3 Two Grid(Correction) Scheme sLIDE 1 One cycle [ G(uh, fn) r+1/3 Relat v iterations of An uh=fn with initial guess uh +up ComputeR=f-An uT+1 3, and restrict rh=Ih Solve Ah e2h=r2h on RelaT v iterations of An uh=fn with initial guess ur+/3++uF+I Above we describe one cycle of a two grid correction scheme. The inputs are an initial guess uh, and a forcing vector fh. The output is the new appro imation to the solution ur+I. Here. a Here, any of the relatation, restriction, and prolongation schemes described. can be used W ll th een turns out that writing the coarse grid correction in terms of the error leads to a simpler and more straightforward formulation and v2 are usually referred to as the number of pre-and post-smoothing iter ons, respectively 3.1E We solve u(0)=u(1)=0 l2=-25(sin(5x)+9sin(15mx) Solution: u=sin(5)+sin(15a) Two grid scheme: h、1 Solve using under-relaxed Jacobi with w=3 Initial condition➮ ➱❐✃❮❒Ï❰ÑÐ❹Ò✧Ó Ô⑦Õ◗❒➞Ð❀Ð❀Ö✸×☞Ø✭Ò✧❒➞ÙÛÚÝÜ✒×❦Þ❤Ö➣ßàÖ á❝â❽ã❿ä➁å➃æ✣ç è✎é⑦ê■ë❵ì④ë✧í❃ê î☞ï✯ð✶ñ ò óõô÷ö✖øîïò❞ù✔ú ò❳û ü❩ý✎þ❵ÿ✁￾✄✂✆☎ñ✞✝✠✟ ê✄✡☞☛✟✌✝✁✍é✏✎ ✍✒✑✔✓ò î ò✖✕ úò✘✗✝✠✟✚✙✛✝é ✝✜✟✌✝☛✘í✣✢✥✤⑦ê✦✎✚✎☞îïò★✧ îï✚ð✻ñ✪✩✚✫ ò ✬ ü✮✭✍✥✯✱✰✤ ✟ ê✳✲ ò ✕ ú✵✴ ✓ò îï✚ð✻ñ✪✩✚✫ ò ✶ ☛✘é✸✷✵✹þ✻✺✽✼✹✽✾❀✿ ✼ ✲✸❁ ò ✕❃❂ ò ❁ ò ✲ ò ✬ ü❃❄ ✍ í✁❅✣ê ✓ ❁ ò❇❆ ❁ ò ✕ ✲✸❁ ò ✍é❉❈✒❊ ✬ ü✮❋✹✌● ÿ●■❍❑❏ ￾▲✼✳þ ❆❚ò ✕❃❂ ❁ ò ò ❆ ❁ ò ✶ ☛✣é✸✷❤ë✍ ✡✌✡✚ê➐ë✟ îï✯ð❁ ✩✚✫ ò ✕ îï✯ð✻ñ✌✩✚✫ ò ▼ ❆❚ò ✬ ü❩ý✎þ❵ÿ✁￾✄✂★☎❁ ✝✠✟ ê✄✡☞☛✟✌✝✁✍é✏✎ ✍✒✑✔✓ò î ò✖✕ úò✘✗✝✠✟✚✙✛✝é ✝✜✟✌✝☛✘í✣✢✥✤⑦ê✦✎✚✎☞îï✚ð❁ ✩✌✫ ò ✧ î❧ï✚ð✻ñ ò◆✬ ❖◗P●▲❘ þ❚❙❦þ❱❯✣þ✻✺ ✿✻✹✽✾ Pþ ●■❍þ ✿✄❲✒✿ ÿ❃þ ●❨❳ ￾❩✼❬❙●❚❏✒✹✽✾❯ ✿☞●▲✹✽✹þ ✿ ✼ ✾❀●▲❍ ✺ ✿✚❭ þ✻❪✖þ✦❫✘❴❭ þ ✾❵❍▲❛❝❜✼❞✺❚￾✹ þ❱￾❍ ✾❵❍✸✾✼ ✾￾➒ÿ ❏■❜þ✻✺☞✺ îïò❢❡ ￾❍ ❯❣￾ ❳✻●▲✹✚✿✻✾❵❍❤❏❣❘ þ ✿ ✼●▲✹ úò ❫❩❴❭ þ ●■❜✼❛✐❜✼ ✾✺❚✼❭ þ ❍ þ✄❙❥￾❛✒❛✐✹✌●✂ ✾❪✘￾▲✼✾❀●▲❍ ✼● ✼❭ þ❱✺ ● ÿ❜ ✼ ✾❀●▲❍✩îï✯ð✶ñ ò ❫❧❦➨þ ✹ þ ❡ ￾❍✏❲♠●❨❳ ✼❭ þ ✹ þ✧ÿ✜￾✄✂❤￾■✼✾❀●■❍❡ ✹ þ✽✺✽✼✹✽✾❀✿ ✼ ✾❀●▲❍❡ ￾❍ ❯ ❛❝✹✚● ÿ●■❍❑❏ ￾▲✼✾❀●▲❍ ✺ ✿✚❭ þ✻❪✖þ✻✺❱❯❳þ✽✺ ✿✄✹✽✾ P þ✚❯❡ ✿￾❍ Pþ ❜ ✺✧þ☞❯✒❫ ♥þ ✹ þ ✿￾➒ÿ❿ÿ❇✼❭ þÛþ☞♦❜♣✾❵❘ ￾➒ÿ❃þ❍❝✿þ Pþ✻✼❬❙❦þ✔þ❍ ✺ ● ÿ❘✦✾❵❍❤❏q❳✻●■✹ ✓ î òr✕ ú ò ●▲✹ ✓ ❆ òs✕ ✲ ò ❫✉t✽✼ ✼❜♣✹✽❍✺ ●■❜✼✈✼❭ ￾▲✼✇❙✹✽✾✼ ✾❵❍❑❏ ✼❭ þ ✿☞●￾ ✹ ✺✧þ ❏■✹✽✾❯ ✿☞●▲✹✽✹þ ✿ ✼ ✾❀●■❍❉✾❵❍ ✼➴þ ✹❪✖✺ ●❨❳ ✼❭ þ➨þ ✹✽✹✌●■✹ ÿ❃þ☞￾✥❯①✺◗✼● ￾ ✺ ✾❪❛ ÿ❃þ ✹ ￾❍ ❯✘❪●▲✹þ❱✺✽✼✹￾ ✾✠❏②❭✼❳✻●■✹❙✇￾✹❯ ❳✻●■✹❪❜ ÿ✜￾■✼✾❀●■❍❫ ☎ñ ￾❍ ❯✖☎❁ ￾ ✹ þ ❜ ✺❜ ￾✘ÿ❿ÿ❲✘✹þ❳ þ ✹✽✹þ✚❯❣✼● ￾①✺✳✼❭ þ ❍✸❜❪Pþ ✹❱●❨❳③❛✐✹þ✻④✪￾❍ ❯ ❛❢● ✺✽✼❞④⑤✺✽❪●✦● ✼❭♣✾❵❍❑❏q✾✼➴þ ✹ ④ ￾▲✼✾❀●▲❍✺ ❡ ✹ þ✽✺❛ þ ✿ ✼ ✾❵❘ þ✧ÿ❲ ❫ ⑥⑧⑦✪⑨ ⑩✛❶❧❷✔❸❺❹❚❻❨❼ á❝â❽ã❿ä➁å➃æ ❽ ❾ê❱✎✍ í✜❅❳ê ❿ ø❞➀ û ✕ ❿ ø✪➁ û ✕ ➀ ✴ ❿❝➂②➂ ✕ ✴ ❈✥➃ ø ✎ ✝é ø ➃▲➄➆➅ û ▼➈➇ ✎ ✝é ø✪➁ ➃▲➄➆➅ û û❚➉ ➊é ✝✜✟✌✝☛✘í➆✢✥✤⑦ê✦✎✚✎✄➋✭î③➌ ✕ ➀ ❄✍ í✜✤✟✚✝✜✍é✣➋✔❿ ✕ ✎ ✝é ø ➃■➄➆➅û ▼ ✎ ✝é ø✪➁ ➃■➄➆➅û ➍✗✍ ✢✥✡✝✷q✎✚ë✙ ê✯ê✥➋✇❊ ✕ ➁ ➎ ❈ ù ❈✒❊ ✕ ➁ ➁✦➏ ❄✍ í✜❅❳ê❚✤✏✎ ✝é❢✢✱✤⑦é✸✷④ê✦✡✪➐⑤✡✚ê✧í➑☛▲➒④ê✦✷♠➓✥☛❳ë✍✒➔✸✝ ✗✝✜✟✌✙✛→ ✕ ❁ ✫ á❝â❽ã❿ä➁å ➣♣↔ ➊é ✝✜✟✌✝☛✘í♥ë✍é✸✷✝✜✟✌✝✁✍é ➇