A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media

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Submitted on 3 Apr 2012

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A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous

media

Daniele Antonio Di Pietro, Serge Nicaise

To cite this version:

Daniele Antonio Di Pietro, Serge Nicaise. A locking-free discontinuous Galerkin method for linear

elasticity in locally nearly incompressible heterogeneous media. Applied Numerical Mathematics,

Elsevier, 2012, 63, pp.105-116. �10.1016/j.apnum.2012.09.009�. �hal-00685020�

❆ ❧♦❝❦✐♥❣✲❢r❡❡ ❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❡❧❛st✐❝✐t② ✐♥

❧♦❝❛❧❧② ♥❡❛r❧② ✐♥❝♦♠♣r❡ss✐❜❧❡ ❤❡t❡r♦❣❡♥❡♦✉s ♠❡❞✐❛

❉❛♥✐❡❧❡ ❆✳ ❉✐ P✐❡tr♦

^❛

✱ ❙❡r❣❡ ◆✐❝❛✐s❡

^❜

❛❉❡♣❛rt♠❡♥t ♦❢ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ■❋P ❊♥❡r❣✐❡s ♥♦✉✈❡❧❧❡s✱ ✶ ✫ ✹ ❛✈❡♥✉❡ ❇♦✐s Pré❛✉✱ ✾✷✽✺✷

❘✉❡✐❧✲▼❛❧♠❛✐s♦♥✱ ❋r❛♥❝❡

❜▲❆▼❆❱✱ ❯♥✐✈❡rs✐té ❞❡ ❱❛❧❡♥❝✐❡♥♥❡s ❡t ❞✉ ❍❛✐♥❛✉t ❈❛♠❜rés✐s✱ ▲❡ ▼♦♥t ❍♦✉②✱ ✺✾✸✶✸ ❱❛❧❡♥❝✐❡♥♥❡s ❈❊❉❊❳ ✾✱

❋r❛♥❝❡

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦❢ ♥✉♠❡r✐❝❛❧ ❧♦❝❦✐♥❣ ✐♥ ❝♦♠♣♦s✐t❡ ♠❛t❡r✐❛❧s ❢❡❛t✉r✐♥❣ q✉❛s✐✲

✐♥❝♦♠♣r❡ss✐❜❧❡ ❛♥❞ ❝♦♠♣r❡ss✐❜❧❡ s❡❝t✐♦♥s✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✇❡ st❛rt ❜② ❡①t❡♥❞✐♥❣ ❛ ❝❧❛ss✐❝❛❧

r❡❣✉❧❛r✐t② ❡st✐♠❛t❡ ❢♦r t❤❡ H

¹

✲♥♦r♠ ♦❢ t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✜❡❧❞ t♦ t❤❡ ❤❡t❡r♦❣❡♥❡♦✉s

❝❛s❡✳ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ ❛ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡❧❛st✐❝✐t② ♣r♦❜❧❡♠ ❛s ❛ ❙t♦❦❡s s②st❡♠ ✇✐t❤

♥♦♥③❡r♦ ❞✐✈❡r❣❡♥❝❡ ❝♦♥str❛✐♥t✳ ❚❤✐s r❡s✉❧t ✐s t❤❡♥ ✉s❡❞ t♦ ❞❡s✐❣♥ ❛ ❧♦❝❦✐♥❣✲❢r❡❡ ❞✐s❝♦♥t✐♥✉♦✉s

●❛❧❡r❦✐♥ ♠❡t❤♦❞✳ ❚❤❡ ❦❡② ♣♦✐♥t ✐s t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t ✐♥ t❤❡ ❡st✐♠❛t❡

♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✉♥✐q✉❡❧② ❞❡♣❡♥❞ ♦♥ t❤❡ t❤✐s ❜♦✉♥❞❡❞ q✉❛♥t✐t②✳ ❚❤❛♥❦s t♦ ❛ ✜♥❡ t✉♥✐♥❣ ♦❢

t❤❡ ♣❡♥❛❧t② t❡r♠✱ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ♣❡♥❛❧t② ♣❛r❛♠❡t❡r ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ♠❡t❤♦❞ ✐s s✐♠♣❧②

❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ s♣❛❝❡ ❞✐♠❡♥s✐♦♥✳ ❚♦ ❝♦♥❝❧✉❞❡✱ ♥✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ t❤❡♦r❡t✐❝❛❧

r❡s✉❧ts ✐s ♣r♦✈✐❞❡❞✳

❑❡②✇♦r❞s✿ ▲✐♥❡❛r ❡❧❛st✐❝✐t②✱ ❝♦♠♣♦s✐t❡ ♠❛t❡r✐❛❧s✱ ❧♦❝❦✐♥❣✲❢r❡❡ ♠❡t❤♦❞✱ ❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥

♠❡t❤♦❞

✶✳ ■♥tr♦❞✉❝t✐♦♥

❚❤✐s ✇♦r❦ ❛❞❞r❡ss❡s t❤❡ ♣r♦❜❧❡♠ ♦❢ ♥✉♠❡r✐❝❛❧ ❧♦❝❦✐♥❣ ✐♥ ❤❡t❡r♦❣❡♥❡♦✉s✱ ❧♦❝❛❧❧② ♥❡❛r❧② ✐♥✲

❝♦♠♣r❡ss✐❜❧❡ ♠❡❞✐❛ ❜② r❡✈✐s✐t✐♥❣ t❤❡ ❝❧❛ss✐❝❛❧ ✇♦r❦s ♦❢ ❇r❡♥♥❡r ❛♥❞ ❙✉♥❣ ❬✶❪ ❛♥❞ ❍❛♥s❜♦ ❛♥❞

▲❛rs♦♥ ❬✷✱ ✸❪❀ s❡❡ ❛❧s♦ ❲✐❤❧❡r ❬✹❪✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ❧❡t Ω ⊂ R

^d

✱ d ≥ 2✱ ❜❡ ❛ ❜♦✉♥❞❡❞ ♣♦❧②❣♦♥❛❧

❞♦♠❛✐♥✳ ❲❡ ❞♦ ♥♦t ❛ss✉♠❡ t❤❛t Ω ✐s ❛ ▲✐♣s❝❤✐t③ ❞♦♠❛✐♥ t♦ ✐♥❝❧✉❞❡ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝r❛❝❦s ✐♥ ♦✉r

❛♥❛❧②s✐s✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ❡❧❛st✐❝✐t② ♣r♦❜❧❡♠

− div σ( u ) = f ✐♥ Ω,

u = 0 ♦♥ ∂Ω, ✭✶✮

✇❤❡r❡ f ∈ L

²

(Ω)

^d

✱ u ❞❡♥♦t❡s t❤❡ ✈❡❝t♦r✲✈❛❧✉❡❞ ❞✐s♣❧❛❝❡♠❡♥t ✜❡❧❞✱ ❛♥❞✱ ❢♦r ❛❧❧ v ∈ H

¹

(Ω)

^d

✱ σ( v ) := 2µǫ( v ) + λ div v I

d

, ǫ( v ) := 1

2 ∇v + ∇v

^t

.

❍❡r❡✱ µ ❛♥❞ λ ❛r❡ s❝❛❧❛r✲✈❛❧✉❡❞ ✜❡❧❞s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ▲❛♠é✬s ♣❛r❛♠❡t❡rs s✉❝❤ t❤❛t 0 < µ ≤ µ ≤ µ < +∞, 0 < λ ≤ λ ≤ λ.

❲❡ ❢♦❝✉s ♦♥ ❤❡t❡r♦❣❡♥❡♦✉s ♠❡❞✐❛ ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ♣❛rt✐t✐♦♥ P

Ω

= {Ω

i

}

1≤i≤NΩ

♦❢ Ω ✐♥t♦

♣♦❧②❤❡❞r❛❧ s✉❜❞♦♠❛✐♥s s✉❝❤ t❤❛t µ ❛♥❞ λ ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ♦♥ P

Ω

✳ ❖✉r ❣♦❛❧ ✐s t♦ ❞❡s✐❣♥ ❛

❧♦❝❦✐♥❣✲❢r❡❡ ❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ ❡rr♦r ❡st✐♠❛t❡ s❤♦✉❧❞ ♥♦t ❜❧♦✇ ✉♣

✇❤❡♥ λ ✐s ✜①❡❞ ❛♥❞ ❧❛r❣❡ ❡♥♦✉❣❤ ❛♥❞ λ → +∞ ✳ ❚❤❡ r♦❜✉st♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ µ ✐s ♥♦t ❝♦♥s✐❞❡r❡❞

❤❡r❡✐♥✱ s✐♥❝❡ ❧❛r❣❡ ❤❡t❡r♦❣❡♥❡✐t② r❛t✐♦s µ/µ ❞♦ ♥♦t ❝♦rr❡s♣♦♥❞ t♦ ♣❤②s✐❝❛❧❧② r❡❧❡✈❛♥t s✐t✉❛t✐♦♥s✳

Pr❡♣r✐♥t s✉❜♠✐tt❡❞ t♦ ❊❧s❡✈✐❡r ❆♣r✐❧ ✸✱ ✷✵✶✷

▲❡t H

^l

(P

Ω

) :=

v ∈ L

²

(Ω) | v

|Ωi

∈ H

^l

(Ω

i

), 1 ≤ i ≤ N

Ω

✳ ❆♥ ✐♥str✉♠❡♥t❛❧ r❡s✉❧t t♦ ♣r♦✈❡ ❛

❧♦❝❦✐♥❣✲❢r❡❡ ❡rr♦r ❡st✐♠❛t❡ ✐s t♦ s❤♦✇ t❤❛t t❤❡ q✉❛♥t✐t② N

^u

:=

k u k

²_H²_(PΩ)d

+ |λ div u |

²_H¹_(PΩ)

¹/²

, ✭✷✮

st❛②s ❜♦✉♥❞❡❞ ✇❤❡♥ λ → +∞✳ ❚❤✐s ✐s ♣r♦✈❡❞ ✐♥ ♦❢ ❙❡❝t✳ ✷ ✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ d = 2✳

❲❤✐❧❡ t❤✐s r❡s✉❧t ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❬✶✱ ❡q✳ ✭✷✳✶✽✮❪ t♦ t❤❡ ❤❡t❡r♦❣❡♥❡♦✉s ❝❛s❡✱ t❤❡ t❡❝❤♥✐q✉❡s ✉s❡❞

✐♥ t❤❡ ♣r♦♦❢ ❛r❡ ♥❡✇ ❛♥❞ ❛r❡ ✐♥s♣✐r❡❞ ❜② t❤❡ r❡❝❡♥t ✇♦r❦ ♦❢ ◆✐❝❛✐s❡ ❛♥❞ ▼❡r❝✐❡r ❬✺❪✳ ❚❤❡ ❦❡② ✐❞❡❛

✐s t♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ❡❧❛st✐❝✐t② ♣r♦❜❧❡♠ ✐♥ t❡r♠s ♦❢ ❛ tr❛♥s♠✐ss✐♦♥ ❙t♦❦❡s ♣r♦❜❧❡♠ ❜② ✐♥tr♦❞✉❝✐♥❣

❛ ✜❝t✐t✐♦✉s ♣r❡ss✉r❡ ❞❡✜♥❡❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ✜rst ▲❛♠é ♣❛r❛♠❡t❡r ❜② t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ t❤❡

❞✐s♣❧❛❝❡♠❡♥t ✜❡❧❞✳ ❚❤❡ ❜♦✉♥❞ ❢♦r ✭✷✮ t❤❡♥ ❢♦❧❧♦✇s ❢r♦♠ ❛♥ ❡♥❡r❣② ❡st✐♠❛t❡ ❢♦r t❤❡ ❡❧❛st✐❝✐t②

♣r♦❜❧❡♠ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❧♦❝❛❧ r❡❣✉❧❛r✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ ❙t♦❦❡s ♣r♦❜❧❡♠✳

❲✐t❤ ❛♥ ❡st✐♠❛t❡ ❢♦r N

u

❛t ❤❛♥❞✱ t❤❡ ❣♦❛❧ ♦❢ ❙❡❝t✳ ✸ ✐s t♦ ❞❡s✐❣♥ ❛ ❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥

✭❞●✮ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ♣✐❡❝❡✇✐s❡ ❛✣♥❡ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢②✐♥❣ ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ❢♦r♠

||| u − u

_h

|||

µ,λ

≤ C

µ

N

u

h,

✇❤❡r❡ |||·|||

µ,λ

✐s t❤❡ ❡♥❡r❣②✲❧✐❦❡ ♥♦r♠ ❞❡✜♥❡❞ ✐♥ ✭✶✹✮✱ u

_h

✐s t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥✱ ❛♥❞ C

µ

❞❡♥♦t❡s

❛ ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥ µ ❛♥❞ ♦♥ k f k

L²(Ω)^d

❜✉t ♥♦t ♦♥ λ✳ ❆ ❦❡② ♣♦✐♥t ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ✐s t♦

♠❛❦❡ s✉r❡ t❤❛t λ ♦♥❧② ❛♣♣❡❛rs ❡✐t❤❡r ✐♥ t❡r♠s ✐♥✈♦❧✈✐♥❣ ❛ ♣r♦❞✉❝t ❜② t❤❡ ❞✐✈❡r❣❡♥❝❡ ♦❢ u

_h

✱ ♦r

✐♥ t❡r♠s t❤❛t ❝❛♥ ❜❡ ❝❛♥❝❡❧❧❡❞ ❜② ❛♥ ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t♦r ✐♥ t❤❡ ❡rr♦r ❡st✐♠❛t❡✳

❚❤✐s ✐s ❛❝❤✐❡✈❡❞ ❤❡r❡ ❜② ❞❡s✐❣♥✐♥❣ t❤❡ ♣❡♥❛❧t② t❡r♠ s♦ t❤❛t ✭✐✮ t❤❡ ✇❡❛❦ ❝♦❡r❝✐✈✐t② ✭♥♦♥♥❡❣❛t✐✈✐t②✮

♦❢ t❤❡ ❞✐s❝r❡t❡ ❜✐❧✐♥❡❛r ❢♦r♠ ✐s ♦❜t❛✐♥❡❞ ❜② ♣❡♥❛❧✐③✐♥❣ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❥✉♠♣ ❧✐❢t✐♥❣s ✐♥ ❛ ❧❡❛st✲

sq✉❛r❡ ❢❛s❤✐♦♥ ✇✐t❤ ❛ ✉s❡r✲❞❡♣❡♥❞❡♥t ♣❛r❛♠❡t❡r η❀ ✭✐✐✮ ❢✉❧❧ ❝♦❡r❝✐✈✐t② ✐s ❛❝❤✐❡✈❡❞ ❜② ❛ st❛♥❞❛r❞

✐♥t❡r✐♦r ♣❡♥❛❧t② t❡r♠ ✇✐t❤ ❛ ❝♦❡✣❝✐❡♥t t❤❛t s♦❧❡❧② ❞❡♣❡♥❞s ♦♥ µ ❛♥❞ ♦♥ t❤❡ ♠❡s❤ s✐③❡ h✳ ❚❤✐s t❡r♠ ✐s r❡q✉✐r❡❞ s✐♥❝❡ ♣❡♥❛❧✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❥✉♠♣s ✐s ✐♥s✉✣❝✐❡♥t t♦ ✉s❡ t❤❡

❞✐s❝r❡t❡ ❑♦r♥ ✐♥❡q✉❛❧✐t② ❞❡r✐✈❡❞ ❜② ❇r❡♥♥❡r ❬✻❪✳ ❯♥❧✐❦❡ ❬✸❪✱ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ♣❛r❛♠❡t❡r η

✐s s✐♠♣❧② ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ s♣❛❝❡ ❞✐♠❡♥s✐♦♥✱ ✐✳❡✳✱ ♥♦ ✉♥❞❡t❡r♠✐♥❡❞ tr❛❝❡ ❝♦♥st❛♥t ❛♣♣❡❛rs✳

❚❤❡ ✐❞❡❛ ♦❢ ♣❡♥❛❧✐③✐♥❣ ✉s✐♥❣ ❥✉♠♣ ❧✐❢t✐♥❣s ❝❛♥ ❜❡ tr❛❝❡❞ ❜❛❝❦ t♦ t❤❡ s❡♠✐♥❛❧ ✇♦r❦s ♦❢ ❇❛ss✐ ❛♥❞

❘❡❜❛② ❬✼✱ ✽❪✳ ❆♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ✐s ♣r♦✈✐❞❡❞ t♦ ❡❛s❡ t❤❡ ♣r❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❧✐❢t✐♥❣s✱ ❛♥❞

t❤❡ ✢✉① ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞ ✐s ❜r✐❡✢② ❞✐s❝✉ss❡❞ ❢♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✐♥ ❙❡❝t✳ ✸✳✺✳

▼♦r❡♦✈❡r✱ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞ t♦ ♠✐♥✐♠❛❧ r❡❣✉❧❛r✐t② s♦❧✉t✐♦♥s ✐s ❛❞❞r❡ss❡❞ ✐♥ ❙❡❝t✳ ✸✳✻✳

❚❤❡ ♥✉♠❡r✐❝❛❧ ❛ss❡ss♠❡♥t ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✐s ♣❡r❢♦r♠❡❞ ✐♥ ❙❡❝t✳ ✹ ✉s✐♥❣ ✭✐✮ t❤❡ ❝❧♦s❡❞

❝❛✈✐t② ♣r♦❜❧❡♠ ♦❢ ❬✸❪ ❛♥❞ ✭✐✐✮ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❞❡r✐✈❡❞ ❜② ❍♦♥❣❥✉♥✱ ❩❤✐❢❡✐✱ ❛♥❞ ❚❛♦t❛♦ ❬✾❪ ❢♦r ❛

❝♦♠♣♦s✐t❡ ♠✉❧t✐✲❧❛②❡r ✐♥✜♥✐t❡ ❝②❧✐♥❞❡r✳ ❚❤❡ r❡s✉❧ts ❝♦♥✜r♠ t❤❛t ♥♦ ❧♦ss ♦❢ ❛❝❝✉r❛❝② ✐s ♦❜s❡r✈❡❞ ✐♥

t❤❡ ✐♥❝♦♠♣r❡ss✐❜❧❡ ❧✐♠✐t✳

✷✳ ❆ r❡❣✉❧❛r✐t② r❡s✉❧t

▲❡tt✐♥❣ U := H

₀¹

(Ω)

^d

✱ t❤❡ ✇❡❛❦ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✭✶✮ r❡❛❞s✿ ❋✐♥❞ u ∈ U s✉❝❤ t❤❛t a( u , v ) =

Z

Ω

f · v ∀ v ∈ U , ✭✸✮

✇❤❡r❡ a( u , v ) := R

Ω

σ( u ):ǫ( v ) ❛♥❞✱ ❢♦r t✇♦ s❡❝♦♥❞✲♦r❞❡r t❡♥s♦rs α ❛♥❞ β ✱ ✇❡ ❤❛✈❡ ❞❡♥♦t❡❞ α:β :=

P

1≤i, j≤d

α

ij

β

ij

✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s ✇❡ ♣r♦✈❡ ❛♥ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡ ❢♦r t❤❡ q✉❛♥t✐t② N

u

❞❡✜♥❡❞ ❜② ✭✷✮

✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ d = 2✳

▲❡♠♠❛ ✶ ✭❙t❛❜✐❧✐t② ❡st✐♠❛t❡✮✳ ■❢ u ∈ U ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ ✭✸✮ ✇✐t❤ f ∈ L

²

(Ω)

^d

✱ t❤❡♥ t❤❡

❢♦❧❧♦✇✐♥❣ ❡♥❡r❣② ❡st✐♠❛t❡ ❤♦❧❞s

k(2µ)

^1/2

ǫ( u )k

L²(Ω)^d,d

+ kλ

^1/2

div u k

L²(Ω)

≤ C

Ω

µ

^1/2

k f k

L²(Ω)^d

, ✭✹✮

✇❤❡r❡ C

Ω

✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ P♦✐♥❝❛ré ❛♥❞ ❑♦r♥ ❝♦♥st❛♥ts✳

✷

Pr♦♦❢✳ ❚❛❦✐♥❣ u ❛s ❛ t❡st ❢✉♥❝t✐♦♥ ✐♥ ✭✸✮ ✐t ✐s ✐♥❢❡rr❡❞ k(2µ)

^1/2

ǫ( u )k

²_L²_(Ω)d,d

+ kλ

^1/2

div u k

²_L²_(Ω)

= a( u , u ) = R

Ω

f · u ✳ ❯s✐♥❣ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③✱ P♦✐♥❝❛ré✱ ❑♦r♥✱ ❛♥❞ ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t✐❡s ✐t ✐s

✐♥❢❡rr❡❞

Z

Ω

f · u ≤ k f k

_L²_(Ω)^d

k u k

_L²_(Ω)^d

≤ C

Ω

k f k

_L²_(Ω)^d

kǫ( u )k

_L²_(Ω)^d,d

≤ 1 2

C

_Ω²

2µ k f k

²_L²_(Ω)d

+ 1

2 k(2µ)

^1/2

ǫ( u )k

²_L²_(Ω)d,d

.

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s✳

❚♦ ♣r♦❝❡❡❞✱ ✇❡ ❞❡r✐✈❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❡❛❦ ❢♦r♠✉❧❛t✐♦♥ ✇❤✐❝❤ r❡❧❛t❡s t❤❡ s♦❧✉t✐♦♥ ♦❢ ✭✸✮ t♦ t❤❛t

♦❢ ❛ tr❛♥s♠✐ss✐♦♥ ❙t♦❦❡s ♣r♦❜❧❡♠✳ ❚♦ t❤✐s ♣✉r♣♦s❡✱ ❧❡t L

²₀

(Ω) := {p ∈ L

²

(Ω) : hpi

Ω

= 0}✱ ✇❤❡r❡✱ ❢♦r

❛ ❢✉♥❝t✐♦♥ φ ✐♥t❡❣r❛❜❧❡ ♦♥ Ω✱ hφi

Ω

:=

_|Ω|¹

d

R

Ω

φ✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠s a

0

∈ L( U × U , R )

❛♥❞ b

0

∈ L( U × L

²₀

(Ω), R ) ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

a

0

( w , v ) :=

Z

Ω

2µǫ( w ):ǫ( v ), b

0

( w , q) := − Z

Ω

div w q.

❚❤❡♥✱ ❧❡tt✐♥❣

p := −λ div u ∈ L

²

(Ω), p ˜ := p + hλ div u i

Ω

∈ L

²₀

(Ω), ✭✺✮

✇❡ ✐♥❢❡r t❤❛t ( u , p) ˜ ∈ U × L

²₀

(Ω) ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ tr❛♥s♠✐ss✐♦♥ ❙t♦❦❡s ♣r♦❜❧❡♠

a

0

( u , v ) + b

0

( v , p) = ˜ Z

Ω

f · v ∀ v ∈ U , b

0

( u , q) =

Z

Ω

gq ∀q ∈ L

²₀

(Ω),

✭✻✮

✇✐t❤ g = − div u ✳ ❲❡ ❛ss✉♠❡ t❤❛t u ∈ H

²

(P

Ω

)

^d

✱ ❛♥❞ t❤❛t ♣r♦❜❧❡♠ ✭✻✮ s❛t✐s✜❡s t❤❡ ♦♣t✐♠❛❧

r❡❣✉❧❛r✐t② s❤✐❢t✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ r❡❣✉❧❛r✐t② ( f , g) ∈ L

²

(Ω)

^d

× H

¹

(P

Ω

) ✐♠♣❧✐❡s t❤❡ r❡❣✉❧❛r✐t② ( u , p) ˜ ∈ H

²

(P

Ω

)

^d

× H

¹

(P

Ω

) ✇✐t❤ t❤❡ ❡st✐♠❛t❡

k u k

H²(PΩ)

+ k pk ˜

H¹(PΩ)

≤ C

S

k f k

L²(Ω)^d

+ kgk

H¹(PΩ)

. ✭✼✮

❚❤❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C

S

> 0 ✐s ❝❧❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ λ s✐♥❝❡ ♣r♦❜❧❡♠ ✭✻✮ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥

t❤✐s ♣❛r❛♠❡t❡r✳ ❆♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭✼✮ ✐s t❤❛t

k u k

_H²_(PΩ)

+ k pk ˜

_H¹_(PΩ)

≤ C

S

k f k

_L²_(Ω)d

+ k div u k

_H¹_(PΩ)

. ✭✽✮

◆♦t❡ t❤❛t t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② s❤✐❢t ✐s s❛t✐s✜❡s ✐❢ ❢♦r ❡❛❝❤ ❝♦r♥❡r ♦❢ P

Ω

✭♥❛♠❡❧② ❛ ❝♦♠♠♦♥

❝♦r♥❡r ♦❢ s♦♠❡ Ω

i

✮✱ t❤❡r❡ ✐s ♥♦ s✐♥❣✉❧❛r ❡①♣♦♥❡♥t ✐♥ t❤❡ str✐♣ (0, 1]✳ ❲❡ r❡❢❡r t♦ ❬✶✵✱ ✶✶❪ ❢♦r t❤❡

st❛♥❞❛r❞ ❙t♦❦❡s s②st❡♠ ❛♥❞ t♦ ❬✶✷✱ ✶✸✱ ✺❪ ❢♦r t❤❡ ❡①t❡♥s✐♦♥ t♦ tr❛♥s♠✐ss✐♦♥ ❙t♦❦❡s ♣r♦❜❧❡♠✳ ❚❤✐s

❝♦♥❞✐t✐♦♥ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❣❡♦♠❡tr✐❝❛❧ ♦♥❡ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ s✐♥❣✉❧❛r ❡①♣♦♥❡♥t ❞❡♣❡♥❞s

♦♥ t❤❡ ✈❛❧✉❡s ♦❢ µ ❛♥❞ t❤❡ ❛♥❣❧❡ ♦❢ t❤❡ s✉❜❞♦♠❛✐♥s Ω

i

♥❡❛r t❤❡ ❝♦r♥❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ µ ✐s

❝♦♥st❛♥t ✐♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥✱ t❤✐s ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② s❤✐❢t ❤♦❧❞s ✐❢ Ω ✐s ❝♦♥✈❡①✳ ❙✐♠✐❧❛r❧②✱ t❤❡

❛ss✉♠♣t✐♦♥ u ∈ H

²

(P

Ω

)

^d

✐s r❡❧❛t❡❞ t♦ t❤❡ ♦♣t✐♠❛❧ r❡❣✉❧❛r✐t② s❤✐❢t ❢♦r t❤❡ ❡❧❛st✐❝ tr❛♥s♠✐ss✐♦♥

♣r♦❜❧❡♠ ✭✸✮✱ ✇❡ ❛❣❛✐♥ r❡❢❡r t♦ ❬✶✵✱ ✶✶✱ ✶✹✱ ✶✸✱ ✺❪✳

❚❤❡♦r❡♠ ✷ ✭❘❡❣✉❧❛r✐t②✮✳ ❚❤❡r❡ ❤♦❧❞s ✇✐t❤ N

u

❞❡✜♥❡❞ ❜② ✭✷✮✱ ❛ss✉♠✐♥❣ ✭✽✮ ❛♥❞ ♣r♦✈✐❞❡❞ λ > C

S

✱ N

u

≤ C

λ,µ

k f k

_L²_(Ω)d

,

✇✐t❤ C

λ,µ

❞❡♣❡♥❞❡♥t ♦♥ Ω✱ λ✱ ❛♥❞ µ ❜✉t ♥♦t ♦♥ λ✳

Pr♦♦❢✳ ❇② ✭✽✮✱ t❤❡r❡ ❤♦❧❞s ✇✐t❤ p ˜ ❞❡✜♥❡❞ ❜② ✭✺✮✱

|λ div u |

_H¹_(PΩ)

= |˜ p|

_H¹_(PΩ)

≤ C

S

k f k

_L²_(Ω)d

+ k div u k

_H¹_(PΩ)

≤ C

S

λ

^1/2

kλ

^1/2

div u k

L²(Ω)

+ 1

λ

^1/2

|λ div u |

H¹(PΩ)

+ C

S

k f k

L²(Ω)^d

.

✸

❍❡♥❝❡✱ ✉s✐♥❣ t❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ✭✹✮✱

1 − C

S

λ

|λ div u |

H¹(PΩ)

≤ C

S

k f k

L²(Ω)^d

1 + C

Ω

(µλ)

^1/2

! ,

❛♥❞ t❤❡ r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ λ > C

S

✳

✸✳ ❉✐s❝r❡t❡ s❡tt✐♥❣

✸✳✶✳ ◆♦t❛t✐♦♥

▲❡t H ⊂ R

⁺_∗

❜❡ ❛ ❝♦✉♥t❛❜❧❡ s❡t ♦❢ ♠❡s❤ s✐③❡s ❤❛✈✐♥❣ ✵ ❛s ✐ts ✉♥✐q✉❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t✱ ❛♥❞

❞❡♥♦t❡ ❜② (T

h

)

h∈H

❛ r❡✜♥❡❞ s❡q✉❡♥❝❡ ♦❢ ♠❛t❝❤✐♥❣ s✐♠♣❧✐❝✐❛❧ ♠❡s❤❡s T

h

= {T } ♦❢ Ω✳ ▲❡t h ∈ H

❜❡ ❛r❜✐tr❛r②✳ ❚❤❡ ❞✐❛♠❡t❡r ♦❢ ❛♥ ❡❧❡♠❡♥t T ∈ T

h

✐s ❞❡♥♦t❡❞ ❜② h

T

❛♥❞ t❤❡ ♠❡s❤ ✐♥❞❡① ✐s s✉❝❤

t❤❛t h = max

T∈Th

h

T

✳ ❚❤❡ s❡t ♦❢ ❢❛❝❡s ♦❢ T

h

✐s ❞❡♥♦t❡❞ ❜② F

h

❀ ❜♦✉♥❞❛r② ❢❛❝❡s ❛r❡ ❝♦❧❧❡❝t❡❞ ✐♥

t❤❡ s❡t F

_h^b

❛♥❞ ✇❡ s❡t F

_hⁱ

:= F

h

\ F

_h^b

✳ ❋♦r ❛❧❧ F ∈ F

h

✇❡ ❧❡t T

F

:= {T ∈ T

h

| F ⊂ ∂T }✳ ❋♦r ❡✈❡r②

✐♥t❡r❢❛❝❡ F ∈ F

_hⁱ

✱ ✇❡ s❡❧❡❝t ❛♥ ❛r❜✐tr❛r② ❜✉t ✜①❡❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧ n

_F

❛♥❞ ♥✉♠❜❡r t❤❡

❡❧❡♠❡♥ts ♦❢ T

F

✐♥ s✉❝❤ ❛ ✇❛② t❤❛t n

_F

♣♦✐♥ts ♦✉t ♦❢ T

1

❀ ♦♥ ❜♦✉♥❞❛r② ❢❛❝❡s F ∈ F

_h^b

t❤❡ ♥♦r♠❛❧ n

_F

✐s ♦✉t✇❛r❞ t♦ Ω✳ ❋♦r ❛❧❧ F ∈ F

h

✇❡ ❞❡♥♦t❡ ❜② h

F

✐ts ❞✐❛♠❡t❡r✳

■t ✐s ❛ss✉♠❡❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s t❤❛t t❤❡ ♠❡s❤ s❡q✉❡♥❝❡ {T

h

}

h∈H

✐s s❤❛♣❡✲r❡❣✉❧❛r ✐♥ t❤❡ ✉s✉❛❧

s❡♥s❡ ♦❢ ❈✐❛r❧❡t ❬✶✺❪✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡r❡ ❡①✐sts ρ > 0 s✉❝❤ t❤❛t max

h∈H

max

T∈Th

h

T

r

T

≤ ρ,

✇❤❡r❡✱ ❢♦r ❛❧❧ T ∈ T

h

✱ h ∈ H✱ r

T

❞❡♥♦t❡s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❧❛r❣❡st ❜❛❧❧ ✐♥s❝r✐❜❡❞ ✐♥ T ✳ ❋♦r ❛❧❧

h ∈ H ❛♥❞ ❛❧❧ ✐♥t❡❣❡rs k ≥ 0 ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❜r♦❦❡♥ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡s✿

P

^k

d

(T

h

) :=

v

h

∈ L

²

(Ω) | v

h|T

∈ P

^k

d

(T ), ∀T ∈ T

h

,

✇❤❡r❡ P

^k

d

(T ) ❞❡♥♦t❡s t❤❡ r❡str✐❝t✐♦♥ t♦ T ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ≤ k ✐♥ ❞✐♠❡♥s✐♦♥ d✳ ❙✐♠✐❧❛r❧②✱

❜r♦❦❡♥ ❙♦❜♦❧❡✈ s♣❛❝❡s ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛♥ ✐♥t❡❣❡r m ≥ 0 ❛s H

^m

(T

h

) :=

v ∈ L

²

(Ω) | v

|T

∈ H

^m

(T), ∀T ∈ T

h

.

❚❤❡ ❜r♦❦❡♥ ❣r❛❞✐❡♥t ❛❝t✐♥❣ ♦♥ ❢✉♥❝t✐♦♥s ✐♥ H

¹

(T

h

) ✐s ❞❡♥♦t❡❞ ❜② ∇

_h

✱ t❤❡ ❜r♦❦❡♥ ❞✐✈❡r❣❡♥❝❡ ❛❝t✐♥❣

♦♥ ❢✉♥❝t✐♦♥s ✐♥ H

¹

(T

h

)

^d

✐s ❞❡♥♦t❡❞ ❜② div

h

✳ ❙✐♠✐❧❛r❧②✱ t❤❡ ❜r♦❦❡♥ ✈❡rs✐♦♥s ♦❢ t❤❡ s②♠♠❡tr✐❝

❣r❛❞✐❡♥t ❛♥❞ ❡❧❛st✐❝✐t② ♦♣❡r❛t♦rs ǫ ❛♥❞ σ ❛r❡ ❞❡✜♥❡❞ ❢♦r ❛❧❧ v ∈ H

¹

(T

h

)

^d

❜② s❡tt✐♥❣

σ

h

( v ) := 2µǫ

h

( v ) + λ div

h

v I

d

, ǫ

h

( v ) := 1

2 ∇

_h

v + ∇

_h

v

^t

.

■t ✐s ❛ss✉♠❡❞ ❤❡r❡ t❤❛t✱ ❢♦r ❛❧❧ h ∈ H✱ T

h

✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ♣❛rt✐t✐♦♥ P

Ω

✱ ♠❡❛♥✐♥❣ t❤❛t ❢♦r

❛❧❧ T ∈ T

h

t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ Ω

i

✱ 1 ≤ i ≤ N

Ω

✱ s✉❝❤ t❤❛t T ⊂ Ω

i

✳ ❚❤✐s ✐♠♣❧✐❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱

t❤❛t ❢♦r ❛❧❧ h ∈ H✱

µ ∈ P

⁰_d

(T

h

), λ ∈ P

⁰_d

(T

h

).

❲❡ ❝❧♦s❡ t❤✐s s❡❝t✐♦♥ ❜② ❞❡✜♥✐♥❣ s♦♠❡ tr❛❝❡ ♦♣❡r❛t♦rs ❝♦♠♠♦♥❧② ✉s❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞●

♠❡t❤♦❞s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❢♦r ❛♥② s❝❛❧❛r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ v ❞❡✜♥❡❞ ♦♥ Ω ❛♥❞ s♠♦♦t❤ ❡♥♦✉❣❤ t♦

❛❞♠✐t ♦♥ ❛❧❧ F ∈ F

h

❛ ♣♦ss✐❜❧② t✇♦✲✈❛❧✉❡❞ tr❛❝❡ ♦♥ F ✇❡ ❧❡t ❢♦r ❛❧❧ F ∈ F

_hⁱ

✱ JvK( x ) := v

|T1

( x ) − v

|T2

( x ), {v}( x ) := 1

2 v

|T1

( x ) + v

|T2

( x ) .

❋♦r ❛❧❧ F ∈ F

_h^b

s✉❝❤ t❤❛t F = ∂T ∩ ∂Ω ✇❡ ❝♦♥✈❡♥t✐♦♥❛❧❧② s❡t {v}( x ) = JvK( x ) = v

|T

( x )✳ ❲❤❡♥

❛♣♣❧✐❡❞ t♦ ✈❡❝t♦r✲ ♦r t❡♥s♦r✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s✱ t❤❡ ❥✉♠♣ ❛♥❞ ❛✈❡r❛❣❡ ♦♣❡r❛t♦rs ❛❝t ❝♦♠♣♦♥❡♥t✲✇✐s❡✳

■❢ ♥♦ ❝♦♥❢✉s✐♦♥ ❝❛♥ ❛r✐s❡✱ t❤❡ ✈❛r✐❛❜❧❡ x ✐s ♦♠✐tt❡❞✱ ❛♥❞ ✇❡ s✐♠♣❧② ✇r✐t❡ {v} ❛♥❞ JvK✳

✹

✸✳✷✳ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ r❡❝❛❧❧ s♦♠❡ ♣r❡❧✐♠✐♥❛r② r❡s✉❧ts✱ ♥❛♠❡❧② t❤❡ ❞✐s❝r❡t❡ ❑♦r♥ ✐♥❡q✉❛❧✐t② ✐♥ ❜r♦❦❡♥

♣♦❧②♥♦♠✐❛❧ s♣❛❝❡s ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❥✉♠♣ ❧✐❢t✐♥❣s✳

✸✳✷✳✶✳ ❉✐s❝r❡t❡ ❑♦r♥✬s ✐♥❡q✉❛❧✐t②

❲❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✉s✉❛❧ H

₀¹

✲❧✐❦❡ ♥♦r♠ ♦♥ H

₀¹

(T

h

)✿

kvk

²_1,h

:= k ∇

_h

vk

²_L²_(Ω)d

+ |v|

²_J

, |v

h

|

²_J

:= X

F∈Fh

1 h

F

kJv

h

Kk

²_L²_(F)

.

❆♥ ✐♠♣♦rt❛♥t ✐♥❣r❡❞✐❡♥t ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r ❡❧❛st✐❝✐t② ♣r♦❜❧❡♠ ✐s t❤❡ ❞✐s❝r❡t❡

❝♦✉♥t❡r♣❛rt ♦❢ ❑♦r♥✬s ✐♥❡q✉❛❧✐t②✱ ✇❤✐❝❤ st❛t❡s t❤❛t t❤❡ k·k

1,h

✲♥♦r♠ ❝❛♥ ❜❡ ❝♦♥tr♦❧❧❡❞ ✐♥ t❡r♠s ♦❢

t❤❡ L

²

✲♥♦r♠ ♦❢ t❤❡ s②♠♠❡tr✐❝ ♣❛rt ♦❢ t❤❡ ❣r❛❞✐❡♥t ♣❧✉s t❤❡ ❥✉♠♣ s❡♠✐♥♦r♠ |·|

J

✳ ❑♦r♥✬s ✐♥❡q✉❛❧✐t✐❡s

❢♦r ♣✐❡❝❡✇✐s❡ H

¹

❢✉♥❝t✐♦♥s ♦♥ ❢❛✐r❧② ❣❡♥❡r❛❧ ♠❡s❤❡s ❛r❡ ♣r♦✈❡❞ ❜② ❇r❡♥♥❡r ❬✻❪✳ ■♥ t❤✐s ✇♦r❦ ✇❡

♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛♥t ♦❢ ❬✻✱ ✭✶✳✶✶✮❪✳

❚❤❡♦r❡♠ ✸ ✭❉✐s❝r❡t❡ ❑♦r♥✬s ✐♥❡q✉❛❧✐t②✮✳ ❚❤❡r❡ ✐s C

K

✉♥✐q✉❡❧② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♠❡s❤ r❡❣✉❧❛r✐t②

♣❛r❛♠❡t❡r ❛♥❞ ♦♥ Ω s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v

_h

∈ P

^k

d

(T

h

)

^d

✱ k ≥ 1✱

k v

_h

k

1,h

≤ C

K

kǫ

h

( v

_h

)k

²_L²_(Ω)d,d

+ | v

_h

|

²_J

¹/²

. ✭✾✮

✸✳✷✳✷✳ ▲✐❢t✐♥❣s

❋♦r ❛♥ ✐♥t❡❣❡r ♣♦❧②♥♦♠✐❛❧ ❞❡❣r❡❡ l ≥ 0 ✇❡ ❞❡✜♥❡ ❛ ❢❛❝❡ ❧✐❢t✐♥❣ ♦♣❡r❛t♦r ✐♥s♣✐r❡❞ ❜② ❇r❡③③✐

❡✳❛✳ ❬✶✻❪ ❛s ❢♦❧❧♦✇s✿ ❋♦r ❛❧❧ F ∈ F

h

❛♥❞ ❛❧❧ v ∈ L

²

(F)

^d

✱ r

^l_F

( v ) ∈ P

^l

d

(T

h

)

^d,d

✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥

t♦ Z

Ω

r

_F^l

( v ):τ

h

= Z

F

v ⊗ n

_F

:{τ

h

} ∀τ

h

∈ P

^l

d

(T

h

)

^d,d

, ✭✶✵✮

✇❤❡r❡✱ ❢♦r t✇♦ ✈❡❝t♦rs a ❛♥❞ b ✇❡ ❤❛✈❡ ❧❡t a ⊗ b := [a

_i

b

j

]

1≤i, j≤d

∈ R

^d,d

✳ ❋♦r t❤❡ s❛❦❡ ♦❢ ❜r❡✈✐t②

✇❡ ❛❧s♦ ✐♥tr♦❞✉❝❡ ❛ s②♠❜♦❧ ❢♦r t❤❡ tr❛❝❡ ♦❢ t❤❡ ❢❛❝❡ ❧✐❢t✐♥❣✱

r

^l_F

( v ) := tr(r

^l_F

( v )) ∈ P

^l

d

(T

h

).

❚❤❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❧✐❢t✐♥❣s ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ l = 0 ❝❛♥ ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ❛✈❡r❛❣❡ ♦❢ v

❛❝r♦ss ♦♥❡ ❢❛❝❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❛❦✐♥❣ τ

h

= χ

T

m

ij

❢♦r 1 ≤ i, j ≤ d ❛♥❞ T ∈ T

F

✐♥ ✭✶✵✮ ✇❤❡r❡

(m

ij

)

i^′j^′

= δ

ii^′

δ

jj^′

❛♥❞ χ

T

❞❡♥♦t❡s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ T ✱ ✐t ✐s ✐♥❢❡rr❡❞

r

_F⁰

( v )

|T

≡ |F |

d−1

card(T

F

)|T |

d

h v i

F

⊗ n

_F

, r

⁰_F

( v )

|T

≡ |F|

d−1

card(T

F

)|T|

d

h v i

F

· n

_F

, ✭✶✶✮

✇❤❡r❡✱ ❢♦r ❛❧❧ ❢✉♥❝t✐♦♥s ϕ ✐♥t❡❣r❛❜❧❡ ♦♥ F✱ ✇❡ ❤❛✈❡ ❧❡t hϕi

F

:= R

F

ϕ/|F|

d−1

✳ ❚❤❡ r❡❧❛t✐♦♥s ✭✶✶✮

❝❛♥ ❜❡ r❡♣❧❛❝❡ t❤❡ ❧✐❢t✐♥❣s ✐♥ ♣r❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳ ❋✐♥❛❧❧②✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ v ∈ H

₀¹

(Ω)

^d

✱ t❤❡

❣❧♦❜❛❧ ❧✐❢t✐♥❣ ♦❢ t❤❡ ❥✉♠♣s ♦❢ v ❛♥❞ ✐ts tr❛❝❡ ❛r❡ r❡s♣❡❝t✐✈❡❧② ❞❡♥♦t❡❞ ❜② R

_h^l

( v ) := X

F∈Fh

r

_F^l

(J v K) ∈ P

^l

d

(T

h

)

^d,d

, R

^l_h

( v ) := tr(R

^l_h

( v )) = X

F∈Fh

r

_F^l

(J v K) ∈ P

^l

d

(T

h

).

✸✳✸✳ ❚❤❡ ❞✐s❝r❡t❡ ♣r♦❜❧❡♠

❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛❝❡s✿

U

_h

:= P

¹

d

(T

h

)

^d

, U

_∗

:= U ∩ H

²

(P

Ω

)

^d

U

_∗h

:= U

_∗

+ U

_h

.

✺

❚❤❡ ❛❞❞✐t✐♦♥❛❧ r❡❣✉❧❛r✐t② ✐♥ U

_∗

❡♥s✉r❡s t❤❛t t❤❡ tr❛❝❡s ♦❢ ❣r❛❞✐❡♥ts ♦♥ ♠❡s❤ ❢❛❝❡s ❛r❡ sq✉❛r❡✲

✐♥t❡❣r❛❜❧❡✳ ❚❤✐s r❡❣✉❧❛r✐t② ❛ss✉♠♣t✐♦♥ ❝❛♥ ❜❡ r❡❧❛①❡❞ ✉s✐♥❣ t❤❡ t❡❝❤♥✐q✉❡s ♦❢ ❬✶✼❪✱ ❚♦ ✇❤✐❝❤ ✇❡

r❡❢❡r ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧s✳ ❲❡ ❞❡✜♥❡ t❤❡ ❞✐s❝r❡t❡ ❜✐❧✐♥❡❛r ❢♦r♠ a

h

∈ L( U

_∗h

× U

_∗h

, R ) s✉❝❤ t❤❛t a

h

( w , v ) :=

Z

Ω

σ

h

( w ):ǫ

h

( v )

− X

F∈Fh

Z

F

{σ

h

( w )}:hJ v Ki

F

⊗ n

_F

+ hJ w Ki

F

⊗ n

_F

:{σ

h

( v )}

+ X

F∈Fh

Z

Ω

η

2µr

⁰_F

(J w K):r

⁰_F

(J v K) + λr

_F⁰

(J w K)r

_F⁰

(J v K)

+ X

F∈Fh

Z

F

γ

µ,F

h

F

J w K·J v K,

✭✶✷✮

✇❤❡r❡ η > 0 ❞❡♥♦t❡s ❛ ✉s❡r✲❞❡♣❡♥❞❡♥t ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡r ❛♥❞✱ ❢♦r ❛❧❧ F ∈ F

h

✱ γ

µ,F

:= max

T∈TF

µ

|T

.

❚❤❡ ❞✐s❝r❡t❡ ♣r♦❜❧❡♠ r❡❛❞s

❋✐♥❞ u

_h

∈ U

_h

s✳t✳ a

h

( u

_h

, v

_h

) = Z

Ω

f · v

_h

❢♦r ❛❧❧ v

_h

∈ U

_h

✳ ✭✶✸✮

❚❤❡ t❡r♠s ✐♥ t❤❡ s❡❝♦♥❞ ❧✐♥❡ ♦❢ ✭✶✷✮ ❛r❡ r❡s♣♦♥s✐❜❧❡✱ r❡s♣❡❝t✐✈❡❧②✱ ❢♦r t❤❡ ✭✇❡❛❦✮ ❝♦♥s✐st❡♥❝② ❛♥❞

s②♠♠❡tr② ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ a

h

❀ t❤❡ t❡r♠s ✐♥ t❤❡ t❤✐r❞ ❧✐♥❡ ❡♥s✉r❡ ♥♦♥♥❡❣❛t✐✈✐t② ❜② ♣❡♥❛❧✐③✐♥❣

t❤❡ ❥✉♠♣ ❧✐❢t✐♥❣s ❛❝r♦ss ♠❡s❤ ❢❛❝❡s t♦ ❝♦♠♣❡♥s❛t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥tr✐❜✉t✐♦♥ ❢r♦♠ t❤❡ ✇❡❛❦

❝♦♥s✐st❡♥❝② ❛♥❞ s②♠♠❡tr② t❡r♠s✳ ❯s✐♥❣ ✭✶✶✮✱ ✐t ✐s ❛ s✐♠♣❧❡ ♠❛tt❡r t♦ r❡❛❧✐③❡ t❤❛t t❤❡s❡ ❝♦♥tr✐❜✉✲

t✐♦♥s ♦♥❧② ♣❡♥❛❧✐③❡ t❤❡ ♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ ❥✉♠♣s❀ ✜♥❛❧❧②✱ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❑♦r♥ ✐♥❡q✉❛❧✐t②

✭✾✮✱ t❤❡ t❡r♠ ✐♥ t❤❡ ❢♦✉rt❤ ❧✐♥❡ ❝♦♥t❛✐♥s ❛ ❢✉❧❧ ♣❡♥❛❧✐③❛t✐♦♥ ♦❢ ❥✉♠♣s t♦ ❡♥s✉r❡ ❝♦❡r❝✐✈✐t②✳

❘❡♠❛r❦ ✹ ✭▲✐❢t✐♥❣✲❜❛s❡❞ ♣❡♥❛❧t② t❡r♠s✮✳ ❈♦♥s✐❞❡r✐♥❣ ❧✐❢t✐♥❣✲❜❛s❡❞ ♣❡♥❛❧t② t❡r♠s ✐♥ t❤❡ t❤✐r❞

❧✐♥❡ ♦❢ ✭✶✷✮ ❛❧❧♦✇s t♦ ❞❡r✐✈❡ ❛ tr✐✈✐❛❧ ❧♦✇❡r ❜♦✉♥❞ ❢♦r t❤❡ ✉s❡r✲❞❡♣❡♥❞❡♥t ♣❛r❛♠❡t❡r η t♦ ❛❝❤✐❡✈❡

st❛❜✐❧✐t② ✭❝❢✳ ▲❡♠♠❛ ✼✮✳ ■♥ ♣r❛❝t✐❝❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ❡①♣r❡ss✐♦♥ ❜❛s❡❞ ♦♥ ✭✶✶✮ ❝❛♥ ❜❡ ✉s❡❞✿

X

F∈Fh

Z

F

(η

µ,F

h w i

F

·h v i

F

+ η

λ,F

(h w i

F

· n

_F

)(h v i

F

· n

_F

)) ,

✇❤❡r❡ η

µ,F

:= η P

T∈TF

(2µ)_|T|F|d−1

card(TF)²|T|d

❛♥❞ η

λ,F

:= η P

T∈TF

λ_|T|F|d−1

card(TF)²|T|d

✳

❘❡♠❛r❦ ✺ ✭❘❡str✐❝t✐♦♥ ♦❢ a

h

t♦ U

_h

× U

_h

❛♥❞ ❈r♦✉③❡✐①✕❘❛✈✐❛rt ✜♥✐t❡ ❡❧❡♠❡♥ts✮✳ ❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t σ

h

( v

_h

) ∈ P

⁰

d

(T

h

)

^d,d

❢♦r ❛❧❧ v

_h

∈ U

_h

✱ ✐t ✐s ✐♥❢❡rr❡❞ t❤❛t✱ ❢♦r ❛❧❧ ( w

_h

, v

_h

) ∈ U

²

h

✱ t❤❡ ❛s②♠♣t♦t✐❝

❝♦♥s✐st❡♥❝② ❛♥❞ s②♠♠❡tr② t❡r♠s ✐♥ t❤❡ s❡❝♦♥❞ ❧✐♥❡ ♦❢ ✭✶✷✮ ❛r❡ ❡q✉✐✈❛❧❡♥t t♦

− X

F∈Fh

Z

F

{σ

h

( w

_h

)}:J v

_h

K⊗ n

_F

+ J w

_h

K⊗ n

_F

:{σ

h

( v

_h

)}

.

❯s✐♥❣ t❤✐s ❢♦r♠✉❧❛t✐♦♥ t♦ ❡①t❡♥❞ a

h

t♦ U

_∗h

× U

_∗h

✇♦✉❧❞ ②✐❡❧❞ ❛ ❝♦♥s✐st❡♥t ❜✐❧✐♥❡❛r ❢♦r♠✳ ❍♦✇❡✈❡r✱

✇❡ ❤❛✈❡ ♣r❡❢❡rr❡❞ t♦ ✉s❡ t❤❡ ❛s✐♠♣t♦t✐❝❛❧❧② ❝♦♥s✐st❡♥t ❢♦r♠✉❧❛t✐♦♥ ✭✶✷✮ s✐♥❝❡ ✐t ♠❛❦❡s ✐t ❡❛s✐❡r t♦

tr❛❝❦ t❤❡ ❞❡♣❡♥❞❡♥❝② ♦♥ λ ❛♥❞ µ ♦❢ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♣❛r❛♠❡t❡r ✐♥ t❤❡ ❡rr♦r ❡st✐♠❛t❡✱ ❛s ❞❡t❛✐❧❡❞

✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡❝t✐♦♥✳ ▼♦r❡♦✈❡r✱ ❛s ✐s t❤❡ ❝❛s❡ ✐♥ ❬✸❪✱ t❤❡ ❛♥❛❧②s✐s r❡❛❞✐❧② ❛♣♣❧✐❡s ✇❤❡♥ t❤❡ ❞●

s♣❛❝❡ U

_h

✐s r❡♣❧❛❝❡❞ ❜② t❤❡ ❈r♦✉③❡✐①✕❘❛✈✐❛rt ✜♥✐t❡ ❡❧❡♠❡♥t s♣❛❝❡ CR (T

h

)

^d

❞❡✜♥❡❞ ❜② ✭✶✻✮✳

✸✳✹✳ ❊♥❡r❣② ❡rr♦r ❡st✐♠❛t❡

▲❡♠♠❛ ✻ ✭❲❡❛❦ ❝♦♥s✐st❡♥❝②✮✳ ▲❡t u ∈ U ❞❡♥♦t❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❝♦♥t✐♥✉♦✉s ♣r♦❜❧❡♠ ✭✶✮ ❛♥❞

❢✉rt❤❡r ❛ss✉♠❡ t❤❛t u ∈ U

_∗

✳ ❚❤❡♥✱

∀ v

_h

∈ U

_h

, a

h

( u , v

_h

) = Z

Ω

f · v

_h

− E

^u

( v

_h

),

✇✐t❤ ❝♦♥s✐st❡♥❝② ❡rr♦r E

u

( v

_h

) := P

F∈Fh

R

F

{σ( u )}: (hJ v

_h

Ki

F

− J v

_h

K) ⊗ n

_F

✳

✻

Pr♦♦❢✳ ❯s✐♥❣ t❤❡ s②♠♠❡tr② ♦❢ σ( u )✱ ✐♥t❡❣r❛t✐♥❣ ❜② ♣❛rts✱ ❛♥❞ r❡❛rr❛♥❣✐♥❣ t❤❡ ❜♦✉♥❞❛r② t❡r♠s ✐t

✐s ✐♥❢❡rr❡❞ ❢♦r ❛❧❧ v

_h

∈ U

_h

✱ Z

Ω

σ( u ):ǫ

h

( v

_h

) = Z

Ω

f · v

_h

+ X

F∈Fh

Z

F

{σ( u )}:J v

_h

K⊗ n

_F

+ X

F∈F_hⁱ

Z

F

Jσ( u )K:{ v

_h

}⊗ n

_F

.

❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s s✐♥❝❡ Jσ( u ) n

_F

K

F

= 0 ❢♦r ❛❧❧ F ∈ F

_hⁱ

❛♥❞ J u K

F

= 0 ❢♦r ❛❧❧ F ∈ F

h

✳

❙t❛❜✐❧✐t② ✐s ❡①♣r❡ss❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡♥❡r❣②✲❧✐❦❡ ♥♦r♠ ❞❡✜♥❡❞ ♦♥ H

¹

(T

h

)

^d

✿

||| v |||

²_µ,λ

:= k(2µ)

^1/2

ǫ

h

( v )k

²_L²_(Ω)d,d

+ kλ

^1/2

div

h

v k

²_L²_(Ω)

+ | v |

²_r,µ

+ | v |

²_r,λ

+ | v |

²_J,µ

✭✶✹✮

✇✐t❤ s❡♠✐♥♦r♠s | v |

²_r,µ

:= P

F∈Fh

k(2µ)

^1/2

r

_F⁰

(J v K)k

²_L²_(Ω)d,d

✱ | v |

²_r,λ

:= P

F∈Fh

kλ

^1/2

r

_F⁰

(J v K)k

²_L²_(Ω)

✱

❛♥❞ | v |

²_J,µ

:= P

F∈Fh

k(γ

µ,F

/h

F

)

^1/2

J v Kk

²_L2(F)^d

.

▲❡♠♠❛ ✼ ✭❈♦❡r❝✐✈✐t②✮✳ ❋♦r ❛❧❧ η > N

∂

= d + 1 t❤❡r❡ ❤♦❧❞s ✇✐t❤ α

η

:= (η − N

∂

)/(1 + η)✱

∀ v

_h

∈ U

_h

, a

h

( v

_h

, v

_h

) ≥ α

η

||| v

_h

|||

²_µ,λ

. ≥ α

η

µ

²

C

K

k v

_h

k

²_1,h

, ✭✶✺✮

✇❤❡r❡ C

K

✐s t❤❡ ❞✐s❝r❡t❡ ❑♦r♥ ❝♦♥st❛♥t ✐♥tr♦❞✉❝❡❞ ✐♥ ✭✾✮✳

Pr♦♦❢✳ ❚❛❦✐♥❣ ( w , v ) = ( v

_h

, v

_h

) ✐♥ ✭✶✷✮ ✐t ✐s ❛ s✐♠♣❧❡ ♠❛tt❡r t♦ r❡❛❧✐③❡ t❤❛t ♦♥❧② t❤❡ ✇❡❛❦ ❝♦♥s✐s✲

t❡♥❝② ❛♥❞ s②♠♠❡tr② t❡r♠s ✐♥ t❤❡ s❡❝♦♥❞ ❧✐♥❡ ❞♦ ♥♦t ❤❛✈❡ ❛ s✐❣♥ ❛ ♣r✐♦r✐✳ ❚♦ ❜♦✉♥❞ t❤❡s❡ t❡r♠s✱

✇❡ ✉s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❧✐❢t✐♥❣ ♦♣❡r❛t♦rs ❣✐✈❡♥ ✐♥ ❙❡❝t✳ ✸✳✷✳✷ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③

✐♥❡q✉❛❧✐t② t♦ ✐♥❢❡r

X

F∈Fh

Z

F

{σ

h

( v

_h

)}:hJ v

_h

Ki

F

⊗ n

_F

= Z

Ω

2µǫ

h

( v

_h

)+λ div

h

v

_h

:R

⁰_h

( v

_h

)

≤ N

∂

k(2µ)

^1/2

ǫ

h

( v

_h

)k

_L²_(Ω)^d,d

| v

_h

|

r,µ

+kλ

^1/2

div

h

v

_h

k

L²(Ω)

| v

_h

|

r,λ

,

✇❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢❛❝t t❤❛t✱ ❢♦r ❛❧❧ F ∈ F

h

✱ t❤❡ ❧♦❝❛❧ ❧✐❢t✐♥❣ ✐s s✉♣♣♦rt❡❞ ✐♥ t❤❡ ❡❧❡♠❡♥ts ♦❢

T

F

t♦ ✐♥❢❡r k(2µ)

^1/2

R

^l_h

( v

_h

)k

²_L2(Ω)^d,d

≤ N

∂

| v

_h

|

²_r,µ

❛♥❞ kλ

^1/2

R

^l_h

( v

_h

)k

²_L²_(Ω)

≤ N

∂

| v

_h

|

²_r,λ

✭❝❢✳✱ ❡✳❣✳✱ ❬✶✽✱

▲❡♠♠❛ ✹✳✸✹❪ ❢♦r ❞❡t❛✐❧s✮✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s ✉s✐♥❣ t✇✐❝❡ t❤❡ ✐♥❡q✉❛❧✐t② x

²

− 2N

_∂^1/2

xy + ηy

²

≥

η−N∂

1+η

(x

²

+ y

²

) ✇✐t❤ ✭✐✮ x = k(2µ)

^1/2

ǫ

h

( v

_h

)k

_L²_(Ω)^d,d

❛♥❞ y = | v

_h

|

r,µ

❀ ✭✐✐✮ x = kλ

^1/2

div

h

v

_h

k

L²(Ω)

❛♥❞ y = | v

_h

|

r,λ

✳

❚❤❡ ❧❛st ✐♥❣r❡❞✐❡♥t t♦ ♣r♦✈❡ ❛♥ ❡rr♦r ❡st✐♠❛t❡ ✐s t♦ s❤♦✇ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ a

h

✐♥ U

_∗h

× U

_h

✳ ❚♦

t❤✐s ❡♥❞✱ ✇❡ ❞❡✜♥❡ ❛♥ ❛✉❣♠❡♥t❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❡♥❡r❣② ♥♦r♠ ♦♥ H

¹

(T

h

)

^d

❛s ❢♦❧❧♦✇s✿

||| v |||

²_µ,λ,∗

:= ||| v |||

²_µ,λ

+ X

T∈Th

h

T

k((2µ)

^1/2

ǫ

h

( v

_h

))

|T

k

²_L²_(∂T)d,d

+ k(λ

^1/2

div

h

v

_h

)

|T

k

²_L²_(∂T)

.

❈❧❡❛r❧②✱ ||| v |||

µ,λ,∗

≥ ||| v |||

µ,λ

❢♦r ❛❧❧ v ∈ H

¹

(T

h

)

^d

✱ ❛♥❞ t❤❡ t✇♦ ♥♦r♠s ❛r❡ ✉♥✐❢♦r♠❧② ❡q✉✐✈❛❧❡♥t ♦♥

U

_h

✳

▲❡♠♠❛ ✽ ✭❇♦✉♥❞❡❞♥❡ss✮✳ ❚❤❡r❡ ❤♦❧❞s ✇✐t❤ β

ρ,η

:= 2 + η + 2ρ

¹²

✱

∀( w , v

_h

) ∈ U

_∗h

× U

_h

, a

h

( w , v

_h

) ≤ β

ρ,η

||| w |||

µ,λ,∗

||| v

_h

|||

µ,λ

.

Pr♦♦❢✳ ❉❡♥♦t❡ ❜② T

₁

, . . . , T

₆

t❤❡ ❛❞❞❡♥❞s ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✶✷✮ ✇✐t❤ ( w , v ) = ( w , v

_h

)✳

▼✉❧t✐♣❧❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t② ②✐❡❧❞

| T

₁

| + | T

₄

| + | T

₅

| + | T

₆

| ≤ (1 + η)||| w |||

µ,λ

||| v

_h

|||

µ,λ

≤ (1 + η)||| w |||

µ,λ,∗

||| v

_h

|||

µ,λ

.

✼

❯s✐♥❣ t❤❡ ❡q✉✐✈❛❧❡♥t ❢♦r♠ T

₃

= P

F∈Fh

R

F

r

⁰_F

(J w K):σ

h

( v

_h

)✱ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③

✐♥❡q✉❛❧✐t② r❡❛❞✐❧② ②✐❡❧❞s | T

₃

| ≤ ||| w |||

µ,λ,∗

||| v

_h

|||

µ,λ

✳ ❋✐♥❛❧❧②✱ t♦ ❡st✐♠❛t❡ T

₂

✇❡ ✉s❡ t❤❡ ❈❛✉❝❤②✕

❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t② t♦ ✐♥❢❡r

| T

₂

| ≤ ||| w |||

µ,λ,∗

× (

X

F∈Fh

1 h

F

X

T∈TF

k(2µ

|T

)

^1/2

hJ v

_h

Ki

F

k

²_L²_(F)d,d

+ kλ

¹_|T^/2

hJ v

_h

Ki

F

· n

_F

k

²_L²_(F)

)

1 2

.

❯s✐♥❣ ✭✶✶✮ ❛♥❞ s✐♥❝❡✱ ♦✇✐♥❣ t♦ ♠❡s❤ r❡❣✉❧❛r✐t②✱ ❢♦r ❛❧❧ T ∈ T

h

❛♥❞ ❛❧❧ F ∈ F

T

✱ t❤❡r❡ ❤♦❧❞s

card(TF)²|T|d

hF|F|d−1

≤ 4ρ✱ t❤❡ t❡r♠ ✐♥ ❜r❛❝❡s ✐s ❜♦✉♥❞❡❞ ❜② 4ρ

| v

_h

|

²_r,µ

+ | v

_h

|

²_r,λ

✳ ❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡

♣r♦♦❢✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✐s ❛ ❝❧❛ss✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛t❛ ✻✱ ✼✱ ❛♥❞ ✽✳

▲❡♠♠❛ ✾ ✭❊rr♦r ❡st✐♠❛t❡✮✳ ▲❡t u ❞❡♥♦t❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ✭✸✮ ❛♥❞ ❢✉rt❤❡r ❛ss✉♠❡ u ∈ U

_∗

✳

❚❤❡♥✱ ❞❡♥♦t✐♥❣ ❜② u

_h

t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ✭✶✸✮✱ t❤❡r❡ ❤♦❧❞s

||| u − u

_h

|||

µ,λ

≤

1 + β

ρ,η

α

η

v_h

inf

∈U_h

||| u − v

_h

|||

µ,λ,∗

+ sup

v_h∈U_h

E

u

( v

_h

)

||| v

_h

|||

µ,λ

.

❋♦❧❧♦✇✐♥❣ ❍❛♥s❜♦ ❛♥❞ ▲❛rs♦♥ ❬✸❪✱ ✇❡ ❞❡r✐✈❡ ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✉s✐♥❣ t❤❡

❈r♦✉③❡✐①✕❘❛✈✐❛rt ✐♥t❡r♣♦❧❛t♦r I

CR

: H

²

(T

h

) → CR (T

h

)✱ ✇❤❡r❡

CR (T

h

) :=

v

h

∈ P

¹_d

(T

h

) | hJv

h

Ki

F

= 0, ∀F ∈ F

h

. ✭✶✻✮

❲❤❡♥ ❛♣♣❧✐❡❞ t♦ ✈❡❝t♦r ❢✉♥❝t✐♦♥s✱ t❤❡ ✐♥t❡r♣♦❧❛t♦r ❛❝ts ❝♦♠♣♦♥❡♥t✇✐s❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦①✐✲

♠❛t✐♦♥ ❡st✐♠❛t❡s ❝❧❛ss✐❝❛❧❧② ❤♦❧❞ ✭❝❢✳ ❬✸✱ ▲❡♠♠❛ ✷✳✸❪ ❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮✿ ❋♦r ❛❧❧ v ∈ H

²

(T

h

)

^d

❛♥❞ ❛❧❧ T ∈ T

h

k v − I

CR

v k

_L²_(T)d

+ h

T

| v − I

CR

v |

_H¹_(T)d

≤ C

CR

h

²_T

k v k

_H²_(T)d

, ✭✶✼❛✮

k div( v − I

CR

v )k

_L²_(T)

+ h

T

| div( v − I

CR

v )|

_H¹_(T)

≤ C

CR

h

T

| div v |

_H¹_(T)

, ✭✶✼❜✮

✇❤❡r❡ C

CR

♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❡s❤ r❡❣✉❧❛r✐t② ♣❛r❛♠❡t❡r ρ✳ ❖❜s❡r✈❡ t❤❛t t❤❡ ❜r♦❦❡♥ H

¹

s❡♠✐✲

♥♦r♠ ♦❢ t❤❡ ❞✐✈❡r❣❡♥❝❡ ❛♣♣❡❛rs ✐♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✭✶✼❜✮ ❝♦❤❡r❡♥t❧② ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥

♦❢ N

u

✭✷✮✳ ■♥❞❡❡❞✱ ✭✶✼❜✮ r❡s✉❧ts ❢r♦♠ ❛ s✐♠♣❧❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ P♦✐♥❝❛ré✲❲✐rt✐♥❣❡r ✐♥❡q✉❛❧✐t②

♦❜s❡r✈✐♥❣ t❤❛t div(I

CR

v ) = hdiv v i

Ω

✳ ❚❤✐s ✐s ❛♥ ✐♠♣♦rt❛♥t ♣r♦♣❡rt② ♠❡❛♥✐♥❣ t❤❛t t❤❡ ❞✐s❝r❡t❡

s♣❛❝❡ ❛❧❧♦✇s t♦ ❛❝❝✉r❛t❡❧② ❛♣♣r♦①✐♠❛t❡ ♥♦♥tr✐✈✐❛❧ ❢✉♥❝t✐♦♥s ✇✐t❤ ③❡r♦ ❞✐✈❡r❣❡♥❝❡✳

❚❤❡♦r❡♠ ✶✵ ✭❈♦♥✈❡r❣❡♥❝❡ r❛t❡✮✳ ❚❤❡r❡ ❤♦❧❞s

||| u − u

_h

|||

µ,λ

≤ χh, ✭✶✽✮

✇✐t❤ χ := C

ρ,µ

N

^u

✇❤❡r❡ N

^u

✐s ❞❡✜♥❡❞ ❜② ✭✷✮ ❛♥❞ C

ρ,µ

♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠❡s❤ r❡❣✉❧❛r✐t②

♣❛r❛♠❡t❡r ρ ❛♥❞ ♦♥ µ✳

Pr♦♦❢✳ ❋♦r t❤❡ s❛❦❡ ♦❢ ❜r❡✈✐t② ✇❡ ❞❡♥♦t❡ ❜② a . b t❤❡ ✐♥❡q✉❛❧✐t② a ≤ Cb ✇✐t❤ C ❣❡♥❡r✐❝ ❝♦♥st❛♥t

♦♥❧② ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♠❡s❤ r❡❣✉❧❛r✐t② ♣❛r❛♠❡t❡r ρ ❛♥❞ ♦♥ µ✳

✭✐✮ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r✳ ▲❡t w

_h

:= I

CR

u ✳ ❚❤❡ ❢❛❝t t❤❛t ||| u − w

_h

|||

µ,λ,∗

. N

u

h ❢♦❧❧♦✇s ❢r♦♠ ✭✶✼✮

✉♣♦♥ ♦❜s❡r✈✐♥❣ t❤❛t | u − w

_h

|

r,µ

= | u − w

_h

|

r,λ

= 0 ❛♥❞ ❛♣♣❧②✐♥❣ s❡✈❡r❛❧ t✐♠❡s t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛❝❡

✐♥❡q✉❛❧✐t② ✭s❡❡✱ ❡✳❣✳✱ ▼♦♥❦ ❛♥❞ ❙ü❧✐ ❬✶✾❪ ♦r ❈❛rst❡♥s❡♥ ❛♥❞ ❋✉♥❦❡♥ ❬✷✵❪✮✿ ❋♦r ❛❧❧ v ∈ H

¹

(T

h

)✱

∀T ∈ T

h

, h

T

kvk

²_L²_(∂T)

. kvk

²_L²_(T)

+ h

²_T

k ∇ vk

²_L²_(T)d

. ✭✶✾✮

✭✐✐✮ ❈♦♥s✐st❡♥❝② ❡rr♦r✳ ▲❡t v

_h

∈ U

_h

❛♥❞ ❞❡♥♦t❡ ❜② π

⁰_h

t❤❡ L

²

✲♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t♦r ♦♥t♦ P

⁰

d

(T

h

)

^d,d

✳

✽

❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t {π

⁰_h

σ( u )}

F

✐s ❝♦♥st❛♥t ♦✈❡r F ∈ F

h

t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③ ✐♥✲

❡q✉❛❧✐t②✱ ✐t ✐s ✐♥❢❡rr❡❞

E

^u

( v

_h

) = X

F∈Fh

Z

F

{σ( u ) − π

_h⁰

σ( u )}:(hJ v

_h

Ki

F

− J v

_h

K)

. (

X

T∈Th

h

T

k 2µ(ǫ( u ) − π

_h⁰

ǫ( u ))

|T

k

²_L²_(∂T)d,d

+ k λ(div u − π

_h⁰

div u )

|T

k

²_L²_(∂T)

)

¹2

× (

X

F∈Fh

h

⁻¹_F

khJ v

_h

Ki

F

− J v

_h

Kk

²_L²_(F)d

)

¹2

:= T

₁

× T

₂

.

❯s✐♥❣ ✭✶✾✮✱ ✐t ✐s r❡❛❞✐❧② ✐♥❢❡rr❡❞ t❤❛t T

₁

. N

^u

h✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ❣❡♥❡r❛❧✐③❡❞ P♦✐♥❝❛ré✕

❋r✐❡❞r✐❝❤s ✐♥❡q✉❛❧✐t② kv − JvKk

_L²_(F)

. h

¹_F^/2

k ∇ vk

_L²_(TF)d

✈❛❧✐❞ ❢♦r ❛❧❧ v ∈ H

¹

(T

h

) ∩ H

₀¹

(Ω) ✭❝❢✳ ❬✷✶❪✮

②✐❡❧❞s T

₂

. k ∇v

_h

k

_L²_(Ω)d,d

✳ ❋✐♥❛❧❧②✱ ✉s✐♥❣ ✭✾✮✱ T

₂

. ||| v

_h

|||

µ,λ

✱ ❛♥❞ t❤❡ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s✳

❘❡♠❛r❦ ✶✶ ✭◆✉♠❡r✐❝❛❧ ❧♦❝❦✐♥❣✮✳ ❚❤❡ ❡st✐♠❛t❡ ✭✶✽✮ s❤♦✇s t❤❛t t❤❡ ❞✐s❝r❡t❡ ♠❡t❤♦❞ ✭✶✸✮ ✐s ❧♦❝❦✐♥❣✲

❢r❡❡ ❢♦r d = 2✱ s✐♥❝❡✱ ♦✇✐♥❣ t♦ ❚❤❡♦r❡♠ ✷✱ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♣❛r❛♠❡t❡r χ ♦♥❧② ❞❡♣❡♥❞s ♦♥ q✉❛♥t✐t✐❡s t❤❛t st❛② ❜♦✉♥❞❡❞ ❢♦r λ → +∞✳

✸✳✺✳ ❋❧✉① ❢♦r♠✉❧❛t✐♦♥

■t ✐s ❛ ❝♦♠♠♦♥ ♣r❛❝t✐❝❡ t♦ ❡①♣r❡ss ❞● ♠❡t❤♦❞s ✐♥ t❡r♠s ♦❢ ♥✉♠❡r✐❝❛❧ ✢✉①❡s✳ ❚♦ t❤✐s ♣✉r♣♦s❡✱

✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐❢t✐♥❣✲❝♦rr❡❝t❡❞ ❣r❛❞✐❡♥t✿ ❋♦r ❛❧❧ v

_h

∈ U

_h

✱ G

⁰_h

( v

_h

) := ∇

_h

v

_h

− R

⁰_h

( v

_h

).

❉✐s❝r❡t❡ s②♠♠❡tr✐❝ ❣r❛❞✐❡♥t✱ ❞✐✈❡r❣❡♥❝❡✱ ❛♥❞ ❡❧❛st✐❝✐t② ♦♣❡r❛t♦rs ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❢r♦♠ G

⁰_h

( v

_h

)✿

E

_h⁰

( v

_h

) := 1 2

G

⁰_h

( v

_h

) + G

⁰_h

( v

_h

)

^t

, D

_h⁰

( v

_h

) := tr(G

⁰_h

( v

_h

)) = div

h

v

_h

− R

⁰_h

( v

_h

), Σ

⁰_h

( v

_h

) := 2µE

_h⁰

( v

_h

) + λD

⁰_h

( v

_h

)I

d

.

▲❡♠♠❛ ✶✷ ✭❋❧✉① ❢♦r♠✉❧❛t✐♦♥✮✳ ❚❤❡r❡ ❤♦❧❞s ❢♦r ❛❧❧ ( w

_h

, v

_h

) ∈ U

²

h

✱ a

h

( w

_h

, v

_h

) =

Z

Ω

Σ

⁰_h

( w

_h

):ǫ

⁰_h

( v

_h

) + X

F∈Fh

Z

F

Φ

F

( w

_h

):J v

_h

K⊗ n

_F

,

✇❤❡r❡

Φ

F

( w

_h

) = −{2µ(ǫ

h

( w

_h

) − r

_F⁰

( w

_h

)) + λ(div

h

w

_h

− r

_F⁰

( w

_h

))I

d

} + γ

µ,F

h

F

J w

_h

K⊗ n

_F

.

✸✳✻✳ ❈♦♥✈❡r❣❡♥❝❡ t♦ ♠✐♥✐♠❛❧ r❡❣✉❧❛r✐t② s♦❧✉t✐♦♥s

❋♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠♣❧❡t❡♥❡ss✱ t❤✐s s❡❝t✐♦♥ ❜r✐❡✢② ❛❞❞r❡ss❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t♦ ♠✐♥✐♠❛❧ r❡❣✉❧❛r✲

✐t② s♦❧✉t✐♦♥s✱ ✐✳❡✳✱ s♦❧✉t✐♦♥s t❤❛t ❜❛r❡❧② s✐t ✐♥ H

₀¹

(Ω)

^d

✳ ■♥ ✈✐❡✇ ♦❢ ❛♣♣❧②✐♥❣ t❤❡ ❛r❣✉♠❡♥ts ♦❢ ❬✷✷❪✱

✇❡ r❡❝♦r❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ a

h

✇✐t❤ ❞✐s❝r❡t❡ ❛r❣✉♠❡♥ts✿

❋♦r ❛❧❧ ( w

_h

, v

_h

) ∈ U

_h

× U

_h

✱

a

h

( w

_h

, v

_h

) = Z

Ω

Σ

⁰_h

( w

_h

):E

⁰_h

( v

_h

) + j

h

( w

_h

, v

_h

), ✭✷✵✮

✾