Daniele Antonio

Di Pietro

Serge Nicaise

div σ(u) = f ✐♥ Ω, u = 0

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

▲❡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 := u 2 H 2 (PΩ) d + |λ div u| 2 H 1 (PΩ)
♣•❛❝❡♠❡♥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 ❛♥❞ ♦♥ f L 2 (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❛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 1 0 (Ω) d ✱ t❤❡ ✇❡❛❦ ❢♦r♠✉•❛t✐♦♥ ♦❢ ♣r♦❜•❡♠ ✭✶✮ r❡❛❞s✿ ❋✐♥❞ u ∈ U s✉❝❤ t❤❛t a(u, v) = Ω f •v ∀v ∈ U, ✭✸✮ ✇❤❡r❡ a(u, v) := Ω σ(u):ǫ(v) ❛♥❞✱ ❢♦r t✇♦ s❡❝♦♥❞✲♦r❞❡r t❡♥s♦rs α ❛♥❞ β✱ ✇❡ ❤❛✈❡ ❞❡♥♦t❡❞ α:β := 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 (2µ) 1 /2 ǫ(u) L 2 (Ω) d,d + λ 1 /2 div u L 2 (Ω) ≤ C Ω µ 1 /2 f L 2 (Ω) d , ✭✹✮ ✇❤❡r❡ C Ω ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ P♦✐♥❝❛ré ❛♥❞ ❑♦r♥ ❝♦♥st❛♥ts✳ ✷ Pr♦♦❢✳ ❚❛❦✐♥❣ u ❛s ❛ t❡st ❢✉♥❝t✐♦♥ ✐♥ ✭✸✮ ✐t ✐s ✐♥❢❡rr❡❞ (2µ) 1 /2 ǫ(u) 2 L 2 (Ω) d,d + λ 1 /2 div u 2 L 2 (Ω) = a(u, u) = Ω f •u✳ ❯s✐♥❣ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③✱ P♦✐♥❝❛ré✱ ❑♦r♥✱ ❛♥❞ (♦✉♥❣✬s ✐♥❡q✉❛•✐t✐❡s ✐t ✐s ✐♥❢❡rr❡❞ Ω f •u ≤ f L 2 (Ω) d u L 2 (Ω) d ≤ C Ω f L 2 (Ω) d ǫ(u) L 2 (Ω) d,d ≤ 1 2 C 2 Ω 2µ f 2 L 2 (Ω) d + 1 2 (2µ) 1 /2 ǫ(u) 2 L 2 (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 2 0 (Ω) := {p ∈ L 2 (Ω) : p Ω = 0}✱ ✇❤❡r❡✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ φ ✐♥t❡❣r❛❜•❡ ♦♥ Ω✱ φ Ω := 1 |Ω| d Ω φ✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ❜✐•✐♥❡❛r ❢♦r♠s a 0 ∈ L(U × U, R) ❛♥❞ b 0 ∈ L(U × L 2 0 (Ω), R) ❞❡✜♥❡❞ ❛s ❢♦••♦✇s✿ a 0 (w, v) := Ω 2µǫ(w):ǫ(v), b 0 (w, q) := - Ω div wq. ❚❤❡♥✱ •❡tt✐♥❣ p := -λ div u ∈ L 2 (Ω), p := p + λ div u Ω ∈ L 2 0 (Ω), ✭✺✮ ✇❡ ✐♥❢❡r t❤❛t (u, p) ∈ U × L 2 0 (Ω) ✐s t❤❡ ✉♥✐q✉❡ s♦•✉t✐♦♥ ♦❢ t❤❡ tr❛♥s♠✐ss✐♦♥ ❙t♦❦❡s ♣r♦❜•❡♠ a 0 (u, v) + b 0 (v, p) = Ω f •v ∀v ∈ U, b 0 (u, q) = Ω gq ∀q ∈ L 2 0 (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❡ u H 2 (PΩ) + p H 1 (PΩ) ≤ C S f L 2 (Ω) d + g H 1 (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 u H 2 (PΩ) + p H 1 (PΩ) ≤ C S f L 2 (Ω) d + div u H 1 (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 f L 2 (Ω) d , ✇✐t❤ C λ,µ ❞❡♣❡♥❞❡♥t ♦♥ Ω✱ λ✱ ❛♥❞ µ ❜✉t ♥♦t ♦♥ λ✳ Pr♦♦❢✳ ❇② ✭✽✮✱ t❤❡r❡ ❤♦•❞s ✇✐t❤ p ❞❡✜♥❡❞ ❜② ✭✺✮✱ |λ div u| H 1 (PΩ) = |p| H 1 (PΩ) ≤ C S f L 2 (Ω) d + div u H 1 (PΩ) ≤ C S λ 1 /2 λ 1 /2 div u L 2 (Ω) + 1 λ 1 /2 |λ div u| H 1 (PΩ) + C S f L 2 (Ω) d . ✸ ❍❡♥❝❡✱ ✉s✐♥❣ t❤❡ ❡♥❡r❣② ❡st✐♠❛t❡ ✭✹✮✱ 1 - C S λ |λ div u| H 1 (PΩ) ≤ C S f L 2 (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 ∈T h h T ✳ ❚❤❡ s❡t ♦❢ ❢❛❝❡s ♦❢ T h ✐s ❞❡♥♦t❡❞ ❜② F h ❀ ❜♦✉♥❞❛r② ❢❛❝❡s ❛r❡ ❝♦••❡❝t❡❞ ✐♥ t❤❡ s❡t F b h ❛♥❞ ✇❡ s❡t F i h := F h \ F b h ✳ ❋♦r ❛•• F ∈ F h ✇❡ •❡t T F := {T ∈ T h | F ⊂ ∂T }✳ ❋♦r ❡✈❡r② ✐♥t❡r❢❛❝❡ F ∈ F i h ✱ ✇❡ 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 b h 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 ∈T h 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 vI 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 i h ✱ v (x) := v |T1 (x) -v |T2 (x), {v}(x) := 1 2 v |T1 (x) + v |T2 (x) . ❋♦r ❛•• F ∈ F b h s✉❝❤ t❤❛t F = ∂T ∩ ∂Ω ✇❡ ❝♦♥✈❡♥t✐♦♥❛••② s❡t {v}(x) = v (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} ❛♥❞ v ✳ ✹ ✸✳✷✳ 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 1 0 ✲•✐❦❡ ♥♦r♠ ♦♥ H 1 0 (T h )✿ v 2 1,h := ∇ h v 2 L 2 (Ω) d + |v| 2 J , |v h | 2 J := F ∈F h 1 h F v h 2 L 2 (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❤❡ • 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✱ v h 1,h ≤ C K ǫ h (v h ) 2 L 2 (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♦ Ω r l F (v):τ h = 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 0 F (v) |T ≡ |F | d-1 card(T F )|T | d v F ⊗n F , r 0 F (v) |T ≡ |F | d-1 card(T F )|T | d v F •n F , ✭✶✶✮ ✇❤❡r❡✱ ❢♦r ❛•• ❢✉♥❝t✐♦♥s ϕ ✐♥t❡❣r❛❜•❡ ♦♥ F ✱ ✇❡ ❤❛✈❡ •❡t ϕ F := F ϕ/|F | d-1 ✳ ❚❤❡ r❡•❛t✐♦♥s ✭✶✶✮ ❝❛♥ ❜❡ r❡♣•❛❝❡ t❤❡ •✐❢t✐♥❣s ✐♥ ♣r❛❝t✐❝❛• ✐♠♣•❡♠❡♥t❛t✐♦♥s✳ ❋✐♥❛••②✱ ❢♦r ❛ ❢✉♥❝t✐♦♥ v ∈ H 1 0 (Ω) d ✱ t❤❡ ❣•♦❜❛• •✐❢t✐♥❣ ♦❢ t❤❡ •✉♠♣s ♦❢ v ❛♥❞ ✐ts tr❛❝❡ ❛r❡ r❡s♣❡❝t✐✈❡•② ❞❡♥♦t❡❞ ❜② R l h (v) := F ∈F h r l F ( v ) ∈ P l d (T h ) d,d , R l h (v) := tr(R l h (v)) = F ∈F h r l F ( v ) ∈ 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) := Ω σ h (w):ǫ h (v) - F ∈F h F {σ h (w)}: v F ⊗n F + w F ⊗n F :{σ h (v)} + F ∈F h Ω η 2µr 0 F ( w ):r 0 F ( v ) + λr 0 F ( w )r 0 F ( v ) + F ∈F h F γ µ,F h F w • v , ✭✶✷✮ ✇❤❡r❡ η > 0 ❞❡♥♦t❡s ❛ ✉s❡r✲❞❡♣❡♥❞❡♥t ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡r ❛♥❞✱ ❢♦r ❛•• F ∈ F h ✱ γ µ,F := max T ∈T F µ |T . ❚❤❡ ❞✐s❝r❡t❡ ♣r♦❜•❡♠ r❡❛❞s ❋✐♥❞ u h ∈ U h s✳t✳ a h (u h , v h ) = 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❡❞✿ F ∈F h F (η µ,F w F • v F + η λ,F ( w F •n F )( v F •n F )) , ✇❤❡r❡ η µ,F := η T ∈T F (2µ) |T |F | d-1 card(T F ) 2 |T | d ❛♥❞ η λ,F := η T ∈T F λ |T |F | d-1 card(T F ) 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♦ - F ∈F h F {σ h (w h )}: v h ⊗n F + w h ⊗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 ) = Ω f •v h -E u (v h ), ✇✐t❤ ❝♦♥s✐st❡♥❝② ❡rr♦r E u (v h ) := F ∈F h F {σ(u)}: ( v h F -v h ) ⊗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 ✱ Ω σ(u):ǫ h (v h ) = Ω f •v h + F ∈F h F {σ(u)}: v h ⊗n F + F ∈F i h F σ(u) :{v h }⊗n F . ❚❤❡ ❝♦♥❝•✉s✐♦♥ ❢♦••♦✇s s✐♥❝❡ σ(u)n F F = 0 ❢♦r ❛•• F ∈ F i h ❛♥❞ u 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 := (2µ) 1 /2 ǫ h (v) 2 L 2 (Ω) d,d + λ 1 /2 div h v 2 L 2 (Ω) + |v| 2 r,µ + |v| 2 r,λ + |v| 2 J,µ ✭✶✹✮ ✇✐t❤ s❡♠✐♥♦r♠s |v| 2 r,µ := F ∈F h (2µ) 1 /2 r 0 F ( v ) 2 L 2 (Ω) d,d ✱ |v| 2 r,λ := F ∈F h λ 1 /2 r 0 F ( v ) 2 L 2 (Ω) ✱ ❛♥❞ |v| 2 J,µ := F ∈F h (γ µ,F /h F ) 1 /2 v 2 L 2 (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 v h 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 F ∈F h F {σ h (v h )}: v h F ⊗n F = Ω 2µǫ h (v h )+λ div h v h :R 0 h (v h ) ≤ N ∂ (2µ) 1 /2 ǫ h (v h ) L 2 (Ω) d,d |v h | r,µ + λ 1 /2 div h v h L 2 (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 (2µ) 1 /2 R l h (v h ) 2 L 2 (Ω) d,d ≤ N ∂ |v h | 2 r,µ ❛♥❞ λ 1 /2 R l h (v h ) 2 L 2 (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 = (2µ) 1 /2 ǫ h (v h ) L 2 (Ω) d,d ❛♥❞ y = |v h | r,µ ❀ ✭✐✐✮ x = λ 1 /2 div h v h L 2 (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 + T ∈T h h T ((2µ) 1 /2 ǫ h (v h )) |T 2 L 2 (∂T ) d,d + (λ 1 /2 div h v h ) |T 2 L 2 (∂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 1 2 ✱ ∀(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 = F ∈F h F r 0 F ( w ):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, * × F ∈F h 1 h F T ∈T F (2µ |T ) 1 /2 v h F 2 L 2 (F ) d,d + λ 1 /2 |T v h F •n F 2 L 2 (F ) 1 2 . ❯s✐♥❣ ✭✶✶✮ ❛♥❞ s✐♥❝❡✱ ♦✇✐♥❣ t♦ ♠❡s❤ r❡❣✉•❛r✐t②✱ ❢♦r ❛•• T ∈ T h ❛♥❞ ❛•• F ∈ F T ✱ t❤❡r❡ ❤♦•❞s card(T F ) 2 |T | d h F |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 inf v h ∈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 ) | v h 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 v -I CR v L 2 (T ) d + h T |v -I CR v| H 1 (T ) d ≤ C CR h 2 T v H 2 (T ) d , ✭✶✼❛✮ div(v -I CR v) L 2 (T ) + h T | div(v -I CR v)| H 1 (T ) ≤ C CR h T | div v| H 1 (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) = div v 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 v 2 L 2 (∂T ) v 2 L 2 (T ) + h 2 T ∇v 2 L 2 (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 ) = F ∈F h F {σ(u) -π 0 h σ(u)}:( v h F -v h ) T ∈T h h T 2µ(ǫ(u) -π 0 h ǫ(u)) |T 2 L 2 (∂T ) d,d + λ(div u -π 0 h div u) |T 2 L 2 (∂T ) 1 2 × F ∈F h h -1 F v h F -v h 2 L 2 (F ) d 1 2 := 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 v -v L 2 (F ) h 1 /2 F ∇v L 2 (T F ) d ✈❛•✐❞ ❢♦r ❛•• v ∈ H 1 (T h ) ∩ H 1 0 (Ω) ✭❝❢✳ ❬✷✶(✮ ②✐❡•❞s T 2 ∇v h L 2 (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 0 h (v h ) := 1 2 G 0 h (v h ) + G 0 h (v h ) t , D 0 h (v h ) := tr(G 0 h (v h )) = div h v h -R 0 h (v h ), Σ 0 h (v h ) := 2µE 0 h (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 ) = Ω Σ 0 h (w h ):ǫ 0 h (v h ) + F ∈F h F Φ F (w h ): v h ⊗n F , ✇❤❡r❡ Φ F (w h ) = -{2µ(ǫ h (w h ) -r 0 F (w h )) + λ(div h w h -r 0 F (w h ))I d } + γ µ,F h F w h ⊗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 1 0 (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 ) = Ω Σ 0 h (w h ):E 0 h (v h ) + j h (w h , v h ), ✭✷✵✮ ✾ ✇✐t❤ j h (w h , v h ) := - Ω 2µR 0 h (w h ):R 0 h (v h ) + λR 0 h (w h )R 0 h (v h ) + F ∈F h Ω η 2µr 0 F ( w h ):r 0 F ( v h ) + λr 0 F ( w h )r 0 F ( v h ) + F ∈F h F γ µ,F h F w h • v h . |❡♠❛r❦ ✶✸ ✭❊①t❡♥s✐♦♥ t♦ U * h × U * h ✮✳ ❆♥ ✐♠♣♦rt❛♥t r❡♠❛r❦ ✐s t❤❛t t❤❡ ❡①♣r❡ss✐♦♥s ✭✶✷✮ ❛♥❞ ✭✷✵✮ ❛r❡ ❡q✉✐✈❛•❡♥t ❛t t❤❡ ❞✐s❝r❡t❡ •❡✈❡• ♦♥•②✱ ✐✳❡✳✱ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ ✭✷✵✮ t♦ U * h × U * h ❞♦❡s ♥♦t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✭✶✷✮✳ ❙✉❝❤ ❛♥ ❡①t❡♥s✐♦♥ ✐s ♥♦t r❡q✉✐r❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✺✳ |❡♠❛r❦ ✶✹ ✭❈♦❡r❝✐✈✐t②✮✳ ▲❡tt✐♥❣ w h = v h ✐♥ ✭✷✵✮ ❛♥❞ ❛ss✉♠✐♥❣ η > N ∂ ✱ ❝♦❡r❝✐✈✐t② ❤♦•❞s ✐♥ t❤❡ ❢♦r♠ a h (v h , v h ) ≥ (2µ) 1 /2 E 0 h (v h ) 2 L 2 (Ω) d,d + λ 1 /2 D 0 h (v h ) 2 L 2 (Ω) + (η -N ∂ ) |v h | 2 r,µ + |v h | 2 r,λ + |v h | 2 J,µ := |]v h [| 2 µ,λ + (η -N ∂ ) |v h | 2 r,µ + |v h | 2 r,λ + |v h | 2 J,µ . ✭✷✶✮ ❚❤❡♦r❡♠ ✶✺ ✭❈♦♥✈❡r❣❡♥❝❡ t♦ ♠✐♥✐♠❛• r❡❣✉•❛r✐t② s♦•✉t✐♦♥s✮✳ ▲❡t (T h ) h∈H ❜❡ ❛♥ ❛❞♠✐ss✐❜•❡ ♠❡s❤ s❡q✉❡♥❝❡✱ ❛♥❞ ❞❡♥♦t❡ ❜② (u h ) h∈H t❤❡ s❡q✉❡♥❝❡ ♦❢ ❞✐s❝r❡t❡ s♦•✉t✐♦♥s t♦ ♣r♦❜•❡♠ ✭✶✸✮✳ ❆ss✉♠❡ η > N ∂ = d + 1✳ ❚❤❡♥✱ ✇✐t❤ u ∈ H 1 0 (Ω) d ✉♥✐q✉❡ s♦•✉t✐♦♥ t♦ ✭✸✮✱ t❤❡r❡ ❤♦•❞s u h → u str♦♥❣•② ✐♥ L 2 (Ω) d , ✭✷✷❛✮ ǫ h (u h ) → ǫ(u) str♦♥❣•② ✐♥ L 2 (Ω) d,d , ✭✷✷❜✮ div h u h → div u str♦♥❣•② ✐♥ L 2 (Ω), ✭✷✷❝✮ |u h | r,µ + |u h | r,λ + |u h | J,µ → 0. ✭✷✷❞✮ Pr♦♦❢✳ ✭✐✮ ❆ ♣r✐♦r✐ ❡st✐♠❛t❡✳ ❯s✐♥❣ ✭✶✺✮ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❈❛✉❝❤②✕❙❝❤✇❛r③ ❛♥❞ ❞✐s❝r❡t❡ P♦✐♥❝❛ré ✐♥❡q✉❛•✐t✐❡s ✐t ✐s ✐♥❢❡rr❡❞ α η µ 2 C K u h 2 1,h ≤ a h (u h , u h ) = Ω f •u h ≤ C P f L 2 (Ω) d u 1,h , ❤❡♥❝❡ t❤❡ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡ u h 1,h ≤ CP αηµ 2 CK f L 2 (Ω) d ❤♦•❞s✳ ❙✐♥❝❡ u h 1,h ✐s ❜♦✉♥❞❡❞ ✉♥✐❢♦r♠•② ✐♥ h✱ ♦✇✐♥❣ t♦ ❬✶✽✱ ❚❤❡♦r❡♠ ✺✳✼(✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ ✭♥♦t r❡♥✉♠❜❡r❡❞ t♦ ❛••❡✈✐❛t❡ t❤❡ ♥♦t❛t✐♦♥✮ ❛♥❞ ❛♥ ❡•❡♠❡♥t w ∈ H 1 0 (Ω) d s✉❝❤ t❤❛t u h → w str♦♥❣•② ✐♥ L 2 (Ω) d ❛♥❞ G 0 h (u h ) ⇀ ∇w ✇❡❛❦•② ✐♥ L 2 (Ω) d,d ✳ ❍❡♥❝❡✱ E 0 h (u h ) ⇀ ǫ(w) ✇❡❛❦•② ✐♥ L 2 (Ω) d,d ❛♥❞ D 0 h (w) ⇀ div w ✇❡❛❦•② ✐♥ L 2 (Ω)✳ ✭✐✐✮ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ u h ✳ ▲❡t v ∈ C ∞ (Ω) d ❛♥❞✱ ❢♦r h ∈ H✱ ❞❡♥♦t❡ ❜② v h t❤❡ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦•❛t❡ ♦❢ v ♦♥t♦ T h ✳ ❚❤❡♥✱ ✉s✐♥❣ v h ❛s ❛ t❡st ❢✉♥❝t✐♦♥ ✐♥ ✭✶✸✮✱ s✐♥❝❡ E 0 h (v h ) → ǫ(v) str♦♥❣•② ✐♥ L 2 (Ω) d,d ✱ D 0 h (v h ) → div v str♦♥❣•② ✐♥ L 2 (Ω)✱ ❛♥❞ j h (w h , v h ) = 0✱ ✐t ✐s ✐♥❢❡rr❡❞ ❢r♦♠ ✭✷✵✮ Ω σ(w):ǫ(v) ← a h (w h , v h ) = Ω f •v h → Ω f •v. ❇② ❞❡♥s✐t② ♦❢ C ∞ (Ω) d ✇❡ ❝♦♥❝•✉❞❡ t❤❛t w ❝♦✐♥❝✐❞❡s ✇✐t❤ u ✉♥✐q✉❡ s♦•✉t✐♦♥ t♦ ✭✶✸✮✳ ❚❤✐s ✐♠♣•✐❡s t❤❛t t❤❡ ✇❤♦•❡ s❡q✉❡♥❝❡ (u h ) h∈H ❝♦♥✈❡r❣❡s ❛♥❞ ♣r♦✈❡s ✭✷✷❛✮✳ ✭✐✐✐✮ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ǫ h (u h )✱ div h u h ✱ ❛♥❞ ♦❢ t❤❡ •✉♠♣s✳ ❯s✐♥❣ ✭✷✶✮ ✐t ✐s ✐♥❢❡rr❡❞ lim sup h→0 |]u h [| 2 µ,λ ≤ lim h→0 a h (u h , u h ) = lim h→0 Ω f •u h = Ω f •u = a(u, u). u = (1, 0) ✭❛✮ ❈•♦s❡❞ ❝❛✈✐t② R 0 R 1 R 2 Ω 1 Ω 2 ✭❜✮ ❈♦♥❝❡♥tr✐❝ ❝②•✐♥❞r✐❝❛• •❛②❡rs ❋✐❣✉r❡ ✶✿ ❈♦♥✜❣✉r❛t✐♦♥s ❢♦r t❤❡ ♥✉♠❡r✐❝❛• ❡①❛♠♣•❡s ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♦✇✐♥❣ t♦ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✱ lim inf h→0 |]u h [| µ,λ ≥ (2µ) 1 /2 ǫ(u) 2 L 2 (Ω) d,d + λ 1 /2 div u 2 L 2 (Ω) = a(u, u). ❍❡♥❝❡✱ ✇❡ ❝♦♥❝•✉❞❡ lim h→0 |]u h [| µ,λ = a h (u, u)✱ ✇❤✐❝❤✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ E 0 h (u h ) ❛♥❞ D 0 h (u h ) ♣r♦✈❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣♦✐♥t✱ ❝•❛ss✐❝❛••② ②✐❡•❞s E 0 h (u h ) → ǫ(u) str♦♥❣•② ✐♥ L 2 (Ω) d,d ❛♥❞ D 0 h (u h ) → div u str♦♥❣•② ✐♥ L 2 (Ω)✳ ❯s✐♥❣ ❛❣❛✐♥ ✭✷✶✮ ✐t ✐s ✐♥❢❡rr❡❞ (η -N ∂ ) (|u h | r,µ + |u h | r,λ ) + |u h | J,µ ≤ a h (u h , u h ) -|]u h [| 2 µ,λ , ❛♥❞ ✭✷✷❞✮ ❢♦••♦✇s✳ ❚♦ ♣r♦✈❡ ✭✷✷❜✮ ❛♥❞ ✭✷✷❝✮ ♦❜s❡r✈❡ t❤❛t✱ ♦✇✐♥❣ t♦ t❤❡ tr✐❛♥❣✉•❛r ✐♥❡q✉❛•✐t②✱ E 0 h (u h ) L 2 (Ω) d,d -R 0 h (u h ) L 2 (Ω) d,d ≤ ǫ h (u h ) L 2 (Ω) d,d ≤ E 0 h (u h ) L 2 (Ω) d,d + R 0 h (u h ) L 2 (Ω) d,d , D 0 h (u h ) L 2 (Ω) -R 0 h (u h ) L 2 (Ω) ≤ div h u h L 2 (Ω) ≤ D 0 h (u h ) L 2 (Ω) + R 0 h (u h ) L 2 (Ω) . ❙✐♥❝❡ R 0 h (u h ) L 2 (Ω) d ≤ (2µ) -1 /2 N ∂ |u h | r,µ ❛♥❞ R 0 h (u h ) L 2 (Ω) ≤ λ -1 /2 |u h | r,λ ✱ t❤❡ ❝♦♥❝•✉s✐♦♥ ❢♦•✲ •♦✇s •❡tt✐♥❣ h → 0 ❛♥❞ ✉s✐♥❣ ✭✷✷❞✮✳ |❡♠❛r❦ ✶✻ ✭•❡♥❡r❛• ♠❡s❤❡s✮✳ ❚❤❡ ❛❜♦✈❡ ♣r♦♦❢ ❡①t❡♥❞s t♦ ❣❡♥❡r❛• ♣♦•②❤❡❞r❛• ♠❡s❤❡s ❜② r❡♣•❛❝✐♥❣ t❤❡ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦•❛t♦r ❜② t❤❡ L 2 ✲♦rt❤♦❣♦♥❛• ♣r♦•❡❝t♦r✳ ❋♦r ❢✉rt❤❡r ❞❡t❛✐•s s❡❡ ❬✶✽✱ ❙❡❝t✳ ✺✳✷( ❛♥❞ ❬✶✽✱ ❈❤❛♣t❡r ✶(✳ ✹✳ ◆✉♠❡r✐❝❛• ❡①❛♠♣•❡s ✹✳✶✳ ❈•♦s❡❞ ❝❛✈✐t② ❖✉r ✜rst t❡st ❝❛s❡ ✐s t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❝•♦s❡❞ ❝❛✈✐t② ♣r♦❜•❡♠ ♣r♦♣♦s❡❞ ❜② ❍❛♥s❜♦ ❛♥❞ ▲❛r✲ s♦♥ ❬✸(✳ ❚❤❡ ❞♦♠❛✐♥ ✐s t❤❡ ✉♥✐t sq✉❛r❡ Ω = (0, 1) 2 ✳ ❆ ❤♦r✐③♦♥t❛• ❞✐s♣•❛❝❡♠❡♥t ✐s ✐♠♣♦s❡❞ ♦♥ t❤❡ ✉♣♣❡r s✐❞❡✱ ✇❤❡r❡❛s t❤❡ ♦t❤❡rs ❛r❡ ❦❡♣t ✜①❡❞❀ s❡❡ ❋✐❣✉r❡ ✶✭❛✮✳ ❚❤❡ ❡•❛st✐❝ ♠♦❞✉•✉s ❛♥❞ P♦✐ss♦♥✬s r❛t✐♦ ❛r❡ ❝❤♦s❡♥ ❛s E = 1000 ❛♥❞ ν = 0.4999✳ ❚❤❡ ✈❛•✉❡s ♦❢ t❤❡ ▲❛♠é ♣❛r❛♠❡t❡rs ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ r❡•❛t✐♦♥s λ = νE (1 + ν)(1 -2ν) , µ = E 2(1 + ν) . ✭✷✸✮ ❚✇♦ ✉♥✐❢♦r♠ tr✐❛♥❣✉•❛r ♠❡s❤❡s ✇✐t❤ ♠❡s❤s✐③❡ h ∈ {6.25 • 10 -2 , 1.5625 • 10 -2 } ❛r❡ ✉s❡❞✱ ❛♥❞ ❛ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ st❛❜✐•✐③❡❞ ❈r♦✉③❡✐①✕|❛✈✐❛rt ♠❡t❤♦❞ ♦❢ ❬✸( ✐s ♣r♦✈✐❞❡❞ ❢♦r t❤❡ s❛❦❡ ♦❢ ❝♦♠✲ ♣•❡t❡♥❡ss✳ ❚❤❡ ✈❛•✉❡s ♦❢ t❤❡ ❤♦r✐③♦♥t❛• ❞✐s♣•❛❝❡♠❡♥t ❛•♦♥❣ t❤❡ ✈❡rt✐❝❛• ❝❡♥t❡r•✐♥❡ ❛♥❞ ♦❢ t❤❡ ✈❡rt✐❝❛• ✶✶ ❞• ♠❡t❤♦❞ ✭✶✸✮ ❙t❛❜✐•✐③❡❞ ❈r♦✉③✐❡①✕|❛✈✐❛rt 0
u x y ✭❜✮ ❋✐♥❡ ♠❡s❤ ❋✐❣✉r❡ ✷✿ |❡s✉•ts ❢♦r t❤❡ ❝•♦s❡❞ ❝❛✈✐t② ♣r♦❜•❡♠✳ ❙♦•✐❞ •✐♥❡s✿ x✲❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❞✐s♣•❛❝❡♠❡♥t ❛•♦♥❣ t❤❡ ✈❡rt✐❝❛• ❝❡♥t❡r•✐♥❡✳ ❉❛s❤❡❞ •✐♥❡s✿ y✲❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❞✐s♣•❛❝❡♠❡♥t ❛•♦♥❣ t❤❡ ❤♦r✐③♦♥t❛• ❝❡♥t❡r•✐♥❡✳ ❞✐s♣•❛❝❡♠❡♥t ❛•♦♥❣ t❤❡ ❤♦r✐③♦♥t❛• ❝❡♥t❡r•✐♥❡ ❛r❡ ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✷✳ ❚❤❡ ♠❡t❤♦❞s s❤♦✇ ❣♦♦❞ ❛❣r❡❡♠❡♥t✱ ❛♥❞ ♥♦ s✐❣♥ ♦❢ •♦❝❦✐♥❣ ✐s ♣r❡s❡♥t ✐♥ ❜♦t❤ ❝❛s❡s✳ ✹✳✷✳ ❈♦♥❝❡♥tr✐❝ ❝②•✐♥❞r✐❝❛• •❛②❡rs ❚♦ t❡st t❤❡ r♦❜✉st♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❤❡t❡r♦❣❡♥❡✐t② ♦❢ t❤❡ ♠❡❞✐✉♠ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❡①❛❝t s♦•✉t✐♦♥ ❞❡r✐✈❡❞ ✐♥ ❬✾( ❢♦r ❛ ❝♦♥✜❣✉r❛t✐♦♥ ❝♦♠♣♦s❡❞ ♦❢ ❝♦♥❝❡♥tr✐❝ ❤♦••♦✇ ❝②•✐♥❞r✐❝❛• •❛②❡rs s✉❜•❡❝t t♦ ❛ ❤♦♠♦❣❡♥❡♦✉s ♣r❡ss✉r❡ q ♦♥ t❤❡ ✐♥♥❡r s✉r❢❛❝❡ ❛♥❞ Q ♦♥ t❤❡ ♦✉t❡r s✉r❢❛❝❡✳ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✉r❡ ✶✭❜✮✳ ❉❡♥♦t❡ ❜② B R t❤❡ ❜❛•• ♦❢ r❛❞✐✉s R > 0 ❝❡♥t❡r❡❞ ✐♥ t❤❡ ♦r✐❣✐♥✳ ❋♦r t❤r❡❡ r❡❛• ♥✉♠❜❡rs 0 < R 0 < R 1 < R 2 ✱ ✇❡ s❡t Ω := B R2 \ B R0 ❛♥❞ P Ω := Ω i := B Ri \ B Ri-1 1≤i≤2 ✳ ❚❤❡ ❡①❛❝t s♦•✉t✐♦♥ ✐♥ ❝②•✐♥❞r✐❝❛• ❝♦♦r❞✐♥❛t❡s (r, ϑ) ✐s s✉❝❤ t❤❛t u |Ωi = α i A i r + β i B i r r, 1 ≤ i ≤ 2, ✇❤❡r❡ r ❞❡♥♦t❡s t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ r❛❞✐❛• ❞✐r❡❝t✐♦♥ ❛♥❞✱ ❢♦r ❛ ❣✐✈❡♥ ♣r❡ss✉r❡ ♠♦❞✉•❡ q✱ t❤❡ ❝♦❡✣❝✐❡♥ts {α i , β i } 1≤i≤2 ❛r❡ s✉❝❤ t❤❛t α i = - 1 + ν i E i , β i = (1 -2ν i )(1 + ν i ) E i , 1 ≤ i ≤ 2. ❍❡r❡✱ E i ❛♥❞ ν i ❞❡♥♦t❡✱ r❡s♣❡❝t✐✈❡•②✱ t❤❡ ❝♦♥st❛♥t ✈❛•✉❡s ♦❢ t❤❡ ❡•❛st✐❝ ♠♦❞✉•✉s ❛♥❞ P♦✐ss♦♥✬s r❛t✐♦✱ ✇❤✐•❡ t❤❡ ❝♦❡✣❝✐❡♥ts {A i , B i } 1≤i≤2 s♦•✈❡ A 1 R 2 0 + B 1 = -q, α 2 -α 1 α 2 -β 2 A 1 R 2 1 + α 2 -β 1 α 2 -β 2 B 1 -B 2 = 0, A 2 R 2 2 + B 2 = -Q, α 1 -β 2 α 2 -β 2 A 1 -A 2 + β 1 -β 2 α 2 -β 2 B 1 R 2 1 = 0. ❚❤❡ ❝♦♥st❛♥t ✈❛•✉❡s ♦❢ t❤❡ ▲❛♠é ♣❛r❛♠❡t❡rs λ i ❛♥❞ µ i ✐♥ ❡❛❝❤ •❛②❡r Ω i ✱ i ∈ {1, 2} ❝❛♥ ❜❡ ✐♥❢❡rr❡❞ ❢r♦♠ ✭✷✸✮✳ ❋♦r t❤❡ t❡sts ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢♦••♦✇✐♥❣ ✈❛•✉❡s✿ R 0 = 0.5✱ R 1 = 0.75✱ R 2 = 1✱ E i = 1 ❢♦r 1 ≤ i ≤ 2✱ ν 1 = 1✱ ν 2 ∈

  ✉♥❞❡❞ ✇❤❡♥ 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❤❡ ♠❡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✐❝❛•

  {0.499, 0.4999, 0.49999}✳ ❚❤❡ st❛❜✐•✐③❛t✐♦♥ ♣❛r❛♠❡t❡r η ✐s t❛❦❡♥ ❡q✉❛• t♦ ✶✳ ❚♦ st✉❞② t❤❡ ❝♦♥✈❡r❣❡♥❝❡✱ ✇❡ ✉s❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✉♥✐❢♦r♠•② r❡✜♥❡❞ ✉♥str✉❝t✉r❡❞ tr✐❛♥❣✉•❛r ♠❡s❤❡s (T hj ) 0≤j≤4 ✇✐t❤ hj := h 0 /2 j ❛♥❞ h 0 = 0.25✳ ❚❤❡ r❡s✉•ts ❞✐s♣•❛②❡❞ ✐♥ ❋✐❣✉r❡ ✸ s❤♦✇ t❤❛t t❤❡ ♣r❡❝✐s✐♦♥ ✐s ♥♦t ❛✛❡❝t❡❞ ✇❤❡♥ t❤❡ ♦✉t❡r •❛②❡r t❡♥❞s t♦ t❤❡ ✐♥❝♦♠♣r❡ss✐❜•❡ •✐♠✐t✳ ✶✷ 10 -2.2 10 -2 10 -1.8 10 -1.6 10 -1✭❛✮ |||uu h ||| µ,λ ✈s✳ h 10 -2.2 10 -2 10 -1.8 10 -1.6 10 -1✭❜✮ uu h L 2 (Ω) d ✈s✳ h ❋✐❣✉r❡ ✸✿ ❊rr♦rs ✈❡rs✉s ♠❡s❤ s✐③❡s |❡♠❛r❦ ✶✼ ✭❙♠♦♦t❤ ❞♦♠❛✐♥s✮✳ ❚❤❡ t❤❡♦r② ♦❢ ❙❡❝t✳ ✷ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ s♠♦♦t❤ ❞♦♠❛✐♥ ♦❢ ❋✐❣✉r❡ ✶✭❜✮✳ ■♥ ♣r❛❝t✐❝❡✱ ❤♦✇❡✈❡r✱ s♠♦♦t❤ ❞♦♠❛✐♥s ❛r❡ ♦❢t❡♥ ❛♣♣r♦①✐♠❛t❡❞ ❜② ♣♦•②❣♦♥❛• ❞♦♠❛✐♥s ❛t t❤❡ ♥✉♠❡r✐❝❛• •❡✈❡•s✱ s♦ t❤❛t t❤❡ ❝❛s❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❙❡❝t✳ ✷ t✉r♥s ♦✉t t♦ ❜❡ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ✐♥ ♣r❛❝t✐❝❡✳ ❬✶( ❙✳ ❇r❡♥♥❡r✱ ▲✳✲(✳ ❙✉♥❣✱ ▲✐♥❡❛r ✜♥✐t❡ ❡•❡♠❡♥t ♠❡t❤♦❞s ❢♦r ♣•❛♥❛r •✐♥❡❛r ❡•❛st✐❝✐t②✱ ▼❛t❤✳ ❈♦♠♣✳ ✹✾ ✭✷✵✵✮ ✭✶✾✾✷✮ ✸✷✶✕✸✸✽✳ ❬✷( P✳ ❍❛♥s❜♦✱ ▼✳ •✳ ▲❛rs♦♥✱ ❉✐s❝♦♥t✐♥✉♦✉s •❛•❡r❦✐♥ ♠❡t❤♦❞s ❢♦r ✐♥❝♦♠♣r❡ss✐❜•❡ ❛♥❞ ♥❡❛r•② ✐♥❝♦♠♣r❡ss✐❜•❡ ❡•❛st✐❝✐t② ❜② ◆✐ts❝❤❡✬s ♠❡t❤♦❞✱ ❈♦♠♣✉t✳ ▼❡t❤♦❞s ❆♣♣•✳ ▼❡❝❤✳ ❊♥❣r❣✳ ✶✾✶ ✭✷✵✵✷✮ ✶✽✾✺✕✶✾✵✽✳ ❬✸( P✳ ❍❛♥s❜♦✱ ▼✳ •✳ ▲❛rs♦♥✱ ❉✐s❝♦♥t✐♥✉♦✉s •❛•❡r❦✐♥ ❛♥❞ t❤❡ ❈r♦✉③❡✐①✕|❛✈✐❛rt ❡•❡♠❡♥t✿ ❆♣♣•✐✲ ❝❛t✐♦♥ t♦ ❡•❛st✐❝✐t②✱ ▼✷❆◆ ▼❛t❤✳ ▼♦❞❡•✳ ◆✉♠❡r✳ ❆♥❛•✳ ✸✼ ✭✶✮ ✭✷✵✵✸✮ ✻✸✕✼✷✳ ❬✹( ❚✳ P✳ ❲✐❤•❡r✱ ▲♦❝❦✐♥❣✲❢r❡❡ ❛❞❛♣t✐✈❡ ❞✐s❝♦♥t✐♥✉♦✉s •❛•❡r❦✐♥ ❋❊▼ ❢♦r •✐♥❡❛r ❡•❛st✐❝✐t② ♣r♦❜•❡♠s✱ ▼❛t❤✳ ❈♦♠♣✳ ✼✺ ✭✷✺✺✮ ✭✷✵✵✻✮ ✶✵✽✼✕✶✶✵✷✳ ❬✺( ❙✳ ◆✐❝❛✐s❡✱ ❉✳ ▼❡r❝✐❡r✱ |❡❣✉•❛r✐t② r❡s✉•ts ♦❢ ❙t♦❦❡s✴▲❛♠é ✐♥t❡r❢❛❝❡ ♣r♦❜•❡♠s✱ ▼❛t❤✳ ◆❛❝❤r✐❝❤t❡♥ ✷✽✺ ✭✷✵✶✷✮ ✸✸✷✕✸✹✽✳ ❬✻( ❙✳ ❈✳ ❇r❡♥♥❡r✱ ❑♦r♥✬s ✐♥❡q✉❛•✐t✐❡s ❢♦r ♣✐❡❝❡✇✐s❡ H 1 ✈❡❝t♦r ✜❡•❞s✱ ▼❛t❤✳ ❈♦♠♣✳ ✼✸ ✭✷✹✼✮ ✭✷✵✵✹✮ ✶✵✻✼✕✶✵✽✼✳ ❬✼( ❋✳ ❇❛ss✐✱ ❙✳ |❡❜❛②✱ ❆ ❤✐❣❤✲♦r❞❡r ❛❝❝✉r❛t❡ ❞✐s❝♦♥t✐♥✉♦✉s ✜♥✐t❡ ❡•❡♠❡♥t ♠❡t❤♦❞ ❢♦r t❤❡ ♥✉♠❡r✲ ✐❝❛• s♦•✉t✐♦♥ ♦❢ t❤❡ ❝♦♠♣r❡ss✐❜•❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ❏✳ ❈♦♠♣✉t✳ P❤②s✳ ✶✸✶ ✭✷✮ ✭✶✾✾✼✮ ✷✻✼✕✷✼✾✳ ❬✽( ❋✳ ❇❛ss✐✱ ❙✳ |❡❜❛②✱ •✳ ▼❛r✐♦tt✐✱ ❙✳ P❡❞✐♥♦tt✐✱ ▼✳ ❙❛✈✐♥✐✱ ❆ ❤✐❣❤✲♦r❞❡r ❛❝❝✉r❛t❡ ❞✐s❝♦♥t✐♥✉✲ ♦✉s ✜♥✐t❡ ❡•❡♠❡♥t ♠❡t❤♦❞ ❢♦r ✐♥✈✐s❝✐❞ ❛♥❞ ✈✐s❝♦✉s t✉r❜♦♠❛❝❤✐♥❡r② ✢♦✇s✱ ✐♥✿ |✳ ❉❡❝✉②♣❡r❡✱ •✳ ❉✐❜❡•✐✉s ✭❊❞s✳✮✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ 2 nd ❊✉r♦♣❡❛♥ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❚✉r❜♦♠❛❝❤✐♥❡r② ❋•✉✐❞ ❉②♥❛♠✐❝s ❛♥❞ ❚❤❡r♠♦❞②♥❛♠✐❝s✱ ✶✾✾✼✱ ♣♣✳ ✾✾✕✶✵✾✳ ❬✾( ❳✳ ❍♦♥❣•✉♥✱ ❙✳ )❤✐❢❡✐✱ )✳ ❚❛♦t❛♦✱ ❊•❛st✐❝ ❛♥❛•②s❡s ♦❢ ❤❡t❡r♦❣❡♥❡♦✉s ❤♦••♦✇ ❝②•✐♥❞❡rs✱ ▼❡❝❤❛♥✲ ✐❝s |❡s❡❛r❝❤ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✸✸ ✭✷✵✵✻✮ ✻✽✶✕✻✾✶✳ ❬✶✵( P✳ •r✐s✈❛r❞✱ ❊••✐♣t✐❝ ♣r♦❜•❡♠s ✐♥ ♥♦♥s♠♦♦t❤ ❞♦♠❛✐♥s✱ ❱♦•✳ ✷✹ ♦❢ ▼♦♥♦❣r❛♣❤s ❛♥❞ ❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ P✐t♠❛♥✱ ❇♦st♦♥✕▲♦♥❞♦♥✕▼❡•❜♦✉r♥❡✱ ✶✾✽✺✳


Adeline Derouaux

Samo Turk

Nick K Olrichs

Stanislav Gobec

Eefjan Breukink

Ana Amoroso

Julien Offant

Julieanne Bostock

Katherine Mariner

Ian Chopra

Thierry Vernet

Astrid Zervosen

Bernard Joris

Jean-Marie Frère

Martine Nguyen-Distèche

Mohammed Terrak
email: mterrak@ulg.ac.be.

Small molecule inhibitors of peptidoglycan synthesis targeting the lipid II precursor

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A c c e p t e d M a n u s c r i p t structure-based virtual screening to the National Cancer Institute library using eHits program 5 and the structure of the glycosyltransferase domain of the Staphylococcus aureus penicillin-6 binding protein 2 resulted in the identification of two small molecules analogues 5, a 2-[1-[(2-7 chlorophenyl)methyl]-2-methyl-5-methylsulfanylindol-3-yl]ethanamine and 5b, a 2-[1-[(3,4-8 dichlorophenyl)methyl]-2-methyl-5-methylsulfanylindol-3-yl]ethanamine that exhibit 9 antibacterial activity against several Gram-positive bacteria but were less active on Gram-10 negative bacteria. The two compounds inhibit the activity of five GTs in the micromolar 11 range. Investigation of the mechanism of action shows that the compounds specifically target 12 peptidoglycan synthesis. Unexpectedly, despite the fact that the compounds were predicted to 13 bind to the GT active site, compound 5b was found to interact with the lipid II substrate via 14 the pyrophosphate motif. In addition, this compound showed a negatively charged 15 phospholipid-dependent membrane depolarization and disruption activity. These small 16 molecules are promising leads for the development of more active and specific compounds to 17 target the essential GT step in cell wall synthesis. 18 The increase of bacterial resistance to antibiotics has resulted in a decline of available 3 efficient antibacterial treatments. Therefore, the discovery and development of new antibiotic 4 classes able to cure infections due to resistant pathogens are urgently needed. 5

Peptidoglycan is an essential polymer and the main constituent of the bacterial cell 6 wall. Its biosynthesis requires several steps and offers many unexplored targets for the 7 development of new antibacterial drugs [1,2]. The last two reactions in peptidoglycan 8 biosynthesis are performed by bifunctional penicillin-binding proteins (PBPs) and result in the 9 assembly of the cell wall polymer from the monomeric intermediates [3]. This takes place 10 outside the plasma membrane and relies on the activity of the glycosyltransferase (GT) 11 module of bifunctional PBPs which uses the lipid II precursor to synthesize glycan chains and 12 their transpeptidase module which catalyzes the cross-linking of two glycan chains via the 13 peptide side chains. Inhibition of either of these two reactions leads to bacterial cell death. β-14 Lactam antibiotics target the transpeptidation reaction but antibiotic therapy based on 15 inhibition of the GTs has not yet been developed. The only well characterized inhibitor 16 targeting the GTs is the natural product moenomycin, a potent antibacterial phosphoglycolipid 17 active at nanomolar concentrations [4]. Despite intensive studies of its structure-function 18

properties it has not been developed for use in human chemotherapy because of poor 19 pharmacokinetic properties related to its C25 lipid chain. A delipidated moenomycin is 20 inactive and the sugar moiety can only be reduced to three saccharide units while retaining 21 good antibacterial activity [4]. 22 Recently the X-ray structures of PBPs or GT modules and their complexes with 23 moenomycin A have been determined [5][6][7][8]. They shed light on the GT domain fold, which 24 exhibits some similarities with that of -lysozyme and confirmed the catalytic mechanism of 25 A c c e p t e d M a n u s c r i p t 4 glycan chains elongation and the mode of action of moenomycin. The GT domain contains an 1 extended enzymatic cleft which can accommodate six sugar units. It is divided into two sub-2 sites, a donor site for the elongating chain and an acceptor site for the lipid II substrate. 3

Between the sub-sites a flexible region is proposed to play an important role in the 4 translocation of the product from the acceptor site to the donor site through a folding and 5 unfolding process [9]. The enzymatic cavity is bordered by several conserved residues and 6 harbors the two glutamate residues essential for catalysis [10]. 7

Inhibition of the GT reaction can be accomplished either by a compound binding to 8 the enzyme, like moenomycin which occupies the donor site mimicking the elongating glycan 9 chain [5], or by agent binding to the lipid II substrate [11]. Natural products such as nisin are 10 known to target the peptidoglycan precursor. Nisin binds to the pyrophosphate motif of lipid 11 II, sequestrates the substrate and induces pore formation in the bacterial membrane. 12

The potential of membrane damaging agents as antibacterials has been validated in the case of 13 the cyclic lipopeptide daptomycin, a drug used in the treatment of certain infections caused by 14 Gram-positive bacteria. Its antibacterial activity against Staphylococcus aureus including 15 methicillin-resistant strains has been shown to be the result of membrane potential disruption 16 [12]. 17

In this paper we report the discovery of new peptidoglycan GT inhibitors through the 18 use of structure-based virtual screening of small molecules from the National Cancer Institute 19 library. Selected hits were evaluated for their ability to inhibit in vitro the GTs activity of 20 Escherichia coli PBP1b in the presence of lipid II substrate, followed by the determination of 21 their effect on bacterial growth. The active compounds were then submitted to several assays 22 to demonstrate their specificities and modes of action. Avanti Polar Lipids Inc. Nisin A was produced, isolated, and purified as described [13]. 7

Moenomycin A, was a gift from Aventis (France). The fluorescent dye 3,3"-8 diethylthiodicarbocyanine iodide (DiSC 2 (5)) was from Molecular Probes Inc. Lipid I, Lipid II 9 and Dansyl-lipid II were synthesized and purified as described elsewhere [14]. 10 Undecaprenylphosphate and undecaprenylpyrophosphate were obtained by phosphorylation 11 of undecaprenol [15] that was isolated from Laurus nobilis as described [16]. Radiolabeled 12 [ 14 C]meso-diaminopimelic acid (A 2 pm)-labelled lipid II was prepared essentially as 13 previously described [17]. 14

The small molecules tested in this study were obtained from the National Cancer Institute and 15 solubilised in DMSO at a concentration of 100 mM or 10 mM depending on the solubility of 16 the compound. The identities of the active compounds 5 and 5b (NSC n° 17383, and NSC n° 17 17382 respectively) were verified by mass spectrometry and NMR. The data (not shown) 18

were consistent with the expected mass and structure for both compounds. 19 20

Proteins expression and purification 21

Five model proteins of the GT51 family were prepared and used in this study. E. coli PBP1b, 22 S. aureus MtgA and PBP2, and the PBP1a from Thermotoga maritima MSB8 were produced 23 and purified as previously described [5,[17][18][19]. The gene encoding (A68-Q723) PBP2 of 24

Enterococcus hirae was cloned into pET28a (+) expression plasmid (Novagen) and the His 25

A c c e p t e d M a n u s c r i p t 6 Tag PBP2 was expressed in E. coli BL21 (DE3). The cells were grown at 37°C to an optical 1 density of 0.8 at 600 nm, protein expression was induced with 0.5 mM isopropyl β-D-1-2 thiogalactopyranoside and incubation was continued overnight at 18°C. The cells were 3 resuspended in 25 mM HEPES pH 7.5, 500 mM NaCl and the PBP2 was purified in one step 4 on a HisTrap column (GE, Heathcare) (unpublished results). The total protein yield was about 5 15 mg per litre of culture and the purity was over 90% as judged by on SDS-PAGE. 6 7

Virtual screening 8

The virtual screening was performed using the programme eHiTS 6.0 from SimBioSys Inc. 9 eHiTS (http://www.simbiosys.com/ehits) [20, 21] was used for the active site detection and 10 docking. Open Babel (http://openbabel.org) was used for manipulating the ligands with 11 various chemical formats. PyMol from DeLano Scientific was used for visual inspection of 12 the results and the graphical representations. 13

The 3D structures of the compounds from the NCI Diversity Set were obtained from the NCI 14 webpage (http://dtp.nci.nih.gov/dw/testmasters/chem3d.html). The 1990 highly diverse 15 compounds of the NCI Diversity Set represented a broad chemical spectrum of the whole NCI 16 database. However, no special preparation of the 3D structures was applied since eHiTS 17 automatically evaluates all the possible protonation states for the ligands and enzymes. The 18 crystal structure of S. aureus PBP2 as a complex with moenomycin (PDB entry 2OLV) [5] 19 was initially prepared with eHiTS. The program automatically detected the ligand in the 20 complex and selected the part of the enzyme within a 7Å margin around the ligand as the 21 active site. The NCI Diversity Set was then docked into the active site. The scoring was 22 according to the eHiTS_Score that is included in the eHiTS software package. Two-23 dimensional similarities search was performed using a ZINC built-in engine and the entire 24 NCI database containing more than 250 000 compounds (Tanimoto similarity index was set to 25 GT activity was monitored using the continuous fluorescence assay method [22]. For the 5 simultaneous screening of numerous conditions this technique was adapted to a 96-well plate 6 format (Greiner Bio-One). The standard reaction was carried out at 30°C in 50 µl of 50 mM 7 Hepes pH 7.5, 200 mM NaCl, 0.2% decyl PEG, 10 mM CaCl 2 , 20% DMSO, 10 µM dansyl-8 lipid II, 1 unit of muramidase (Cellosyl) and 100 nM E. coli PBP1b. The data were collected 9 for 30 min using a Victor 3 fluorimeter (Perkin Elmer) with excitation at 355 nm and 10 emission at 536 nm. 11

For the other GT enzymes the reaction conditions were adapted for optimal activity as 12 follows, with only variable conditions given: 2 µM S. aureus MtgA was used in the presence 13 of 20 µM dansyl-lipid II, 10% DMSO and 10 mM MnCl 2 . 2.5 µM S. aureus PBP2 was used 14 in the presence of 20 µM dansyl-lipid II and 50 mM sodium acetate pH 5. E. hirae PBP2 and 15 T. maritima PBP1a were used at 300 nM and 150 nM respectively. 16

Radioactive assay. 17

The radioactive assay was performed in 30 µl in the same conditions as the fluorescence assay 18 using 4 µM [ 14 C]lipid II (0.126 µCi nmol -1 ) instead of the fluorescent substrate and by 19 omitting the muramidase. The reaction was stopped with moenomycin (10 µM) and the 20 products were separated by TLC in 2-propanol-ammonium hydroxide-H 2 O (6:3:1; V/V/V) 21 and analyzed using a Typhoon Trio+ Imager and the ImageQuant TL software (GE 22 Healthcare). 23

GT inhibition assay. 24

The IC 50 values of compounds 5 and 5b were determined using a fluorescence based assay. The initial rate of the reaction was determined in the presence of various concentrations of 1 inhibitor (20-1500 µM) and plotted versus the inhibitor concentration using the Sigma plot 2 program (Systas Software). The IC 50 value is the inhibitor concentration that decreases the 3 initial rate by a factor of 2. 4 5

MIC determination and effect of 5b on bacterial viability 6

The S. aureus, Enterococcus faecalis, Listeria innocua, Micrococcus luteus, Streptococcus 7 epidermis, Streptococcus pneumoniae and Pseudomonas aeruginosa strains listed in table 2 8 were obtained from the American Type Culture Collection (ATCC). The E. coli strains 1411 9 and SM 1411 were reported previously [START_REF]network, and the Fonds de la Recherche Fondamentale Collective[END_REF]. 10 Minimum inhibitory concentration (MIC) determinations were carried out using the Clinical 11 and Laboratory Standard Institute (CLSI, 2009) broth microdilution method. Compounds 12 were solubilised in 100% DMSO at a concentration of 10 mg/ml, and twenty-fold diluted in 13 Mueller-Hinton broth (MHB), just before utilization. 14 For bacterial viability experiments S. aureus ATCC 25923 colonies were grown 15 overnight on Mueller-Hinton agar (MHA) plates and suspended in MHB (1.10 8 CFU/mL). 16

Compound 5b was added at 1, 2 and 4 times the MIC and mixed. Samples were taken at 17 different time intervals of incubation at 37° C, serially diluted and plated on MHA in 18 triplicate. The colonies were counted after incubation at 37° C for 18h. 19 20

Pathway reporter assays in Bacillus subtilis 21

We have employed a Bacillus subtilis reporter system which differentiates inhibitors of cell 22 envelope, DNA, RNA, protein and fatty acid biosynthesis by quantifying the upregulation of a 23 specific luciferase reporter construct in each case as described [24,25]. 24

The assay based on the LiaRS two-component system in Bacillus subtilis that senses 25 reporter gene under the control of liaI promoter. The effect of antibiotics or compounds 3 spotted on a Whatman disc is revealed by blue/white selection method after overnight 4 incubation at 37°C on agar-plate containing 5-bromo-4-chloro-3-indolyl-beta-D-5 galactopyranoside (X-Gal). 6 7

LIVE/DEAD bacterial viability assay 8

The membrane damage inflicted by 10 minutes exposure to a panel of agents on S. aureus 9 SH1000 was analysed using the LIVE/DEAD Baclight™ bacterial viability kit as reported 10

[46]. Compound 5b and several control antimicrobial agents were tested at 4x MIC in 11 comparison to a drug free control. 12 13

Preparation of large unilamellar vesicles (LUVs) 14

Large unilamellar vesicles were prepared essentially as described [28]. Desired amounts of 15 lipid solutions in chloroform were mixed and evaporated under a gentle stream of nitrogen. 16

The lipid film was subsequently dried for 20 min under vacuum. The film was hydrated by the 17 addition of the buffer of choice under mechanical agitation and submitted to ten freeze-thaw 18 cycles using liquid nitrogen and a water bath. The lipid suspension was then extruded ten 19 times through a polycarbonate membrane filter with a pore size of 200 nm (Whatman 20 International) [29]. The final phospholipid concentration was determined by phosphate 21 analysis according to Rouse et al. [30]. 22 23 2.9. Vesicles binding assay 24 DOPC with or without 1 mol% Lipid II vesicles were prepared as described above in 10 mM 25 A c c e p t e d M a n u s c r i p t 10 MES-KOH, 15 mM K 2 SO 4 at pH 7. Vesicles (1 mM lipid Pi) were incubated with 5 μM and 1 20 μM 5b, respectively, for 15 min at room temperature. The mixture was centrifuged in a 2 TLA 120.2 rotor using a Beckman Ultracentrifuge (TL-100) for 1.5 h at 100 krpm and 20 °C. 3

The amount of 5b before centrifugation and in the supernatant and pellet was determined by 4 monitoring its intrinsic fluorescence on a Cary Eclipse fluorescence spectrophotometer 5 (Varian Inc.). The percentage of 5b in the supernatant and pellet was determined by 6 comparing the maximal value (350 nm) of fluorescence emission intensity. 7 8

Carboxyfluorescein leakage assay in large unilamellar vesicles 9

Carboxyfluorescein (CF)-loaded LUVs were prepared in 50 mM MES-KOH, 100 mM K 2 SO 4 , 10 pH 6.5 (K+-buffer) as described above with the addition of 50 mM CF. Following the 11 extrusion step, the vesicle suspension was applied to Sephadex G-50 spin columns 12 equilibrated with K+-buffer to remove free CF. The final phospholipid concentration was 13 determined by phosphate analysis according to Rouser et al. [30]. Vesicles were resuspended 14 in K + -buffer to a concentration of 25 μM and 5b was added at the desired concentration 1 min 15 prior to the addition of nisin A (50 nM, final conc.). High-salt experiments were conducted in 16 K + -buffer supplemented with 0.5M NaCl. The nisin-induced CF leakage from the vesicles 17 was monitored with excitation and emission wavelengths set at 430 nm and 513 nm, 18 respectively. Triton X-100 was added 1 min after the addition of nisin, to a final concentration 19 of 0.2% (w/v) to fully disrupt the lipid vesicles and the corresponding fluorescence was taken 20 as 100% leakage. The 5b-induced reduction of leakage is expressed as a percentage of the 21 leakage observed in the absence of 5b relative to the total release of carboxyfluorescein after 22 addition of Triton X-100. The leakage results in an increase of fluorescence as the dye self-23 quenching is reduced. 24 The cytoplasmic membrane depolarization activity of 5b was determined with the membrane 2 potential-sensitive dye DiSC 2 (5) using Staphylococcus simulans and Micrococcus flavus 3 grown in LB broth at 37ºC and 30ºC, respectively. Bacterial cells in the mid-logarithmic 4 phase were centrifuged at 5000 rpm, washed in 5 mM HEPES (pH 7.8), and resuspended in 5 the same buffer to an optical density at 600 nm of 0.05 in a 1 cm cuvette. A stock solution of 6 DiSC 2 (5) was added to a final concentration of 0.4 μM and quenching was allowed to occur at 7 room temperature for approximately 1 min. The desired concentration of 5b was added and 8 changes in fluorescence due to the disruption of the membrane potential gradient across the 9 cytoplasmic membrane were continuously recorded with a Cary Eclipse spectrofluorometer at 10 an excitation and emission wavelengths of 622 and 670 nm, respectively. 11 12 13 3. RESULTS 14 15

Structure-based virtual screening and identification of GT inhibitors hits 16

We set out using a virtual screening approach based on the published S. aureus PBP2 17 structure in complex with moenomycin (PDB entry: 2OLV) [5] to identify small molecule 18 inhibitors of the GT activity and to reduce the number of compounds to be tested using a 19 direct in vitro assay based on lipid II substrate (see Materials and Methods). This approach 20 has been used successfully in the identification of inhibitors of other targets of the cell wall 21 synthesis pathway [31,32]. 22

The selected compounds were first evaluated with E. coli PBP1b then with other GT 23 including S. aureus PBP2 [5, 17]. In the first step of screening, the 30 highest ranked 24 compounds (predicted pKd values between -3.61 and -4.37) were selected for biochemical 25 One compound (5, ranked as the fifth best by eHits) (Figure 1A), a 2-[1-[(2-6 chlorophenyl)methyl]-2-methyl-5-methylsulfanylindol-3-yl]ethanamine, was found to inhibit 7 the GT activity of E. coli PBP1b (IC 50 of 59 µM). In the second step, a search based on two-8 dimensional similarities to compound 5 identified an analogue of compound 5 called 5b 9 (Figure 1A), about 2 times more active than compound 5 against E. coli PBP1b (IC 50 of 29 10 µM) and was the focus of further characterisation. 11

The eHiTS-predicted binding mode of compound 5b is presented in Figure 1B. The 12 model shows that compound 5b partly overlaps with co-crystallized moenomycin rings C and 13 E, but is buried deeper because of favourable hydrophobic interactions. On the other hand 14 moenomycin does not form hydrophobic interactions in the hydrophobic shelf due to its 15 hydrophilic character. 16

Interestingly, some indole derivatives were shown to inhibit lysozyme, a structural 17 values of compound 5b for E. coli PBP1b, S. aureus PBP2 and MtgA, T. maritima PBP1a and 3 E. hirae PBP2 were determined using the fluorescence assay and were in the micromolar 4 range (21-56 µM) (Table 1). 5 Furthermore, both 5 and 5b exhibit antibacterial activity against several important 6 human Gram-positive pathogens with MIC values as low as 4 µg/ml (Table 2). Interestingly 7 the compounds displayed MIC values of 4-8 µg/ml against methicillin-resistant S. aureus 8 strains (MRSA) which pose a major problem in hospitals. The two GT inhibitors were less 9 active against Gram-negative bacteria, as observed for other indole and tryptamine containing 10 compounds [38][39][40]. The MIC values of compounds 5 and 5b against E. coli 1411 were 64 11 and 16 µg/ml respectively. They decreased to the values observed in Gram-positive organisms 12 after polymyxin B nonapeptide (PMBN) treatment, or in an acrAB efflux mutant (Table 2). 13

These data suggest that uptake of the compounds into E. coli is affected both by efflux and 14 poor permeation through the outer membrane. 15

The effect of different concentrations of compound 5b on the growth and survival of S. 16 aureus was evaluated by measuring the number of living cells and turbidity over time. The 17 killing curves at 1x MIC (10 µM) and 2x MIC show that 5b causes an approximately 1 log 10 18 and 1.3 log 10 reduction of viability respectively after 90 min (Figure 2). At time zero (during 19 sample mixing) the cell count was slightly lower than the control. At 4x MIC, the killing 20 curve shows that 5b causes an immediate decrease in the bacterial count, more important than 21 that observed at 1x and 2x MIC (1x log 10 approximately), followed by a slower phase yielding 22 an approximately 5.4 log10 reduction of viability after 90 minutes (Figure 2) and no viable 23 cells were observed after 120 min (data not shown) showing that the compound is 24 bactericidal. Similarly, the cell turbidity (Absorbance at 600nm) at 4x MIC also decreased 25 A c c e p t e d M a n u s c r i p t 14 (36%) immediately after addition of 5b, then remained unchanged (data not shown). It seems 1 that a two-phase process is taking place upon addition of 5b to the cells: a rapid phase occurs 2 immediately after addition of the compound followed by a slower one. 3 4 3.3. Mode of action of compounds 5 and 5b 5

Compounds 5 and 5b induce cell wall biosensors in Bacillus subtilis. 6

Compounds 5 and 5b were predicted to specifically target peptidoglycan synthesis in bacterial 7 cells. In order to characterize their mode of action we have employed a Bacillus subtilis 8 reporter assay which differentiates inhibitors of cell wall, DNA, RNA, protein and fatty acid 9 biosynthesis by quantifying the upregulation of specific luciferase reporter gene constructs in 10 each pathway [24,25]. Using this system, only the cell wall biosensor expression was induced 11 in the presence of compounds 5 and 5b showing that they specifically inhibit cell wall 12 biosynthesis as initially hypothesised (Table 3). 13 Two possible mechanisms for inhibition of the GT reaction include direct binding to 14 the enzyme, or binding to the lipid II substrate. Experiments using fluorescence and 15 microcalorimetry were not conclusive on whether 5b binds a GT enzyme (data not shown). 16 Therefore, we checked whether 5 and 5b could interfere with the lipid II. First we used the 17 Bacillus subtilis reporter strain BFS2470 (β-galactosidase reporter system) sensing cell wall 18 antibiotics that interfere with the lipid II cycle [26,27]. Using a disk diffusion assay with 19 different antibiotics or compounds 5 or 5b on agar-plates containing X-Gal, the following 20 observations were made (Figure 3A): compounds 5, 5b and nisin (as a positive control) 21 produced a blue colour at the edge of the inhibition halo indicating an interaction of 22 compounds 5 and 5b with the lipid II. In contrast, moenomycin and ampicillin did not induce 23 β-galactosidase and the blue colour around the inhibition zone. These results indicate that 24 compounds 5 and 5b could interact with lipid II. 25

Compound 5b interacts with lipid II. 1

To test for a possible specific interaction between 5b and lipid II, a competition assay was set 2 up with the lantibiotic nisin, which uses lipid II for its pore-forming activity [14,41,42]. This 3 experiment was performed with compound 5b which is more active than 5 on GT activity. A 4 similar approach had previously been used with vancomycin in the study that determined the 5 inhibition of the lipid II-dependent pore-forming activity of nisin by vancomycin in intact 6 cells [43] and in a model membrane setup as is used here (data not shown) [44]. The addition 7 of nisin to carboxyfluorescein-loaded vesicles containing lipid II causes leakage of the 8 fluorescent dye from the vesicles, due to lipid II-dependent pore formation by nisin (CF 9 loaded vesicles without lipid II showed negligible leakage after adding nisin, Figure 4). When 10 50 nM nisin was added to 25 μM DOPC vesicles containing 0.1 mol% lipid II, a leakage of 11 ~75% was observed. In the presence of 2.5 μM 5b, CF leakage was significantly decreased by 12 ~36%. Moreover, increasing concentrations of 5b resulted in higher reductions and almost 13 complete inhibition of leakage at 15 μM (1.5x MIC) (Figure 3B). These results suggest that 14 5b binds to lipid II, thereby preventing an interaction of the latter with nisin. 15

To get insight into the binding site of 5b on lipid II, we first tested whether 5b could 16 also inhibit the lipid I-dependent pore-forming activity of nisin. Nisin does not discriminate 17 between lipid I (lacking the GlcNAc moiety) and lipid II. Leakage due to pore formation of 18 nisin with lipid I was diminished to an equal extent by the presence of 5b as with lipid II (data 19 not shown). This suggests that the GlcNAc moiety of lipid II is not an essential requirement 20 for the interaction. If 5b specifically interacts with lipid II, structurally related compounds 21 should compete for binding. In order to test possible interaction of 5b with the 22 (pyro)phosphate moiety of lipid II, the inhibition effect of 5b on nisin-lipid II pore formation 23 was analyzed in the presence of the peptidoglycan biosynthetic intermediates 24 undecaprenylphosphate (11-P) or undecaprenylpyrophosphate (11-PP), respectively (Figure 25 A c c e p t e d M a n u s c r i p t 16 3C). Nisin activity in the presence and absence of 5b was analyzed in CF-loaded DOPC 1 vesicles containing 0.1 mol% lipid II and 2 mol% of either 11-P, 11-PP or, as a control, the 2 anionic lipid DOPG. Strikingly, in the 11-PP containing vesicles, nisin induced CF-release 3 was not significantly reduced in the presence of 2.5 μM 5b. In comparison, 5b was still able to 4 cause significant inhibition of nisin activity in the presence of 11-P or DOPG, albeit 5 somewhat less profound than in their absence (Figure 3C). This indicates that the excess of 6 11-PP competes with lipid II for binding 5b and that the pyrophosphate moiety is important 7 for the interaction. To investigate this in more detail, NMR measurements were performed 8 using undecaprenylpyrophosphate in detergent micelles in the presence and absence of 5b. 31 P 9 NMR revealed a change in chemical shift (ΔδP=-0.4 ppm) for one of the phosphate groups in 10 the presence of 5b (Figure 3D). No significant shifts were observed when 5b was added to 11 undecaprenylphosphate-containing Triton-X100 micelles (data not shown). These results 12 confirm the importance of the pyrophosphate moiety for the interaction of 5b with lipid II. 13 14

Membrane disruptive effects of 5b 15

The vesicle binding assay using ultracentrifugation showed that 5b was present exclusively in 16 the pellet in both lipid II-containing and pure DOPC large unilamellar vesicles (LUVs) while 17 5b alone was present in the supernatant (data not shown). This shows that 5b has an affinity 18 for the lipid vesicles, regardless of whether lipid II is present or not. To gain insight into the 19 effect of 5b on membrane, leakage experiments were subsequently conducted with LUVs 20 containing anionic phospholipids in the absence of lipid II. Vesicles composed of only the 21 zwitterionic lipid DOPC displayed negligible leakage even in the presence of 40 µM 5b 22 (Figure 4A). Strikingly, in the presence of the negatively charged phospholipid DOPG 23 leakage was detected at 5 µM concentration and almost total CF-leakage was obtained at 20 24 µM of 5b (Figure 4A). This suggests that 5b has a general membrane effect independent of To investigate if 5b exerts similar effects on bacterial membranes that are known to 2 contain relatively large amounts of negatively charged lipids, the depolarization effect of 5b 3 on the plasma membrane in intact cells of Staphylococcus simulans and Micrococcus flavus 4 was determined using the membrane-potential sensitive fluorophore DiSC 2 (5). This dye 5 inserts into the cytoplasmic membrane in the presence of a membrane potential resulting in 6 self-quenching of fluorescence. The addition of 5b to a final concentration of 1 μM caused a 7 major release of the dye from the membrane in M. flavus (Figure 4b) showing that the 8 membrane potential was dissipated upon binding of 5b. S. simulans also showed significant 9 membrane depolarization at 5µM concentration of 5b. Nisin was more effective in dissipating 10 the membrane potential in M. flavus (Figure 4b), which is in relative agreement with the 11 difference in MIC values for both compounds (~5 nM nisin for M. flavus versus ~10 μM 5b 12 for most Gram-positive strains tested). 13

These results are supported by experiments performed with the LIVE/DEAD 14 BacLight™ bacterial viability assay [45]. The membrane damage inflicted by a 10 minute 15 exposure to compound 5b on S. aureus was analysed. Cells treated with the test agent 5b at 4x 16 MIC only maintained 33% membrane integrity (67% damage), which does suggest that the 17 membranes of S. aureus cells were permeabilized by the addition of 5b. 18 GTs in vitro and having good antibacterial activity in the micromolar range. These 2 compounds specifically induce peptidoglycan pathway stress sensors in Bacillus, confirming 3 that bacterial cell wall synthesis is the target. 4

Moreover, we have found that compound 5b binds to the lipid II precursor. This is 5 reminiscent of several lipid II-targeting antibiotics such as vancomycin and related (second 6 generation) glycopeptides [46], and the lantibiotics mersacidin and nisin [11]. 7

Undecaprenylpyrophosphate competes with lipid II for interaction with 5b. As 8 undecaprenylphosphate and DOPG did not significantly compete with lipid II for 5b, the 9 pyrophosphate moiety appears to be essential for the interaction with 5b. Nisin also binds the 10 pyrophosphate of lipid II, and this could explain the effective inhibition of nisin-lipid II pore 11 assembly by 5b. The data obtained from NMR analysis of undecaprenylpyrophosphate in the 12 presence and absence of 5b further strengthens the hypothesis that the pyrophosphate moiety 13 is critical for complex formation of 5b with lipid II. The amine residue of 5b has 14 independently been shown to be essential for its glycosyltransferase inhibiting activity, 15 suggesting its crucial involvement in the interaction with the pyrophosphate group of lipid II. 16

The bacitracin antibiotic has also been shown to inhibit PG synthesis by sequestration of 17 undecaprenyl pyrophosphate and was known to specifically bind to the pyrophosphate moiety 18 of the lipid carrier [47]. 19 Compound 5b at low concentration caused membrane depolarization in M. flavus and 20 S. simulans as did nisin. Several lines of evidence also indicate that compound 5b affects in 21 vitro membrane integrity in a negatively charged lipid dependent way in the absence of lipid 22 II. These results suggest that membrane disruption could contribute to its antibacterial 23 activity. Indeed, the effect of 5b on S. aureus viability shows that a two phase process 24 contributes to its bactericidal activity. The first and immediate effect could be the result of 25 A c c e p t e d M a n u s c r i p t 19 membrane integrity disruption. The second and slower phase (4x MIC) is more likely the 1 consequence of cell wall synthesis inhibition as was previously observed for the kinetics of 2 inhibition of S. aureus by moenomycin [48]. Therefore, it appears from our data that the 3 identified compounds have two modes of action, namely, inhibition of cell wall synthesis and 4 membrane depolarization/disruption. This mode of action resembles that of nisin which binds 5 and sequesters lipid II preventing its use by the peptidoglycan polymerases and inducing 6 pores formation in the bacterial membrane and membrane depolarization. The properties 7 described for 5b could also be shared by compound 5 due to close similarity between the two 8 compounds. 9

Bacterial resistance to antibiotics requires a continuing effort to find new antibacterial 10 compounds able to cope with emerging resistant strains. The glycosyltransferase step in 11 peptidoglycan biosynthesis is a validated target for new antibacterial development. The small 12 antibacterial molecules described here are much simpler agents compared to known natural 13 products targeting lipid II. Considering the importance of lipid II and the pyrophosphate 14 motif for the whole enzymatic machinery of the bacterial cell wall synthesis pathway, this 15 simple mode of binding to lipid II may limit the occurrence of resistance to such promising 16 compounds. Finally, this work provides the basis for the development of more active and 17 specific analogues of compounds 5/5b to target the essential GT step in cell wall synthesis 18 using small molecule inhibitors.

  dioleoyl-sn-glycero-3-phosphocholine (DOPC), 1,2-dioleoyl-sn-glycero-3-[phospho-rac-4(1-glycerol)] (DOPG), 1,2-dioleoyl-sn-glycero-3-[phospho-rac-(3-lysyl(1-glycerol))] (lysyl-5 DOPG) and 1,2-dioleoyl-3-trimethylammonium-propane (DOTAP) were purchased from 6

  25

  cell wall caused by compounds interfering with lipid II cycle was used according 1 to published methods [26, 27]. The B. subtilis strain BFS2470 carries a β-galactosidase 2

  25

  A c c e p t e d M a n u s c r i p t 11 2.11. Membrane depolarization assay 1

  A c c e p t e d M a n u s c r i p t 12 testing. The number of compounds tested was further reduced to 21 as the remaining 1 compounds were insoluble. Soluble compounds were then tested in vitro at 0.2 and 1 mM 2 final concentrations for their ability to inhibit E. coli PBP1b GT activity in a 96-well 3 microtiter plate assay with fluorescent lipid II. Compounds which inhibited PBP1b activity or 4 interfered with the fluorescence assay were additionally tested in a radioactivity-based assay. 5

  homologue of GT [33]. The docking model of indole 3-carbinol into the active site of 18 lysozyme [34] shows some similarities with the predicted binding of compounds 5/5b to the 19 GT. Compounds 5/5b at 1mM concentrations do not inhibit lysozyme and have no effect on 20 the penicillin-binding activity of the second domain of E. coli PBP1b or the Actinomadura 21 R39 DD-peptidase (data not shown). This indicates that compounds 5 and 5b do not exhibit a 22 promiscuous behaviour. Two other assays based on the use of a detergent [35, 36] and 23 aggregation of the green fluorescent protein (GFP) [37] also support this observation (data not 24 shown). 25 A c c e p t e d M a n u s c r i p t 13 3.2. Evaluation of compounds 5 and 5b for GTs inhibition and antibacterial activity. 1 Both 5 and 5b were able to inhibit the GT reaction catalyzed by the five GTs tested. The IC 50 2

A

  c c e p t e d M a n u s c r i p t lipid II, and this disruption of membrane integrity is anionic lipid-dependent. 1

  be efficiently disrupted either through the inhibition of 22 the glycosyltransferase or the transpeptidase enzymatic activities or by sequestration of the 23 lipid II, thus, preventing its use as a substrate. 24In our effort to identify small molecules inhibitors of the GT reaction, we have 25 A c c e p t e d M a n u s c r i p t 18 discovered two structurally related indole derivative compounds (5 and 5b) inhibiting several 1

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Figure legends 1 2 Figure 1 .Figure 2 .Figure 3 .

 2123 Figure legends

A 8 Figure 4 .

 84 Figure 4. Membrane disruption effect of compound 5b. 9

  Figure1Click here to download high resolution image

Table 1 .

 1 IC 50 values of compound 5b for various GTs.The IC50 of compound 5 for E. coli PBP1b was 59 ± 3.3 µM. For comparison, the IC 50 value of moenomycin for S. aureus PBP2 was 1.3 ± 0.3 µM.

	Table1
	IC 50 (µM)
		Compound 5b
	E. coli PBP1b	29 ± 12
	S. aureus PBP2	20.7 ± 1
	T. maritima PBP1a	56 ± 7
	E. hirae PBP2	33 ± 6
	S. aureus MtgA	27 ± 9
	1

Table 2 .

 2 Susceptibility of a panel of bacteria to compounds 5, 5b and moenomycin.

	MIC (µg/ml)
	5	5b	moenomycin

Med Chem 51(23): 7442-8, 2008.

C. Competition of undecaprenylpyrophosphate (11-PP), undecaprenylphosphate (11-P) and

Acknowledgments 21

This work was supported by the European Commission within the "EUR-INTAFAR"

We thank Jennifer Griffiths and Marie Schloesser for technical assistance. 4 5