØÓÒ ÃÙØÙÓÖÒØØÓÒÑØØÙ×ÔÖÙ×ØÙÙÐÑÒÓØÙÒÒÔÔÖÒÙÚÒ 

µÑÔ×ÐÖ×ÓÐÙÙØÓÐÐ×Ø 

ÑÑÓÑÓÙÐÒ×ÈÓ××ÓÒÒÐÙÙÒÖÚÓØËÙÖÚ×××ØØÒÚØ ÙÐÑ×ÙÙÖØÆÒÐ××ØÖÚØÒÐÓÐ××ÓÓÖÒØ××ÐÙÙ Ø××℄ÝØØØÝÑÐÐÐÓÐÒÖÚÓÒÑÖØØÑ×Ò ÚÙÐÐÐÓÔÙÐØÑØÖÐÔÖÑØÖÒÑÖØØÑ××ØÖÚØØÚØÒ×ÓØÖÓÔ 

ÒÒÔÓØØÒÒÑÑÓÑÓÙÐEMDECDÓ×ØÐ×ØÒÒ×ÓØÖÓÔÒ ÈÔÖÒÑØÖÐÔÖÑØÖ×ØÑÖØØÒÝÐ××ØÚÒÓÒÒ×ÙÙÒØ ÃÑÑÓÑÓÙÐ 

ÚÙÐÐÐÓÐØÙÙÒ×ÙÙÒØÒÒ×ØÚ×ØÒÓØ×ÙÓÖÑÑÓÑÓÙÐ 

ÚÒÑØÖÐÔÖÑØÖÒÑÖØØÑ×Ò ÑØØÙ×ÐÙ××ÙØÙÓÒ×ÒØÖØÓÓÒÝØ×ÙÙÖ×ÖØØÚÒ×ÓÐÙÓØØØ Ó×ØÙ×Øβ×Ò×ÓØÖÓÔÒ×Ø×ØξÌÓ××ÓÐØØÒØØ×× Ò×ÒÒÓÐØØÒØØÐÓÐØÑÑÓÑÓÙÐØE1E2ÖÔÔÙÚØÒÓ×ØÒ 

ÖÓÔÚÓÒ××ÖÓØØÐÓÐÒÑÑÓÑÓÙÐÒÚÙÐÐÑÙÓØÓÓÒ ÐÓØØÑÐÐØÔÙ××ØÓ××ÓÒÚÒ×ÓØ×ÙÓÖ××ÓÐÚÙØÙ 

2ÒÒÐÐÒÌ××Ý×ÒÖØ×××ØÔÙ×××Ò×ÓØ 

1Î×ØÚ×ØÑÐn1×ÙÙÒÒ××ÓÒ×ÙØÙ ØÙÐÔØE1/E2 = 

= ÔØE1/E2 

ÔÝ×ÝÝÐ×ÚÓÒÒ×ÓØÖÓÔÒÚØÐÙ×ØÖÔÔÙÑØØÓØÒÐÓÐÐÐ 

= ÌØÒ℄ØØÑÑÓÑÓÙÐÒÓÑØÖÒÒ×ÖÚÓEg 

H(α) = 

|∇f(m,n)|δα,α(m,n) 

m,n 

E1 

E2 

= ξ. 

√ EMDECD

ÅÄÄÁÆËÇÎÄÄÍËÈÈÊÁÊÃÁÆÃÍÈÊÍÁÄÍÍÆ ÑÑÓÑÓÙÐÐÐÔØØÐÐÒÝØÝ× 

E = ÝÚ×ÝØÝ× ÌØÒÐÓÐÒÑÑÓÑÓÙÐÒÐÙ×ØÚÓÒÖÓØØÝØØÑÐÐ 

ÙÐÒEÚÙÐÐÑÙÓØÓÓÒ Ò×ÓØÖÓÔÒξ×ÑÖ×ÒÑÑÓÑÓ 

E1E2. 

×ÒØÒ×Ú××ÑÑÓÑÓÙÐØEMDECDÔÒÒÚØÄ××ÖÔÔÖ ÐÙÐÐØÝØÑØØÙ×ØÓ×ÓØØÚØØØÓÑØÖ×Ò×ÖÚÓÒEgÖÔ Í×ÒØÙØÑÙ×ØÒÔÖÙ×ØÐÐØØÒ℄ØØÔÔÖÒÓ×ØÙ×ÔÖÓ 

E1 

ÓØÒ×ÑÖ×ÐÐÑÑÓÑÓÙÐÐÐÚÓÒÑÔÖ×ØÒÓÒÔÖÙ× ÔÙÚÙÙ×Ó×ØÙ×ØÓÒ××ØÔÙ×××Ö×ØÖÚÓØÙÒÐÒÖ×Ø 

Ñ××βÓÒÔÔÖÒÓ×ØÙ×ÔÖÓ×ÒØØÌÐÐÒÐÓÐØÑÑÓÑÓÙÐØÓÚØ 

ØÐÐÚÐØÖÔÔÙÚÙÙ×℄E = 

ÌÙÐÙØÒÒÙÓÑØØØÑÐÐÚÓÒÝØØÚÒÐ×ÐÐÓ× 

ÐÑÐÐÑÙÙØØÙÙÔ×ØÐ× 26%×ÑÑÓÑÓÙÐÒØÚ×ÖÚÓÑØÖ 

ÈÓ××ÓÒÒÐÙÙÒÓÐÐÓÐ×ÐÚÓÒÓÒ×ÓØÖÓÔÒÈÓ××ÓÒÒÐÙ ÙÒÚÐÐÐ×ÙÓÖÝØÝØØÀÝÚ×ÚÓÒÙØÒÒÝØØÐ×Ø×ÙÙ× ÈÓ××ÓÒÒÐÙÚÙØ ØÙ×ÐÐ×ÐÐÙÒβ> 

ØÓÖ××ÔØÚÅÜÛÐÐÒÝØÝØØ℄ 

ÓÓÒ×ÓØØÑÐÐÑÑÓÑÓÙÐÒÐÙ×Ø×ÒÈÓ××ÓÒÒÐÙÙÒ Ò×ÓØÖÓÔÒÚÐÐÐÝØÝ× 

 

, 

= E ξ, 

E2 = E √ ξ . 

−0.25β + 6.5 GPa, 

E1 = ξ (−0.25β + 6.5) GPa, 

E2 = 1 √ ξ (−0.25β + 6.5) GPa. 

µ12 

µ21 

= E1 

E2

ÅÄÄÁÆËÇÎÄÄÍËÈÈÊÁÊÃÁÆÃÍÈÊÍÁÄÍÍÆ 

25 

20 

8 

15 

6 

10 

4 

5 

2 

ÃÙÚÃÑÑÓÑÓÙÐØE1E2Ò×ÓØÖÓÔÒÓ×ØÙÒÙÒØÓÒ 

0 

0 

30 

30 

10 

10 

20 

20 

10 

5 

10 

5 

 

Kosteusprosentti 0 0 

Anisotropia 


Anisotropia 

ÑÙÓØÓÓÒ ÚÙØÚÓÒÖÓØØØÓÐÐ×ÒÈÓ××ÓÒÒÐÙÚÙÒµÒ×ÓØÖÓÔÒξÚÙÐÐ 

µ12µ21ÓÒÖ×ØÓØØÒÚÓÒ×ÓØÖÓÔÒ×ÙØÒÓØÒÈÓ××ÓÒÒÐÙ 

µ12 

= ξ. 

µ21 Î×ØÚ×ØÙÒÑÑÓÑÓÙÐÒØÔÙ×××ÓÑØÖÒÒ×ÖÚÓµ= 

√ 

µ12 = µ ξ, 

µ21 = µ ÒÒÐÙÚÙÐÐÐÒÖÒÒÚ×ØÚÙÙ×Ó×ØÙ×ÔÖÓ×ÒØØÒØ××ØÝ××ÝØØÒ ÚØØ××℄×ØØØÝÑÔÖ×ÒØÙÐÓ×ÒÔÖÙ×ØÙÚÖÚÓØ ÂÐÐÒÓÐÐ×ØÒØÙÐÓ×ØÒÔÖÙ×ØÐÐÚÓÒÚÐØØÓÐÐ×ÐÐÈÓ××Ó 

√ . 

ξ 

 

Ý×××ÑÐÐ××ØÓØÙØÙÙØ××ØÔÙ×××ÐÙÙÙ×ÓÔÐÐ×ÐØÒÒÒÐØ ÖÒØØÓØÚÓÑÖØØ×ÐÐØÐÐÒÑØÖÐÓÒ×ÓØÖÓÓÔÔ×ØÆÝ ÀÙÓÑÓÒÖÚÓ×ØÓÒØØÒ×ÓØÖÓÔÒξ×××ÖÚÓÒÝ×ÙØÙÓ 

××ÅÜÛÐÐÒÝØÝ×ÓÒÈÓ××ÓÒÒÐÙÙÒÑÖØÐÑÒÚÙÓ×Ò 

µ = 0.015β + 0.150. 

ÚÓÑ×× 

µ21×ÓØÖÓÓÔÔ×ÐÐÑØÖÐÐÐÐ 

= ×ÐÚØÚØÑÙ×ØE1 E2µ12 = 

12 

10


2 

1.5 

0.6 

1 

0.4 

0.5 

0.2 

ÃÙÚÈÓ××ÓÒÒÐÙÚÙØµ12µ21Ò×ÓØÖÓÔÒÓ×ØÙÒÙÒØÓÒ 

0 

0 

30 

30 

10 

10 

20 

20 

10 

5 

10 

5 


Anisotropia 


ÃÝØØÝ××ÑÐÐ××ÓÐØØÒÐØÒÔ×ÙÙ××ÙÙÒØ×ØÐÙÙÑÓÙÐØÖÔÔÙ ÄÙÙÑÓÙÐØ 

Anisotropia 

ÐÙÓÖÚÓÅÐÒÒØÓÒÒ×ÙÙÖÓÒ×Ò×Ò(x,y)Ø×ÓÒÐÙÙÑÓÙÐ ÑØØÓÑ×Ò×ÓØÖÓÔ×ØÓ×ØÙ×ØÝØØÒÒÐÐÑØØØÙØÙ 

G12Ó×ÓØÖÓÓÔÔ×××ÑØÖÐ××ÓÒ E 

G12 = 

2(1 + µ) . ÃÙØÒÐÐØÓØØÒÓÑØÖ×ØÒ×ÖÚÓÒÚÙÐÐÑÖØÐØÝØÓÐÐ ÒÒÑÑÓÑÓÙÐÈÓ××ÓÒÒÐÙÙÚØÖÔÙÒ×ÓØÖÓÔ×ØÌØÒÐÙÙ 

ÑÓÙÐÚÓÒÖÓØØÑÙÓØÓÓÒ Eg 

G12 = 

2(1 + µg) = 

√ 

E1E2 

2(1 + √ µ12µ21) = 

E 

2(1 + µ) , ÓÓÒ×ÓØØÑÐÐÐÙ×Ø ×ÒÐÙÙÑÓÙÐ× 

6.5 − 0.25β 

G12 = 

2.3 + 0.03β GPa. ÐÙÓÖÚÓ È×ÙÙ××ÙÙÒØ×ÐÐÐÙÙÑÓÙÐÐÐÝØØØÒÚØØ××℄×ØØØÝØÙ 

G23 

1 

0.8 

= 0.09 GPa, 

G31 = 0.34 GPa.


ÃÓ×ØÙÒÙØØÑÑÙÓÓÒÑÙÙØÓ× 

3 

2.5 

2 

1.5 

0.6 

ÃÙÚÎ×ÑÑÐÐÐÙÙÑÓÙÐG12Ó×ØÙÒÙÒØÓÒÓÐÐÖØÓ 

1 

0.4 

0.5 

0.2 

0 

0 

0 5 10 15 20 25 

0 2 4 6 8 10 

ØÝÚÑÙÓÓÒÑÙÙØÓ×ÃÓ×ÔÔÖÒÙØÙÖÒÒÓÒÚÓÑ×ØÒ×ÓØ ÃÙÔÖÙÐÙÝÖ×ØÝÑ×ØÔÔÖ××ÙØØÓ×ØÙÒÚØÐÙ×Ø×ÝÒ 

Kosteusprosentti 

Anisotropia 

Ø××ÑÙÓÓÒÑÙÙØÓ×ØØÙÐØÙØÖÖÓ×ØØÒÓ×××ÓÖÒØØÓ ÖÓÓÔÔÒÒØÝØÝÝØÑÓØØÙÓÑÓÓÒÓ×ØÙÒÚÙØÙ×ÑÐÐÒÒØ ÐÙ××Ö×ÒÌÐÐÒÚÓÒÝØØÚØØ××℄×ØØØÝÖ×Ø ÑÔÖ×ÒØÙÐÓ×ÒÔÖÙ×ØÙÚÑÐÐÓ××ÝÒÐÙÒÓ×ØÙ×ØÓ ØÙÚÐÐÚÒÝÑÐÐÔØε [12] ÃÖØÓÑØÓÚØÒ×ÓØÖÓÔÒξÙØÓÒ 

 

α1(ξ) 0 

0 = 

0 α2(ξ) 

α1(ξ) = 0.0006 − 0.00015 ξ − 1, 

α2(ξ) = 0.0006 + 7 × 10−8 ÚÓÒÐÓÔÙÐØÐ×ÝØÒÐÑÒØÒÔ×ÙÙÒÝÐÀÙÓÑÓÒÖÚÓ×Ø ÌØÒ××Ó×ØÙÒÙØØÑÐÙÚÒÝÑØÙÐÐ×ÓÐÑÒ 

ÓÒØØÒ×ÓØÖÓÔξÓÒÒÖÚÓÐØÒ×ÙÙÖÑÔØÝØ×ÙÙÖÙÒÝ ÑÒØØØÒÓÒ×ØØÙØÚ×ÒÝØÝÒÚÙÐÐÚ×ØÚ×ÒÒØÝ××ÓØ Ø××ÓÖÖÓ×××Ö×ÒÌÑÒÐÒÚÒÝÑØÑÙÙÒÒØÒÐ 

(ξ − 1). 

ÐÒÑÒÒØÔØÙÙÝØÚÓÑÒÑÓÐÑÔÒ×ÙÙÒØÒ 

×ÓØÒÖØÓÑØÔÝ×ÝÚØÖÐ×Ò×ÓØÖÓÓÔÔ×××ØÔÙ×××ξ= 1 

G 12 

Kosteuslaajenemiskertoimet 

x 10−3 

1.4 

ÑØα1×ÒÒÒα2ÔÙÒÒÒÒ×ÓØÖÓÔÒÙÒØÓÒ 

1.2 

1 

0.8 

.

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ÚÙØØ×ÐÙØØ×Ó×ÖØÝÑØÐÐÖÙÒÓÐÐÊÙÒØÓÒÚÙØÙ×Ø ÙÔÖÙÐÙÙÒØÖ×ØÐÐÒÒÙÑÖ××Ø×ÙÖÚ××Ó×Ó××Ä××ÔÖÓ× ÒÐÖÓØØÑØØÑÝ×Ø×Ó×ÖØÝÑÓØÒØÐÐØÔÙ×ÐÐÖÚÐØ Ò××ÌÓ×ÐØÝ×ØØÙØØÙÒÖÙÒÓÒØÓØÙØØÑÒÒÝØÒÒ××ÓÒ 

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×ØØÝÒÝØÙÔÐÐÒÒÐ×ÒØÓÐÑ×ØÓÓÓÒ×Ý×ÝÒ ÖÒÓÐÑ×ØÓÒ℄ÚÙÐÐÆÙÑÖÖÒÓÒÖØÝ××ØÐÑÒØØÑÒØÐÑÒ ÃÓÑÔÓ×ØØÐØØÑÐÐÒÒÙÑÖÒÒØÓØÙØÙ×ØØÒÆÙÑÖÓÐÇÝÒÆÙÑÖ 

ÐØØÑÐÐÒÑÙÒÒÅÁÌÐÑÒØØÒÔÖÙ×ØÙÚÖØ×ÓØÒÐÙ× ØÒÓÐÑ×ØÓÓÐÒØÙÙ×ØØÙØØÙ ØÙØØØÒÝØ×ØÝ××ÌÒÐÐ×ÒÓÖÓÙÐÙÒÒ××Ê××ÒÖÒÅÒÐÒÒ ÒØÓ 

ÝØÑÒØÑÒÑÙÙØÑÑÙÙØÓ×ÐØØÐÚÝØØÚÒÝØÒÒØÓ ÐÑ×ØÓÚÖØÒØØÝÒÆÙÑÖÖÒÐÒÚÙÐÐÎÅÁÌÐÑÒØØ ÓÐÚØÓÚÐÑ×ÓÐÑ×ØÓ××ÓÙÙØØÒÑÝ×ÓÐÑÒÓÖØÖÒÐ×Ò ÆÙÑÖÖÒÓÐÑ×ØÓÒÝØØÓÒÒ×ØÙÙÖ×ÒÝØØÐØØÝÑÒÓ 

Ñ×× 

ÇÐÑÒØÓØÙØÙ×××ÔÝØØÒÝØØÑÒÐÓÔÙÐØÓÐÑ×ØÓÒÑÙÓ Ù×ØÖÔÒÚÒØÑ××ÒÒÙØ×ÙØØÙÒ×ÒÖÒÓØØÓØÒÒØÝ× ÌÒ×ÓÖÒÚØÓÖÒÓØØÓ 

ÚÒÝÑØÒ×ÓÖÒ×ØØÑ×ÒÚØÓÖÑÙÓÓ××ÓÐÐÓÒÒÐÒÒÒÖØÐÙ ÚÙÒØÒ×ÓÖØ×ØØØÒÓÖÚØÑØÖ××ÙÙÖÐÐ ÒÑÙÙØØÙÒØÔÙ×××ÓÐÑÖÔÔÙÑØÓÒØÓÑÔÓÒÒØØÌØÒ×× ÃÓ×ÚÒÝÑÒÒØÝ×ØÒ×ÓÖØÓÚØÒ×ÝÑÑØÖ×ÓÒÒ×× 

ÒÒØÝ×ÚØÓÖØ˜ε˜σÐÙÔÖ×ØÒØÒ×ÓÖ×ÙÙÖÒÓÑÔÓÒÒØØÒÚÙÐ ØÓÑÙÓÓ××ÓÒÚÒÝÑÒÒØÝ×ØÒ×ÓÖÒÚÐÒÒØÒ×ÓÖØÙÐÓÓÐÙ ØÒÒÝØÓÖÚØÙÙ×ÒÚØÓÖ×ÙÙÖÒÔ×ØØÙÐÓÐÐÅÖØÐÐÒÚÒÝÑ 

2ÑØÖ××ÙÙÖØÚØÓÖÐÐÇÐÐÐÒÒ×ÙÙÖÚÖ 

Ð×ÙÖÚ×Ø 

ÐÙØÒÓÖÚØ2 × 

 

 

ÀÙÓÑÓÑÐÐØÒ×ÓÖÒ×ÝÑÑØÖ×ÝÝ×ÔØ 

T 

˜ε 

ÑÐÐÓÒØÖØÓÓ×ÒÒ×Ò×ÙØÒÑÙÙÒÒÓ×ÑØÖ×ÒTÒ××Ñ× ÌÙÒÒØÙ×ØØÒ×ÓÖ×ÙÙÖØÑÙÙØØÙÚØÓÓÖÒØ×ØÓÑÙÙÒÒÓ×××ÓØØ 

˜σ . 

ÒØ×ØÓÒÖÖÓÒÙÐÑÐÐΘØÔÙ××××× 

×TÓÒØÚÒÓÑÒÒÓÓÖÒØ×ØÓÑÙÙÒÒÓ×ÒÂÓÒÑØÖ×ÓÓÖ 

 

= 

ǫ11 ǫ22 2ǫ12 

T = σ11 σ22 σ12 

ε : σ = ǫijσij = ǫ11σ11 + ǫ22σ22 + 2ǫ12σ12 = ˜ε· ˜σ.

ÅÄÄÁÆÌÇÌÍÌÍËÆÍÅÊÊÁÆÇÀÂÄÅÁËÌÇÇÆ 

ÌÐÐÒÖÖØÝ××12ÓÓÖÒØ×××ØØØÝÚÒÝÑØÒ×ÓÖε [12]ÓÒ 

Ó×Ø×ÙÑÑÙ×ÙÖÓØØÑÐÐÚÓÒÑÙÙÒÒÓ×ÖÓØØ3×3ÑØÖ×Ò 

RÚÙÐÐÑÙÓÓ×× 

Ñ××RÑØÖ×ÒÓÑÔÓÒÒØØÓÚØ 

jl , 

˜ε 

ÌØÒÒÒØÝ×ÒÑÙÙÒÒÓ×ÑØÖ×××Ò ÐÚÑ×××ÓÑÔÓÒÒØ××ÓÐÖÖÓÒØ×ÙØÒÚÒÝÑÚØÓÖ×× ÀÙÓÑÓÒÖÚÓ×ØÓÒØØÒÒØÝ×ÚØÓÖÒÑÙÙÒÒÓ×ÑØÖ×ÓÒÖÐÒÒ×Ð 

ÒØÖÑÒÑÖÑÙÙØØÙÙÐÐÑÒØÙ××ÑØÖ×××ØØÑØÖÐ Ø×ÑÙÙÒÒÓ×Ø×ÒÚØÑÐÐÙÐÑΘ→−ΘÓÐÐÓÒÒÓ×ØÒ× ÓÐÐÐεσÄ××ÙÓÑØÒØØÑÙÙÒÒÓ×ÑØÖ×Ú×ØÚØÒ ÂØÓ××ÑÖØÒ×ÐÚÝÝÒÚÙÓ×Ú×ØÚÚØÓÖ×ÙÙÖØÑÝ××ÝÑ 

ÑÐÐ××ÝØØØÝÓÖÒØØÓÙÐÑφÓÒÙØÙ×ÙÙÒÒÒÅÒÚÐÒÒÙÐ 

ØØÙØÚÒÒÝØÝ× Ó×Ø×ÒÓÑÔÓÒÒØØÙÖÓØØÑÐÐÚØÓÖ×ÙÙÖÐÐÚ×ØÚÓÒ× ÂÒÒØÝ×ÚÒÝÑØÒ×ÓÖÒÚÐÐÐÔØÓÒ×ØØÙØÚÒÒÝØÝ× ÑÓØÒΘ = 

Ñ××ÑØÖ×CÓÒÐÙÔÖ×ÒÒÐÒÒÒ×ØÒÓÒ×ØØÙØÚ×ÒØÒ×ÓÖÒ 

CijklÓÑÔÓÒÒØØÒÚÙÐÐ×ØØØÝÒÑÑÒ×ØØÝØØÒ×ÓÖÒ×ÝÑÑØ ÖÓÑÒ×ÙÙØ×ÓÖØÓØÖÓÓÔÔ×ÒÑØÖÐÒØÔÙ×××ÓÑÒ×ÙÙÒ 

ε 

 

T = 

cos Θ − sin Θ 

sin Θ cos Θ 

. 

[xy] 

ij = TikTjlǫ [12] = R ˜σ [xy] 

, 

ǫ [12] 

⎡ 

⎢ 

R = ⎢ 

⎣ 

cos2 Θ sin 2 sin 

Θ − cos Θ sin Θ 

2 Θ cos2 Θ cos Θ sin Θ 

2 cos Θ sin Θ −2 cos Θ sin Θ cos2 Θ − sin2 ⎤ 

⎥ 

⎦ 

Θ 

. 

⎡ 

⎢ 

⎣ 

cos2 Θ sin2 sin 

Θ −2 cos Θ sin Θ 

2 Θ cos2 Θ 2 cos Θ sin Θ 

cos Θ sin Θ − cos Θ sin Θ cos2 Θ − sin2 ⎤ 

⎥ 

⎦ 

Θ 

. 

π/2 − φ 

C1112 = C1222 = 0ÙÓÑÓÒ 

= C σ,


ÓÓÖÒØ×ØÓ××ÑÙÓØÓÓÒ 

ÓÐÐÓÒ××ÓÒ×ØØÙØÚÒÒÑØÖ×xyÓÓÖÒØ××ÓÒ 

ÌÐÐÒÓÒ×ØØÙØÚÒÒÝØÝ×ÚÓÒÖÓØØÓÓÐØÐÐÝØ×××xy 

ÑÙÙÒÒÓ×ØÒ×ÓÖÐÐ ÃÓ×ÓÓÖÒØ×ØÓÑÙÙÒÒÓ×ØÔØÙÙÚÒxyØ×Ó××ÔØØÐÐÒ 

Ø Ë×ÔÓÒ×ØØÙØÚ×ÒØÒ×ÓÖÒÚÑ×ØÓÑÔÓÒÒØØÑÙÙÒØÙÚØ×ÙÖÚ× 

Ó×ØÖÓØØÑÐÐ×ÙÑÑÙ×ÙÙÓÑÓÑÐÐÓÖØÓØÖÓÓÔÔ×ÐÐÑØÖ 

T3i 

C 

 

= ÐÐÐÔØÚÓÑÒ×ÙÙ×C3132 

Ø×ÙÑÑÑÐÐÐØÒÖÖÓ×ØÒÔ×ÙÙÒÝÐÅØÖ×ÚÓÒ×ÙÓ ÑÒØÓØÓÖÒØÒ×ÓÖÒ ÖÒ×ÓÚÐØÆÙÑÖÖÒÐ××ÓÒËÝÑÖÖÙØÒØØÙÓØØÚØÑÖ ÆÒÑØÖ×ÒÚÙÐÐÔÝ×ØÝØÒÑÙÓÓ×ØÑÒÒÐ×××ÒÐ ÚØÓÖÒÓØØÓÚ×ØÒØÚ×ØÚ× 

. 

ØÐÑÒ ÆÙÑÖÖÒÐÒÒÑÐÐ ÑÙ×ÒÚÒÝÑÚØÓÖÒ 

ÅÐÐÒØÓØÙØÙ×ÆÙÑÖÖÒÐÐÐÓÒÒ×ØÙÑÐÓ×ÙÓÖÚÚ××ØÙÒÒ ØÖÚØØÚØÓ××Ø×ØÒØÓÑÑÒÓÐÑÒÝØÑ××ÆÙÑÖÖÒÐÒ ÔÖÙ×ØÙ×ÒÑÙ××ØÑÐÐÚÓÒÓÐÑÓÖÓØØÑÐÐØØÚÒ 

C [xy] 

3i3j = 

 

⎡ 

⎢ 

C = ⎢ 

⎣ 

C1111 C1122 C1112 

C1122 C2222 C1222 

C1112 C1222 C1212, 

⎤ 

⎡ 

C11 C12 0 

⎤ 

⎥ 

⎦ = 

⎢ 

⎣C12 

C22 

⎥ 

0 

⎥ 

⎦ 

0 0 C33 

. 

σ [xy] = C [xy] ε [xy] = R T C [12] Rε [xy] , 

C [xy] = R T C [12] R. 

= δ3i. 

[12] 

3i3j = T3pTiqT3rTjsCpqrs 0×ÒØÒ×ÓÖÐÐxyÓÓÖÒØ×× 

= TiqTjsC3q3s, 

C3231 = 

C3131 cos 2 Θ + C3232 sin 2 Θ (C3131 − C3232) cos Θ sin Θ 

(C3131 − C3232) cos Θ sin Θ C3131 sin 2 Θ + C3232 cos 2 Θ

ÅÄÄÁÆÌÇÌÍÌÍËÆÍÅÊÊÁÆÇÀÂÄÅÁËÌÇÇÆ ÓÑÙÓØÓÖ×ÙÐÑÙÓØÓÓÒÚØÑÐÐØÑÒÓÐÐ×ÁØ×ÝÝÝ× 

ÚÖØÒØØÝÖÙØÓÓÔÖØØÓÖRhØÓÑÓÐÑ×ØÓ××ØÝ×ÒÙØÓÑØØ ÑØÖ×ÒÓÓÑ×ÒÝØØÒØÖÚØ×ÚÙØØ×ÐÐÐØØÐÑÒØØ ×ÒÚÐÒÒÒ×ØÔÙÑÐÐØØÖØÝÑÐÐ ××ØÑÖØØÐÝ××ÐØÙØÓÑØØ××ØÝØÒ×ÓÔÚÒÙÒØÓÚÖÙÙ 

ÚÒ×ÓØÖÓÔÓÖÒØØÓØØÓÃÓ×Ò×ÓØÖÓÔÓÖÒØØÓØØÓ ÑÓÒÑÙØÒÒÓÔÖØÓÓØÒØØÚÖØÒÖÓØØØÒÖÐÐÒÒÐÑÒ ØØØÒ×ÙÓÖØØØÚÐÓÐÑÓÝØØÝÚ×ÒØÓ×ØÓ×ØÐÙØØ ÃÓÒ×ØØÙØÚ×ØÒØÒ×ÓÖÒÑÖØØÑÒÒÓÒÝØØÝ××ÑÐÐ××ÑÐÓ 

Ø××ÖØØÚÒØÚÖÓÖÐ×Ø×ÐÐÚÖÓÒØÝÐÐØØÚ××ÓÒ ÓÙÙØÒÐÙÑÒÓÐÑÒØ××Ö×ÒÓÒÓÒ×ØØÙØÚ×ØÒØÒ×ÓÖ 

Ù×ÒÑÙÙØÑ×ØÓØÙÒ×ÚÔÙ××ØØ Ö×ÚÒÓÔ×ØÚÖ×Ò×ÙÙÖ×ÓÒ×ÑÑ×Ò×ØÒÐÑÒØÐÐÝØØ ÒÑÙÓÓ×ØÑÒÒÑÝ×ÚÚÓÔÖØÓÄ××ÚÔÙ××ØÒÑ 

ØÒÓÐÐÙØÑÒØØÚ ÑÐÐÒ×ÒØÒ×ÓÖØÚÐÑ×××ØÐÙ××ÑÙØØÚÙØÙ×ÒÓÔÙ ÃÓÒ×ØØÙØÚ×ÒØÒ×ÓÖÒÑÙÓÓ×ØÑ×ØÝÖØØØÒÑÝ×ÒÓÔÙØØÐ× 

ÙÔÖÙÐÙÙÒÃÓ××ÝØØØÒÐØØØØÚÐÐÓÓÝ×ÒÖØ××ØØÙØ ÆÙÑÖ××ØÐ×ÑÐÐØ×ØØØÒÖÙÒÓÒÚÐÒÒÒÚÙØÙ×ØÔÔÖÒ ÊÙÒØÓÒÚÙØÙ× 

ØÚÔ×ÒÐÚÝØØÚÒÖÙÒØÓÒÌÙÐÓ×ØÓÒ×ØØØÝÃÙÚ×× ØÙØÝ×ØØÙØØÙÖÙÒØÓÝ×ØØØÒØÑÓÓÒÒØØØÝÒ 

ÑÒÖÙÒÓÒÚÐÒÒÐÐÙÓÑØØÚ×Ø×ÙÙÖÑÔÚÙØÙ×ÙÔÖÙÐÙÙÒ ÃÙØÒ×ÑÙÐØÓÒØÙÐÓ××ØÓÒ×ÐÚ×ØÙÓÑØØÚ××ÓÒØ×Ó×ÖØÝ 

Ó×ÐØØÖÑÔÚÖØÐÙÖØÝ××ØÓÐÐ×ÒØÙÐÓ×ÒÓÐ×ØÖÔÒÓÒ ÐÒØÚÙØØÐÒÒÖØ×ÙÒÝ×ØÝ×ÓØÒÑÙØØÚÐØØÚ××ØÖØ ×ÙØÓÚØÐÔØÒ×ÑÒÐ×ÝØ×ÐØÒÄØØØØÚÒÖÙÒÓÒ ÙÒÐØØØØÚÒÖÙÒÓÒÚÐÒÒÐÐÄØØØØÚÒÖÙÒÓÒÚ ÃÝØØÝÒÒÐÚÖÓÒÓÓÓÒ96 × 96Ó×ØÙ×ÔÖÓ×ÒØØβ= 

ÚÐÒÒÒÚÚ×ØÑ××Ì×Ó×ÖØÝÑÒ×ØÑÒÒØ×ÒÑÙÙØØÖØ 

ÓÐÐ×××ØØØÝÙØÙÙÓ×ØÙ×ÐÒÑ××ØÐØÒØ×ÓÒ×ÙÙÒØ×Ò ×ÙÒÐÙÓÒÒØØØÝ×ÒÎÖØÑÐÐÚØØ××℄×ØØØÝÒÓÐÐ×Ò ×ÑÙÐÓØÙÒØÙÐÓ×ÒÒÝØØ×ÐØØØÚÔÖÙÒØÓÓÒÓÚÐÒ 

ÖØÝÑÒÆÑØ×ÒÙØØÚØÐÓÐ×ØÐØØÒÑÓÑÒØØÙÓÖÑ ØØ×ÓØØÚÐÐÌÑÓÒÑÝ×Ý×Ð××ØÖÚ×ÐÐØ×Ó×ÖØÝÑÒ 

ÐÒØÑ×××Ý×Ð××ØÖÚ 

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