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Aug 23, 2023 187 likes •811 views

rsss rs t s ts P t

slide-1
SLIDE 1

❚r❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ✇✐t❤ s✐❣♥✲❝❤❛♥❣✐♥❣ ❝♦❡✣❝✐❡♥ts

P■❈❖❋ ✷✵✶✷

❆✳✲❙✳ ❇♦♥♥❡t✲❇❡♥ ❉❤✐❛†✱ ▲✳ ❈❤❡s♥❡❧†✱ P✳ ❈✐❛r❧❡t†✱ ❍✳ ❍❛❞❞❛r‡

†P❖❡♠s t❡❛♠✱ ❊♥st❛✱ P❛r✐s✱ ❋r❛♥❝❡ ‡❉❡❋■ t❡❛♠✱ ➱❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡

➱❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡✱ ❆♣r✐❧ ✷✕✹✱ ✷✵✶✷

✶ ✴ ✶✺

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SLIDE 2

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ ✌ ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ❞✐✈ ✐♥ D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

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SLIDE 3

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ ✌ ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ❞✐✈ ✐♥ ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

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SLIDE 4

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ ❞✐✈ (A∇u) + k2nu = ✐♥ D ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

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SLIDE 5

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❞✐✈ (A∇u) + k2nu = ✐♥ D ∆w + k2w = ✐♥ D ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

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SLIDE 6

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❞✐✈ (A∇u) + k2nu = ✐♥ D ∆w + k2w = ✐♥ D u − w = ♦♥ ∂D ν · A∇u − ν · ∇w = ♦♥ ∂D. ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

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SLIDE 7

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❋✐♥❞ (u, w) ∈ H1(D) × H1(D) s✉❝❤ t❤❛t✿ ❞✐✈ (A∇u) + k2nu = ✐♥ D ∆w + k2w = ✐♥ D u − w = ♦♥ ∂D ν · A∇u − ν · ∇w = ♦♥ ∂D. ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

slide-8
SLIDE 8

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❋✐♥❞ (u, w) ∈ H1(D) × H1(D) s✉❝❤ t❤❛t✿ ❞✐✈ (A∇u) + k2nu = ✐♥ D ∆w + k2w = ✐♥ D u − w = ♦♥ ∂D ν · A∇u − ν · ∇w = ♦♥ ∂D. ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ k ∈ C ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ (u, w) ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

slide-9
SLIDE 9

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ■❚❊P

◮ ❙❝❛tt❡r✐♥❣ ✐♥ t✐♠❡✲❤❛r♠♦♥✐❝ r❡❣✐♠❡ ❜② ❛♥ ✐♥❝❧✉s✐♦♥ D ✭❝♦❡✣❝✐❡♥ts A ❛♥❞ n✮ ✐♥ R2✿ ✇❡ ❧♦♦❦ ❢♦r ❛♥ ✐♥❝✐❞❡♥t ✇❛✈❡ t❤❛t ❞♦❡s ♥♦t s❝❛tt❡r✳ ◮ ❚❤✐s ❧❡❛❞s t♦ st✉❞② t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❝❢✳ ❋✳ ❈❛❦♦♥✐✬s t❛❧❦✮✿ ✌ u ✐s t❤❡ t♦t❛❧ ✜❡❧❞ ✐♥ D ✌ w ✐s t❤❡ ✐♥❝✐❞❡♥t ✜❡❧❞ ✐♥ D ❋✐♥❞ (u, w) ∈ H1(D) × H1(D) s✉❝❤ t❤❛t✿ ❞✐✈ (A∇u) + k2nu = ✐♥ D ∆w + k2w = ✐♥ D u − w = ♦♥ ∂D ν · A∇u − ν · ∇w = ♦♥ ∂D. ν ν D A = Id✱ n = 1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ ∂D

❉❡❢✐♥✐t✐♦♥✳ ❱❛❧✉❡s ♦❢ k ∈ C ❢♦r ✇❤✐❝❤ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❛ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ (u, w) ❛r❡ ❝❛❧❧❡❞ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ◮ ❚❤❡ ❣♦❛❧ ✐♥ t❤✐s t❛❧❦ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ s❡t ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❛t ♠♦st ❞✐s❝r❡t❡✳

✷ ✴ ✶✺

slide-10
SLIDE 10

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ✇✐t❤ ✳ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-11
SLIDE 11

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-12
SLIDE 12

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-13
SLIDE 13

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ X ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-14
SLIDE 14

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ X ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ X ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-15
SLIDE 15

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ X ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ X ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ◮ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-16
SLIDE 16

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ X ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ X ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ◮ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

◮ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-17
SLIDE 17

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ■❚❊P

◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts (u, w) ∈ X\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ (u′, w′) ∈ X✱

  • D

A∇u · ∇u′ ✲ ∇w · ∇w′ = k2

  • D

(nuu′ ✲ ww′), ♥♦t ❝♦❡r❝✐✈❡ ♦♥ X ♥♦t ❛♥ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ X ✇✐t❤ X = {(u, w) ∈ H1(D) × H1(D) | u − w ∈ H1

0(D)}✳

◮ ❚❤✐s ✐s ❛ ♥♦♥ st❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✳ ◮ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ s✐♥❝❡ ✶✾✽✻✲✶✾✽✽ ✭❇❡❧❧✐s✱ ❈❛❦♦♥✐✱

❈♦❧t♦♥✱ ●✐♥t✐❞❡s✱ ●✉③✐♥❛✱ ❍❛❞❞❛r✱ ❑✐rs❝❤✱ ❑r❡ss✱ ▼♦♥❦✱ ❙②❧✈❡st❡r✱ P❛ï✈är✐♥t❛✱ ❘②♥♥❡✱ ❙❧❡❡♠❛♥✳✳✳✮

◮ ■♥ t❤✐s t❛❧❦✱ ✇❡ ✇❛♥t t♦ ❤✐❣❤❧✐❣❤t ❛♥ ■❞❡❛ ✶✿ ❆♥❛❧♦❣② ✇✐t❤ ❛♥♦t❤❡r ♥♦♥ st❛♥❞❛r❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✳✳✳

✸ ✴ ✶✺

slide-18
SLIDE 18

❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧ ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❉▼❚❊P✮

◮ ❚✐♠❡✲❤❛r♠♦♥✐❝ ♣r♦❜❧❡♠ ✐♥ ❡❧❡❝tr♦♠❛❣♥❡t✐s♠ ✭❛t ❛ ❣✐✈❡♥ ❢r❡q✉❡♥❝②✮ s❡t ✐♥ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ♦❢ R2✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ❊✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r ✐♥ ✷❉✿ ❋✐♥❞ s✉❝❤ t❤❛t✿ ❞✐✈ ✐♥ ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ ✱ ✲ ✲

✹ ✴ ✶✺

slide-19
SLIDE 19

❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧ ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❉▼❚❊P✮

◮ ❚✐♠❡✲❤❛r♠♦♥✐❝ ♣r♦❜❧❡♠ ✐♥ ❡❧❡❝tr♦♠❛❣♥❡t✐s♠ ✭❛t ❛ ❣✐✈❡♥ ❢r❡q✉❡♥❝②✮ s❡t ✐♥ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ♦❢ R2✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 := ε|Ω1 > 0 µ1 := µ|Ω1 > 0 ε2 := ε|Ω2 < 0 µ2 := µ|Ω2 < 0 ❊✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r ✐♥ ✷❉✿ ❋✐♥❞ s✉❝❤ t❤❛t✿ ❞✐✈ ✐♥ ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ ✱ ✲ ✲

✹ ✴ ✶✺

slide-20
SLIDE 20

❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧ ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❉▼❚❊P✮

◮ ❚✐♠❡✲❤❛r♠♦♥✐❝ ♣r♦❜❧❡♠ ✐♥ ❡❧❡❝tr♦♠❛❣♥❡t✐s♠ ✭❛t ❛ ❣✐✈❡♥ ❢r❡q✉❡♥❝②✮ s❡t ✐♥ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ♦❢ R2✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 := ε|Ω1 > 0 µ1 := µ|Ω1 > 0 ε2 := ε|Ω2 < 0 µ2 := µ|Ω2 < 0 ◮ ❊✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r Ez ✐♥ ✷❉✿ ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

❞✐✈(µ−1 ∇v) + k2εv = 0 ✐♥ Ω. ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ ✱ ✲ ✲

✹ ✴ ✶✺

slide-21
SLIDE 21

❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧ ❚r❛♥s♠✐ss✐♦♥ ❊✐❣❡♥✈❛❧✉❡ Pr♦❜❧❡♠ ✭❉▼❚❊P✮

◮ ❚✐♠❡✲❤❛r♠♦♥✐❝ ♣r♦❜❧❡♠ ✐♥ ❡❧❡❝tr♦♠❛❣♥❡t✐s♠ ✭❛t ❛ ❣✐✈❡♥ ❢r❡q✉❡♥❝②✮ s❡t ✐♥ ❛ ❤❡t❡r♦❣❡♥❡♦✉s ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ Ω ♦❢ R2✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 := ε|Ω1 > 0 µ1 := µ|Ω1 > 0 ε2 := ε|Ω2 < 0 µ2 := µ|Ω2 < 0 ◮ ❊✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r Ez ✐♥ ✷❉✿ ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

❞✐✈(µ−1 ∇v) + k2εv = 0 ✐♥ Ω. ◮ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts v ∈ H1

0(Ω)\{0}

s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v′ ∈ H1

0(Ω)✱

  • Ω1

µ−1

1 ∇v · ∇v′ ✲

  • Ω2

|µ2|−1∇v · ∇v′ = k2

  • Ω1

ε1vv′ ✲

  • Ω2

|ε2|vv′

  • .

✹ ✴ ✶✺

slide-22
SLIDE 22

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ ✿

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❚❤❡ ✐♥t❡r❢❛❝❡ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-23
SLIDE 23

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ ✿

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❚❤❡ ✐♥t❡r❢❛❝❡ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-24
SLIDE 24

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ Σ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ ✿

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❚❤❡ ✐♥t❡r❢❛❝❡ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-25
SLIDE 25

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ Σ ◮ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ Ω1✿ Ω1 Σ ν

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥

❚❤❡ ✐♥t❡r❢❛❝❡ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-26
SLIDE 26

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ Σ ◮ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ Ω1✿ Ω1 Σ ν

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❚❤❡ ✐♥t❡r❢❛❝❡ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-27
SLIDE 27

❊q✉✐✈❛❧❡♥❝❡ ❉▼❚❊P✴■❚❊P

◮ ❉▼❚❊P ✐♥ t❤❡ ❞♦♠❛✐♥ Ω✿ Ω1 ❉✐❡❧❡❝tr✐❝ Ω2 ▼❡t❛♠❛t❡r✐❛❧ Σ ν ε1 = n µ1 = A ε2 = −1 µ2 = −1

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

❙②♠♠❡tr② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✐♥t❡r❢❛❝❡ Σ ◮ ❲❡ ♦❜t❛✐♥ ❛ ♣r♦❜❧❡♠ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ■❚❊P ✐♥ Ω1✿ Ω1 Σ ν

❚r❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ♦♥ Σ

◮ ❚❤❡ ✐♥t❡r❢❛❝❡ Σ ✐♥ t❤❡ ❉▼❚❊P ♣❧❛②s t❤❡ r♦❧❡ ♦❢ t❤❡ ❜♦✉♥❞❛r② ∂D ✐♥ t❤❡ ■❚❊P✳

✺ ✴ ✶✺

slide-28
SLIDE 28

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦✿ t❤r❡❡ st❡♣s

✶ ❆♥ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ t✇♦ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠s ✷ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧

❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✸ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✻ ✴ ✶✺

slide-29
SLIDE 29

✶ ❆♥ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ t✇♦ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠s ✷ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧

❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✸ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✼ ✴ ✶✺

slide-30
SLIDE 30

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ ❋✐♥❞ s✉❝❤ t❤❛t✿ ❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❚❤❡ ❢♦r♠ ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r ✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ ✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-31
SLIDE 31

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ ◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❚❤❡ ❢♦r♠ ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r ✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ ✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-32
SLIDE 32

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ µ1 > 0 µ2 < 0

✭µ1 ❛♥❞ µ2 ❝♦♥st❛♥t t♦ s✐♠♣❧✐❢②✮

◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✳ ❚❤❡ ❢♦r♠ ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r ✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ ✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-33
SLIDE 33

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ µ1 > 0 µ2 < 0

✭µ1 ❛♥❞ µ2 ❝♦♥st❛♥t t♦ s✐♠♣❧✐❢②✮

◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ (PV ) ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦ H−1(Ω)✳

❚❤❡ ❢♦r♠ ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r ✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ ✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-34
SLIDE 34

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ µ1 > 0 µ2 < 0

✭µ1 ❛♥❞ µ2 ❝♦♥st❛♥t t♦ s✐♠♣❧✐❢②✮

◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ (PV ) ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦ H−1(Ω)✳

❚❤❡ ❢♦r♠ a ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r ✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ ✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-35
SLIDE 35

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ µ1 > 0 µ2 < 0

✭µ1 ❛♥❞ µ2 ❝♦♥st❛♥t t♦ s✐♠♣❧✐❢②✮

◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ (PV ) ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦ H−1(Ω)✳

❚❤❡ ❢♦r♠ a ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r µ2 = −µ1✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ (PV )✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ ✳

✽ ✴ ✶✺

slide-36
SLIDE 36

❙t✉❞② ♦❢ t❤❡ ❉▼❚P

◮ Pr♦❜❧❡♠ ❢♦r Ez ✐♥ ❛ s②♠♠❡tr✐❝ ✷❉ ❞♦♠❛✐♥✿ Ω2 Ω1 Σ µ1 > 0 µ2 < 0

✭µ1 ❛♥❞ µ2 ❝♦♥st❛♥t t♦ s✐♠♣❧✐❢②✮

◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

  • Ω

µ−1 ∇v · ∇v′

  • a(v,v′)

= f, v′Ω

l(v′)

, ∀v′ ∈ H1

0(Ω).

❉❡❢✐♥✐t✐♦♥✳ ❲❡ ✇✐❧❧ s❛② t❤❛t t❤❡ ♣r♦❜❧❡♠ (PV ) ✐s ✇❡❧❧✲♣♦s❡❞ ✐❢ t❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦ H−1(Ω)✳

❚❤❡ ❢♦r♠ a ✐s ♥♦t ❝♦❡r❝✐✈❡✳ ❋♦r µ2 = −µ1✱ ✇❡ ❝❛♥ ❜✉✐❧❞ ❛ ❦❡r♥❡❧ ♦❢ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ t♦ (PV )✳ ■❞❡❛ ✷✿ ❯s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠ (PV )✳

✽ ✴ ✶✺

slide-37
SLIDE 37

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, v′) = l(v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t ✐s ❚✲❝♦❡r❝✐✈❡✿ ❚ ✳ ■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠

❚

✭❛♥❞ s♦ ✮ ✐s ✇❡❧❧✲♣♦s❡❞✳ ✶ ❉❡✜♥❡ ✷ ❚ ❚ s♦ ❚ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢

✾ ✴ ✶✺

slide-38
SLIDE 38

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ⇔ (P❚

V )

❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, ❚v′) = l(❚v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t ✐s ❚✲❝♦❡r❝✐✈❡✿ ❚ ✳ ■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠

❚

✭❛♥❞ s♦ ✮ ✐s ✇❡❧❧✲♣♦s❡❞✳ ✶ ❉❡✜♥❡ ✷ ❚ ❚ s♦ ❚ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢

✾ ✴ ✶✺

slide-39
SLIDE 39

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ⇔ (P❚

V )

❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, ❚v′) = l(❚v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t a ✐s ❚✲❝♦❡r❝✐✈❡✿

  • Ω

µ−1 ∇v · ∇(❚v) ≥ C v2

H1

0(Ω)✳

■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠ ⇒ (P❚

V ) ✭❛♥❞ s♦ (PV )✮ ✐s ✇❡❧❧✲♣♦s❡❞✳

✶ ❉❡✜♥❡ ✷ ❚ ❚ s♦ ❚ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢

✾ ✴ ✶✺

slide-40
SLIDE 40

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ⇔ (P❚

V )

❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, ❚v′) = l(❚v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t a ✐s ❚✲❝♦❡r❝✐✈❡✿

  • Ω

µ−1 ∇v · ∇(❚v) ≥ C v2

H1

0(Ω)✳

■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠ ⇒ (P❚

V ) ✭❛♥❞ s♦ (PV )✮ ✐s ✇❡❧❧✲♣♦s❡❞✳

✶ ❉❡✜♥❡ ❚1 v = v1 ✐♥ Ω1 −v2 + ... ✐♥ Ω2 ✷ ❚ ❚ s♦ ❚ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢

✾ ✴ ✶✺

slide-41
SLIDE 41

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ⇔ (P❚

V )

❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, ❚v′) = l(❚v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t a ✐s ❚✲❝♦❡r❝✐✈❡✿

  • Ω

µ−1 ∇v · ∇(❚v) ≥ C v2

H1

0(Ω)✳

■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠ ⇒ (P❚

V ) ✭❛♥❞ s♦ (PV )✮ ✐s ✇❡❧❧✲♣♦s❡❞✳

✶ ❉❡✜♥❡ ❚1 v = v1 ✐♥ Ω1 −v2 + 2SΣv1 ✐♥ Ω2 , ✇❤❡r❡ SΣ ✐s t❤❡ s②♠♠❡tr②✳ Ω1 Ω2 Σ SΣ ✷ ❚ ❚ s♦ ❚ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢

✾ ✴ ✶✺

slide-42
SLIDE 42

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✶✴✷

▲❡t ❚ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)✳

(PV ) ⇔ (P❚

V )

❋✐♥❞ v ∈ H1

0(Ω) s✉❝❤ t❤❛t✿

a(v, ❚v′) = l(❚v′), ∀v′ ∈ H1

0(Ω).

  • ♦❛❧✿

❋✐♥❞ ❚ s✉❝❤ t❤❛t a ✐s ❚✲❝♦❡r❝✐✈❡✿

  • Ω

µ−1 ∇v · ∇(❚v) ≥ C v2

H1

0(Ω)✳

■♥ t❤✐s ❝❛s❡✱ ▲❛①✲▼✐❧❣r❛♠ ⇒ (P❚

V ) ✭❛♥❞ s♦ (PV )✮ ✐s ✇❡❧❧✲♣♦s❡❞✳

✶ ❉❡✜♥❡ ❚1 v = v1 ✐♥ Ω1 −v2 + 2SΣv1 ✐♥ Ω2 , ✇❤❡r❡ SΣ ✐s t❤❡ s②♠♠❡tr②✳ Ω1 Ω2 Σ SΣ ✷ ❚1 ◦ ❚1 = Id s♦ ❚1 ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ H1

0(Ω)

✾ ✴ ✶✺

slide-43
SLIDE 43

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ ✳ ✹ ❲✐t❤ ❚ ✐♥ ✐♥ ✱ ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st s❛t✐s✜❡s ✳ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ ❛♥❞ ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢ ✐♥❢ ♦r s✉♣ s✉♣ ✇❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-44
SLIDE 44

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ SΣ = 1 ⇒ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ |µ2| > µ1 ✳ ✹ ❲✐t❤ ❚ ✐♥ ✐♥ ✱ ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st s❛t✐s✜❡s ✳ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ ❛♥❞ ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢ ✐♥❢ ♦r s✉♣ s✉♣ ✇❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-45
SLIDE 45

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ SΣ = 1 ⇒ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ |µ2| > µ1 ✳ ✹ ❲✐t❤ ❚2v = v1 − 2SΣv2 ✐♥ Ω1 −v2 ✐♥ Ω2 ✱ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ µ1 > |µ2| ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ t♦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st s❛t✐s✜❡s ✳ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ ❛♥❞ ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢ ✐♥❢ ♦r s✉♣ s✉♣ ✇❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-46
SLIDE 46

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ SΣ = 1 ⇒ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ |µ2| > µ1 ✳ ✹ ❲✐t❤ ❚2v = v1 − 2SΣv2 ✐♥ Ω1 −v2 ✐♥ Ω2 ✱ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ µ1 > |µ2| ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1 ∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦

H−1(Ω) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st κµ = µ1/µ2 s❛t✐s✜❡s κµ = −1✳ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ ❛♥❞ ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢ ✐♥❢ ♦r s✉♣ s✉♣ ✇❤❡r❡ ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-47
SLIDE 47

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ SΣ = 1 ⇒ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ |µ2| > µ1 ✳ ✹ ❲✐t❤ ❚2v = v1 − 2SΣv2 ✐♥ Ω1 −v2 ✐♥ Ω2 ✱ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ µ1 > |µ2| ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1 ∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦

H−1(Ω) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st κµ = µ1/µ2 s❛t✐s✜❡s κµ = −1✳ ◮ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ µ1 ❛♥❞ µ2 ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ (µ−1 ∇·) ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢

Ω1∩Vµ1/ ✐♥❢ Ω2∩Vµ2 < −1

♦r s✉♣

Ω1∩V

µ1/ s✉♣

Ω2∩V

µ2 > −1 ✇❤❡r❡ V ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ Σ✳ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-48
SLIDE 48

■❞❡❛ ♦❢ t❤❡ ❚✲❝♦❡r❝✐✈✐t② ✷✴✷

✸ ❖♥❡ ❤❛s a(v, ❚1v) =

  • Ω

|µ|−1|∇v|2 − 2

  • Ω2

µ−1

2

∇v · ∇(SΣ v1) ❨♦✉♥❣✬s ✐♥❡q✉❛❧✐t② ✰ SΣ = 1 ⇒ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ |µ2| > µ1 ✳ ✹ ❲✐t❤ ❚2v = v1 − 2SΣv2 ✐♥ Ω1 −v2 ✐♥ Ω2 ✱ a ✐s ❚✲❝♦❡r❝✐✈❡ ✇❤❡♥ µ1 > |µ2| ✳ ✺ ❈♦♥❝❧✉s✐♦♥✿ ❚❤❡♦r❡♠✳ ❚❤❡ ♦♣❡r❛t♦r ❞✐✈ (µ−1 ∇·) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❢r♦♠ H1

0(Ω) t♦

H−1(Ω) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦♥tr❛st κµ = µ1/µ2 s❛t✐s✜❡s κµ = −1✳ ◮ ❇② ❛ ❧♦❝❛❧✐③❛t✐♦♥ ♣r♦❝❡ss✱ ✇❤❡♥ µ1 ❛♥❞ µ2 ❛r❡ ♥♦t ❝♦♥st❛♥t✱ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❞✐✈ (µ−1 ∇·) ✐s ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✇❤❡♥ ✐♥❢

Ω1∩Vµ1/ ✐♥❢ Ω2∩Vµ2 < −1

♦r s✉♣

Ω1∩V

µ1/ s✉♣

Ω2∩V

µ2 > −1 ✇❤❡r❡ V ✐s ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ Σ✳ ◮ ❚❤✐s t❡❝❤♥✐q✉❡ ❛❧s♦ ❛❧❧♦✇s t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥ s②♠♠❡tr✐❝ ❝♦♥✜❣✉r❛t✐♦♥s✳

✶✵ ✴ ✶✺

slide-49
SLIDE 49

✶ ❆♥ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ t✇♦ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠s ✷ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ❉✐❡❧❡❝tr✐❝✴▼❡t❛♠❛t❡r✐❛❧

❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✸ ❚❤❡ ❚✲❝♦❡r❝✐✈✐t② ♠❡t❤♦❞ ❢♦r t❤❡ ■♥t❡r✐♦r ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠

✶✶ ✴ ✶✺

slide-50
SLIDE 50

❙t✉❞② ♦❢ t❤❡ ■❚❊P

◮ ❉❡✜♥❡ ♦♥ X × X t❤❡ s❡sq✉✐❧✐♥❡❛r ❢♦r♠ a((u, w), (u′, w′)) =

  • Ω

A∇u · ∇u′ ✲ ∇w · ∇w′ − k2(nuu′ ✲ ww′), ✇✐t❤ X = {(u, w) ∈ H1(Ω) × H1(Ω) | u − w ∈ H1

0(Ω)}✳

■♥tr♦❞✉❝❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❚ ✳ ❋♦r ✱ ❛♥❞ ✱ ♦♥❡ ✜♥❞s ❚ ❯s✐♥❣ t❤❡ ❛♥❛❧②t✐❝ ❋r❡❞❤♦❧♠ t❤❡♦r❡♠✱ ♦♥❡ ❞❡❞✉❝❡s t❤❡ Pr♦♣♦s✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ❛♥❞ ✳ ❚❤❡♥ t❤❡ s❡t ♦❢ tr❛♥s♠✐s✲ s✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❞✐s❝r❡t❡ ❛♥❞ ❝♦✉♥t❛❜❧❡✳ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✐t✉❛t✐♦♥s ✇❤❡r❡ ❛♥❞ ❝❤❛♥❣❡ s✐❣♥ ✐♥ ✇♦r❦✐♥❣ ✇✐t❤ ❚ ✮✳

✶✷ ✴ ✶✺

slide-51
SLIDE 51

❙t✉❞② ♦❢ t❤❡ ■❚❊P

◮ ❉❡✜♥❡ ♦♥ X × X t❤❡ s❡sq✉✐❧✐♥❡❛r ❢♦r♠ a((u, w), (u′, w′)) =

  • Ω

A∇u · ∇u′ ✲ ∇w · ∇w′ − k2(nuu′ ✲ ww′), ✇✐t❤ X = {(u, w) ∈ H1(Ω) × H1(Ω) | u − w ∈ H1

0(Ω)}✳

◮ ■♥tr♦❞✉❝❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❚(u, w) = (u − 2w, −w)✳ ❋♦r ✱ ❛♥❞ ✱ ♦♥❡ ✜♥❞s ❚ ❯s✐♥❣ t❤❡ ❛♥❛❧②t✐❝ ❋r❡❞❤♦❧♠ t❤❡♦r❡♠✱ ♦♥❡ ❞❡❞✉❝❡s t❤❡ Pr♦♣♦s✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ❛♥❞ ✳ ❚❤❡♥ t❤❡ s❡t ♦❢ tr❛♥s♠✐s✲ s✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❞✐s❝r❡t❡ ❛♥❞ ❝♦✉♥t❛❜❧❡✳ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✐t✉❛t✐♦♥s ✇❤❡r❡ ❛♥❞ ❝❤❛♥❣❡ s✐❣♥ ✐♥ ✇♦r❦✐♥❣ ✇✐t❤ ❚ ✮✳

✶✷ ✴ ✶✺

slide-52
SLIDE 52

❙t✉❞② ♦❢ t❤❡ ■❚❊P

◮ ❉❡✜♥❡ ♦♥ X × X t❤❡ s❡sq✉✐❧✐♥❡❛r ❢♦r♠ a((u, w), (u′, w′)) =

  • Ω

A∇u · ∇u′ ✲ ∇w · ∇w′ − k2(nuu′ ✲ ww′), ✇✐t❤ X = {(u, w) ∈ H1(Ω) × H1(Ω) | u − w ∈ H1

0(Ω)}✳

◮ ■♥tr♦❞✉❝❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❚(u, w) = (u − 2w, −w)✳ ◮ ❋♦r k ∈ Ri\{0}✱ A > Id ❛♥❞ n > 1✱ ♦♥❡ ✜♥❞s ℜe a((u, w), ❚(u, w)) ≥ C (u2

H1(Ω) + w2 H1(Ω)),

∀(u, w) ∈ X. ❯s✐♥❣ t❤❡ ❛♥❛❧②t✐❝ ❋r❡❞❤♦❧♠ t❤❡♦r❡♠✱ ♦♥❡ ❞❡❞✉❝❡s t❤❡ Pr♦♣♦s✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ❛♥❞ ✳ ❚❤❡♥ t❤❡ s❡t ♦❢ tr❛♥s♠✐s✲ s✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❞✐s❝r❡t❡ ❛♥❞ ❝♦✉♥t❛❜❧❡✳ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✐t✉❛t✐♦♥s ✇❤❡r❡ ❛♥❞ ❝❤❛♥❣❡ s✐❣♥ ✐♥ ✇♦r❦✐♥❣ ✇✐t❤ ❚ ✮✳

✶✷ ✴ ✶✺

slide-53
SLIDE 53

❙t✉❞② ♦❢ t❤❡ ■❚❊P

◮ ❉❡✜♥❡ ♦♥ X × X t❤❡ s❡sq✉✐❧✐♥❡❛r ❢♦r♠ a((u, w), (u′, w′)) =

  • Ω

A∇u · ∇u′ ✲ ∇w · ∇w′ − k2(nuu′ ✲ ww′), ✇✐t❤ X = {(u, w) ∈ H1(Ω) × H1(Ω) | u − w ∈ H1

0(Ω)}✳

◮ ■♥tr♦❞✉❝❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❚(u, w) = (u − 2w, −w)✳ ◮ ❋♦r k ∈ Ri\{0}✱ A > Id ❛♥❞ n > 1✱ ♦♥❡ ✜♥❞s ℜe a((u, w), ❚(u, w)) ≥ C (u2

H1(Ω) + w2 H1(Ω)),

∀(u, w) ∈ X. ◮ ❯s✐♥❣ t❤❡ ❛♥❛❧②t✐❝ ❋r❡❞❤♦❧♠ t❤❡♦r❡♠✱ ♦♥❡ ❞❡❞✉❝❡s t❤❡ Pr♦♣♦s✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t A > Id ❛♥❞ n > 1✳ ❚❤❡♥ t❤❡ s❡t ♦❢ tr❛♥s♠✐s✲ s✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❞✐s❝r❡t❡ ❛♥❞ ❝♦✉♥t❛❜❧❡✳ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✐t✉❛t✐♦♥s ✇❤❡r❡ ❛♥❞ ❝❤❛♥❣❡ s✐❣♥ ✐♥ ✇♦r❦✐♥❣ ✇✐t❤ ❚ ✮✳

✶✷ ✴ ✶✺

slide-54
SLIDE 54

❙t✉❞② ♦❢ t❤❡ ■❚❊P

◮ ❉❡✜♥❡ ♦♥ X × X t❤❡ s❡sq✉✐❧✐♥❡❛r ❢♦r♠ a((u, w), (u′, w′)) =

  • Ω

A∇u · ∇u′ ✲ ∇w · ∇w′ − k2(nuu′ ✲ ww′), ✇✐t❤ X = {(u, w) ∈ H1(Ω) × H1(Ω) | u − w ∈ H1

0(Ω)}✳

◮ ■♥tr♦❞✉❝❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❚(u, w) = (u − 2w, −w)✳ ◮ ❋♦r k ∈ Ri\{0}✱ A > Id ❛♥❞ n > 1✱ ♦♥❡ ✜♥❞s ℜe a((u, w), ❚(u, w)) ≥ C (u2

H1(Ω) + w2 H1(Ω)),

∀(u, w) ∈ X. ◮ ❯s✐♥❣ t❤❡ ❛♥❛❧②t✐❝ ❋r❡❞❤♦❧♠ t❤❡♦r❡♠✱ ♦♥❡ ❞❡❞✉❝❡s t❤❡ Pr♦♣♦s✐t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t A > Id ❛♥❞ n > 1✳ ❚❤❡♥ t❤❡ s❡t ♦❢ tr❛♥s♠✐s✲ s✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐s ❞✐s❝r❡t❡ ❛♥❞ ❝♦✉♥t❛❜❧❡✳ ◮ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ s✐t✉❛t✐♦♥s ✇❤❡r❡ A − Id ❛♥❞ n − 1 ❝❤❛♥❣❡ s✐❣♥ ✐♥ Ω ✇♦r❦✐♥❣ ✇✐t❤ ❚(u, w) = (u − 2χw, w)✮✳

✶✷ ✴ ✶✺

slide-55
SLIDE 55

■❚❊P ✇❤❡♥ A = Id

◮ ❲❤❡♥ A = Id✱ t❤❡ ■❚P ✐s ♥♦t ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✐♥ X ❧✐❦❡✇✐s❡ t❤❡ ❉▼❚P ✐s ♥♦t ♦❢ ❋r❡❞❤♦❧♠ t②♣❡ ✐♥ H1

0(Ω) ✇❤❡♥ µ1 = −µ2✳

❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ ❋✐♥❞ s✉❝❤ t❤❛t✿

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

■❞❡❛ ✸✿ ❚❤✐s tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ❉▼❚P✳ ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ ❋r❡❞❤♦❧♠ s❡♥s❡ ❛s s♦♦♥ ❛s ❞♦❡s ♥♦t ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ Pr♦♦❢✿ ❚✲❝♦❡r❝✐✈✐t② ♦r s❡❡ ❏✳ ❙②❧✈❡st❡r✬s ✇♦r❦ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st✉❞②✳

✶✸ ✴ ✶✺

slide-56
SLIDE 56

■❚❊P ✇❤❡♥ A = Id

◮ ❲❡ ❝❤❛♥❣❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ✇♦r❦✐♥❣ ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ v := u − w ∈ H2

0(D)✿ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts

v ∈ H2

0(D)\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v′ ∈ H2 0(D)✱

  • D

1 1 − n (∆v + k2nv)(∆v′ + k2v′) = 0. ◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (FV ) ❋✐♥❞ v ∈ H2

0(D) s✉❝❤ t❤❛t✿

  • D

1 1 − n ∆v∆v′

  • a(v,v′)

= f, v′D

  • l(v′)

, ∀v′ ∈ H2

0(D).

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

■❞❡❛ ✸✿ ❚❤✐s tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ❉▼❚P✳ ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ ❋r❡❞❤♦❧♠ s❡♥s❡ ❛s s♦♦♥ ❛s ❞♦❡s ♥♦t ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ Pr♦♦❢✿ ❚✲❝♦❡r❝✐✈✐t② ♦r s❡❡ ❏✳ ❙②❧✈❡st❡r✬s ✇♦r❦ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st✉❞②✳

✶✸ ✴ ✶✺

slide-57
SLIDE 57

■❚❊P ✇❤❡♥ A = Id

◮ ❲❡ ❝❤❛♥❣❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ✇♦r❦✐♥❣ ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ v := u − w ∈ H2

0(D)✿ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts

v ∈ H2

0(D)\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v′ ∈ H2 0(D)✱

  • D

1 1 − n (∆v + k2nv)(∆v′ + k2v′) = 0. ◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (FV ) ❋✐♥❞ v ∈ H2

0(D) s✉❝❤ t❤❛t✿

  • D

1 1 − n ∆v∆v′

  • a(v,v′)

= f, v′D

  • l(v′)

, ∀v′ ∈ H2

0(D).

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

■❞❡❛ ✸✿ ❚❤✐s tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ❉▼❚P✳ ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ ❋r❡❞❤♦❧♠ s❡♥s❡ ❛s s♦♦♥ ❛s ❞♦❡s ♥♦t ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ Pr♦♦❢✿ ❚✲❝♦❡r❝✐✈✐t② ♦r s❡❡ ❏✳ ❙②❧✈❡st❡r✬s ✇♦r❦ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st✉❞②✳

✶✸ ✴ ✶✺

slide-58
SLIDE 58

■❚❊P ✇❤❡♥ A = Id

◮ ❲❡ ❝❤❛♥❣❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ✇♦r❦✐♥❣ ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ v := u − w ∈ H2

0(D)✿ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts

v ∈ H2

0(D)\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v′ ∈ H2 0(D)✱

  • D

1 1 − n (∆v + k2nv)(∆v′ + k2v′) = 0. ◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (FV ) ❋✐♥❞ v ∈ H2

0(D) s✉❝❤ t❤❛t✿

  • D

1 1 − n ∆v∆v′

  • a(v,v′)

= f, v′D

  • l(v′)

, ∀v′ ∈ H2

0(D).

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

■❞❡❛ ✸✿ ❚❤✐s tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ❉▼❚P✳ ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ ❋r❡❞❤♦❧♠ s❡♥s❡ ❛s s♦♦♥ ❛s ❞♦❡s ♥♦t ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✳ Pr♦♦❢✿ ❚✲❝♦❡r❝✐✈✐t② ♦r s❡❡ ❏✳ ❙②❧✈❡st❡r✬s ✇♦r❦ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st✉❞②✳

✶✸ ✴ ✶✺

slide-59
SLIDE 59

■❚❊P ✇❤❡♥ A = Id

◮ ❲❡ ❝❤❛♥❣❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ✇♦r❦✐♥❣ ♦♥ t❤❡ ❞✐✛❡r❡♥❝❡ v := u − w ∈ H2

0(D)✿ k ✐s ❛ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts

v ∈ H2

0(D)\{0} s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ v′ ∈ H2 0(D)✱

  • D

1 1 − n (∆v + k2nv)(∆v′ + k2v′) = 0. ◮ ❲❡ ❢♦❝✉s ♦♥ t❤❡ ♣r✐♥❝✐♣❛❧ ♣❛rt✿ (FV ) ❋✐♥❞ v ∈ H2

0(D) s✉❝❤ t❤❛t✿

  • D

1 1 − n ∆v∆v′

  • a(v,v′)

= f, v′D

  • l(v′)

, ∀v′ ∈ H2

0(D).

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

■❞❡❛ ✸✿ ❚❤✐s tr❛♥s♠✐ss✐♦♥ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ❉▼❚P✳ ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ (FV ) ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ t❤❡ ❋r❡❞❤♦❧♠ s❡♥s❡ ❛s s♦♦♥ ❛s 1 − n ❞♦❡s ♥♦t ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ∂D✳ ◮ Pr♦♦❢✿ ❚✲❝♦❡r❝✐✈✐t② ♦r s❡❡ ❏✳ ❙②❧✈❡st❡r✬s ✇♦r❦ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st✉❞②✳ ✶✸ ✴ ✶✺

slide-60
SLIDE 60
  • ❡♥❡r❛❧✐③❛t✐♦♥s

✧ ❚✲❝♦❡r❝✐✈✐t② ❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ♥♦♥✲❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ✭L∞✮ ❛♥❞ ♦t❤❡r ♣r♦❜❧❡♠s ✭▼❛①✇❡❧❧✬s ❡q✉❛t✐♦♥s✱ ❡❧❛st✐❝✐t②✱ ✳✳✳✮✳ ✧ ■t ❛❧❧♦✇s t♦ ❥✉st✐❢② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ st❛♥❞❛r❞ ✜♥✐t❡ ❡❧❡♠❡♥t ♠❡t❤♦❞s✳ ♠ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ A − Id ❝❤❛♥❣❡ s✐❣♥ ✐♥ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ❜♦✉♥❞❛r②❄ ✌ ❋♦r t❤❡ ❡q✉✐✈❛❧❡♥t ❉▼❚P✱ str♦♥❣ s✐♥❣✉❧❛r✐t✐❡s ❛♣♣❡❛r ❛t t❤❡ ✐♥t❡r❢❛❝❡ ❛♥❞ H1 ✐s ♥♦ ❧♦♥❣❡r t❤❡ ❛♣♣r♦♣r✐❛t❡ ❢✉♥❝t✐♦♥❛❧ ❢r❛♠❡✇♦r❦✳ ❲❡ ♦❜s❡r✈❡ ❛ ❜❧❛❝❦ ❤♦❧❡ ♣❤❡♥♦♠❡♥♦♥ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❳✳ ❈❧❛❡②s✮✳ ♠ ❲❡ ❛r❡ ♥♦t ❛❜❧❡ t♦ ✉s❡ t❤❡ ❚✲❝♦❡r❝✐✈✐t② t❡❝❤♥✐q✉❡ t♦ ♣r♦✈❡ ❡①✐st❡♥❝❡ ♦❢ tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s✳ ⇒ ❚✲❝♦❡r❝✐✈✐t② ❣✐✈❡s ♣♦s✐t✐✈✐t② ❜✉t ♦♣❡r❛t♦rs ❛r❡ ♥♦ ❧♦♥❣❡r s②♠♠❡tr✐❝✳

✶✹ ✴ ✶✺

slide-61
SLIDE 61

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

❆✳✲❙✳ ❇♦♥♥❡t✲❇❡♥ ❉❤✐❛✱ ▲✳ ❈❤❡s♥❡❧✱ P✳ ❈✐❛r❧❡t ❏r✳✱ T✲❝♦❡r❝✐✈✐t② ❢♦r s❝❛❧❛r ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠s ❜❡t✇❡❡♥ ❞✐❡❧❡❝tr✐❝s ❛♥❞ ♠❡t❛♠❛t❡r✐❛❧s✱ ▼✷❆◆✱ t♦ ❛♣♣❡❛r✱ ✷✵✶✷✳ ❆✳✲❙✳ ❇♦♥♥❡t✲❇❡♥ ❉❤✐❛✱ ▲✳ ❈❤❡s♥❡❧✱ ❍✳ ❍❛❞❞❛r✱ ❖♥ t❤❡ ✉s❡ ♦❢ T✲❝♦❡r❝✐✈✐t② t♦ st✉❞② t❤❡ ✐♥t❡r✐♦r tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s✱ ❙❡r✳ ■✱ ✸✹✾✿✻✹✼✕✻✺✶✱ ✷✵✶✶✳ ❆✳✲❙✳ ❇♦♥♥❡t✲❇❡♥ ❉❤✐❛✱ P✳ ❈✐❛r❧❡t ❏r✳✱ ❈✳▼✳ ❩✇ö❧❢✱ ❚✐♠❡ ❤❛r♠♦♥✐❝ ✇❛✈❡ ❞✐✛r❛❝t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ♠❛t❡r✐❛❧s ✇✐t❤ s✐❣♥✲s❤✐❢t✐♥❣ ❝♦❡✣❝✐❡♥ts✱ ❏✳ ❈♦♠♣✉t✳ ❆♣♣❧✳ ▼❛t❤✱ ✷✸✹✿✶✾✶✷✕✶✾✶✾✱ ✷✵✶✵✱ ❈♦rr✐❣❡♥❞✉♠ ❏✳ ❈♦♠♣✉t✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✷✸✹✿✷✻✶✻✱ ✷✵✶✵✳ ❋✳ ❈❛❦♦♥✐✱ ❍✳ ❍❛❞❞❛r✱ ❚r❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡s ✐♥ ✐♥✈❡rs❡ s❝❛tt❡r✐♥❣ t❤❡♦r②✱ s✉❜♠✐tt❡❞✱ ✷✵✶✷✳ ▲✳ ❈❤❡s♥❡❧✱ ■♥t❡r✐♦r tr❛♥s♠✐ss✐♦♥ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r ▼❛①✇❡❧❧✬s ❡q✉❛t✐♦♥s✿ t❤❡ T✲❝♦❡r❝✐✈✐t② ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤✱ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✱ t♦ ❛♣♣❡❛r✱ ✷✵✶✷✳

✶✺ ✴ ✶✺