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■t❡r❛t✐✈❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❛ s♣❛❝❡✇✐s❡ ❞❡♣❡♥❞❡♥t s♦✉r❝❡ ✐♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s

P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉❝❧❡❛r ❙❛❢❡t② ■♥st✐t✉t❡ ♦❢ ❘❆❙✱ ▼♦s❝♦✇✱ ❘✉ss✐❛ ◆♦rt❤✲❊❛st❡r♥ ❋❡❞❡r❛❧ ❯♥✐✈❡rs✐t②✱ ❨❛❦✉ts❦✱ ❘✉ss✐❛

◗✉❛s✐❧✐♥❡❛r ❊q✉❛t✐♦♥s✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ❉♦❧❣♦♣r✉❞♥②✱ ❘✉ss✐❛✱ ❙❡♣t❡♠❜❡r ✶✷✕✶✺✱ ✷✵✶✻

✶✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

❖✉t❧♦♦❦

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥ ■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠ ■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

• ❡♥❡r❛❧✐③❛t✐♦♥s

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✷✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

■♥✈❡rs❡ ♣r♦❜❧❡♠s

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ ♦❢ ♠❛♥② ❛♣♣❧✐❡❞ ♣r♦❜❧❡♠s ♦❢ s❝✐❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣ ❧❡❛❞s t♦ t❤❡ ♥❡❡❞ ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s✳ ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r❡ ♣❛rt✐❝✉❧❛r❧② ♥♦t❡✇♦rt❤②✳ ■♥✈❡rs❡ ♣r♦❜❧❡♠s ❛r❡ ❢♦r♠✉❧❛t❡❞ ❛s ♥♦♥✲❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡② ❛r❡ ♦❢t❡♥ ❝❧❛ss✐✜❡❞ ❛s ✐❧❧✲♣♦s❡❞ ✭❝♦♥❞✐t✐♦♥❛❧❧② ✇❡❧❧✲♣♦s❡❞✮ ♣r♦❜❧❡♠s✳ ■♥ t❤❡ t❤❡♦r❡t✐❝❛❧ st✉❞②✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ q✉❡st✐♦♥s ♦❢ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛♥❞ ✐ts st❛❜✐❧✐t② ❛r❡ ♣r✐♠❛r✐❧② ❝♦♥s✐❞❡r❡❞✳

❆❧✐❢❛♥♦✈ ❖▼✳ ■♥✈❡rs❡ ❍❡❛t ❚r❛♥s❢❡r Pr♦❜❧❡♠s✳ ❙♣r✐♥❣❡r❀ ✷✵✶✶✳ ▲❛✈r❡♥t✬❡✈ ▼▼✱ ❘♦♠❛♥♦✈ ❱●✱ ❙❤✐s❤❛ts❦✐✐ ❙P✳ ■❧❧✲♣♦s❡❞ Pr♦❜❧❡♠s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s ❛♥❞ ❆♥❛❧②s✐s✳ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②❀ ✶✾✽✻✳

✸✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

❈♦❡✣❝✐❡♥t ✐♥✈❡rs❡ ♣r♦❜❧❡♠s

❈♦❡✣❝✐❡♥t ✐♥✈❡rs❡ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ ✐❞❡♥t✐❢②✐♥❣ ❝♦❡✣❝✐❡♥t ❛♥❞✴♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ✉s❡ ♦❢ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ✐♥❢♦r♠❛t✐♦♥ ✐s ♦❢ ✐♥t❡r❡st ❛♠♦♥❣ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ♥♦♥✲st❛t✐♦♥❛r② ♣r♦❜❧❡♠s✱ t❛s❦s ♦❢ r❡❝♦✈❡r✐♥❣ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦♥ t✐♠❡ ♦r s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡ ✉s✉❛❧❧② tr❡❛t❡❞ ❛s ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡s❡ t❛s❦s r❡❧❛t❡ t♦ ❛ ❝❧❛ss ♦❢ ❧✐♥❡❛r ✐♥✈❡rs❡ ♣r♦❜❧❡♠s✱ ✇❤✐❝❤ s✉✣❝✐❡♥t❧② s✐♠♣❧✐✜❡s t❤❡✐r st✉❞②✳ ❖♥❧② ✐♥ s♦♠❡ ❝❛s❡s ✇❡ ❤❛✈❡ ❧✐♥❡❛r ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ✖ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❡q✉❛t✐♦♥✱ ❖t❤❡r ❝♦❡✣❝✐❡♥t ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❛r❡ ♥♦♥❧✐♥❡❛r✱ t❤❛t s✐❣♥✐✜❝❛♥t❧② ❝♦♠♣❧✐❝❛t❡❞ t❤❡✐r st✉❞②✳

■s❛❦♦✈ ❱✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ❢♦r P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r❀ ✷✵✻✳ Pr✐❧❡♣❦♦ ❆■✱ ❖r❧♦✈s❦② ❉●✱ ❱❛s✐♥ ■❆✳ ▼❡t❤♦❞s ❢♦r ❙♦❧✈✐♥❣ ■♥✈❡rs❡ Pr♦❜❧❡♠s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✳ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ■♥❝❀ ✷✵✵✵✳

✹✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

❆❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s

❚❤❡ t❛s❦ ♦❢ ✐❞❡♥t✐❢②✐♥❣ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦♥ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠s✳ ❆❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡ ♦❢t❡♥ ❢♦r♠✉❧❛t❡❞ ✉s✐♥❣ t❤❡ s♦❧✉t✐♦♥ ❛t t❤❡ ✜♥❛❧ ♠♦♠❡♥t ♦❢ t✐♠❡ ✖ ✜♥❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥✳ ■♥ ♠♦r❡ ❣❡♥❡r❛❧ ❝❛s❡ t❤❡ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ✐s st❛t❡❞ ❛s s♦♠❡ t✐♠❡ ✐♥t❡❣r❛❧ ❛✈❡r❛❣❡ ✖ ✐♥t❡❣r❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥✳ ❚❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ t♦ s✉❝❤ ❛♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ❛♥❞ ✇❡❧❧✲♣♦s❡❞♥❡ss ♦❢ t❤✐s ♣r♦❜❧❡♠ ✐♥ ✈❛r✐♦✉s ❢✉♥❝t✐♦♥❛❧ ❝❧❛ss❡s ❛r❡ ❡①❛♠✐♥❡❞ ✐♥ t❤❡ ♠❛♥② ✇♦r❦s✳

❘✉♥❞❡❧❧ ❲✳ ❆♣♣❧✐❝❛❜❧❡ ❆♥❛❧②s✐s✳ ✶✾✽✵❀✶✵✭✸✮✿✷✸✶✕✷✹✷✳ Pr✐❧❡♣❦♦ ❆■✱ ❙♦❧♦✈✬❡✈ ❱❱✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ✶✾✽✼❀✷✸✭✶✶✮✿✶✾✼✶✕✶✾✽✵✳ ■s❛❦♦✈ ❱✳ ❈♦♠♠✉♥✐❝❛t✐♦♥s ♦♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✳ ✶✾✾✶❀ ✹✹✭✷✮✿✶✽✺✕✷✵✾✳ Pr✐❧❡♣❦♦ ❆■✱ ❑♦st✐♥ ❆❇✳ ❙❜♦r♥✐❦✿ ▼❛t❤❡♠❛t✐❝s✳ ✶✾✾✸❀✼✺✭✷✮✿✹✼✸✕✹✾✵✳ ❑❛♠②♥✐♥ ❱▲✳ ▼❛t❤❡♠❛t✐❝❛❧ ◆♦t❡s✳ ✷✵✵✺❀✼✼✭✹✮✿✹✽✷✕✹✾✸✳

✺✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s

■♥ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ t❤❡ ♠❛✐♥ ❢♦❝✉s ✐s ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ st❛❜❧❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s t❤❛t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♣❡❝✉❧✐❛r ♣r♦♣❡rt✐❡s ♦❢ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s✳ ■♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ❛r❡ ❜❛s❡❞ ♦♥ ✉s✐♥❣ ❣r❛❞✐❡♥t ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✐❞✉❛❧ ❢✉♥❝t✐♦♥❛❧✳ ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ s✉❝❤ ❛♣♣r♦❛❝❤❡s r❡❧❛t❡s t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ✐♥✐t✐❛❧✲❜♦✉♥❞❛r② ♣r♦❜❧❡♠s ❢♦r t❤❡ ♦r✐❣✐♥❛❧ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ❛♥❞ ✐ts ❝♦♥❥✉❣❛t❡✳

❱♦❣❡❧ ❈❘✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ❢♦r ■♥✈❡rs❡ Pr♦❜❧❡♠s✳ ❙♦❝✐❡t② ❢♦r ■♥❞✉str✐❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s❀ ✷✵✵✷✳ ❙❛♠❛rs❦✐✐ ❆❆✱ ❱❛❜✐s❤❝❤❡✈✐❝❤ P◆✳ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙♦❧✈✐♥❣ ■♥✈❡rs❡ Pr♦❜❧❡♠s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✳ ❉❡ ●r✉②t❡r❀ ✷✵✵✼✳ ▲✐♦♥s ❏▲✳ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ♦❢ ❙②st❡♠s ●♦✈❡r♥❡❞ ❜② P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r❀ ✶✾✼✶✳ ▼❛❦s✐♠♦✈ ❱■✳ ❉②♥❛♠✐❝❛❧ ■♥✈❡rs❡ Pr♦❜❧❡♠s ♦❢ ❉✐str✐❜✉t❡❞ ❙②st❡♠s✳ ❱❙P❀ ✷✵✵✷✳

✻✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

■♥✈❡rs❡ ♣r♦❜❧❡♠ ✇✐t❤ ✜♥❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥

❋♦r t❤❡ r❡q✉✐r❡❞ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❛ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✐♠❡✱ ❛♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✇✐t❤ ✜♥❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❛ ❜♦✉♥❞❛r② ♣r♦❜❧❡♠ ❢♦r ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♦r❞❡r✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ❝❛♥ ✉s❡ st❛♥❞❛r❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ st❛t✐♦♥❛r② ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✳ ❙✉❝❤ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ✜♥✐t❡✲❞✐✛❡r❡♥❝❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❙❱ ❜♦♦❦ ✭s❡❝t✐♦♥ ✻✳✹✮✳ ■♥ t❤❡ ❳❨❏ ✇♦r❦ t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ tr❛♥s✐t✐♦♥ t♦ ❛ ❡✈♦❧✉t✐♦♥❛r② ♣r♦❜❧❡♠ ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡✱ ♣❡❝✉❧✐❛r✐t② ♦❢ ✇❤✐❝❤ ✐s t❤❡ ♥♦♥✲❧♦❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥✳

❙❛♠❛rs❦✐✐ ❆❆✱ ❱❛❜✐s❤❝❤❡✈✐❝❤ P◆✳ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙♦❧✈✐♥❣ ■♥✈❡rs❡ Pr♦❜❧❡♠s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✳ ❉❡ ●r✉②t❡r❀ ✷✵✵✼✳ ❳✐❛♥❣t✉❛♥ ❳✱ ❨❛♦♠❡✐ ❨✱ ❏✉♥①✐❛ ❲✳ ■♥✿ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✿ ❈♦♥❢❡r❡♥❝❡ ❙❡r✐❡s❀ ❱♦❧✳ ✷✾✵✳ ■❖P P✉❜❧✐s❤✐♥❣❀ ✷✵✶✶✳ ♣✳ ✵✶✷✵✶✼✳

✼✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■♥tr♦❞✉❝t✐♦♥

❚❤✐s ✇♦r❦

❲❡ ❝♦♥str✉❝t s♣❡❝✐❛❧ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ♦❢ ✐❞❡♥t✐✜❝❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ ❛ s♣❛❝❡✇✐s❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ s♦✉r❝❡ ✐♥ ❛ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s✳ ❚❤❡② ❢✉❧❧② t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t ❝♦♥s✐❞❡r❡❞ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢❡❛t✉r❡s✱ ✇❤✐❝❤ r❡❧❛t❡ t♦ t❤❡✐r ❡✈♦❧✉t✐♦♥❛r② ❝❤❛r❛❝t❡r✳ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ ❈❛✉❝❤② ♣r♦❜❧❡♠s ♦♥ ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❚❤❡ ✜rst ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ ♠❡t❤♦❞ r❡❧❛t❡s t♦ t❤❡ ✐t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦♥ t❤❡ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s✳ ❙✉❝❤ ❛♣♣r♦❛❝❤ ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❜❡❢♦r❡✳

Pr✐❧❡♣❦♦ ❆■✱ ❑♦st✐♥ ❆❇✳ ❖♥ ❝❡rt❛✐♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ✜♥❛❧ ❛♥❞ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥✳ ❙❜♦r♥✐❦✿ ▼❛t❤❡♠❛t✐❝s✳ ✶✾✾✸❀✼✺✭✷✮✿✹✼✸✕✹✾✵✳ Pr✐❧❡♣❦♦ ❆■✱ ❑♦st✐♥ ❆❇✳ ▼❛t❤❡♠❛t✐❝❛❧ ◆♦t❡s✳ ✶✾✾✸❀✺✸✭✶✮✿✻✸✕✻✻✳ ❑♦st✐♥ ❆❇✳ ❙❜♦r♥✐❦✿ ▼❛t❤❡♠❛t✐❝s✳ ✷✵✶✸❀✷✵✹✭✶✵✮✿✶✸✾✶✕✶✹✸✹✳

✽✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

✷❉ ♣r♦❜❧❡♠

▲❡t x = (x1, x2) ❛♥❞ Ω ❜❡ ❛ ❜♦✉♥❞❡❞ ♣♦❧②❣♦♥✳ ❚❤❡ ❞✐r❡❝t ♣r♦❜❧❡♠ ✐s ❢♦r♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✳ ❲❡ s❡❛r❝❤ u(x, t)✱ 0 ≤ t ≤ T, T > 0 s✉❝❤ t❤❛t ✐t ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r✿ ∂u ∂t − div(k(x)gradu) + c(x)u = f(x), x ∈ Ω, 0 < t ≤ T, ✇✐t❤ ❝♦❡✣❝✐❡♥ts 0 < k1 ≤ k(x) ≤ k2✱ c(x) ≥ 0✳ ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ❛❧s♦ s♣❡❝✐✜❡❞✿ k(x)∂u ∂n + µ(x)u = 0, x ∈ ∂Ω, 0 < t ≤ T, ✇❤❡r❡ µ(x) ≥ µ1 > 0, x ∈ ∂Ω ❛♥❞ n ✐s t❤❡ ♥♦r♠❛❧ t♦ Ω✳ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡ u(x, 0) = u0(x), x ∈ Ω. ❚❤❡ ❢♦r♠✉❧❛t✐♦♥ ♣r❡s❡♥ts t❤❡ ❞✐r❡❝t ♣r♦❜❧❡♠✱ ✇❤❡r❡ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛s ✇❡❧❧ ❛s t❤❡ ❜♦✉♥❞❛r② ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❛r❡ s♣❡❝✐✜❡❞✳

✾✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

■♥✈❡rs❡ ♣r♦❜❧❡♠

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✱ ✇❤❡r❡ ✐♥ ❡q✉❛t✐♦♥✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ f(x) ✐s ✉♥❦♥♦✇♥✳ ❆♥ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥ ✐s ♦❢t❡♥ ❢♦r♠✉❧❛t❡❞ ❛s u(x, T) = uT (x), x ∈ Ω. ■♥ t❤✐s ❝❛s❡✱ ✇❡ s♣❡❛❦ ❛❜♦✉t t❤❡ ✜♥❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❛❜♦✈❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ ❛ ♣❛✐r ♦❢ u(x, t), f(x) ❢r♦♠ ❡q✉❛t✐♦♥s ❛♥❞ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ✐s ✇❡❧❧✲♣♦s❡❞✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥❞✐t✐♦♥s ❢♦r ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❛❜♦✈❡✲♠❡♥t✐♦♥❡❞ ✇♦r❦s✳

✶✵✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

❇✐❧✐♥❡❛r ❢♦r♠

■♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ H = L2(Ω)✱ ✇❡ ❞❡✜♥❡ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ❛♥❞ ♥♦r♠ ✐♥ t❤❡ st❛♥❞❛r❞ ✇❛②✿ (u, v) =

• Ω

u(x)v(x)dx, u = (u, u)1/2. ❚♦ s♦❧✈❡ ♥✉♠❡r✐❝❛❧❧② t❤❡ ♣r♦❜❧❡♠✱ ✇❡ ❡♠♣❧♦② ✜♥✐t❡✲❡❧❡♠❡♥t ❛♣♣r♦①✐♠❛t✐♦♥s ✐♥ s♣❛❝❡✳ ❲❡ ❞❡✜♥❡ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ a(u, v) =

• Ω

(k grad u grad v + c uv) dx +

• ∂Ω

µ uvdx. ❲❡ ❤❛✈❡ a(u, u) ≥ δu2, δ > 0.

❇r❡♥♥❡r ❙❈✱ ❙❝♦tt ▲❘✳ ❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ♦❢ ✜♥✐t❡ ❡❧❡♠❡♥t ♠❡t❤♦❞s✳ ❙♣r✐♥❣❡r❀ ✷✵✵✽✳ ❚❤♦♠✁ ❡❡ ❱✳ ●❛❧❡r❦✐♥ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞s ❢♦r P❛r❛❜♦❧✐❝ Pr♦❜❧❡♠s✳ ❇❡r❧✐♥✿ ❙♣r✐♥❣❡r ❱❡r❧❛❣❀ ✷✵✵✻✳

✶✶✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

❋✐♥✐t❡ ❡❧❡♠❡♥ts

❉❡✜♥❡ ❛ s✉❜s♣❛❝❡ ♦❢ ✜♥✐t❡ ❡❧❡♠❡♥ts V h ⊂ H1(Ω)✳ ▲❡t xi, i = 1, 2, ..., Mh ❜❡ tr✐❛♥❣✉❧❛t✐♦♥ ♣♦✐♥ts ❢♦r t❤❡ ❞♦♠❛✐♥ Ω✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❡♥ ✉s✐♥❣ ▲❛❣r❛♥❣❡ ✜♥✐t❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ✜rst ♦r❞❡r ✭♣✐❡❝❡✲✇✐s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥✮ ✇❡ ❝❛♥ ❞❡✜♥❡ ♣②r❛♠✐❞ ❢✉♥❝t✐♦♥ χi(x) ⊂ V h, i = 1, 2, ..., Mh✱ ✇❤❡r❡ χi(xj) = 1, if i = j, 0, if i = j. ❋♦r v ∈ Vh✱ ✇❡ ❤❛✈❡ v(x) =

Mh

• i=i

viχi(x), ✇❤❡r❡ vi = v(xi), i = 1, 2, ..., Mh✳

✶✷✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

❉✐s❝r❡t❡ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r

❲❡ ❞❡✜♥❡ t❤❡ ❞✐s❝r❡t❡ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r A ❛s (Ay, v) = a(y, v), ∀ y, v ∈ V h. ❚❤❡ ♦♣❡r❛t♦r A ❛❝ts ♦♥ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ V h ❛♥❞ A = A∗ ≥ δI, δ > 0, ✭✶✮ ✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ✐♥ V h✳

✶✸✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

❙❡♠✐❞✐s❝r❡t❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠

❋♦r t❤❡ ♣r♦❜❧❡♠✱ ✇❡ ♣✉t ✐♥t♦ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ t❤❡ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥ ❢♦r w(t) ∈ V h✿ dw dt + Aw = ϕ, 0 < t ≤ T, w(0) = φ, ✇❤❡r❡ ϕ = Pf✱ φ = Pu0 ✇✐t❤ P ❞❡♥♦t✐♥❣ L2✲♣r♦❥❡❝t✐♦♥ ♦♥t♦ V h✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ❛ss✉♠❡ w(T) = ψ, ✇❤❡r❡ ψ = PuT ✳

✶✹✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❊q✉❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♦r❞❡r

❋♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✇✐t❤ ✜♥❞✐♥❣ w(t), ϕ t❤❡ s✐♠♣❧❡st ❛♣♣r♦❛❝❤ ✐s t♦ ❡❧✐♠✐♥❛t❡ ✈❛r✐❛❜❧❡ ϕ ❬✵✱ ✵❪✳ ❉✐✛❡r❡♥t✐❛t✐♥❣ ❡q✉❛t✐♦♥ ♦♥ t✐♠❡✱ ✇❡ ♦❜t❛✐♥ d2w dt2 + Adw dt = 0, 0 < t ≤ T. ❋✉rt❤❡r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤❡ ❝♦rr❡❝t♥❡ss ♦❢ s✉❝❤ ♣r♦❜❧❡♠✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠ ❛♥❞ ❡①❛♠♣❧❡s ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❱✳ ❚❤❡ ✇❡❛❦♥❡ss ♦❢ s✉❝❤ ❛♣♣r♦❛❝❤ ✐s ❝❛✉s❡❞ ❜② t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ♣r♦❜❧❡♠✳ ❲❡ ♣r❛❝t✐❝❛❧❧② ❧♦s❡ t❤❡ ❡✈♦❧✉t✐♦♥❛r② ❝❤❛r❛❝t❡r ♦❢ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ❛♥❞ ♠✉st st♦r❡ ❞❛t❛ ✐♥ ❡❛❝❤ t✐♠❡ st❡♣✳

❙❛♠❛rs❦✐✐ ❆❆✱ ❱❛❜✐s❤❝❤❡✈✐❝❤ P◆✳ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙♦❧✈✐♥❣ ■♥✈❡rs❡ Pr♦❜❧❡♠s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✳ ❉❡ ●r✉②t❡r❀ ✷✵✵✼✳

✶✺✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

◆♦♥✲❧♦❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

❚❤❡ s❡❝♦♥❞ ❛♣♣r♦❛❝❤ ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❳❨❏✮ ✐s ❜❛s❡❞ ♦♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡✳ ▲❡t v = dw dt ✱ t❤❡♥ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s dv dt + Av = 0, 0 < t ≤ T. ❲❡ ❢♦r♠✉❧❛t❡ ♥♦♥✲❧♦❝❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❤❛✈❡ v(0) + Aw(0) = ϕ, v(T) + Aw(T) = ϕ. ❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ②✐❡❧❞s v(T) − v(0) = χ, χ = A(φ − ψ).

❳✐❛♥❣t✉❛♥ ❳✱ ❨❛♦♠❡✐ ❨✱ ❏✉♥①✐❛ ❲✳ ■♥✿ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✿ ❈♦♥❢❡r❡♥❝❡ ❙❡r✐❡s❀ ❱♦❧✳ ✷✾✵✳ ■❖P P✉❜❧✐s❤✐♥❣❀ ✷✵✶✶✳ ♣✳ ✵✶✷✵✶✼✳

✶✻✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

■t❡r❛t✐✈❡ ♣r♦❝❡ss

❋♦r ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✇❡ ✉s❡ t❤❡ s✐♠♣❧❡st ✐t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❢♦r ❡q✉❛t✐♦♥✳ ❚❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ❚❤❡ ♥❡✇ ❛♣♣r♦①✐♠❛t✐♦♥ k + 1 ✐s ❢♦✉♥❞ ❜② s♦❧✈✐♥❣ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠✿ vk+1(0) = vk(T) − χ, dvk+1 dt + Avk+1 = 0, 0 < t ≤ T, k = 0, 1, ..., ✇✐t❤ s♦♠❡ ❣✐✈❡♥ v0(0)✳ ❚❤❡ ❞❡s✐r❡❞ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ❡q✉❛t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ✉s✐♥❣ vk+1(0)✱ ❢♦r ❡①❛♠♣❧❡✱ ❢r♦♠ t❤❡ ❡q✉❛❧✐t② ϕk+1 = φ + vk+1(0).

✶✼✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✲ ✶✴✷

❲❡ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ❢♦r ❡rr♦r zk+1(t) = vk+1(t) − v(t)✿ zk+1(0) = zk(T), dzk+1 dt + Azk+1 = 0, 0 < t ≤ T, k = 0, 1, ..., ✇✐t❤ ❣✐✈❡♥ z0(0)✳ ▼✉❧t✐♣❧②✐♥❣ ❡q✉❛t✐♦♥ ❢♦r zk ✐♥ V h ❜② zk✱ ✇❡ ♦❜t❛✐♥ dzk dt , zk

• + (Azk, zk) = 0.

❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t dzk dt , zk

• = zk d

dtzk, ②✐❡❧❞s d dtzk + δzk ≤ 0.

✶✽✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✲ ✷✴✷

❚❤✉s✱ zk(t) ≤ exp(−δt)zk(0). ❲❡ ❤❛✈❡ zk+1(0) = zk(T) ≤ exp(−δT)zk(0). ❚❤✐s ❣✐✈❡s t❤❡ ❞❡s✐r❡❞ ❡st✐♠❛t❡ vk+1(0) − v(0) ≤ ̺ vk(0) − v(0), ̺ = exp(−δT), ❢♦r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✇✐t❤ ❧✐♥❡❛r s♣❡❡❞ ̺ < 1✳ ❋♦r t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✇❡ ❤❛✈❡ ϕk+1 − ϕ ≤ ̺ vk(0) − v(0).

✶✾✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❚✇♦✲❧❡✈❡❧ s❝❤❡♠❡

❋♦r ❝♦♠♣✉t❛t✐♦♥❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ t❤❡ t✐♠❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡s❡r✈❡s s♣❡❝✐❛❧ ❛tt❡♥t✐♦♥✳ ▲❡t ✉s ❞❡✜♥❡ ❛ ✉♥✐❢♦r♠ ❣r✐❞ ✐♥ t✐♠❡ tn = nτ, n = 0, 1, ..., N, τN = T ❛♥❞ ❞❡♥♦t❡ yn = y(tn), tn = nτ✳ ❋♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✇❡ ✉s❡❞ ❢✉❧❧② ✐♠♣❧✐❝✐t t✇♦✲❧❡✈❡❧ s❝❤❡♠❡✱ ✇❤❡♥ vn+1 − vn τ + Avn+1 = 0, n = 0, 1, ..., N − 1, vN − v0 = χ.

❙❛♠❛rs❦✐✐ ❆❆✳ ❚❤❡ t❤❡♦r② ♦❢ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡s✳ ◆❡✇ ❨♦r❦✿ ▼❛r❝❡❧ ❉❡❦❦❡r❀ ✷✵✵✶✳

✷✵✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

■t❡r❛t✐✈❡ ♣r♦❝❡ss

❚❤❡ ❣r✐❞ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss✿ vk+1 = vk

N − χ,

vk+1

n+1 − vk+1 n

τ + Avk+1

n+1 = 0,

n = 0, 1, ..., N − 1, k = 0, 1, ..., ✇❤❡r❡ ϕk+1 = φ + vk+1 .

✷✶✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❙t✉❞② ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✲ ✶✴✸

❚❤❡ st✉❞② ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✐s ❝♦♥❞✉❝t❡❞ ✉s✐♥❣ t❤❡ s❛♠❡ ❛♣♣r♦❛❝❤ ❛s ❢♦r t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r t❤❡ s❡♠✐❞✐s❝r❡t❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✳ ▲❡t ♥♦✇ zk+1

n+1 = vk+1 n+1 − vn+1✱ t❤❡♥

zk+1 = zk

N,

zk+1

n+1 − zk+1 n

τ + Azk+1

n+1 = 0,

n = 0, 1, ..., N − 1, k = 0, 1, ... . ❚❤❡ ❦❡② ♠♦♠❡♥t ♦❢ ♦✉r ❝♦♥s✐❞❡r❛t✐♦♥ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ ❛♥ ❡st✐♠❛t❡ ♥♦r♠ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦✈❡r t✐♠❡✳

✷✷✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❙t✉❞② ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✲ ✷✴✸

❲❡ ♠✉❧t✐♣❧② ❡q✉❛t✐♦♥ ❢♦r zk

n+1 ✐♥ V h ❜② τzk n+1 ❛♥❞ ♦❜t❛✐♥

zk

n+12 + τ(Azk n+1, zk n+1) = (zk n, zk n+1).

❲❡ ❤❛✈❡ (1 + τδ)zk

n+1 ≤ zk n,

n = 0, 1, ..., N − 1, zk

n ≤ (1 + τδ)−nzk 0,

n = 1, 2, ..., N. ❆ ♣r✐♦r✐ ❡st✐♠❛t❡ ❛❧❧♦✇s ✉s t♦ ♦❜t❛✐♥ zk+1 = zk

N ≤ (1 + τδ)−Nzk 0.

❚❤❡r❡❜② vk+1 − v0 ≤ ¯ ̺ vk

0 − v0,

¯ ̺ = (1 + τδ)−N, ✇❤✐❝❤ ♣r♦✈✐❞❡s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✭¯ ̺ < 1✮✳

✷✸✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ♦❢ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♣r♦❜❧❡♠

❙t✉❞② ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ✲ ✸✴✸

❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❡♥s✉r❡❞ ❜② t❤❡ ❡st✐♠❛t❡ ϕk+1 − ϕ ≤ ¯ ̺ vk

0 − v0.

❚❤✐s ❛❧❧♦✇ ✉s t♦ ❢♦r♠✉❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛✐♥ ❛ss❡rt✐♦♥✳

❚❤❡♦r❡♠

❚❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❝♦♥✈❡r❣❡s ❧✐♥❡❛r❧② ✇✐t❤ s♣❡❡❞ ¯ ̺ < 1✳

✷✹✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

■t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

❲❤❡♥ st✉❞②✐♥❣ t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ t❤❡ ❝♦♥str✉❝t✐✈❡ ♠❡t❤♦❞ ♦❢ ✐t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ♦❢t❡♥ ✉s❡❞✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤ ❢♦r t❤❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠✳

Pr✐❧❡♣❦♦ ❆■✱ ❑♦st✐♥ ❆❇✳ ❖♥ ❝❡rt❛✐♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✇✐t❤ ✜♥❛❧ ❛♥❞ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥✳ ❙❜♦r♥✐❦✿ ▼❛t❤❡♠❛t✐❝s✳ ✶✾✾✸❀✼✺✭✷✮✿✹✼✸✕✹✾✵✳ Pr✐❧❡♣❦♦ ❆■✱ ❑♦st✐♥ ❆❇✳ ▼❛t❤❡♠❛t✐❝❛❧ ◆♦t❡s✳ ✶✾✾✸❀✺✸✭✶✮✿✻✸✕✻✻✳ ❑♦st✐♥ ❆❇✳ ❙❜♦r♥✐❦✿ ▼❛t❤❡♠❛t✐❝s✳ ✷✵✶✸❀✷✵✹✭✶✵✮✿✶✸✾✶✕✶✹✸✹✳

✷✺✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

■t❡r❛t✐✈❡ ♣r♦❝❡ss

■♥ t❤❡ ♥❡✇ ✐t❡r❛t✐✈❡ st❡♣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❞❡t❡r♠✐♥❡❞ ❢♦r t = T✿ ϕk+1 = dwk dt (T) + Aψ, k = 0, 1, ..., ✇✐t❤ s♦♠❡ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❛ss✉♠♣t✐♦♥ ϕ0✳ ❚❤❡♥✱ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞✿ dwk+1 dt + Awk+1 = ϕk+1, 0 < t ≤ T, wk+1(0) = φ.

✷✻✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

❚✐♠❡ ❞✐s❝r❡t✐③❛t✐♦♥

❚❤❡ t✐♠❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✐s ❛❣❛✐♥ ❢♦r♠✉❧❛t❡❞ ✐♥ t❤❡ ❜❛s✐s ♦❢ ✐♠♣❧✐❝✐t ❛♣♣r♦①✐♠❛t✐♦♥✳ ❋♦r♠❛❧❧②✱ ✇❡ ❞❡✜♥❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦♥ ❡①♣❛♥❞❡❞ ❣r✐❞✿ tn = nτ, n = −1, 0, ..., N, τN = T. ❲❡ ❝♦♠❡ t♦ t❤❡ ♣r♦❜❧❡♠ wn+1 − wn τ + Awn+1 = ϕ, n = −1, 0, ..., N − 1 w0 = φ, wN = ψ.

✷✼✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

■t❡r❛t✐✈❡ ♣r♦❝❡ss

❋♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ t❤❡ ❣r✐❞ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r t❤❡ s❡♠✐❞✐s❝r❡t❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐s ❛♣♣❧✐❡❞✿ ϕk+1 = wk

N − wk N−1

τ + Aψ, k = 0, 1, .... ❚♦ ❛♣♣r♦①✐♠❛t❡ ❡q✉❛t✐♦♥ t❤❡ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ ✐s ✉s❡❞ wk+1

n+1 − wk+1 n

τ + Awk+1

n+1 = ϕk+1,

n = −1, 0, ..., N − 1, ✉♥❞❡r ❝♦♥❞✐t✐♦♥ wk+1 = φ.

✷✽✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠

❚❤❡ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ❝♦♥✈❡r❣❡s ❧✐♥❡❛r❧② ✇✐t❤ s♣❡❡❞ ¯ ̺ = (1 + τδ)−N, ❛♥❞ t❤❡ ❡st✐♠❛t❡ ϕk+1 − ϕ ≤ ¯ ̺ ϕk − ϕ. ✐s ✈❛❧✐❞✳

✷✾✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

• ❡♥❡r❛❧✐③❛t✐♦♥s

■♥t❡❣r❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥

❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ♦❢ ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✱ t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥ ✐s ♦❢t❡♥ ✉s❡❞ ✐♥st❡❛❞ ♦❢ t❤❡ ✜♥❛❧ ♦✈❡r❞❡t❡r♠✐♥❛t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ✐s ✐♥✈♦❧✈❡❞ T ω(t)u(x, t)dt = uT (x), x ∈ Ω, ✇❤❡r❡ ω(t) ✖ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ ❛♥❞ ω(t) ≥ 0, T ω(t)dt = 1. ❋♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐t❡r❛t✐✈❡ ♣r♦❝❡ss ❝♦♥s✐❞❡r❡❞ ❛❜♦✈❡ ❝❛♥ ❜❡ ✉s❡❞✳

✸✵✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

• ❡♥❡r❛❧✐③❛t✐♦♥s

■t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t

■♥t❡❣r❛t✐♥❣ ❡q✉❛t✐♦♥ ✇✐t❤ ✇❡✐❣❤t ω(t) ♦✈❡r t ❢r♦♠ 0 t♦ T✱ ✇❡ ♦❜t❛✐♥ T ω(t)dw dt (t)dt + Aψ = ϕ. ❚❤❡ ✐t❡r❛t✐✈❡ r❡✜♥❡♠❡♥t ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✿ ϕk+1 = T ω(t)dwk dt (t)dt + Aψ, k = 0, 1, ... . ❲❡ ❤❛✈❡ ϕk+1 − ϕ ≤ ̺ ϕk − ϕ, k = 0, 1, ..., ❛t t❤❛t ̺ = T ω(t) exp(−δt)dt.

✸✶✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

• ❡♥❡r❛❧✐③❛t✐♦♥s

▼♦r❡ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠s

❲❡ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ t❤❡ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✱ ✇❤❡♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t✐♠❡✳ ■♥ ♠♦r❡ ❣❡♥❡r❛❧ ❝❛s❡✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐❞❡♥t✐❢②✐♥❣ ♠✉❧t✐♣❧✐❝❛t✐✈❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✱ ✇❤❡♥ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦♥ t✐♠❡ ✐s ❦♥♦✇♥ ❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ s♣❛t✐❛❧ ✈❛r✐❛❜❧❡s ✐s ✉♥❦♥♦✇♥✱ ✐s st❛t❡❞✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ∂u ∂t − div(k(x)gradu) + c(x)u = β(t)f(x), x ∈ Ω, 0 < t ≤ T, ✇❤❡r❡ β(t) ✖ s♦♠❡ ❣✐✈❡♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ t❤❡ ♣❛✐r u(x, t), f(x)✳ ❲❡ ❛ss✉♠❡ t❤❛t β(t) > 0, dβ dt ≥ 0, 0 ≤ t ≤ T, β(T) = 1.

✸✷✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

✷❉ ♠♦❞❡❧ ♣r♦❜❧❡♠

❲❡ ❝♦♥s✐❞❡r ♠♦❞❡❧ ♣r♦❜❧❡♠✱ ✇❤❡♥ ∂u ∂t − div grad u + cu = f(x), x ∈ Ω, 0 < t ≤ T, ∂u ∂n = 0, x ∈ ∂Ω, 0 < t ≤ T, u(x, 0) = 0, x ∈ Ω. ❚❤❡ ❢♦r✇❛r❞ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✐♥ t❤❡ ✉♥✐t sq✉❛r❡ Ω = {x = (x1, x2) | 0 < x1 < 1, 0 < x2 < 1} ✇✐t❤ ❣✐✈❡♥ r✐❣❤t✲❤❛♥❞ s✐❞❡ f(x) ❛♥❞ uT (x) = u(x, T). ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✇❤❡♥ uT (x) ✐s ❦♥♦✇♥✱ ❜✉t ✇❡ ♥❡❡❞ t♦ ✜♥❞ f(x)✳

✸✸✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❘✐❣❤t✲❤❛♥❞ s✐❞❡

❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s t❛❦❡♥ ❛s f(x) = 1 1 + exp(γ(x1 − x2)). ❋✉♥❝t✐♦♥ g(s) = (1 + exp(γs))−1 ❛t ❞✐✛❡r❡♥t ✈❛❧✉❡s γ

✸✹✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

◗✉❛s✐✲r❡❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥t

❚❤❡ ❢♦r✇❛r❞ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✇✐t❤✐♥ t❤❡ ✜rst q✉❛s✐✲r❡❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥t✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠ ❛t t❤❡ ✜♥✐t❡ t✐♠❡ ✭t❤❡ ❢✉♥❝t✐♦♥ u(x, T)✮ ✐s ✉s❡❞ ❛s ✐♥♣✉t ❞❛t❛ ❢♦r t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✳ ❲❡ ♣❡r❢♦r♠ t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❡✛❡❝t ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡rr♦rs ♦♥ t❤❡ ❜❛s✐s ♦❢ ❝❛❧❝✉❧❛t✐♦♥s ♦♥ ❞✐✛❡r❡♥t t✐♠❡ ❣r✐❞s✱ ✇❤❡♥ ✉s✐♥❣ t❤❡ ✐♥♣✉t ❞❛t❛ ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ♣r♦❜❧❡♠ ♦♥ ♠♦r❡ ❞❡t❛✐❧❡❞ t✐♠❡ ❣r✐❞ ❛♥❞ ✇✐t❤ ❛ ♠♦r❡ ❛❝❝✉r❛t❡ ❛♣♣r♦①✐♠❛t✐♦♥s ✐♥ t✐♠❡✳ ❋♦r t❤❡ ❜❛s❡ ❝❛s❡ ✇❡ s❡t c = 10✱ T = 0.1✱ γ = 10✳ ❲❤❡♥ s♦❧✈✐♥❣ t❤❡ ❢♦r✇❛r❞ ♣r♦❜❧❡♠ ✇❡ ✉s❡ t❤❡ ❈r❛♥❦✲◆✐❝♦❧s♦♥ s❝❤❡♠❡ ❢♦r t✐♠❡ ❞✐s❝r❡t✐③❛t✐♦♥✱ t❤❡ t✐♠❡ st❡♣ ✐s τ = 1 · 10−4✳ ❚❤❡ ✉♥✐❢♦r♠ ♠❡s❤ ✇✐t❤ t❤❡ ❞✐✈✐s✐♦♥ ✐♥t♦ ✺✵ ✐♥t❡r✈❛❧s ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥ ✐s ✉s❡❞✱ t❤❡ ▲❛❣r❛♥❣✐❛♥ ✜♥✐t❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ s❡❝♦♥❞ ❞❡❣r❡❡ ❛r❡ ❛♣♣❧✐❡❞✳

✸✺✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❚❤❡ s♦❧✉t✐♦♥ ❛t t❤❡ ✜♥✐t❡ t✐♠❡

❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ♣r♦❜❧❡♠ uT (x) = u(x, T)

✸✻✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❊rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥

❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✉s✐♥❣ ❢✉❧❧② ✐♠♣❧✐❝✐t s❝❤❡♠❡✳ ❚❤❡ ❡rr♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦♥ ❛ s✐♥❣❧❡ ✐t❡r❛t✐♦♥ ✐s ❡✈❛❧✉❛t❡❞ ❛s ❢♦❧❧♦✇s ε∞(k) = max

x∈Ω |ϕk(x) − f(x)|,

ε2(k) = ϕk(x) − f(x), ✇❤❡r❡ ϕ(x) ✖ t❤❡ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥✱ ❛♥❞ f(x) ✖ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✳

✸✼✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐♦♥s ✲ ✶✴✷

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

✸✽✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤

◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts

❈♦♥✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐♦♥s ✲ ✷✴✷

■t❡r❛t✐✈❡ ♣r♦❝❡ss ❢♦r ✐❞❡♥t✐❢②✐♥❣ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡

✸✾✴✸✾ ❝ P❡tr ◆✳ ❱❛❜✐s❤❝❤❡✈✐❝❤