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❖♥ ♠✉❧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 = −∆sκ ♦♥ 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
• uγ(·, t) ∈ K := [−1, 1]✱ t ∈ [0, T] ❝♦♥s❡r✈❡❞ ♦r❞❡r ♣❛r❛♠❡t❡r❀ uγ(·, t) = −1 ✈♦✐❞✱

uγ(·, t) = 1 ❝♦♥❞✉❝t♦r

• wγ(·, t) ❝❤❡♠✐❝❛❧ ♣♦t❡♥t✐❛❧
• φγ(·, t) ❡❧❡❝tr✐❝ ♣♦t❡♥t✐❛❧

P❤❛s❡ ✜❡❧❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s✉r❢❛❝❡ ❞✐✛✉s✐♦♥ ✭❞✐✛✉s❡ ✐♥t❡r❢❛❝❡✮ γ ∂uγ

∂t − ∇.( b(uγ) ∇wγ ) = 0

✐♥ ΩT := Ω × (0, T], wγ = −γ ∆uγ + γ−1 Ψ′(uγ) ✐♥ ΩT, ✇❤❡r❡ |uγ| < 1✱ ✰ ■✳❈✳ ✰ ❇✳❈✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✺✴✸✾

✻

P❤❛s❡✲✜❡❧❞ ♠♦❞❡❧

❉❡❣❡♥❡r❛t❡ ❝♦❡✣❝✐❡♥ts b(s) := 1 − s2✱ ∀ s ∈ K ❖❜st❛❝❧❡✲❢r❡❡ ❡♥❡r❣② Ψ(s) := 1

2

• 1 − s2

✐❢ s ∈ K, ∞ ✐❢ s ∈ K, r❡str✐❝ts uγ(·, ·) ∈ K✳ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s❤❛r♣ ✐♥t❡r❢❛❝❡ ♠♦❞❡❧ γ → 0 t❤❡♥ {x; uγ(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✐♦♥

U n

ε ∈ Sh ⇔ uγ

W n

ε ∈ Kh ⇔ wγ

Φn

ε ∈ Sh ⇔ φγ

❉♦✉❜❧❡ ♦❜st❛❝❧❡ ❢♦r♠✉❧❛t✐♦♥ ✭ε r❡❣✉❧❛r✐s❛t✐♦♥ ♣❛r❛♠❡t❡r✮ γ U n

ε − U n−1 ε

τn , χ h + (Ξε(U n−1

ε

) ∇ W n

ε , ∇χ) = 0

∀ χ ∈ Sh, γ (∇U n

ε , ∇[χ − U n ε ]) ≥ (W n ε + γ−1 U n−1 ε

, χ − U n

ε )h

∀ χ ∈ Kh, ❞✐s❝r❡t❡ ✐♥♥❡r ♣r♦❞✉❝t ✭♠❛ss ❧✉♠♣✐♥❣✮ (η1, η2)h :=

• Ω πh(η1(x) η2(x)) dx

Ξε(·) ≈ b(·) ❈♦♥✈❡r❣❡♥❝❡ ✭❊①✐st❡♥❝❡✮ ✷❉✿ ❇❛rr❡tt✱ ◆ür♥❜❡r❣✱ ❙t②❧❡s ✭✷✵✵✹✮✱ ✸❉✿ ❇❛➡❛s✱ ◆ür♥❜❡r❣ ✭✷✵✵✻✮ h → 0✱ ε → 0✱ τ = O(h2)

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✼✴✸✾

✽

▼❛tr✐① ❢♦r♠✉❧❛t✐♦♥

❚❤❡ ❞✐s❝r❡t❡ s②st❡♠ ❋✐♥❞ {U n

ε, W n ε} ∈ KJ × RJ s✉❝❤ t❤❛t

γ (V − U n

ε)T B U n ε − (V − U n ε)T M W n ε

≥ (V − U n

ε)T s

∀ V ∈ KJ , γ M U n

ε + τn An−1 W n ε

= r Mij := (χi, χj)h, Bij := (∇χi, ∇χj), An−1

ij

:= (Ξε(U n−1

ε

) ∇χi, ∇χj) r := γ M U n−1

ε

− α τn An−1 Φn

ε ∈ RJ ,

s := γ−1 M U n−1

ε

∈ RJ .

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✽✴✸✾

✾

❇❧♦❝❦ ●❛✉ss✲❙❡✐❞❡❧ ❛❧❣♦r✐t❤♠ ✇✐t❤ ♣r♦❥❡❝t✐♦♥

Pr♦❥❡❝t❡❞ ❜❧♦❝❦ ●❛✉ss✲❙❡✐❞❡❧ (V − U n,k

ε

)T (γ (BD − BL) U n,k

ε

− M W n,k

ε

) ≥ (V − U n,k

ε

)T (s + γ BT

L U n,k−1 ε

) γ M U n,k

ε

+ τn (AD − AL) W n,k

ε

= r + τn AT

L W n,k−1 ε

2 × 2 s②st❡♠ ❢♦r ❡✈❡r② ✈❡rt❡①❀ ❡①♣❧✐❝✐t s♦❧✉t✐♦♥

• U n,k

ε

• j =
• Mjj

rj + τn An−1

jj

• sj

γ [Mjj]2 + τn γ An−1

jj

Bjj

• K
• W n,k

ε

• j =

rj − γ Mjj [U n,k

ε

]j τn An−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

• γ (V − U n,k

ε

)T B U n,k

ε

≥ (V − U n,k

ε

)T s + (V − U n,k

ε

)T M W n,k−1

ε

∀ V ∈ KJ

• W n,k

ε

= W n,k−1

ε

+ S−1 −γ M U n,k

ε

− τn An−1 W n,k−1

ε

+ r

• 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

ε) =

• j ∈ J :
• [U n

ε]j

• = 1
• ,

t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ❧✐♥❡❛r

• γ

B(U n

ε)

− M(U n

ε)

γ M τn An−1 U n

ε

W n

ε

• =
• s(U n

ε)

r

• .

✇✐t❤

• Bij =
• δij

i ∈ J Bij ❡❧s❡ ,

• Mij =
• i ∈

J Mij ❡❧s❡ , j ∈ J, ❛♥❞

• si =
• γ [U n

ε]i

i ∈ J si ❡❧s❡ .

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✶✴✸✾

✶✷

❯③❛✇❛ ❛❧❣♦r✐t❤♠

❖♣t✐♠❛❧ ❝❤♦✐❝❡ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥t S(U n

ε) = M

B(U n

ε)−1

M(U n

ε) + τn An−1

❆♣♣r♦①✐♠❛t✐♦♥ U n,k

ε

≈ U n

ε

S = S(U n,k

ε

) = M B(U n,k

ε

)−1 M(U n,k

ε

) + τn An−1 ❯③❛✇❛ ✇✐t❤ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r S(U k) γ (V − U n,k

ε

)T B U n,k

ε

≥ (V − U n,k

ε

)T s + (V − U n,k

ε

)T M W n,k−1

ε

∀ V ∈ KJ , W n,k

ε

= S(U n,k

ε

)−1 −M B(U n,k

ε

)−1 s(U n,k

ε

) + r

• .

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✷✴✸✾

✶✸

❯③❛✇❛ ❛❧❣♦r✐t❤♠

❙♦❧✉t✐♦♥ ♦❢ t❤❡ s✉❜♣r♦❜❧❡♠s

• ✜rst st❡♣✱ ❡❧❧✐♣t✐❝ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✇✐t❤ ❞♦✉❜❧❡ ♦❜st❛❝❧❡✱ ✇❡ ❝❛♥ ✉s❡ st❛♥❞❛r❞

♠❡t❤♦❞s✿ ♣r♦❥❡❝t❡❞ ●❛✉ss✲❙❡✐❞❡❧ ♦r ▼♦♥♦t♦♥❡ ♠✉❧t✐❣r✐❞❀ ✐t❡r❛t✐♦♥s ❝❛♥ ❜❡ st♦♣♣❡❞ ✇❤❡♥ ✇❡ ♦❜t❛✐♥ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ ❝♦✐♥❝✐❞❡♥❝❡ st❡♣ ✲ ♦♥❧② ❢❡✇ ✐t❡r❛t✐♦♥s✳ ✐♥♣✉t W n,k−1

ε

✱ ♦✉t♣✉t U n,k

ε

• s❡❝♦♥❞ st❡♣ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❧✐♥❡❛r s②♠♠❡tr✐❝ s❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠
• γ2

B −γ M(U n,k

ε

) −γ M(U n,k

ε

) −τn An−1

• U k

W n,k

ε

• =
• γ

s − r

• st❛♥❞❛r❞ W✲❝②❝❧❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ❢♦r s❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠s ✭❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ♠✐①❡❞

❋❊▼✮✱ ❝❛♥♦♥✐❝❛❧ r❡str✐❝t✐♦♥ ❛♥❞ ♣r♦❧♦♥❣❛t✐♦♥✱ ❜❧♦❝❦ ●❛✉ss✲❙❡✐❞❡❧ s♠♦♦t❤❡r ✭1 s♠♦♦t❤✐♥❣ st❡♣✮✱ ❛❧t❡r♥❛t✐✈❡ ✭❱❛♥❦❛ t②♣❡ ✭✶✾✽✻✮✮ s♠♦♦t❤❡r ❙❝❤rö❜❡r❧✱ ❩✉❧❡❤♥❡r ✭✷✵✵✸✮✳ ✐♥♣✉t Jk = J(U n,k

ε

)✱ ♦✉t♣✉t W n,k

ε

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✸✴✸✾

✶✹

❯③❛✇❛ ❛❧❣♦r✐t❤♠

◆✉♠❡r✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭♠♦r❡ ♥❛t✉r❛❧✱ ❜✉t ♥♦ ♣r♦♦❢ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✮ Ξε(U n−1

ε

) ↔ πh[b(U n−1

ε

)]✱ π[b(U n−1

ε

)] ≡ 0 ♦♥ Jdeg An−1 ✲ ❤❛s ③❡r♦ r♦✇s ❞✉❡ t♦ t❤❡ ❞❡❣❡♥❡r❛❝② ♦❢ b(·) ❉❡❣❡♥❡r❛t❡ s❡t j ∈ Jdeg :=

• j ∈ J : πh

b(U n−1

ε

)

• ≡ 0 ♦♥ s✉♣♣(χj)
• ❙♦❧✉t✐♦♥

❲❡ ✉s❡ t❤❡ ❢❛❝t ✭Jdeg ⊂ Jk✮ U n,k

j

= U n−1

j

❢♦r ❛❧❧ j ∈ Jdeg ❲❡ ♦❜t❛✐♥ ❛♥ ❡q✉✐✈❛❧❡♥t s❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠ ✇✐t❤ r❡❣✉❧❛r ♠❛tr✐① A−1

deg

• γ2

B −γ M −γ M −τn An−1

deg

• U k
• W k
• =
• γ

s − rdeg

• ,

✇❤❡r❡ W j = W n−1

j

❢♦r j ∈ Jdeg✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✹✴✸✾

✶✺

❯③❛✇❛ ❛❧❣♦r✐t❤♠

❈♦♠♣❧❡t❡ ❛❧❣♦r✐t❤♠ ✶✳ ■♥✐t✐❛❧✐③❛t✐♦♥✿ ❙t❛rt ✇✐t❤ ✐♥✐t❛❧ ❣✉❡ss U n,0

ε

= U n−1

ε

✱ s❡t J0 = J(U n,0

ε ) ❛♥❞ ❝♦♠♣✉t❡

W n,0

ε

❜② s♦❧✈✐♥❣ t❤❡ ❧✐♥❡❛r s❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠ ✇✐t❤ ❝♦✐♥❝✐❞❡♥❝❡ s❡t J0✳ ✷✳ ❯③❛✇❛ ✐t❡r❛t✐♦♥s✿ ❢♦r k = 1, . . . ❞♦

• ✶st s✉❜✲st❡♣ ❈♦♠♣✉t❡ t❤❡ ❛♣♣r♦①✐♠❛t❡ ❝♦✐♥❝✐❞❡♥❝❡ s❡t

Jk = J(U n,k

ε

)✱ ✇❤❡r❡ U n,k

ε

✐s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❡❧❧✐♣t✐❝ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❜② P●❙ ♦r ▼▼●✳

• ■❢

Jk = Jk−1 ❣♦ t♦ st❡♣ ✸✳

• ✷♥❞ s✉❜✲st❡♣ ❙♦❧✈❡ ❛ ❧✐♥❡❛r s②♠♠❡tr✐❝ s❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠ ❜② t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞

✇✐t❤ ❜❧♦❝❦ ●❛✉ss✕❙❡✐❞❡❧ s♠♦♦t❤❡r t♦ ♦❜t❛✐♥ W n,k

ε

✳

• ■❢ max

j∈J |

• W n,k

ε

• j −
• W n,k−1

ε

• j | < tol✱ ✇✐t❤ tol ❜❡✐♥❣ t❤❡ ♣r❡s❝r✐❜❡❞ t♦❧❡r❛♥❝❡✱ ❣♦

t♦ st❡♣ ✸✳ ✸✳ ❯③❛✇❛ ✐t❡r❛t✐♦♥s ❤❛✈❡ ❝♦♥✈❡r❣❡❞✿ ❈♦♠♣✉t❡ U n,k+1

ε

✉♣ t♦ t❤❡ ❞❡s✐r❡❞ ❛❝❝✉r❛❝② ❢r♦♠ t❤❡ ❡❧❧✐♣t✐❝ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❢r♦♠ t❤❡ ✶st s✉❜✲st❡♣ ✉s✐♥❣ W n,k

ε

✳ ✹✳ ❙❡t U n

ε = U n,k+1 ε

✱ W n

ε = W n,k ε

✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✺✴✸✾

✶✻

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❋❊▼ ❝♦❞❡ ❆❧❜❡rt❛❀ ❛❞❛♣t✐✈❡ ♠❡s❤❡s✿ ✐❢ |U n| < 1 ✭✐✳❡✳ ✐♥ t❤❡ ✐♥t❡r❢❛❝✐❛❧ r❡❣✐♦♥✮ s❡t h = hmin ≈

1 Nf ❡❧s❡ ✭|U n| = 1✮ s❡t

h = hmax =

1 Nc✳

❈♦♠♣❛r✐s♦♥ ♦❢ ❯③❛✇❛ ❛♥❞ ●❛✉ss✲❙❡✐❞❡❧ ♠❡t❤♦❞s γ =

1 12π

Nf τ

• ❙

❯③❛✇❛✲▼● r❛t✐♦ ✶✷✽ ✶❡✲✻ ✶✹✷✷✼♠ ✸✹✹✺♠ ✹✳✶✸ ✻✹ ✹❡✲✻ ✷✺✷♠ ✶✹✻♠ ✶✳✼✷ ✸✷ ✶✳❡✲✺ ✾♠✹✵s ✶✶♠✷✵s ✵✳✽✺ ❚❛❜❧❡ ✶✿ ❈♦♠♣✉t❛t✐♦♥ t✐♠❡s ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ h γ Nf τ

• ❙

❯③❛✇❛✲▼● r❛t✐♦ 1/12π ✶✷✽ ✶❡✲✻ ✶✹✷✷✼♠ ✸✹✹✺♠ ✹✳✶✸ 1/6π ✻✹ ✹❡✲✻ ✽✺✸♠ ✷✺✾♠ ✸✳✷✾ 1/3π ✸✷ ✶✳❡✲✺ ✾✸♠ ✸✵♠ ✸✳✶ ❚❛❜❧❡ ✷✿ ❈♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡s ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ γ

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✻✴✸✾

✶✼

▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ✲ ♥♦t❛t✐♦♥

Pr♦❜❧❡♠ ♠❛tr✐① ♦♥ ✜♥❡ ♠❡s❤ A =

• B

−M M A

• ❙❛❞❞❧❡ ♣♦✐♥t ♣r♦❜❧❡♠ ✇✐t❤ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②

A

• U

W

• ≥
• r

s

• U ∈ KJ

■♥t❡r❣r✐❞ tr❛♥s❢❡r ♦♣❡r❛t♦rs ✭❝❛♥♦♥✐❝❛❧ r❡str✐❝t✐♦♥ ❛♥❞ ♣r♦❧♦♥❣❛t✐♦♥✮ Ic

f✱ If c

❈♦❛rs❡ ♠❛tr✐① Ac Ac = Ic

f B If c

−Ic

f M If c

Ic

f M If c

Ic

f A If c

• ❽✉❜♦♠ír ❇❛➡❛s

❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✼✴✸✾

✶✽

▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠

❚✇♦✲❣r✐❞ s❝❤❡♠❡ ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ A

• U

W

• ≥
• r

s

• U ∈ KJ
• ♣r❡ s♠♦♦t❤✐♥❣

m ✐t❡r❛t✐♦♥ ♦❢ ♣r♦❥❡❝t❡❞ ●❛✉ss✲❙❡✐❞❡❧ (U 0, W 0) → (U m, W m)

• ❝♦❛rs❡ ❣r✐❞ ❝♦rr❡❝t✐♦♥ s♦❧✈❡ t❤❡ ❝♦❛rs❡ ♣r♦❜❧❡♠ ❡①❛❝t❧②

✶✳ ❝♦♠♣✉t❡ r❡s✐❞✉❛❧ (Qu, Qw) = (r, s) − A(U m, W m) ✷✳ Ac

• V u

V w

• ≥

Ic

fQu

Ic

fQw

• V u ∈ KJc

∗

✸✳ ✉♣❞❛t❡ s♦❧✉t✐♦♥ (U m+1, W m+1) = (U m + If

c V u, W m + If c V w)

• ♣♦st s♠♦♦t❤✐♥❣

m ✐t❡r❛t✐♦♥ ♦❢ ♣r♦❥❡❝t❡❞ ●❛✉ss✲❙❡✐❞❡❧ (U m+1, W m+1) → (U 2m+1, W 2m+1)

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✽✴✸✾

✶✾

❈♦❛rs❡ ❣r✐❞ ❝♦rr❡❝t✐♦♥

❲❡ r❡q✉✐r❡ |U m+1 + If

c V u| ≤ 1

◆❡✇ ♦❜st❛❝❧❡ ❢♦r V u −1 − U m+1 ≤ If

c V u ≤ 1 − U m+1

❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ✜♥❡ ♠❡s❤✱ ❜✉t t♦ ❝♦♠♣✉t❡ V u ♦♥ t❤❡ ❝♦❛rs❡ ❣r✐❞ ✇❡ ♥❡❡❞ ❛ ✏❝♦❛rs❡ ♦❜st❛❝❧❡✑✳ ❙♦❧✉t✐♦♥ ▼❛♥❞❡❧ ✭✶✾✽✹✮ ❧♦♦❦ ❢♦r V u ∈ KJc

∗

✇❤❡r❡ KJc

∗ =

• V ∈ RJc; Qc

f(−1 − U m+1) ≤ V ≤ Rc f(−1 − U m+1)

• ✇✐t❤ ✉♣♣❡r✴❧♦✇❡r

♦❜st❛❝❧❡ r❡str✐❝t✐♦♥ ♦♣❡r❛t♦rs ❞❡✜♥❡❞ ❛s

• Qc

fv

• (p)

= max

• v(q); q ∈ N f ∩ int supp χp, χp ∈ V c

h

• ,
• Rc

fv

• (p)

= min

• v(q); q ∈ N f ∩ int supp χp, χp ∈ V c

h

• ,

✇✐t❤ p ∈ N k−1, v ∈ V f

h ✳

❑♦r♥❤✉❜❡r ✭✶✾✾✹✮ s❧✐❣❤t❧② ❜❡tt❡r ♦❜st❛❝❧❡ r❡str✐❝t✐♦♥ ✭s✉✐t❛❜❧❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ✐♥st❡❛❞ ♦❢ ♠✐♥✴♠❛①✮✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✶✾✴✸✾

✷✵

❈♦❛rs❡ ❣r✐❞ ❝♦rr❡❝t✐♦♥ ✲ ♠❧t✐♣❧❡ ❣r✐❞s

❘❡str✐❝t✐♦♥s ❢♦r ♥✉♠❡r✐❝❛❧ ❝♦♥✈❡r❣❡♥❝❡ ❣r✐❞ ♦♥ t❤❡ ❧♦✇❡st ❤❛s t♦ ❜❡ ✜♥❡ ❡♥♦✉❣❤❀ ♥✉♠❜❡r ♦❢ ❣r✐❞ ❧❡✈❡❧s ❞❡♣❡♥❞s ♦♥ γ ❛♥❞ hmin✱ ✇❡ ♥❡❡❞✿

• s♠❛❧❧ γ ❢♦r ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s❤❛r♣ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠
• s♠❛❧❧ hmin ✭❞❡♣❡♥❞✐♥❣ ♦♥ γ✮ ❢♦r ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✉♦✉s ♣❤❛s❡✲✜❡❧❞

♠♦❞❡❧ ❋❡❛s✐❜❧❡ ♣❛r❛♠❡t❡r ❝♦♠❜♥❛t✐♦♥s ✐♥ ✸❉

• γ =

1 12π✱ Nf = 1 128✱ 6 ♠❡s❤ ♣♦✐♥ts ✐♥ t❤❡ ✐♥t❡r❢❛❝❡✱ 2✲❧❡✈❡❧ ♠❡t❤♦❞

• γ =

1 9π✱ Nf = 1 128✱ 8 ♠❡s❤ ♣♦✐♥ts ✐♥ t❤❡ ✐♥t❡r❢❛❝❡✱ 3✲❧❡✈❡❧ ♠❡t❤♦❞

❈♦❛rs❡ ❣r✐❞ s♦❧✈❡r ✐♥❡①❛❝t s♦❧✉t✐♦♥ ✇✐t❤ ♣r♦❥❡❝t❡❞ ●❙✱ ✸✵ ✐t❡r❛t✐♦♥ ❛r❡ ❡♥♦✉❣❤ ❢♦r t❤❡ 3✲❧❡✈❡❧ ♠❡t❤♦❞✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✵✴✸✾

✷✶

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

▼❡s❤❡s ♦♥ ❞✐✛❡r❡♥t ❧❡✈❡❧s

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✶✴✸✾

✷✷

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ▼✉❧t✐❣r✐❞ ❛♥❞ ❯③❛✇❛ ♠❡t❤♦❞s

✸❉ ❝♦♠♣✉t❛t✐♦♥ ✇✐t❤ ❛❜♦✉t 900000 ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ γ =

1 9π

♠❡t❤♦❞ t♦t❛❧ ✐t❡r❛t✐♦♥ ❈P❯ t✐♠❡ ▼● ✷✵✻✸ ✷✷✵✵♠ ❯③❛✇❛✲▼● ✹✼✻✵ ✸✸✾✵♠ ❚❛❜❧❡ ✸✿ 3✲❧❡✈❡❧ ▼● ✈s✳ ❯③❛✇❛ ✭✽✲❧❡✈❡❧✮❀ ❲✲❝②❝❧❡✱ ✶ s♠♦♦t❤✐♥❣ st❡♣ ▼✉❧t✐❣r✐❞ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ❛❜♦✉t 1.5 t✐♠❡s ❢❛st❡r ✇✐t❤ 2× ❧❡ss ✐t❡r❛t✐♦♥s✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✷✴✸✾

✷✸

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.06✱ τ = 10−6✱ Nf = 128✱ Nc = 2

❋✐❣✉r❡ ✶✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 0.001, 0.005✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✸✴✸✾

✷✹

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.06✱ τ = 10−6✱ Nf = 128✱ Nc = 2

❋✐❣✉r❡ ✷✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0.01, 0.015, T = 0.06✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✹✴✸✾

✷✺

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.08✱ τ = 5 × 10−6✱ Nf = 48✱ Nc = 2

❋✐❣✉r❡ ✸✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 0.001, 0.005✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✺✴✸✾

✷✻

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.08✱ τ = 5 × 10−6✱ Nf = 48✱ Nc = 2

❋✐❣✉r❡ ✹✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0.01, 0.015, T = 0.08✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✻✴✸✾

✷✼

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.06✱ τ = 1 × 10−6✱ Nf = 128✱ Nc = 2

❋✐❣✉r❡ ✺✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 0.0015, 0.003✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✼✴✸✾

✷✽

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.06✱ τ = 1 × 10−6✱ Nf = 128✱ Nc = 2

❋✐❣✉r❡ ✻✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0.00505, 0.0051, T = 0.06✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✽✴✸✾

✷✾

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.001✱τ = 10−5✱ Nf = 64✱ Nc = 2

❋✐❣✉r❡ ✼✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 1.5 × 10−4, 3.5 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✷✾✴✸✾

✸✵

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

γ =

1 12π✱ T = 0.001✱τ = 10−5✱ Nf = 64✱ Nc = 2

❋✐❣✉r❡ ✽✿ ✭α = 0✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 4 × 10−4, 4.5 × 10−4, 1 × 10−3✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✵✴✸✾

✸✶

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 114✱ γ =

1 12π✱ T = 5 × 10−4✱ τ = 1 × 10−7✱ Nf = 128✱ Nc = 16

❋✐❣✉r❡ ✾✿ ✭α = 114π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 8 × 10−5, 1.2 × 10−5✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✶✴✸✾

✸✷

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 114✱ γ =

1 12π✱ T = 5 × 10−4✱ τ = 1 × 10−7✱ Nf = 128✱ Nc = 16

❋✐❣✉r❡ ✶✵✿ ✭α = 114π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 2 × 10−4, 2.4 × 10−4, 3.6 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✷✴✸✾

✸✸

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 114✱ γ =

1 12π✱ T = 5 × 10−4✱ τ = 10−7✱ Nf = 128✱ Nc = 16

❋✐❣✉r❡ ✶✶✿ ✭α = 114π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 8 × 10−5, 1.2 × 10−5✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✸✴✸✾

✸✹

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 114✱ γ =

1 12π✱ T = 5 × 10−4✱ τ = 10−7✱ Nf = 128✱ Nc = 16

❋✐❣✉r❡ ✶✷✿ ✭α = 114π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 2 × 10−4, 2.4 × 10−4, 3.6 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✹✴✸✾

✸✺

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 300✱ γ =

1 12π✱ T = 1.25 × 10−4✱ τ = 10−7✱ Nf = 128 ✱ Nc = 16

❋✐❣✉r❡ ✶✸✿ ✭α = 300π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 0, 2.5 × 10−5, 7.5 × 10−5✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✺✴✸✾

✸✻

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 300✱ γ =

1 12π✱ T = 1.25 × 10−4✱ τ = 10−7✱ Nf = 128 ✱ Nc = 16

❋✐❣✉r❡ ✶✹✿ ✭α = 300π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t)✱ ✇✐t❤ ❝✉t t❤r♦✉❣❤ t❤❡ ♠❡s❤ ❛t x3 = 0 ❛t t✐♠❡s t = 1.15 × 10−4, 1.2 × 10−4, T = 1.25 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✻✴✸✾

✸✼

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 120✱ γ =

1 12π✱ T = 2.7 × 10−4✱ τ = 10−7✱ Nf = 128 ✱ Nc = 16

❋✐❣✉r❡ ✶✺✿ ✭α = 120π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t) ❛t t✐♠❡s t = 0, 7 × 10−5, 1.3 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✼✴✸✾

✸✽

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

α = 120✱ γ =

1 12π✱ T = 2.7 × 10−4✱ τ = 10−7✱ Nf = 128 ✱ Nc = 16

❋✐❣✉r❡ ✶✻✿ ✭α = 120π✮ ❩❡r♦ ❧❡✈❡❧ s❡ts ❢♦r Uε(x, t) ❛t t✐♠❡s t = 1.9 × 10−4, 2.3 × 10−4, T = 2.7 × 10−4✳

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✽✴✸✾

✸✾

❋✐♥❛❧ r❡♠❛r❦s

• ❢❛st ❝♦❛rs❡ s♦❧✈❡r ♥❡❡❞❡❞ ❢♦r ❡✣❝✐❡♥❝②
• ❧✐♠✐t❡❞ ✢❡①✐❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ γ
• r♦❜✉st ❡①❝❡♣t ❛❜♦✈❡ r❡♠❛r❦s
• ≈ 2× ❧❡ss ✐t❡r❛t✐♦♥s t❤❛♥ ❯③❛✇❛
• t❤❡♦r②❄

❽✉❜♦♠ír ❇❛➡❛s ❍❛rr❛❝❤♦✈ ✷✵✵✼ ✸✾✴✸✾