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
23 Mode k>(n-1)/2 on grid h becomes(n-k)mode on grid 2h The effect of restricting a solution which has significant high frequencies can have very negative effects, since high modes on h may appear as low modes on 2h and hence exhibit slow convergence 2.3.2 Summary Only low modes in h can be represented well in 2h Low modes on h become higher modes in 2h Hence having a faster convergence rate k=1 LOW HIGH grid h LOW匚HGH 2.4 Prolongation The process of transferring a vector between a fine and a coarse mesh is koum v& Given w2h we obtain wn by prolongation wA=Ah w2h I R h: prolongation operator(matrix)
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