Simplest procedure is injection 2h. i wh2i We shall assume, for simplicity, that n+ l is an even number. Restriction by injection reduces to taking the components of w2h to be the components of wh at every other point. We will see later that other forms of restriction can also SLIDE 11 Intuiti h GOOD D If the soltion is "smooth", the restricted function is a good appro himation to nal grid function on the fine mesh. On the other hand for an oscillatory " function, a lot of information is lost during the restriction SLIDE 12 The concept of“ smooth”or“ oscillatory”/ nction can be made more precise If we write u": eigenvectors of A Only the modes k= 1 by grid 2h 7+1✲✴✳☎✵✡✶✩✷✯✸✺✹✼✻✽✶✿✾❁❀❃❂❄✸✺❅❃❆✩✾✍✸❇✳❈✹❊❉●❋✕❍✖■✍❏✖❑▲❉◆▼✑❋ ❖✌P✧◗✫❘ ❙❯❚❱❖✟◗✫❘ P✍❙ ❲❀❳✾❩❨ ❚❭❬❳❪✖❫✕❫✖❫❄❪✓❴✴❵❜❛ P ❝■❡❞❣❢✿❤✑✐●✐✂❤✑❞✍❞❦❥❃❧♠■❦♥❜♦❄▼✑♣❡❞❦❉●❧✽qr✐s❉◆❏✖❉●❑✙t✉♥✽❑✈❢✩❤✚❑①✇✟② ❬ ❉❈❞♠❤✑❋③■✖④✉■✖❋⑤❋✿❥❃❧♠⑥✧■❄♣✖⑦❊⑧❇■❦❞❦❑▲♣❦❉◆❏❄❑▲❉◆▼✑❋③⑥✖t ❉●❋✖❍✖■✧❏✖❑✙❉◆▼✚❋⑨♣❁■✧⑩✑❥✿❏✧■❦❞❡❑❶▼✟❑❷❤✕❸✚❉●❋✓❹✝❑✈❢✩■♠❏✧▼✑❧❺q✩▼✚❋❻■✖❋✿❑✙❞❼▼✼♦ ❖P✧◗ ❑❶▼☛⑥✧■❼❑✈❢✩■♠❏✧▼✑❧❺q✩▼✚❋❻■✖❋✿❑✙❞❼▼✼♦ ❖◗ ❤✑❑❺■✖④✑■❄♣❦t❽▼✑❑✈❢✩■✖♣❩q✿▼✑❉●❋✢❑❶⑦ ❝■❼❾❿❉●✐●✐➀❞✖■✧■❼✐✯❤✑❑❷■❄♣❡❑●❢✿❤✑❑❺▼✚❑●❢✿■❄♣➁♦❄▼✑♣❦❧❡❞✡▼❣♦✤♣❁■❄❞❦❑✙♣❦❉◆❏✖❑▲❉◆▼✑❋③❏✧❤✑❋③❤✑✐➂❞✖▼ ⑥✧■➃❥✴❞✖■✧⑩✫⑦ ➄❻➅✓➆●➇❯➈➊➉✩➉ ➋❷➌✻❁❆✩✳✯✻❁✳☎➍✫✸✕✷✯➎❳➏ ➐♦❼❑✈❢✩■♠❞❄▼✚✐s❥❃❑▲❉◆▼✑❋③❉❈❞❱➑✙❞✧❧♠▼✺▼✑❑✈❢❯➒❦♥✽❑●❢✿■♠♣❁■❄❞❦❑✙♣❦❉◆❏✖❑❶■✧⑩❊♦✧❥❃❋r❏❄❑▲❉◆▼✑❋➓❉❈❞♠❤➔❹✓▼✺▼✕⑩→❤✍q✫qr♣❁▼✖➣❳❉●❧♠❤✑❑▲❉◆▼✑❋↔❑❶▼ ❑✈❢✩■↕▼✚♣❦❉➙❹✚❉●❋r❤✑✐➛❹✫♣❦❉◆⑩♠♦✧❥❃❋❻❏✖❑▲❉◆▼✑❋➜▼✑❋➜❑●❢✿■❊➝①❋r■☛❧✡■❄❞❁❢✩⑦➟➞➁❋❱❑●❢✿■↕▼✑❑✈❢✩■✖♣✌❢✩❤✚❋❻⑩✉♥❡❾✂■☛❞✖■✧■☛❑●❢✿❤✑❑ ♦❄▼✑♣❼❤✑❋➠➑❷▼✉❞✖❏✖❉●✐●✐✯❤✑❑❷▼✑♣❦t✢➒❩♦✧❥❃❋❻❏✖❑✙❉◆▼✚❋✩♥➡❤➔✐✯▼✑❑✽▼✼♦➃❉●❋✕♦❄▼✑♣❦❧♠❤✑❑▲❉◆▼✑❋⑨❉❈❞➃✐✯▼✑❞❦❑➁⑩✚❥❃♣❦❉●❋✓❹✟❑●❢✿■❡♣❁■❦❞❦❑▲♣❦❉◆❏❄❑▲❉◆▼✑❋ ▼✍q✿■✖♣❁❤✑❑▲❉◆▼✑❋r⑦ ➄❻➅✓➆●➇❯➈➊➉✫➢ ➤❢✩■❼❏✧▼✑❋❻❏✧■❷q❻❑❩▼✼♦⑤➑✙❞❦❧✡▼✺▼✑❑✈❢❯➒❡▼✑♣⑤➑❷▼✉❞✖❏✖❉●✐●✐✯❤✑❑❷▼✑♣❦t✢➒➛♦✧❥❃❋r❏❄❑▲❉◆▼✑❋⑨❏✍❤✚❋➥⑥✧■✤❧✡❤❳⑩✫■✤❧✡▼✚♣❁■➁qr♣❁■✧❏❄❉❈❞✖■✺⑦ ➋❲➀➦✸ ➦✾✍✳➙✻✍✸ ➧❯➨❃➩❿✸✖✳☎➫✫✸➌➍❳✸✕❂❦✻✍❀✫✾✧✹✂❀❲➀➭ ❖◗ ❚ ➯❴ ➨❦➲ ❛ ➳ ➨ ➧ ➨ ➵➌✷☎➎♠✻❁➸✿✸❊✵❼❀❃❅❃✸✺✹✽➺ ❚❭❬❳❪✖❫✖❫✕❫❄❪ ✇✝➻ ❬ ➼ ➽ ✾❁✸☛➾❁➍✓✳❈✹❁✳✯➚✩✷☎✸✕➪❡➚✴➎✌➫❳✾❁✳❈❅ ➼❳➶➘➹ ➴✙➷✕➬➂➮▲➬➂➱✺✃➂❐❣❒➡➱✖❮❺❰❦Ï▲➬➂Ð P✍◗ Ñ Ò❦Ó Ô ❬❳❪ ➼ ❪✖❫✖❫✕❫❄❪ ✇✝➻ ❬ ➼ ❪ Õ✃➂➬ Õ➮✙❐✼Ð Ñ Ò❦Ó Ô ✇➔② ❬ ➼ ❪✕❫✖❫✖❫✕❪ ✇✝➻ ❬❳❪ ✇ Ö