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▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❢♦r ♥♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

▲❛✉r❡♥❝❡ ❍❆▲P❊❘◆

▲❆●❆ ✲ ❯♥✐✈❡rs✐té P❛r✐s ✶✸

❙✐♠✉❧❛t✐♦♥ ♦❢ ❋❧♦✇ ✐♥ P♦r♦✉s ▼❡❞✐❛ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ❲❛st❡ ▼❛♥❛❣❡♠❡♥t ❛♥❞ ❈❖✷ ❙❡q✉❡str❛t✐♦♥ ❘❛❞♦♥ ■♥st✐t✉t❡ ✐♥ ▲✐♥③ ❖❦t♦❜❡r ✷✵✶✶

✶ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❖✉t❧✐♥❡

✶

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✷

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✸

❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠s ❢♦r ❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✹

◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❆ t❤❡♦r❡t✐❝❛❧ st✉❞② ❆ ♠♦❞✐✜❡❞ ❛♣♣r♦❛❝❤

✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❊①❛♠♣❧❡ ✿ ♦❝❡❛♥✲❛t♠♦s♣❤❡r❡ ❝♦✉♣❧✐♥❣

❋❧♦r✐❛♥ ▲❡♠❛r✐é t❤ès❡ ✷✵✵✾ ❤tt♣✿✴✴✇✇✇✳❛t♠♦s✳✉❝❧❛✳❡❞✉✴ ❢❧♦r✐❛♥✴❘❡s❡❛r❝❤✳❤t♠❧

✸ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ✐ss✉❡s

❖♥❡ ❝♦✉♣❧❡s ❞✐✛❡r❡♥t ♠♦❞❡❧s ✐♥ ❞✐✛❡r❡♥t ③♦♥❡s✱ ✇❤✐❝❤ ❧✐✈❡ ♦♥ ❞✐✛❡r❡♥t t✐♠❡✲s❝❛❧❡s✳ ❚❤❡ ❝♦✉♣❧✐♥❣ ♠✉st ❜❡ ✈❡r② ❡✣❝✐❡♥t ✿ ♦♥❡ ♦r t✇♦ ✐t❡r❛t✐♦♥s✳ ❚❤❡ ❞❛t❛ ♦❢ ♦♥❡ ♠♦❞❡❧ ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ❛t ❡❛❝❤ t✐♠❡✲st❡♣ ♦❢ t❤❡ ♥❡✐❣❤❜♦r✐♥❣ ♠♦❞❡❧✳ ❖♥❡ ♠✉st ❛✈♦✐❞ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ t✐♠❡ t♦ ♦✈❡rr✉❧❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✳

✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ✐ss✉❡s

❖♥❡ ❝♦✉♣❧❡s ❞✐✛❡r❡♥t ♠♦❞❡❧s ✐♥ ❞✐✛❡r❡♥t ③♦♥❡s✱ ✇❤✐❝❤ ❧✐✈❡ ♦♥ ❞✐✛❡r❡♥t t✐♠❡✲s❝❛❧❡s✳ ❚❤❡ ❝♦✉♣❧✐♥❣ ♠✉st ❜❡ ✈❡r② ❡✣❝✐❡♥t ✿ ♦♥❡ ♦r t✇♦ ✐t❡r❛t✐♦♥s✳ ❚❤❡ ❞❛t❛ ♦❢ ♦♥❡ ♠♦❞❡❧ ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ❛t ❡❛❝❤ t✐♠❡✲st❡♣ ♦❢ t❤❡ ♥❡✐❣❤❜♦r✐♥❣ ♠♦❞❡❧✳ ❖♥❡ ♠✉st ❛✈♦✐❞ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ t✐♠❡ t♦ ♦✈❡rr✉❧❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✳

✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ✐ss✉❡s

❖♥❡ ❝♦✉♣❧❡s ❞✐✛❡r❡♥t ♠♦❞❡❧s ✐♥ ❞✐✛❡r❡♥t ③♦♥❡s✱ ✇❤✐❝❤ ❧✐✈❡ ♦♥ ❞✐✛❡r❡♥t t✐♠❡✲s❝❛❧❡s✳ ❚❤❡ ❝♦✉♣❧✐♥❣ ♠✉st ❜❡ ✈❡r② ❡✣❝✐❡♥t ✿ ♦♥❡ ♦r t✇♦ ✐t❡r❛t✐♦♥s✳ ❚❤❡ ❞❛t❛ ♦❢ ♦♥❡ ♠♦❞❡❧ ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ❛t ❡❛❝❤ t✐♠❡✲st❡♣ ♦❢ t❤❡ ♥❡✐❣❤❜♦r✐♥❣ ♠♦❞❡❧✳ ❖♥❡ ♠✉st ❛✈♦✐❞ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ t✐♠❡ t♦ ♦✈❡rr✉❧❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✳

✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ✐ss✉❡s

❖♥❡ ❝♦✉♣❧❡s ❞✐✛❡r❡♥t ♠♦❞❡❧s ✐♥ ❞✐✛❡r❡♥t ③♦♥❡s✱ ✇❤✐❝❤ ❧✐✈❡ ♦♥ ❞✐✛❡r❡♥t t✐♠❡✲s❝❛❧❡s✳ ❚❤❡ ❝♦✉♣❧✐♥❣ ♠✉st ❜❡ ✈❡r② ❡✣❝✐❡♥t ✿ ♦♥❡ ♦r t✇♦ ✐t❡r❛t✐♦♥s✳ ❚❤❡ ❞❛t❛ ♦❢ ♦♥❡ ♠♦❞❡❧ ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ❛t ❡❛❝❤ t✐♠❡✲st❡♣ ♦❢ t❤❡ ♥❡✐❣❤❜♦r✐♥❣ ♠♦❞❡❧✳ ❖♥❡ ♠✉st ❛✈♦✐❞ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ t✐♠❡ t♦ ♦✈❡rr✉❧❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✳

✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥t❡♥ts

✶

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✷

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✸

❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠s ❢♦r ❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✹

◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❆ t❤❡♦r❡t✐❝❛❧ st✉❞② ❆ ♠♦❞✐✜❡❞ ❛♣♣r♦❛❝❤

✺ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❖✉t❧✐♥❡

✶

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✷

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✸

❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠s ❢♦r ❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✹

◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❆ t❤❡♦r❡t✐❝❛❧ st✉❞② ❆ ♠♦❞✐✜❡❞ ❛♣♣r♦❛❝❤

✻ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ❛♥❝❡st♦r ✿ ➱♠✐❧❡ P✐❝❛r❞

❏♦✉r♥✳ ❞❡ ▼❛t❤✳ ✭✹è♠❡ sér✐❡✮✱ t♦♠❡ ❱■✳✲❋❛s❝ ■■✱ ✶✽✾✵ ❚r❛✐té ❞✬❆♥❛✲ ❧②s❡✱ t♦♠❡ ✷✱ ✶✽✾✶✱●❛✉t❤✐❡r✲ ❱✐❧❧❛rs✳

✼ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ▼❖❙

❊①❡♠♣❧❡ ♦r✐❣✐♥❛❧ ❞❡ ✶✾✽✷ ♣♦✉r ❧❡s ♠ét❤♦❞❡s ❞❡ r❡❧❛①❛t✐♦♥ ❞✬♦♥❞❡s ❈✐r❝✉✐t ❝♦♠♣❧❡t ❈✐r❝✉✐t ♣❛rt✐t✐♦♥♥é

✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

❘❡✈✐❡✇ ✿ ❇✉rr❛❣❡ ❡t ❛❧✱ ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✶✾✾✻✳ ˙ ②✶ = ❢✶(t, ②✶, ②✷, · · · , ②♣), ˙ ②✷ = ❢✷(t, ②✶, ②✷, · · · , ②♣) ˙ ②❥ = ❢❥(t, ②✶, ②✷, · · · , ②♣) ˙ ②♣ = ❢♣(t, ②✶, ②✷, · · · , ②♣)

✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

❘❡✈✐❡✇ ✿ ❇✉rr❛❣❡ ❡t ❛❧✱ ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✶✾✾✻✳ ˙ ②✶ = ❢✶(t, ②✶, ②✷, · · · , ②♣), ˙ ②✷ = ❢✷(t, ②✶, ②✷, · · · , ②♣) ˙ ②❥ = ❢❥(t, ②✶, ②✷, · · · , ②♣) ˙ ②♣ = ❢♣(t, ②✶, ②✷, · · · , ②♣)

❆♣♣r♦①✐♠❛t✐♦♥s s✉❝❝❡ss✐✈❡s

˙ ②(❦+✶)

✶

= ❢✶(t, ②(❦)

✶

, ②(❦)

✷

, · · · , ②(❦)

♣

), ˙ ②(❦+✶)

✷

= ❢✷(t, ②(❦)

✶

, ②(❦)

✷

, · · · , ②(❦)

♣

) ˙ ②(❦+✶)

❥

= ❢❥(t, ②(❦)

✶

, ②(❦)

✷

, · · · , ②(❦)

♣

) ˙ ②(❦+✶)

♣

= ❢♣(t, ②(❦)

✶

, ②(❦)

✷

, · · · , ②(❦)

♣

) ②(❦+✶)−②∞ ≤ ▲❦(❚ − t✵)❦ ❦! ②(✵)−②∞

▲✐♥❞❡❧ö❢✱ ❏♦✉r♥✳ ❞❡ ▼❛t❤✳ ✭✹è♠❡ sé✲

r✐❡✮✱ t♦♠❡ ❳✳✲❋❛s❝ ■■✱ ✶✽✾✹

✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

❘❡✈✐❡✇ ✿ ❇✉rr❛❣❡ ❡t ❛❧✱ ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✶✾✾✻✳ ˙ ②✶ = ❢✶(t, ②✶, ②✷, · · · , ②♣), ˙ ②✷ = ❢✷(t, ②✶, ②✷, · · · , ②♣) ˙ ②❥ = ❢❥(t, ②✶, ②✷, · · · , ②♣) ˙ ②♣ = ❢♣(t, ②✶, ②✷, · · · , ②♣)

❏❛❝♦❜✐

˙ ②(❦+✶)

✶

= ❢✶(t, ②(❦+✶)

✶

, ②(❦)

✷

, ②(❦)

❥

, · · · , ②(❦)

♣

), ˙ ②(❦+✶)

✷

= ❢✷(t, ②(❦)

✶

, ②(❦+✶)

✷

, ②(❦)

❥

, · · · , ②(❦)

♣

) ˙ ②(❦+✶)

❥

= ❢❥(t, ②(❦)

✶

, ②(❦)

✷

, · · · , ②(❦+✶)

❥

, · · · , ②(❦)

♣

) ˙ ②(❦+✶)

♣

= ❢♣(t, ②(❦)

✶

, ②(❦)

✷

, ②(❦)

❥

, · · · , ②(❦+✶)

♣

)

✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

❘❡✈✐❡✇ ✿ ❇✉rr❛❣❡ ❡t ❛❧✱ ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✶✾✾✻✳ ˙ ②✶ = ❢✶(t, ②✶, ②✷, · · · , ②♣), ˙ ②✷ = ❢✷(t, ②✶, ②✷, · · · , ②♣) ˙ ②❥ = ❢❥(t, ②✶, ②✷, · · · , ②♣) ˙ ②♣ = ❢♣(t, ②✶, ②✷, · · · , ②♣)

• ❛✉ss✲❙❡✐❞❡❧

˙ ②(❦+✶)

✶

= ❢✶(t, ②(❦+✶)

✶

, ②(❦)

✷

, · · · , ②(❦)

❥

, ②(❦)

♣

), ˙ ②(❦+✶)

✷

= ❢✷(t, ②(❦+✶)

✶

, ②(❦+✶)

✷

, · · · , ②(❦)

❥

, ②(❦)

♣

) ˙ ②(❦+✶)

❥

= ❢❥(t, ②(❦+✶)

✶

, ②(❦+✶)

✷

, · · · , ②(❦+✶)

❥

, · · · , ②(❦)

♣

) ˙ ②(❦+✶)

♣

= ❢♣(t, ②(❦+✶)

✶

, ②(❦+✶)

✷

, · · · , ②(❦+✶)

❥

, ②(❦+✶)

♣

)

✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

❘❡✈✐❡✇ ✿ ❇✉rr❛❣❡ ❡t ❛❧✱ ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✶✾✾✻✳ ˙ ②✶ = ❢✶(t, ②✶, ②✷, · · · , ②♣), ˙ ②✷ = ❢✷(t, ②✶, ②✷, · · · , ②♣) ˙ ②❥ = ❢❥(t, ②✶, ②✷, · · · , ②♣) ˙ ②♣ = ❢♣(t, ②✶, ②✷, · · · , ②♣)

❚❤❡♦r❡t✐❝❛❧ st✉❞② ✿ ❖✳ ◆❡✈❛♥❧✐♥♥❛✱ ❘❡♠❛r❦s ♦♥ P✐❝❛r❞✲▲✐♥❞❡❧ö❢ ✐t❡r❛t✐♦♥s✳ ❇■❚ ✶✾✽✾✳ ❚❤❡r❡ ❝❛♥ ❜❡ ❛ s❧♦✇ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ str♦♥❣ ❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ s✉❜❝✐r❝✉✐ts✳ ❋♦rt✉♥❛t❡❧② ❢♦r ✐♥t❡❣r❛t❡❞ ♣r♦❜❧❡♠s✱ t❤✐s str♦♥❣ ❝♦✉♣❧✐♥❣ ♦❝❝✉rs ♦♥❧② ♦♥ s❤♦rt t✐♠❡ ✐♥t❡r✈❛❧s✳

✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❖✉t❧✐♥❡

✶

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✷

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✸

❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠s ❢♦r ❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✹

◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❆ t❤❡♦r❡t✐❝❛❧ st✉❞② ❆ ♠♦❞✐✜❡❞ ❛♣♣r♦❛❝❤

✶✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠

Ω✶ Ω✷ Γ✶ Γ✷ t

   (∂t − ∆)✉❦+✶

✶

= ❢ ✐♥ Ω✶ × (✵, ❚) ✉❦+✶

✶

(·, ✵) = ✉✵ ✐♥ Ω✶ ✉❦+✶

✶

= ✉❦

✷

♦♥ Γ✶ × (✵, ❚)    (∂t − ∆)✉❦+✶

✷

= ❢ ✐♥ Ω✷ × (✵, ❚) ✉❦+✶

✷

(·, ✵) = ✉✵ ✐♥ Ω✷ ✉❦+✶

✷

= ✉❦

✶

♦♥ Γ✷ × (✵, ❚)

• ❛♥❞❡r ✶✾✾✼ ✭t❤❡s✐s✮✱ ●✐❧❛❞✐✲❑❡❧❧❡r ✶✾✾✼ ✭■❝❛s❡ r❡♣♦rt✮✳

✶✶ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

Pr♦♣❡rt✐❡s

• ❛♥❞❡r✲❙t✉❛rt ✶✾✾✼✱ ●❛♥❞❡r✲❩❤❛♦ ✷✵✵✷✳ ●✐❧❛❞✐✲❑❡❧❧❡r ✷✵✵✷

✭❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥✮

2 4 6 8 10 12 14 16 18 20 10

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10 iteration k maximum error 0.1 overlap 0.06 overlap

❚❂✵✳✵✺ ✿ s✉♣❡r❧✐♥❡❛r ❝♦♥✈❡r❣❡♥❝❡ r❡❣✐♠❡

❡✷♥

❥

≤ ❡r❢❝( ♥δ

√ ❚ )❡✵ ❥

▲∞(Γ❥ × (✵, ❚))

2 4 6 8 10 12 14 16 18 20 10

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10

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10 Iteration maximum error 0.1 overlap 0.06 overlap

❚❂✸ ✿ ❧✐♥❡❛r ❝♦♥✈❡r❣❡♥❝❡ r❡❣✐♠❡

❡✷♥

❥

≤ (φ(δ))♥❡✵

❥

▲∞(Γ❥ × R+)

✶✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

Pr♦♣❡rt✐❡s

• ❛♥❞❡r✲❙t✉❛rt ✶✾✾✼✱ ●❛♥❞❡r✲❩❤❛♦ ✷✵✵✷✳ ●✐❧❛❞✐✲❑❡❧❧❡r ✷✵✵✷

✭❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥✮

2 4 6 8 10 12 14 16 18 20 10

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10 iteration k maximum error 0.1 overlap 0.06 overlap

❚❂✵✳✵✺ ✿ s✉♣❡r❧✐♥❡❛r ❝♦♥✈❡r❣❡♥❝❡ r❡❣✐♠❡

❡✷♥

❥

≤ ❡r❢❝( ♥δ

√ ❚ )❡✵ ❥

▲∞(Γ❥ × (✵, ❚))

2 4 6 8 10 12 14 16 18 20 10

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10

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10 Iteration maximum error 0.1 overlap 0.06 overlap

❚❂✸ ✿ ❧✐♥❡❛r ❝♦♥✈❡r❣❡♥❝❡ r❡❣✐♠❡

❡✷♥

❥

≤ (φ(δ))♥❡✵

❥

▲∞(Γ❥ × R+)

▼❛t❤❡♠❛t✐❝❛❧ t♦♦❧s ✿ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❛♥❞ ▲❛♣❧❛❝❡✲❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥ t✐♠❡✴tr❛♥s✈❡rs❡ s♣❛❝❡ ✈❛r✐❛❜❧❡s✳

✶✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❤❡ ▼♦❞✐✜❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠

❏❛❝♦❜✐ ♦r ●❛✉ss✲❙❡✐❞❡❧ ✇❛② ✿ L✉ := ∂t✉ + (❛ · ∇)✉ − ν∆✉ + ❝✉ ✐♥ Ω × (✵, ❚)    L✉❦+✶

✶

= ❢ ✐♥ Ω✶ × (✵, ❚) ✉❦+✶

✶

(·, ✵) = ✉✵ ✐♥ Ω✶ B✶✉❦+✶

✶

= B✶✉❦

✷

♦♥ Γ✶ × (✵, ❚)    L✉❦+✶

✷

= ❢ ✐♥ Ω✷ × (✵, ❚) ✉❦+✶

✷

(·, ✵) = ✉✵ ✐♥ Ω✷ B✷✉❦+✶

✷

= B✷✉❦+✶

✶

♦♥ Γ✷ × (✵, ❚)

✶✸ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

• ❡♥❡r❛❧✐s❛t✐♦♥

❖♣t✐♠❛❧ ❙❝❤✇❛r③ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲■❚❍ ❖❘ ❲■❚❍❖❯❚ ♦✈❡r❧❛♣✳ ❊q✉❛t✐♦♥ L✉ := ∂t✉ + (❛ · ∇)✉ − ν∆✉ + ❝✉ ✐♥ Ω × (✵, ❚) ❇♦✉♥❞❛r② ♦♣❡r❛t♦rs B❥✉ := (ν∇✉ − ❛) · ♥❥ + ♣✉ + q(∂t + ❛ · ∇✉ − ν∆❙✉ + ❝✉) ❚❤❡♦r❡♠s ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s ♦♥ ♣, q✱ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ✇❡❧❧✲♣♦s❡❞ ✐♥ s✉✐t❡❞ ❙♦❜♦❧❡✈ s♣❛❝❡s ❛♥❞ ❝♦♥✈❡r❣❡s ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t ♦✈❡r❧❛♣✱ ✐♥❝❧✉❞✐♥❣ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts ❛♥❞ ❝✉r✈❡❞ ❜♦✉♥❞❛r✐❡s ✭▼❛rt✐♥✱ ❍❛❧♣❡r♥✱ ❏❛♣❤❡t✱ ❙③❡❢t❡❧✱ ●❛♥❞❡r✱ ✇✐t❤♦✉t ♦✈❡r❧❛♣✱ ▼✐♥❤ ❇✐♥❤ ❚r❛♥ ✇✐t❤ ♦✈❡r❧❛♣✮✳

✶✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈❤♦✐❝❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts

♣ ❧

L✉ := ∂t✉ + (❛ · ∇)✉ − ν∆✉ + ❝✉ ✐♥ Ω × (✵, ❚) B❥✉ := (ν∇✉ − ❛) · ♥ + ♣✉ + q(∂t + ❛ · ∇✉ − ν∆❙✉ + ❝✉) ❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❜❧❡♠✱ ♣∗, q∗(❛, ν, ❝, ∆t, ∆①).

✶✺ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❆s②♠♣t♦t✐❝ ❢♦r♠✉❧❛❡ ∆t ∼ ❈ ′❤

❉✐r✐❝❤❧❡t ♦r ◆❡✉♠❛♥♥ tr❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ρ∗ ∼ ✶ − ❈❤ ♦✈❡r❧❛♣ ▲ ≈ ❈❤ ❘♦❜✐♥ tr❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s

• ρ∗ ∼ ✶ − ❈❤✶/✷

✇✐t❤♦✉t ♦✈❡r❧❛♣ ρ∗ ∼ ✶ − ❈❤✶/✸ ✇✐t❤ ♦✈❡r❧❛♣ q = ✵

• ρ∗ ∼ ✶ − ❈❤✶/✹

✇✐t❤♦✉t ♦✈❡r❧❛♣ ρ∗ ∼ ✶ − ❈❤✶/✺ ✇✐t❤ ♦✈❡r❧❛♣

✶✻ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❚❡st ❝❛s❡ ✿ ♣♦r♦s✐t② ✰ r♦t❛t✐♥❣ ✈❡❧♦❝✐t②✰ ♥♦♥❝♦♥❢♦r♠❛❧ ❣r✐❞s

L✉ = ϕ∂✉ ∂t + ∇ · (❛(①)✉ − ν(①)∇✉) + ❝✉ = ❢ ❛✶ = (−s✐♥( π

✷ (② − ✶))❝♦s(π(① − ✶ ✷)), ✸❝♦s( π ✷ (② − ✶))s✐♥(π(① − ✶ ✷)))✱

ν✶ = ✵.✵✵✸✱ ϕ✶ = ✵.✶ ✱ ❛✷ = ❛✶✱ ν✷ = ✵.✵✶ ✱ ϕ✷ = ✶

❈♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ (✵, ✶) × (✵, ✷)✱ ✜♥❛❧ t✐♠❡ ❚ = ✶.✺

−0.2 0.2 0.4 0.6 0.8 1 1.2 −0.5 0.5 1 1.5 2 2.5 x y Velocity field 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 DG−OSWR Solution, At time t=T=0 x y 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

✶✼ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥✈❡r❣❡♥❝❡

♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ (♣∗, q∗) ♦❜t❛✐♥❡❞ ❜② ♦♣t✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❢❛❝t♦r✳

2 4 6 8 10 12 14 16 18 20 10

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Error between momodomain (variational) and DGOSWR solutions versus the iterations Iterations log10(Error) Order2 Robin

✶✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥ ❢♦r ❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s

❘♦❜✐♥ tr❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ❛r❡ ❜❡tt❡r t❤❛♥ ❉✐r✐❝❤❧❡t✱ ❜✉t s❡❝♦♥❞ ♦r❞❡r tr❛♥s♠✐ss✐♦♥ ❝♦♥❞✐t✐♦♥s ✐♠♣r♦✈❡ s✐❣♥✐✜❝❛♥t❧②✳ ♦✈❡r❧❛♣ ✐s ❜❡tt❡r ✐❢ ♣♦ss✐❜❧❡✱ ❜✉t ♥♦♥♦✈❡r❧❛♣♣✐♥❣ ✇✐t❤ s❡❝♦♥❞ ♦r❞❡r s❤♦✉❧❞ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐❢ ♥♦t✳ ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✐s ❛❧♠♦st ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♣❛r❛♠❡t❡rs✳ ❱❡r② r♦❜✉st ✇❤❡♥ ❛♣♣❧✐❡❞ t♦ ✈❛r✐❛❜❧❡ ❝♦❡✣❝✐❡♥ts✳

✶✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❖✉t❧✐♥❡

✶

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✷

❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥

✸

❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❛❧❣♦r✐t❤♠s ❢♦r ❛❞✈❡❝t✐♦♥✲❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✹

◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❆ t❤❡♦r❡t✐❝❛❧ st✉❞② ❆ ♠♦❞✐✜❡❞ ❛♣♣r♦❛❝❤

✷✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❋r❛♠❡

❋r❛♠❡ ✿ ❆◆❘ ❙❍P ❈❖✷✳ ❈♦❧❧❛❜♦r❛t♦rs ✿ ❋✐❧✐♣❛ ❈❛❡t❛♥♦✱ ❆✳ ▼✐❝❤❡❧ ✭■❋P✮✱ ❋❧♦r✐❛♥ ❍❛❡❜❡r❧❡✐♥ ✭❞♦❝t♦r❛♥t✮✳ ▼❛rt✐♥ ●❛♥❞❡r✱ ✱ ❏éré♠✐❡ ❙③❡❢t❡❧✱ ▼✐♥❤ ❇✐♥❤ ❚r❛♥ ✭❞♦❝t♦r❛♥t✮✳

✷✶ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❆ t❤❡♦r❡t✐❝❛❧ st✉❞②

▲✳ ❍❛❧♣❡r♥ ❛♥❞ ❏✳ ❙③❡❢t❡❧✳ ◆♦♥❧✐♥❡❛r ❙❝❤✇❛r③ ❲❛✈❡❢♦r♠ ❘❡❧❛①❛t✐♦♥ ❢♦r ❙❡♠✐❧✐♥❡❛r ❲❛✈❡ Pr♦♣❛❣❛t✐♦♥✳ ▼❛t❤✳ ❈♦♠♣✳ ✷✵✵✾✱ ❋✳ ❈❛❡t❛♥♦✱ ▼✳ ●❛♥❞❡r✱ ▲✳ ❍❛❧♣❡r♥ ❛♥❞ ❏✳ ❙③❡❢t❡❧✳ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r s❡♠✐❧✐♥❡❛r r❡❛❝t✐♦♥✲❞✐✛✉s✐♦♥✳ ❙❡♣t❡♠❜❡r ✷✵✶✵

✉t − ν∆✉ + ❢ (✉) = ✵ ❞❛♥s R❞ × (✵, ❚), ✉(·, ✵) = ✉✵ ❞❛♥s R❞, ❢ ∈ C✷(R), ❢ (✵) = ✵✳ ❚❤❡♦r❡♠ ■❢ ✉✵ ∈ ❍✷(R✷), t❤❡♥ t❤❡r❡ ❡①✐sts ❚ > ✵ s✉❝❤ t❤❛t t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❤❛s ❛ ✉♥✐q✉❡ ✇❡❛❦ s♦❧✉t✐♦♥ ✉ ∈ ▲✷(✵, ❚; ❍✶(R✷))∩ C([✵, ❚]; ▲✷(R✷)). ■♥ ❛❞❞✐t✐♦♥ ✉ ∈ ▲∞(✵, ❚; ❍✷(R✷)).

✷✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

L✉ := ∂t✉ − ν∆✉ + ❢ (✉)    L✉❦+✶

✶

= ✵ ✐♥ Ω✶ × (✵, ❚ ❦+✶

✶

) ✉❦+✶

✶

(·, ✵) = ✉✵ ✐♥ Ω✶ B✶✉❦+✶

✶

= B✶✉❦

✷

♦♥ Γ✶ × (✵, ❚ ❦+✶

✶

)    L✉❦+✶

✷

= ✵ ✐♥ Ω✷ × (✵, ❚ ❦+✶

✷

) ✉❦+✶

✷

(·, ✵) = ✉✵ ✐♥ Ω✷ B✷✉❦+✶

✷

= B✷✉❦+✶

✶

♦♥ Γ✷ × (✵, ❚ ❦+✶

✷

) ❇✐(✉) = ν ∂✉ ∂♥✐ + ♣✉ + q ∂✉ ∂t − ν ∂✷✉ ∂② ✷

• ,

♣ > ✵, q ≥ ✵

✷✸ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❊①✐st❡♥❝❡ ❢♦r t❤❡ ✐t❡r❛t❡s

❚❤❡♦r❡♠ ▲❡t ❣ ✵

✶ ❛♥❞ ❣ ✵ ✷ ✐♥ ❍✶(✵, ❚; ▲✷(Γ)) ∩ ▲∞(✵, ❚; ❍

✶ ✷ (Γ))✱ ✉✵ ∈ ❍✷(R✷),

♣ > ✵ ❛♥❞ q ≥ ✵ ❜❡ ❣✐✈❡♥✳ ❙✉♣♣♦s❡ t❤❛t (ν∂♥✐ ✉✵ + ♣✉✵)|Γ = ❣ ✵

✐ (✵, ·), ✐❢

q = ✵. ❚❤❡♥✱ t❤❡ ❛❧❣♦r✐t❤♠ ❞❡✜♥❡s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ♦❢ ✐t❡r❛t❡s (✉❦

✶, ✉❦ ✷)

q = ✵ : ✉❦

✐ ∈ ❲ ✶,∞(✵, ❚ ❦ ✐ ; ▲✷(Ω✐)) ∩ ▲∞(✵, ❚ ❦ ✐ ; ❍✷(Ω✐)) ∩ ❍✶(✵, ❚ ❦ ✐ ; ❍✶(Ω✐)),

q > ✵ : ✉❦

✐ ∈ ❲ ✶,∞(✵, ❚ ❦ ✐ ; ▲✷(Ω✐)) ∩ ▲∞(✵, ❚ ❦ ✐ ; ❍✷ ✷(Ω✐)) ∩ ❍✶(✵, ❚ ❦ ✐ ; ❍✶(Ω✐)),

❢♦r s♦♠❡ ❚ ❦

✐ ≤ ❚

✷✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❊①✐st❡♥❝❡ ❢♦r t❤❡ ✐t❡r❛t❡s

❚❤❡♦r❡♠ ▲❡t ❣ ✵

✶ ❛♥❞ ❣ ✵ ✷ ✐♥ ❍✶(✵, ❚; ▲✷(Γ)) ∩ ▲∞(✵, ❚; ❍

✶ ✷ (Γ))✱ ✉✵ ∈ ❍✷(R✷),

♣ > ✵ ❛♥❞ q ≥ ✵ ❜❡ ❣✐✈❡♥✳ ❙✉♣♣♦s❡ t❤❛t (ν∂♥✐ ✉✵ + ♣✉✵)|Γ = ❣ ✵

✐ (✵, ·), ✐❢

q = ✵. ❚❤❡♥✱ t❤❡ ❛❧❣♦r✐t❤♠ ❞❡✜♥❡s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ♦❢ ✐t❡r❛t❡s (✉❦

✶, ✉❦ ✷)

q = ✵ : ✉❦

✐ ∈ ❲ ✶,∞(✵, ❚ ❦ ✐ ; ▲✷(Ω✐)) ∩ ▲∞(✵, ❚ ❦ ✐ ; ❍✷(Ω✐)) ∩ ❍✶(✵, ❚ ❦ ✐ ; ❍✶(Ω✐)),

q > ✵ : ✉❦

✐ ∈ ❲ ✶,∞(✵, ❚ ❦ ✐ ; ▲✷(Ω✐)) ∩ ▲∞(✵, ❚ ❦ ✐ ; ❍✷ ✷(Ω✐)) ∩ ❍✶(✵, ❚ ❦ ✐ ; ❍✶(Ω✐)),

❢♦r s♦♠❡ ❚ ❦

✐ ≤ ❚

❚ ❦

✶ ≤ ❚ ❦−✶ ✷

, ❚ ❦

✷ ≤ ❚ ❦−✶ ✶

✷✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

Pr♦♦❢

q = ✵✳ ❇❛♥❛❝❤✬s ✜①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ✐♥ ❛ ❜❛❧❧ ✐♥ H(❚) := ❲ ✶,∞(✵, ❚; ▲✷(Ω✐)) ∩ ▲∞(✵, ❚; ❍✷(Ω✐)) ∩ ❍✶(✵, ❚; ❍✶(Ω✐)). ▲❡♠♠❛ ▲❡t ❚ > ✵✳ ▲❡t ✉✵ ∈ ❍✷(Ω✐)✱ ❣ ∈ ❍✶(✵, ❚; ▲✷(Γ)) ∩ ▲∞(✵, ❚; ❍

✶ ✷ (Γ))✳

❋♦r ❛♥② ✈ ∈ H(❚)✱ t❤❡ ❧✐♥❡❛r ♣r♦❜❧❡♠      ✇t − ν∆✇ = −❢ (✈), ✐♥ Ω✐ × (✵, ❚), ✇(·, ·, ✵) = ✉✵|Ω✐ , ✐♥ Ω✐, ν ∂✇

∂♥✐ + ♣✇ = ❣,

♦✈❡r Γ × (✵, ❚). ✭✶✮ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ H(❚)✱ ❤❡♥❝❡ ❞❡✜♥✐♥❣ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ✇ = T (✈) ✐♥ H(❚)✱ ✇✐t❤ T (✈)✷

H(❚) ≤ ❈❡❚

✉✵✷

❍✷(Ω✐ ) + ❣✷ ❍✶(✵,❚;▲✷(Γ))∩▲∞(✵,❚;❍

✶ ✷ (Γ))

+❚(ϕ(✈▲∞((✵,❚)×Ω✐ )))✷✈✷

❲ ✶,∞(✵,❚;▲✷(Ω✐ ))

• ✭✷✮

✷✺ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❝♦♥t✐♥✉❡

▲❡t ▼ ❜❡ s✉❝❤ t❤❛t ▼✷ ≥ ✹❈(✉✵✷

❍✷(Ω✐ ) + ❣✷ ❍✶(✵,❚;▲✷(Γ))∩▲∞(✵,❚;❍

✶ ✷ (Γ))),

✭✸✮ ✇❤❡r❡ ❈ ✐s t❤❡ ✉♥✐✈❡rs❛❧ ❝♦♥st❛♥t ♦❢ ❡st✐♠❛t❡ ✭✷✮✱ ❛♥❞ ❞❡✜♥❡ t❤❡ t✐♠❡ ❚✵(▼) = s✉♣{❚ ′ ≤ ❚, ♠❛① ❡❚ ′ ✷ , ✷❈❡❚ ′(ϕ(▼))✷❚ ′, ✷❡

❚′ ✷ √

❚ ′ϕ(▼)

• ≤ ✶}.

✭✹✮ ▲❡♠♠❛ ❉❡✜♥❡ B▼ := {✇ ∈ H(❚✵) : ✇H(❚✵) ≤ ▼}. ❚❤❡♥ T (B▼) ⊆ B▼✱ B▼ ✐s ❛ ❝❧♦s❡❞ ♠❡tr✐❝ s✉❜s♣❛❝❡ ♦❢ ▲∞(✵, ❚✵; ▲✷(Ω✐))✱ ❛♥❞ T ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ✐♥ B▼✳

✷✻ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❆ ❝♦♠♠♦♥ t✐♠❡ ♦❢ ❡①✐st❡♥❝❡

❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ▼ ❛♥❞ ❚ s✉❝❤ t❤❛t✱ ✐❢ ✉✵✷

❍✷(Ω✐ ) + ❣ ✵ ✐ ✷ ❍✶(✵,❚;▲✷(Γ))∩▲∞(✵,❚;❍

✶ ✷ (Γ)) ≤ ▼ ✷,

✭✺✮ (✉❦

✶, ✉❦ ✷) ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ✐♥t❡r✈❛❧ [✵, ❚ ] ❢♦r ❛❧❧ ♣♦s✐t✐✈❡ ❦✳

E❑

❙ = ❑

• ❦=✵

E❦, U❑(t) = s✉♣

✵≤❦≤❑

✉❦

✶(t)∞ +

s✉♣

✵≤❦≤❑

✉❦

✷(t)∞.

       E❑

❙ (t) +

t

✵

G❑(s)❞s ≤ ✷ t

✵

ϕ

• U❑(s)
• E❑

❙ (s)❞s +

t

✵

G✵(s)❞s, ♠❛①

✵≤❦≤❑

t

✵

G❦(s)❞s ≤ ✷ t

✵

ϕ

• U❑(s)
• E❑

❙ (s)❞s +

t

✵

G✵(s)❞s. ✭✻✮ U❑(t) ≤ ✹❈✹(✶+E❑

❙ (t)+(ϕ′(U❑(t)))✷E❑ ❙ (t)+ ♠❛① ✵≤❦≤❑

t

✵

G❦(s) ❞s). ✭✼✮ ❋r♦♠ t❤✐s ❣❡t ❚ ♦♥ ✇❤✐❝❤ ❛❧❧ t❤❡ q✉❛♥t✐t✐❡s ❛r❡ ❜♦✉♥❞❡❞✳

✷✼ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥✈❡r❣❡♥❝❡

❚❤❡♦r❡♠ ❲✐t❤ t❤❡ ♥♦t❛t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✺✱ t❤❡ s❡q✉❡♥❝❡ (✉❦

✶, ✉❦ ✷) ❝♦♥✈❡r❣❡s✱ ❛s

❦ → ∞, t♦ (✉|Ω✶, ✉|Ω✷), ✐♥ ▲∞(✵, ❚ ; ❍✶(Ω✐))✳

✷✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❆❧❣♦r✐t❤♠❡ ❞❡ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ♣♦✉r ✉♥❡ éq✉❛t✐♦♥ ♥♦♥ ❧✐♥é❛✐r❡

         ∂t✇ ❦+✶

✐

+ L✇ ❦+✶

✐

+ F(✇ ❦+✶

✐

) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × ((✵, ❚) B✐✇ ❦+✶

✐

= B✐✇ ❦

✸−✐

❞❛♥s Γ × (✵, ❚) B✐✉ := ∂♥✐ ✉ + ♣✉

✷✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

The numerical approximation

Discretization in each sub-domain: ⊲ Finite elements in space, finite differences in time. ⊲ Semi-implicit Euler method. Discretization of the iterative algorithm: ⊲ Construction of the interface operator (u1, u2) − → (B1(u2), B2(u1)). Numerical tests: ⊲ Computational domain: Ω×]0, T[, where Ω =] − 1, 1[×]0, 1[. ⊲ Ω1 =] − 1, 0[×]0, T[, Ω2 =]0, 1[×]0, T[. ⊲ Nonlinear reactions: f (u) = u3, f (u) = eu − 1, . . . ⊲ Numerical solutions: we compare ⊲ The mono-domain solution in Ω; ⊲ The domain decomposition solution.

Filipa Caetano – p. 11

✸✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

Numerical results - Robin transmission conditions

Error, in the L∞ norm, between the DD and the M solutions. Robin conditions: different values of p.

1 2 3 4 5 6 7 8 9 10 10

−5

10

−4

10

−3

10

−2

10

−1

10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations Robin p = 9.5 Robin p = 40 Filipa Caetano – p. 12

✸✶ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

Optimized parameters - nonlinear transmission conditions

Motivation: there exists optimized parameters popt and (popt, qopt) for the linear reaction-diffusion equation ∂tu − ν∆u + bu = 0, which satisfy popt ∼ popt(b, ν, ∆t), qopt ∼ qopt(b, ν, ∆t) ([Bennequin, Gander, Halpern]). − → Replace b by f ′(u): p(u) = popt( f ′(u), ν, ∆t), q(u) = qopt( f ′(u), ν, ∆t). Nonlinear transmission conditions: Robin: Bi(u) = ∂u

∂ni + p(u)u.

Order 2: Bi(u) = ∂u

∂ni + p(u)u + q(u)(∂tu − ν∆yu).

Filipa Caetano – p. 13

✸✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

Nonlinear Robin and nonlinear Order 2

Difference, in the L∞ norm, between the DD solution and the M solutions. Nonlinear Robin and order 2 conditions. f (u) = u3. T = 1. ν = 1. h = 0.125 and h = 0.0625.

1 2 3 4 5 6 7 8 9 10 10

−15

10

−10

10

−5

10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations Nonlinear Robin Popt−num=8.5 Nonlinear Order 2 1 2 3 4 5 6 7 8 9 10 10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations Nonlinear Robin Popt−num=9.5 Nonlinear Order 2

Filipa Caetano – p. 14

✸✸ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

Nonlinear Robin and nonlinear Order 2

Difference, in the L∞ norm, between the DD solution and the M solutions. Nonlinear Robin and order 2 conditions. f (u) = u3. T = 1. ν = 0.2. h = 0.125 and h = 0.0625.

1 2 3 4 5 6 7 8 9 10 10−12 10 −10 10 −8 10 −6 10 −4 10 −2 10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations Nonlinear Robin Popt−num=12.25 Nonlinear Order 2 1 2 3 4 5 6 7 8 9 10 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations Nonlinear Robin Popt−num=14.5 Nonlinear Order 2

Filipa Caetano – p. 15

✸✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

A simple model in geological CO2 storage modelling

The model: reactive chemical system with two types of materials, evolving through equilibrium values ueq

1 and ueq 2 .

Heterogeneous distribution of the materials in the spatial domain. f (x, y, u) = k1S1(x, y)(u − ueq

1 )3 + k2S2(x, y)(u − ueq 2 )3.

k1, k2: reaction speeds. S1, S2: distribution of the materials in the spatial domain.

Filipa Caetano – p. 16

✸✺ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

A simple model in geological CO2 storage modelling

−1 −0.5 0.5 1 0.5 1 1.5 2 0.49 0.495 0.5 0.505 DD solution 0.494 0.496 0.498 0.5 0.502 −1 −0.5 0.5 1 1 2 0.49 0.495 0.5 0.505 Monodomain solution 0.494 0.496 0.498 0.5 0.502 −1 −0.5 0.5 1 0.5 1 1.5 2 Error at time t=1 2 4 6 8 x 10

−5

Filipa Caetano – p. 17

✸✻ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆✉♠❡r✐❝s

A simple model in geological CO2 storage modelling

Difference, in the L∞ norm, between the DD solution and the M solution. Nonlinear Robin conditions: different values of h.

1 2 3 4 5 6 7 8 9 10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 Number of iterations Linf error Linf error between DD and monodomain solutions after 10 iterations h=0.125 h=0.0625 h=0.03125 Filipa Caetano – p. 18

✸✼ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥ ✿ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠

         ∂t✇ ❦+✶

✐

+ L✇ ❦+✶

✐

+ F(✇ ❦+✶

✐

) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × ((✵, ❚) B✐✇ ❦+✶

✐

= B✐✇ ❦

✸−✐

❞❛♥s Γ × (✵, ❚) B✐✉ := ∂♥✐ ✉ + ♣✉

✸✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥ ✿ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠

         ∂t✇ ❦+✶

✐

+ L✇ ❦+✶

✐

+ F(✇ ❦+✶

✐

) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × ((✵, ❚) B✐✇ ❦+✶

✐

= B✐✇ ❦

✸−✐

❞❛♥s Γ × (✵, ❚) M✐ : ( λ , ❢ ) → (✇✐) s♦❧✉t✐♦♥ ❞❡          ∂t✇✐ + L✇✐ + F(✇✐) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × (✵, ❚) B✐✇ ❦+✶

✐

= λ ❞❛♥s Γ × (✵, ❚) λ❦

✶ = −λ❦−✶ ✷

+ ✷♣ M✷(λ❦−✶

✷

, ❢ ) λ❦

✷ = −λ❦−✶ ✶

+ ✷♣ M✶(λ❦−✶

✶

, ❢ ).

✸✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥ ✿ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠

M✐ : ( λ , ❢ ) → (✇✐) s♦❧✉t✐♦♥ ❞❡          ∂t✇✐ + L✇✐ + F(✇✐) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × (✵, ❚) B✐✇ ❦+✶

✐

= λ ❞❛♥s Γ × (✵, ❚) λ❦

✶ = −λ❦−✶ ✷

+ ✷♣ M✷(λ❦−✶

✷

, ❢ ) λ❦

✷ = −λ❦−✶ ✶

+ ✷♣ M✶(λ❦−✶

✶

, ❢ ). ▲✐♥❡❛r ❝❛s❡ ✿ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❂ ❏❛❝♦❜✐ ❢♦r t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ■ ■ ✷♣

✷

✵ ■ ✷♣

✶

✵ ■

✶ ✷

✷♣

✷ ✵ ❢

✷♣

✶ ✵ ❢

✸✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❡q✉❛t✐♦♥ ✿ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠

M✐ : ( λ , ❢ ) → (✇✐) s♦❧✉t✐♦♥ ❞❡          ∂t✇✐ + L✇✐ + F(✇✐) = ❢✇ ❞❛♥s Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ ❞❛♥s Ω✐ G✇✐ = ❣ ❞❛♥s (∂Ω✐ \ Γ) × (✵, ❚) B✐✇ ❦+✶

✐

= λ ❞❛♥s Γ × (✵, ❚) λ❦

✶ = −λ❦−✶ ✷

+ ✷♣ M✷(λ❦−✶

✷

, ❢ ) λ❦

✷ = −λ❦−✶ ✶

+ ✷♣ M✶(λ❦−✶

✶

, ❢ ). ▲✐♥❡❛r ❝❛s❡ ✿ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❂ ❏❛❝♦❜✐ ❢♦r t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠

• ■

■ − ✷♣ M✷(·, ✵) ■ − ✷♣ M✶(·, ✵) ■

• ·
• λ✶

λ✷

• =

✷♣ M✷(✵, ❢ ) ✷♣ M✶(✵, ❢ )

• ✸✾ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) := −λ✷ + ✷♣ M✷(λ✷, ❢ ) −λ✶ + ✷♣ M✶(λ✶, ❢ )

• −

λ✶ λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) := −λ✷ + ✷♣ M✷(λ✷, ❢ ) −λ✶ + ✷♣ M✶(λ✶, ❢ )

• −

λ✶ λ✷

• = ✵.

❖♥❡ ✐t❡r❛t✐♦♥ ❂ r❡s♦❧✉t✐♦♥ ♦❢ ❛ ♥♦♥ ❧✐♥❡❛r ♣r♦❜❧❡♠✳ ❝❢ ❛❜♦✈❡

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) :=

• −λ✷ + ✷♣ M✷(λ✷, ❢ )

−λ✶ + ✷♣ M✶(λ✶, ❢ )

• −
• λ✶

λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤

❋✐①❡❞ ♣♦✐♥t → ◆❡✇t♦♥ ✿ Ψ′(λ♥) · (λ♥+✶ − λ♥) = −Ψ(λ♥) M❧✐♥

✐

: ( ❆ , λ , ❢ ) → ✇✐ s♦❧✉t✐♦♥ ❞❡          ∂t✇✐ + L✇✐ + ❆ ✇✐ = ❢✇ Ω✐ × (✵, ❚) ✇✐(·, ✵) = ✇✵ Ω✐ J ✇✐ = ❣ (∂Ω✐ \ Γ) × (✵, ❚) B✐✇✐ = λ Γ × (✵, ❚)

Ψ′(λ) =

• −■

−■ + ✷♣ M❧✐♥

✷ (F′(M✷(λ✷, ✵)), ·, ✵)

−■ + ✷♣ M❧✐♥

✶ (F′(M✷(λ✶, ✵)), ·, ✵)

−■

• .

❑r②❧♦✈ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠

✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) := −λ✷ + ✷♣ M✷(λ✷, ❢ ) −λ✶ + ✷♣ M✶(λ✶, ❢ )

• −

λ✶ λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) :=

• −λ✷ + ✷♣ M✷(λ✷, ❢ )

−λ✶ + ✷♣ M✶(λ✶, ❢ )

• −
• λ✶

λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✶

◆❡✇t♦♥ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ (∂t + L + F ′(✇ ♥))✇ ♥+✶ = F ′(✇ ♥)✇ ♥ − F(✇ ♥) + ❢✇ ♦♥ Ω × (✵, ❚) ✇ ♥+✶(①, ✵) = ✇✵(①) ♦♥ Ω G(✇ ♥+✶) = ❣(①, t) ♦♥ ∂Ω × (✵, ❚)

✷

❖♥❡ ❣❡t ❛ ❧✐♥❡❛r ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠✱ ✇✐t❤ ♠❛tr✐① Ψ′(λ♥) ✿

• ■❞

■❞ −✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ·, ✵)

■❞ −✷♣ M❧✐♥

✶ (F′(✇ ♥ ✶ ), ·, ✵)

■❞

• ·

λ♥+✶

✶

λ♥+✶

✷

• =

= ✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ✵, (F ′(✇ ♥ ✷ )✇ ♥ ✷ − F(✇ ♥ ✷ ) + q, ✇✵, ❣))

✷♣ M❧✐♥

✶ (F ′(✇ ♥ ✶ ), ✵, (F′(✇ ♥ ✶ )✇ ♥ ✶ − F(✇ ♥ ✶ ) + q, ✇✵, ❣))

• .

✸

❘❡s♦❧✉t✐♦♥ ✇✐t❤ ❑r②❧♦✈

✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) :=

• −λ✷ + ✷♣ M✷(λ✷, ❢ )

−λ✶ + ✷♣ M✶(λ✶, ❢ )

• −
• λ✶

λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✶

◆❡✇t♦♥ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ (∂t + L + F ′(✇ ♥))✇ ♥+✶ = F ′(✇ ♥)✇ ♥ − F(✇ ♥) + ❢✇ ♦♥ Ω × (✵, ❚) ✇ ♥+✶(①, ✵) = ✇✵(①) ♦♥ Ω G(✇ ♥+✶) = ❣(①, t) ♦♥ ∂Ω × (✵, ❚)

✷

❖♥❡ ❣❡t ❛ ❧✐♥❡❛r ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠✱ ✇✐t❤ ♠❛tr✐① Ψ′(λ♥) ✿

• ■❞

■❞ −✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ·, ✵)

■❞ −✷♣ M❧✐♥

✶ (F′(✇ ♥ ✶ ), ·, ✵)

■❞

• ·

λ♥+✶

✶

λ♥+✶

✷

• =

= ✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ✵, (F ′(✇ ♥ ✷ )✇ ♥ ✷ − F(✇ ♥ ✷ ) + q, ✇✵, ❣))

✷♣ M❧✐♥

✶ (F ′(✇ ♥ ✶ ), ✵, (F′(✇ ♥ ✶ )✇ ♥ ✶ − F(✇ ♥ ✶ ) + q, ✇✵, ❣))

• .

✸

❘❡s♦❧✉t✐♦♥ ✇✐t❤ ❑r②❧♦✈

✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

◆❡✇t♦♥✴❙❝❤✇❛r③✴❑r②❧♦✈ ♦r ❙❝❤✇❛r③✴◆❡✇t♦♥✴❑r②❧♦✈ ♦✉ ✳✳✳ ❄

✶ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ t❤❡ ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ❛ ✜①❡❞ ♣♦✐♥t

❛❧❣♦r✐t❤♠ Ψ(λ, ❢ ) :=

• −λ✷ + ✷♣ M✷(λ✷, ❢ )

−λ✶ + ✷♣ M✶(λ✶, ❢ )

• −
• λ✶

λ✷

• = ✵.

✷ ◆❡st❡❞ ❛♣♣r♦❛❝❤ ✸ ❈♦♠♠♦♥ ◆❡✇t♦♥ ❛♣♣r♦❛❝❤ ✶

◆❡✇t♦♥ ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ (∂t + L + F ′(✇ ♥))✇ ♥+✶ = F ′(✇ ♥)✇ ♥ − F(✇ ♥) + ❢✇ ♦♥ Ω × (✵, ❚) ✇ ♥+✶(①, ✵) = ✇✵(①) ♦♥ Ω G(✇ ♥+✶) = ❣(①, t) ♦♥ ∂Ω × (✵, ❚)

✷

❖♥❡ ❣❡t ❛ ❧✐♥❡❛r ✐♥t❡r❢❛❝❡ ♣r♦❜❧❡♠✱ ✇✐t❤ ♠❛tr✐① Ψ′(λ♥) ✿

• ■❞

■❞ −✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ·, ✵)

■❞ −✷♣ M❧✐♥

✶ (F′(✇ ♥ ✶ ), ·, ✵)

■❞

• ·

λ♥+✶

✶

λ♥+✶

✷

• =

= ✷♣ M❧✐♥

✷ (F ′(✇ ♥ ✷ ), ✵, (F ′(✇ ♥ ✷ )✇ ♥ ✷ − F(✇ ♥ ✷ ) + q, ✇✵, ❣))

✷♣ M❧✐♥

✶ (F ′(✇ ♥ ✶ ), ✵, (F′(✇ ♥ ✶ )✇ ♥ ✶ − F(✇ ♥ ✶ ) + q, ✇✵, ❣))

• .

✸

❘❡s♦❧✉t✐♦♥ ✇✐t❤ ❑r②❧♦✈

✹✵ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♠♣❛r❛✐s♦♥

10

1

10

2

10

3

Number of grid cells per dimension, Nx = Ny Number matrix inversions Classical Approach Nested Iteration Approach Common Iteration Approach O(N1/2.75) O(N1/7)

■♠♣❧❡♠❡♥t❛t✐♦♥ ❜② ❋❧♦r✐❛♥ ❍❛❡❜❡r❧❡✐♥✱ ✜♥✐t❡ ✈♦❧✉♠❡s✱ q = ✵✱ ♥♦♥❝♦♥❢♦r♠❛❧ ♠❡s❤❡s✱ t♦ s♦♠❡ r❡❛❝t✐✈❡ s②st❡♠✱ ✇✐t❤ ❝❤❡♠✐str②✳

✹✶ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥

❊①t❡♥s✐♦♥ ♦❢ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❯s❡❞ t♦ r❡✜♥❡ ❧♦❝❛❧❧② t❤❡ ♠❡s❤ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡ P♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇ t❤❡ r❡❛❝t✐✈❡ ❢r♦♥t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❆r❝❛♥❡ ❙❡❡ t❤❡ r❡❛❧ st✉✛ ✐♥ ❆♥t❤♦♥② ▼✐❝❤❡❧✬s t❛❧❦✳

✹✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥

❊①t❡♥s✐♦♥ ♦❢ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❯s❡❞ t♦ r❡✜♥❡ ❧♦❝❛❧❧② t❤❡ ♠❡s❤ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡

♣

P♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇ t❤❡ r❡❛❝t✐✈❡ ❢r♦♥t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❆r❝❛♥❡ ❙❡❡ t❤❡ r❡❛❧ st✉✛ ✐♥ ❆♥t❤♦♥② ▼✐❝❤❡❧✬s t❛❧❦✳

✹✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥

❊①t❡♥s✐♦♥ ♦❢ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❯s❡❞ t♦ r❡✜♥❡ ❧♦❝❛❧❧② t❤❡ ♠❡s❤ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡

♣

P♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇ t❤❡ r❡❛❝t✐✈❡ ❢r♦♥t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❆r❝❛♥❡ ❙❡❡ t❤❡ r❡❛❧ st✉✛ ✐♥ ❆♥t❤♦♥② ▼✐❝❤❡❧✬s t❛❧❦✳

✹✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥

❊①t❡♥s✐♦♥ ♦❢ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❯s❡❞ t♦ r❡✜♥❡ ❧♦❝❛❧❧② t❤❡ ♠❡s❤ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡

♣

P♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇ t❤❡ r❡❛❝t✐✈❡ ❢r♦♥t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❆r❝❛♥❡ ❙❡❡ t❤❡ r❡❛❧ st✉✛ ✐♥ ❆♥t❤♦♥② ▼✐❝❤❡❧✬s t❛❧❦✳

✹✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥❝❧✉s✐♦♥

❊①t❡♥s✐♦♥ ♦❢ ♦♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❯s❡❞ t♦ r❡✜♥❡ ❧♦❝❛❧❧② t❤❡ ♠❡s❤ ✐♥ t✐♠❡ ❛♥❞ s♣❛❝❡

♣

P♦ss✐❜❧❡ t♦ ❢♦❧❧♦✇ t❤❡ r❡❛❝t✐✈❡ ❢r♦♥t ■♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❆r❝❛♥❡ ❙❡❡ t❤❡ r❡❛❧ st✉✛ ✐♥ ❆♥t❤♦♥② ▼✐❝❤❡❧✬s t❛❧❦✳

✹✷ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❆ ❢❡✇ ♥✉♠❜❡rs

∂✉ ∂t + ∇ · (❛(①, ③)✉ − ν①(③) ✵ ✵ ν③(③)

• ∇✉) = ✵.

❖❝❡❛♥ ✿ (✵, ▲) × (−❤❖, ✵)✱ ❛t♠♦s♣❤❡r❡ ✿(✵, ▲) × (✵, ❤❆) ❤❖ = ✺❦♠✱ ❤❆ = ✸✵❦♠✱ ▲ = ✺✵✵✵❦♠✳ ❛(①, ③) = ( ✺✵

✻ s✐♥( ✷π① ▲ ) ❝♦s( π③ ❤❆ ), −✵.✶ ❝♦s( ✷π① ▲ ) s✐♥( π③ ❤❆ )),

−❤❖ ≤ ③ ≤ ✵ (✵.✺ s✐♥( ✷π①

▲ ) ❝♦s( π③ ❤❖ ), −✵.✵✵✶ ❝♦s( ✷π① ▲ ) s✐♥( π③ ❤❖ )),

✵ ≤ ③ ≤ ❤❆ ν①(③) =

• ✶✵✵,

③ ≤ ✵ ✶✵✵✵✵, ③ ≥ ✵ , ν③(③) =

• ✶✵−✺ + (✶✵−✷ − ✶✵−✺)❡

−(③+✶✵✵)✷ ✽✵✵✵

, ③ ≤ ✵ ✶✵−✸ + (✵.✶ − ✶✵−✸)❡

−(③−✸✵)✷ ✶✵✵✵

, ③ ≥ ✵ ■♥✐t✐❛❧ st❛t❡ ✉✵ =    ✶✹❡− ❧♥(✼)

✶✵✺ (③+✹✾✺)✷) + ✷,

−❤❖ ≤ ③ ≤ −✹✾✺, ✶✻, −✹✾✺ ≤ ③ ≤ ✵, ✶✻ −

✹✶③ ✸✵✵✵✵,

✵ ≤ ③ ≤ ❤❆. ❚♦t❛❧ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥ ✿ ✺ ❞❛②s✳ ❍♦♠♦❣❡♥❡♦✉s ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳

✹✸ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❙✐♠✉❧❛t✐♦♥

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

6

−0.5 0.5 1 1.5 2 2.5 3 x 10

4

x y Mesh

◆❆,① = ✻✵, ◆❆,③ = ✼✵, ◆❆,t = ✾✵ ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ P✶ ✐♥ t✐♠❡ ◆❖,① = ✷✹✵, ◆❖,③ = ✺✵, ◆❖,t = ✶✷ P✶ ✜♥✐t❡ ❡❧❡♠❡♥ts ✐♥ s♣❛❝❡✳ ◆✇ = ✷✵ t✐♠❡ ✇✐♥❞♦✇s ♦❢ ✻ ❤♦✉r ❡❛❝❤ ✸ ✐t❡r❛t✐♦♥s ❛r❡ s✉✣❝✐❡♥t t♦ r❡❛❝❤ t❤❡ ❣❧♦❜❛❧ s❝❤❡♠❡ ❛❝❝✉r❛❝②✳

✹✹ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♠♣❧❡①✐t②

◆① = ♠❛①(◆✵,①, ◆❆,①) = ✷✹✵✱ ◆③ = ◆❖,③ + ◆❆,③ = ✶✷✵✱ ◆t = ♠❛①(◆❖,t, ◆❆,t) = ✾✵✳

✶ ◆✶ ✿ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥t❛r② ♦♣❡r❛t✐♦♥s ❢♦r t❤❡ ♠♦♥♦❞♦♠❛✐♥ s♦❧✉t✐♦♥✱ ✷ ◆✷(♣) ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥t❛r② ♦♣❡r❛t✐♦♥s ❢♦r t❤❡ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠

r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✇✐t❤ ♣ ✐t❡r❛t✐♦♥s✳ ◆✶ ∼ ✷◆✸

③ ◆① + ✹◆✷ ③ ◆①◆t◆✇

◆✷(♣) ∼ ✷ ♠❛①(◆✸

❆,③◆❆,①, ◆✸ ❖,③◆❖,①) + ✹♣ ♠❛① ✶≤✐≤✷(◆✷ ✐,②◆✐,①◆✐,t)◆✇

✹✺ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♠♣❧❡①✐t②

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10

10

number of iterations p complexity N2(3) N2(p) N1 2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 x 10

10

number of iterations p complexity N2(p) with N1,t=12, N2,t=90 N2(p) with N1,t=N2,t=90 N1

❢✉❧❧ ✜♥❡ ❝♦♠♣✉t❛t✐♦♥✴ ❝♦♥❢♦r♠❛❧ ✐♥ t✐♠❡✴ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♥♦♥ ❝♦♥❢♦r♠❛❧ ✐♥ t✐♠❡ ✇✐t❤ ♥♦♥ ❝♦♥❢♦r♠❛❧ ♠❡s❤ r❡✜♥❡♠❡♥t ❈♦♠♣✉t❛t✐♦♥s ✿ ❈❛r♦❧✐♥❡ ❏❛♣❤❡t✳

▲❍✫ ❈✳ ❏❛♣❤❡t ✫ ❏✳ ❙③❡❢t❡❧✳ ❙♣❛❝❡✲❚✐♠❡ ◆♦♥❝♦♥❢♦r♠✐♥❣ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ❲❛✈❡❢♦r♠ ❘❡❧❛①❛t✐♦♥ ❢♦r ❍❡t❡r♦❣❡♥❡♦✉s Pr♦❜❧❡♠s ❛♥❞ ●❡♥❡r❛❧ ●❡♦♠❡tr✐❡s✳ ❉♦♠❛✐♥ ❉❡❝♦♠♣♦s✐t✐♦♥ ▼❡t❤♦❞s ✐♥ ❙❝✐❡♥❝❡ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ❳■❳✱ ▲❡❝t✳ ◆♦t❡s ❈♦♠♣✉t✳ ❙❝✐✳ ❊♥❣✳✱ ✼✽✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥ ✷✵✶✶✳ ❍✉❛♥❣✱ ❨✳ ❀ ❑♦r♥❤✉❜❡r✱ ❘✳ ❀ ❲✐❞❧✉♥❞✱ ❖✳ ❀ ❳✉✱ ❏✳ ❡❞s✳ ♣♣ ✼✺✲✽✻✳

♣ ✹✻ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥✈❡r❣❡♥❝❡ ❢❛❝t♦r

❚❤❡ ❝❛s❡ ♦❢ ❤❛❧❢✲s♣❛❝❡s ❛♥❞ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts ✿ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐♥ t✐♠❡ ❛♥❞ tr❛♥s✈❡rs❡ s♣❛❝❡ δ(③) = ❛✷ + ✹ν❝ + ✹ν③, ③ = ✐(ω + ❜ · ❦) + ν|❦|✷, ❈♦♥✈❡r❣❡♥❝❡ ❢❛❝t♦r ρ(ω, ❦, P, ▲) = P − δ✶/✷ P + δ✶/✷ ✷ ❡−✷δ✶/✷▲/ν, P(③) = ♣ + q③.

✹✼ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈❤♦✐❝❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts

φ(③) = ❛✷ + ✹ν❝ + ✹ν③, ③ = ✐(ω + ❜ · ❦) + ν|❦|✷ ρ(③, P, ▲) = P(③) − φ✶/✷(③) P(③) + φ✶/✷(③) ✷ ❡−✷δ✶/✷▲ ❚❛②❧♦r ❡①♣❛♥s✐♦♥✱P ③ ✵ ✷ ③ ✵ ✱ ❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥❢

P P♥ s✉♣ ③ ❑

③ P ▲ ❑ ❚ t ❦❥ ❳❥ ①❥

♣ ✹✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈❤♦✐❝❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts

φ(③) = ❛✷ + ✹ν❝ + ✹ν③, ③ = ✐(ω + ❜ · ❦) + ν|❦|✷ ρ(③, P, ▲) = P(③) − φ✶/✷(③) P(③) + φ✶/✷(③) ✷ ❡−✷δ✶/✷▲ ❚❛②❧♦r ❡①♣❛♥s✐♦♥✱P(③) =

• φ(✵) + ✷ν③/
• φ(✵)✱

❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥❢

P P♥ s✉♣ ③ ❑

③ P ▲ ❑ ❚ t ❦❥ ❳❥ ①❥

♣ ✹✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈❤♦✐❝❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts

φ(③) = ❛✷ + ✹ν❝ + ✹ν③, ③ = ✐(ω + ❜ · ❦) + ν|❦|✷ ρ(③, P, ▲) = P(③) − φ✶/✷(③) P(③) + φ✶/✷(③) ✷ ❡−✷δ✶/✷▲ ❚❛②❧♦r ❡①♣❛♥s✐♦♥✱P(③) =

• φ(✵) + ✷ν③/
• φ(✵)✱

❇❡st ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥❢

P∈P♥ s✉♣ ③∈❑

|ρ(③, P, ▲)|, ❑ = ( π ❚ , π ∆t ), ❦❥ ∈ ( π ❳❥ , π ∆①❥ )

♣ ✹✽ ✴ ✹✾

▼♦t✐✈❛t✐♦♥ ❢♦r ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❲❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❖♣t✐♠✐③❡❞ ❙❝❤✇❛r③ ✇❛✈❡❢♦r♠ r❡❧❛①❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ◆♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s

❈♦♥✈❡r❣❡♥❝❡

♠❡❛♥ ✈❛❧✉❡ ♦❢ t❤❡ (♣∗, q∗) ♦❜t❛✐♥❡❞ ❜② ♦♣t✐♠✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❢❛❝t♦r✳

2 4 6 8 10 12 14 16 18 20 10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

Error between momodomain (variational) and DGOSWR solutions versus the iterations Iterations log10(Error) Order2 Robin

✹✾ ✴ ✹✾