❖♥ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞s ❢♦r t❤❡ ❈❛❤♥✕❍✐❧❧✐❛r❞ ❡q✉❛t✐♦♥ ✇✐t❤ ♦❜st❛❝❧❡ ♣♦t❡♥t✐❛❧ ❽✉❜♦♠ír ❇❛➡❛s ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ■♠♣❡r✐❛❧ ❈♦❧❧❡❣❡ ▲♦♥❞♦♥ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❘♦❜❡rt ◆ür♥❜❡r❣ ❤tt♣✿✴✴✇✇✇✳♠❛✳✐❝✳❛❝✳✉❦✴⑦❧✉❜♦ ❧✉❜♦❅✐♠♣❡r✐❛❧✳❛❝✳✉❦
✶ ❖✈❡r✈✐❡✇ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✷✳ ❈♦♥t✐♥✉♦✉s ♠♦❞❡❧ ✸✳ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞ ✹✳ ❙♦❧✈❡rs ❢♦r t❤❡ ❞✐s❝r❡t❡ s②st❡♠ ✺✳ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✴✸✾
✷ ■♥tr♦❞✉❝t✐♦♥ ❊✈♦❧✉t✐♦♥ ♦❢ s✉r❢❛❝❡s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ♠❛t❡r✐❛❧ s❝✐❡♥❝❡ ✭♠✐❝r♦str✉❝t✉r❡ ♣r❡❞✐❝t✐♦♥✱ ♠❛t❡r✐❛❧ ♣r♦t❡rt✐❡s✱ ✈♦✐❞ ❡❧❡❝tr♦♠✐❣r❛t✐♦♥ ✐♥ s❡♠✐❝♦♥❞✉❝t♦rs✮✱ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ ❡t❝✳ ❖✈❡r✈✐❡✇ ❉❡❝❦❡❦❡❧♥✐❝❦✱ ❉③✉✐❦✱ ❊❧❧✐♦tt ✭✷✵✵✺✮ ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✴✸✾
✸ ■♥tr♦❞✉❝t✐♦♥ ❙✉r❢❛❝❡ ❞✐✛✉s✐♦♥ s❤❛r♣ ✐♥t❡r❢❛❝❡ ♠♦❞❡❧ V = − ∆ s κ ♦♥ Γ( t ) • Γ( t ) ✈♦✐❞ s✉r❢❛❝❡ • ∆ s s✉r❢❛❝❡ ▲❛♣❧❛❝✐❛♥ • V ✈❡❧♦❝✐t② ♦❢ Γ( t ) • κ ❝✉r✈❛t✉r❡ ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✴✸✾
✹ P❤❛s❡✲✜❡❧❞ ♠♦❞❡❧ ❉✐✛✉s❡ ✐♥t❡r❢❛❝❡ ✇✐t❤ ✐♥t❡r❢❛❝❡ ✇✐❞t❤ ≈ γπ ❆❧t❡r♥❛t✐✈❡s t♦ ♣❤❛s❡✲✜❡❧❞ ❛♣♣r♦❛❝❤ • ❉✐r❡❝t ♠❡t❤♦❞s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡ ❞✐✛✉s✐♦♥ ♠♦❞❡❧✱ ♣r♦❜❧❡♠s ✇✐t❤ t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s • ▲❡✈❡❧ s❡t ♠❡t❤♦❞s ❝❛♥ ❤❛♥❞❧❡ t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✹✴✸✾
✺ P❤❛s❡✲✜❡❧❞ ♠♦❞❡❧ • γ > 0 ✐♥t❡r❢❛❝✐❛❧ ♣❛r❛♠❡t❡r • u γ ( · , t ) ∈ K := [ − 1 , 1] ✱ t ∈ [0 , T ] ❝♦♥s❡r✈❡❞ ♦r❞❡r ♣❛r❛♠❡t❡r❀ u γ ( · , t ) = − 1 ✈♦✐❞✱ u γ ( · , t ) = 1 ❝♦♥❞✉❝t♦r • w γ ( · , t ) ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧ • φ γ ( · , t ) ❡❧❡❝tr✐❝ ♣♦t❡♥t✐❛❧ P❤❛s❡ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s✉r❢❛❝❡ ❞✐✛✉s✐♦♥ ✭❞✐✛✉s❡ ✐♥t❡r❢❛❝❡✮ γ ∂u γ ∂t − ∇ . ( b ( u γ ) ∇ w γ ) = 0 ✐♥ Ω T := Ω × (0 , T ] , w γ = − γ ∆ u γ + γ − 1 Ψ ′ ( u γ ) ✐♥ Ω T , ✇❤❡r❡ | u γ | < 1 ✱ ✰ ■✳❈✳ ✰ ❇✳❈✳ ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✺✴✸✾
✻ P❤❛s❡✲✜❡❧❞ ♠♦❞❡❧ ❉❡❣❡♥❡r❛t❡ ❝♦❡✣❝✐❡♥ts b ( s ) := 1 − s 2 ✱ ∀ s ∈ K ❖❜st❛❝❧❡✲❢r❡❡ ❡♥❡r❣② � 1 � 1 − s 2 � ✐❢ s ∈ K , Ψ( s ) := 2 ∞ ✐❢ s �∈ K , r❡str✐❝ts u γ ( · , · ) ∈ K ✳ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s❤❛r♣ ✐♥t❡r❢❛❝❡ ♠♦❞❡❧ γ → 0 t❤❡♥ { x ; u γ ( x, t ) = 0 } → Γ( t ) ✱ Γ( t ) ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ s❤❛r♣ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ❆❞✈❛♥t❛❣❡s ♦❢ ♣❤❛s❡✲✜❡❧❞ ❛♣♣r♦❛❝❤ • ♥♦ ❡①♣❧✐❝✐t tr❛❝❦✐♥❣ ♦❢ t❤❡ ✐♥t❡r❢❛❝❡ ♥❡❡❞❡❞ • ❝❛♥ ❤❛♥❞❧❡ t♦♣♦❧♦❣✐❝❛❧ ❝❤❛♥❣❡s ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✻✴✸✾
✼ ◆✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ε ∈ S h ⇔ u γ U n ε ∈ K h ⇔ w γ W n ε ∈ S h ⇔ φ γ Φ n ❉♦✉❜❧❡ ♦❜st❛❝❧❡ ❢♦r♠✉❧❛t✐♦♥ ✭ ε r❡❣✉❧❛r✐s❛t✐♦♥ ♣❛r❛♠❡t❡r✮ � U n � h ε − U n − 1 ε + (Ξ ε ( U n − 1 ) ∇ W n ∀ χ ∈ S h , γ , χ ε , ∇ χ ) = 0 ε τ n ε + γ − 1 U n − 1 γ ( ∇ U n ε , ∇ [ χ − U n ε ]) ≥ ( W n , χ − U n ε ) h ∀ χ ∈ K h , ε � ❞✐s❝r❡t❡ ✐♥♥❡r ♣r♦❞✉❝t ✭♠❛ss ❧✉♠♣✐♥❣✮ ( η 1 , η 2 ) h := Ω π h ( η 1 ( x ) η 2 ( x )) d x Ξ ε ( · ) ≈ b ( · ) ❈♦♥✈❡r❣❡♥❝❡ ✭❊①✐st❡♥❝❡✮ ✷❉✿ ❇❛rr❡tt✱ ◆ür♥❜❡r❣✱ ❙t②❧❡s ✭✷✵✵✹✮✱ ✸❉✿ ❇❛➡❛s✱ ◆ür♥❜❡r❣ ✭✷✵✵✻✮ h → 0 ✱ ε → 0 ✱ τ = O ( h 2 ) ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✼✴✸✾
✽ ▼❛tr✐① ❢♦r♠✉❧❛t✐♦♥ ❚❤❡ ❞✐s❝r❡t❡ s②st❡♠ ε } ∈ K J × R J s✉❝❤ t❤❛t ❋✐♥❞ { U n ε , W n ε ) T B U n ε ) T M W n ε ) T s ∀ V ∈ K J , γ ( V − U n ε − ( V − U n ( V − U n ≥ ε ε + τ n A n − 1 W n γ M U n = r ε A n − 1 M ij := ( χ i , χ j ) h , := (Ξ ε ( U n − 1 B ij := ( ∇ χ i , ∇ χ j ) , ) ∇ χ i , ∇ χ j ) ε ij − α τ n A n − 1 Φ n ε ∈ R J , s := γ − 1 M U n − 1 ∈ R J . r := γ M U n − 1 ε ε ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✽✴✸✾
✾ ❇❧♦❝❦ ●❛✉ss✲❙❡✐❞❡❧ ❛❧❣♦r✐t❤♠ ✇✐t❤ ♣r♦❥❡❝t✐♦♥ Pr♦❥❡❝t❡❞ ❜❧♦❝❦ ●❛✉ss✲❙❡✐❞❡❧ ) T ( γ ( B D − B L ) U n,k ) T ( s + γ B T ( V − U n,k − M W n,k ( V − U n,k L U n,k − 1 ) ≥ ) ε ε ε ε ε γ M U n,k + τ n ( A D − A L ) W n,k L W n,k − 1 r + τ n A T = ε ε ε 2 × 2 s②st❡♠ ❢♦r ❡✈❡r② ✈❡rt❡①❀ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥ � � � � r j + τ n A n − 1 M jj � s j � jj U n,k j = γ [ M jj ] 2 + τ n γ A n − 1 ε B jj jj K � � r j − γ M jj [ U n,k j = � ] j ε W n,k ε τ n A n − 1 jj ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✾✴✸✾
✶✵ ❯③❛✇❛ ❛❧❣♦r✐t❤♠ ❯③❛✇❛✲▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ●räs❡r✱ ❑♦r♥❤✉❜❡r ✭✷✵✵✺✮✱ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❇❧♦✇❡②✱ ❊❧❧✐♦tt ✭✶✾✾✶✱ ✶✾✾✷✮✱ ❖✉t❡r ❯③❛✇❛✲t②♣❡ ✐t❡r❛t✐♦♥s ❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐s❛t✐♦♥✱ t✇♦ s✉❜✲st❡♣s ) T B U n,k ) T s + ( V − U n,k ) T M W n,k − 1 • γ ( V − U n,k ≥ ( V − U n,k ∀ V ∈ K J ε ε ε ε ε + S − 1 � � − τ n A n − 1 W n,k − 1 • W n,k = W n,k − 1 − γ M U n,k + r ε ε ε ε S − 1 ✲ ♣r❡❝♦♥❞✐t✐♦♥❡r ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✵✴✸✾
✶✶ ❯③❛✇❛ ❛❧❣♦r✐t❤♠ Pr❡❝♦♥❞✐t✐♦♥❡r ■❢ ✇❡ ❦♥♦✇ t❤❡ ❡①❛❝t ❝♦✐♥❝✐❞❡♥❝❡✴❝♦♥t❛❝t s❡t � � � � � � � J ( U n � [ U n j ∈ J : ε ) = ε ] j � = 1 , t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ❧✐♥❡❛r � � � � � � γ � − � U n s ( U n B ( U n M ( U n � ε ) ε ) ε ) ε = . W n τ n A n − 1 r γ M ε ✇✐t❤ � i ∈ � δ ij J � B ij = , B ij ❡❧s❡ � i ∈ � 0 J � j ∈ J, M ij = , M ij ❡❧s❡ ❛♥❞ � i ∈ � γ [ U n ε ] i J s i = � . s i ❡❧s❡ ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✶✴✸✾
✶✷ ❯③❛✇❛ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❝❤♦✐❝❡ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥t ε ) = M � ε ) − 1 � S ( U n B ( U n M ( U n ε ) + τ n A n − 1 ❆♣♣r♦①✐♠❛t✐♦♥ U n,k ≈ U n ε ε ) = M � ) − 1 � S = S ( U n,k B ( U n,k M ( U n,k ) + τ n A n − 1 ε ε ε ❯③❛✇❛ ✇✐t❤ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r S ( U k ) ) T B U n,k ) T s + ( V − U n,k ) T M W n,k − 1 ∀ V ∈ K J , γ ( V − U n,k ( V − U n,k ≥ ) − 1 � � ε ε ε ε ε ) − 1 � − M � W n,k S ( U n,k B ( U n,k s ( U n,k = ) + r . ε ε ε ε ❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✷✴✸✾
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