Iter erative methods Multigrid Techniques Lecture 7
✂✁☎✄✝✆✟✞✠✁☛✡✌☞✍✄✏✎✑✄✒✁☛✓✕✔✗✖✙✘✠✚ ✎✑✛✙✜✌✁☎✡✣✢✤✆✥✡✦✖★✧✩✄✝✪☎✓✕✫✙✡✭✬✍✛✮✄✝✘ ✯✰✄✝✪✒✁☛✛✮✆✱✄✳✲
1 Background Brandt(1973)published first paper SLIDE 1 Offers the possibility of solving a problem with work and storage propor tional to the number of unknowns Well developed for elliptic proble tions is still an active area of research Good Introductory Reference: A Multigrid TutoriaL, W. L. briggs E. Henson, and S.F. McCormick, SIAM Monograph, 2000 2 Basic Principles 2.1 Some ideas ain, the one pre principles of multigrid methods. The ideas presented can be readily extended to multiple dimensions. First, however, we will review some simple facts about iterative methods seen in the last lecture and develop some ideas to accelerate the iterative proces 1. Multigrid is an iterative method - a good initial guess will reduce the number of iterations to solve An un=fn by iteration, we could take ugh Here An, un and fn denote the matriz, vector of unknowns, and force vector, respectively, that resalts from discretizing our model problem on a grid of size h. uh u2h means that the initial guess for the iterative process on grid h is "appro imated from the solution of the problem on grid of size 2h. We use the word approximate because uh and u2n are vectors that have different lengths. We shall see later how this approxi- mation is carried out. It is clear that the procedure outlined is recursive that is, the pro blem on the grid 2h can also be soloed by iteration, urith an initial guess provided by solving A4A wAh =fh, etc.. We point out that teration on coarser meshes is cheaper because n is smaller and therefore
✴ ✵✷✶✹✸✻✺✽✼✤✾❀✿✍❁❃❂✮❄ ❅ ❆❈❇❊❉●❋■❍ ❏✩❑▼▲✭◆✂▲✭❖◗P❙❘❚▲❱❯❲P❳◆❙▲❱❨❬❩✌❭❪▲✗❖◗❫✂❴❵❩❜❛❙❝❡❞❙▲❢❫❣❨❤❴❥✐✑❦❧❨❤❫❣♠♥❯♦❩✗♣rq❱s✉t❳✈✂✇✠❘❪①❪②❪❖◗③④❴r❭❪▲❢❯❲⑤♥❨✌❴r❩✹❘♥❫❣❘❚▲✭❨ ⑥③⑦❩⑧❭✮❘❪❨❤❫❙⑨✦❩⑧③④⑨✭❫❙❖✥❨⑧▲❢❴r①❪❖⑦❩✌❴❱⑩ ❏❷❶❬❸❹▲✭❨❤❴❧❩⑧❭♥▲✹❘❺P❈❴⑧❴⑧③⑦②♥③⑦❖◗③❻❩❵❞❡P❣❼☎❴⑧P❙❖◗◆✉③⑦♠♥❽❾❫✍❘❪❨✌P❙②♥❖⑦▲❱❿ ⑥③❻❩✌❭ ⑥P✂❨⑧➀✽❫❙♠♥❯✙❴r❩⑧P✂❨✌❫❙❽❙▲➁❘❪❨✌P❙❘❚P❙❨⑧➂ ❩✌③⑦P✂♠♥❫❣❖✱❩⑧P✍❩⑧❭❪▲✹♠✉①❪❿➃②❺▲❱❨❥P❣❼✟①❪♠❪➀✉♠❪P⑥♠♥❴❱⑩ ❏➅➄➆▲✭❖◗❖✻❯♦▲❱◆❙▲❱❖⑦P✂❘❺▲❢❯❃❼➇P❙❨➈❖◗③⑦♠♥▲❱❫❣❨➈▲❱❖⑦❖◗③⑦❘❪❩⑧③④⑨❾❘❪❨⑧P✂②❪❖◗▲✭❿✗❴❬✐✑❫❣❘❪❘❪❖◗③④⑨✭❫❳❩✌③⑦P✂♠➉❩✌P✕P❙❩⑧❭❪▲❱❨➈▲❱➊❈①♥❫❳➂ ❩✌③⑦P✂♠♥❴❥③④❴❥❴❵❩✌③⑦❖◗❖●❫❙♠✕❫✂⑨✣❩⑧③◗◆❙▲✹❫❙❨⑧▲❢❫✤P❙❼☎❨✌▲❱❴⑧▲❱❫❙❨✌⑨❤❭✱⑩ ➋❧➌❳➌❣➍✍➎➐➏❣➑➓➒❵➌❣➍❙➔❈→✌➑❵➌✭➒➓➣✤↔✝↕✌➙◗↕✌➒❵↕❤➏✂→❤↕❢➛✟➜➅➝✽➞❙➟➠❊➡❻➢❱➤r➡➇➥✽➦❈➞❣➠➨➧❱➤⑧➡❊➩❢➟➫✂➭➲➯ ➳✱➯✉➵●➒❵➸❻➺❱➺❢➻❤➫ ➼ ➯ ➽❀➯❈➾✝↕❤➏❈➻➓➌❱➏❺➫❈➚❱➏❈➍❾➪❪➯ ➶☎➯✉➹✍→✦➘☎➌❱➒❵➴✹➸➷→r➬♦➫♦➪❣➎➱➮✃➹❐➹✍➌❱➏❈➌❱➺❱➒r➚❱❒✂❮❺➫♦❰❱Ï❢Ï❱Ï❙➯ Ð ✵✷✶✹Ñ✟Ò✦✸ÔÓÕ✾✒Ò✭❂✮✸✠Ò❱Ö❃×✦Ø✠Ñ Ù✃ÚrÛ Ü❧Ý✝ÞÕßáàrâ❬ß✱ã❀ä ❅ ❆❈❇❊❉●❋➆å æ❃ç❃è❀é❊ê❊ê✠ë✉ì✭ç✦í❾î❳ï❹ð❤çòñ✭ó❈ñ❳é❊ï♥í✍ô➇õ❪ç❲î❳ï❹ç❲ö❣é❊÷❾ç✭ï❪ì✣é➨î❳ï❹ñ❳ê✟ø❹ù⑧î❙ú✭ê⑦ç✭÷ûô➱îüé❊ê❊ê❻ë✉ì❤ôýù⑧ñ❣ô➱ç➉ô❊õ♥çòú❤ñþì✣é➨ð ø❚ù✣é❊ï❹ð✦éø❹ê⑦ç✦ì✍î❵ÿ✹÷✍ë♦ê➷ôýé❻ó❣ù✣é➨ö❡÷✗ç✦ô➇õ❪î❢öþì✁✄✂❹õ❪ç➃é➨ö✂ç✌ñ❳ì✃ø❚ù⑧ç✦ì✭ç✦ï❺ô➱ç❤ö✕ð❤ñ❳ï➆ú❤ç✤ù⑧ç❤ñ❙ö❣é❊ê✆☎✕ç✞✝✂ô➱ç✭ï❚ö✂ç✌ö✽ô➱î ÷✍ë♦ê➷ô➐éø❹ê⑦çòö❣é❊÷❾ç✭ï❪ì✣é➨î❣ï❪ì✁✠✟✻é❊ù❤ì❤ô➐í✹õ♥î❳è✻ç✁✡þç✭ù❤í➃è✻ç➉è❀é❊ê❊ê❥ù⑧ç✁✡❱é➨ç✭è ì✭î❳÷✗ç✮ì✣é❊÷✠ø❹ê⑦ç❜ÿ✦ñ✂ð✦ô➐ì➉ñ❙ú❤î❳ë♦ô é❊ô➓ç✦ù⑧ñ❣ô➐é☛✡❳ç❡÷❾ç✭ô❊õ♥î❱ö❳ì❾ì✦ç❤ç✭ï é❊ï ô➇õ❪ç❡ê⑦ñ❳ì❤ô➁ê⑦ç❤ð✭ô➐ë♦ù⑧ç✙ñ❳ï❹öòö❙ç✁✡❳ç✦ê⑦î❤ø✩ì✦î❣÷❾ç❡é➨ö❙ç❤ñ❳ì✽ô➱î➉ñ✂ð❤ð✌ç✭ê⑦ç✭ù⑧ñ❳ô➓ç ô➇õ❪ç✤é❊ô➱ç✭ù⑧ñ❳ôýé☛✡❳ç✃ø❹ù⑧î❱ð❤ç✦ì❤ì☞ q✂⑩✍✌➉①❪❖❻❩✌③⑦❽✂❨⑧③④❯ ③④❴❾❫❣♠ ③⑦❩⑧▲❱❨✌❫❣❩⑧③◗◆❙▲✙❿✍▲✭❩⑧❭❪P♦❯✏✎ ❫ ó✂î❢î❢ö✽é❊ï❺é❊ô➐é➨ñ❣ê✟ó❙ë♥ç✣ì❤ì ⑥③◗❖◗❖❧❨✌▲❱❯❪①♥⑨✦▲✙❩⑧❭❪▲ ♠✉①❪❿➃②❺▲❱❨✠P❣❼☎③❻❩✌▲✭❨❤❫❳❩✌③⑦P✂♠♥❴✁✑ ❩⑧P✗❴⑧P❙❖◗◆❙▲ ✒✔✓✍✕✍✓✗✖✙✘ ✓ ②✉❞✽③❻❩✌▲✭❨❤❫❳❩✌③⑦P✂♠✛✚ ⑥▲❜⑨✭P❙①❪❖④❯❡❩✌❫❣➀✂▲ ✕✢✜✓✗✣ ✕✥✤✦✓★✧ ⑥❭❪▲✭❨✌▲✩✒✪✤✦✓✫✕✥✤✦✓✗✖✬✘✤✞✓ ⑩✭⑩✭⑩ ✭➃ç✦ù✌ç ✒✔✓ í ✕✮✓ ñ❣ï❚ö ✘✓ ö✂ç✦ï❹î❳ô➓ç✕ô❊õ♥ç✮÷✗ñ❳ôýù✣é✯✝❣í✰✡❳ç❤ð✦ô➓î❳ù❃î❵ÿ✙ë♦ï✲✱❣ï❚î❣è❀ï❪ì❤í✤ñ❳ï❹ö❾ÿ✦î❳ù⑧ð❤ç ✡❳ç❤ð✦ô➓î❳ù❤í➃ù⑧ç✣ìýø♥ç❤ð✦ôýé☛✡þç✭ê✆☎þí✤ô➇õ❪ñ❣ô❜ù⑧ç✦ì✣ë♦ê➷ô➨ì➁ÿ❤ù⑧î❳÷ ö❣é④ì✭ð✦ù⑧ç✭ôýé✴✳✦é❊ï✉ó î❣ë♦ù❡÷✗î❱ö✂ç✦ê☎ø❹ù⑧î❙ú✭ê⑦ç✭÷ î❳ï ñ✙ó❣ù✣é➨ö❲î❵ÿ✍ì✣é✴✳❢ç✶✵✷ ✕✮✜✓✸✣ ✕✹✤✞✓ ÷❾ç❤ñ❣ï❪ì❾ô❊õ♥ñ❳ô❬ô➇õ❪ç❡é❊ï♥é❊ôýé➨ñ❳ê☎ó❣ë♥ç✦ì❤ì✃ÿ✦î❣ù✗ô❊õ♥ç✗é❊ô➓ç✦ù⑧ñ❣ô➐é☛✡❳ç ø❹ù⑧î❢ð✌ç✦ì❤ì✗î❣ïòó❣ù✣é➨ö✄✵✩é④ì✻✺➓ñ✌ø✂ø❚ù⑧î✼✝✂é❊÷❾ñ❣ô➱ç❤ö✾✽✠ÿ❤ù⑧î❳÷ ô❊õ♥ç❾ì✭î❳ê❻ë♦ô➐é➨î❣ï î❵ÿ✍ô❊õ♥ç➁ø❹ù⑧î❙ú✭ê⑦ç✭÷ î❳ï ñ✗ó❙ù✣é➨ö❃î❵ÿ➃ì✣é✴✳❢ç ❛ ✵✷üæ❃ç✍ë✉ì✦ç✍ô➇õ❪ç✗è✻î❳ù⑧ö❃ñ❤ø❙ø❚ù✌î✁✝✂é❊÷❾ñ❣ô➱ç✙ú❤ç✌ð❤ñ❣ë✉ì✦ç ✕✓ ñ❳ï❹ö ✕✤✞✓ ñ❳ù⑧ç ✡❳ç❤ð✦ô➓î❳ù❤ì✽ô➇õ❪ñ❣ô❬õ❪ñ✿✡❳ç❃ö❳é❀▼ç✦ù⑧ç✭ï♥ô➁ê⑦ç✭ï❈ó❙ô❊õ✉ì✁ æ❃ç✽ì⑧õ♥ñ❳ê❊ê✝ì✦ç❤ç❡ê⑦ñ❳ô➓ç✭ù✍õ❪î❣è ô➇õ✉é④ì✙ñ✌ø✂ø❚ù⑧î✼✝✂é☛❁ ÷✗ñ❳ôýé➨î❳ï✰é④ì❾ð✌ñ❣ù✣ù✣é➨ç❤ö❃î❳ë♦ô❂✩❃✣ô❥é④ì✍ð✭ê⑦ç❤ñ❳ù✍ô❊õ♥ñ❳ô✃ô➇õ❪ç▼ø❹ù⑧î❱ð❤ç❤ö❳ë♦ù✌ç✗î❳ë♦ôýê➷é❊ï❚ç❤ö✙é④ì✤ù⑧ç❤ð✭ë♦ù❤ì❤é☛✡❳ç☞❄ ô➇õ❪ñ❣ô✒é④ì❤í❀ô❊õ♥ç✝ø❚ù⑧î✂ú✦ê◗ç✦÷ î❳ï❃ô➇õ❪ç✠ó❙ù✣é➨ö ❛ ✵✰ð❤ñ❣ï❃ñ❣êì✭î✽ú❤ç➁ì✭î❳ê✆✡❳ç❤ö✽ú✁☎➃é❊ô➓ç✦ù✌ñ❳ôýé➨î❳ï❪í❀è❀é❊ô➇õ❃ñ❳ï é❊ï❺é❊ô➐é➨ñ❣ê❈ó❙ë♥ç✣ì❤ì☎ø❚ù✌î✿✡❢é➨ö❙ç❤ö➃ú✁☎✹ì✭î❳ê✯✡❱é❊ï✉ó ✒✪❅ ✓ ✕✮❅ ✓ ✖❆✘❅ ✓ í☎ç✦ô➓ð❇✼❈✼✽æ✮ç❀ø♥î❳é❊ï♥ô❀î❳ë♦ô☛ô❊õ♥ñ❳ô é❊ô➓ç✭ù⑧ñ❳ôýé➨î❳ïüî❳ïòð❤î❱ñ❣ù❤ì✭ç✦ù✹÷✗ç✣ì⑧õ♥ç✣ì➈é④ì✤ð✌õ♥ç✌ñ❤ø❪ç✭ù✤ú❤ç❤ð✌ñ❣ë✉ì✭ç✍❉➅é④ì▼ì✣÷✗ñ❳ê❊ê⑦ç✭ù❤í❧ñ❳ï❹ö❾ô➇õ❪ç✭ù⑧ç➨ÿ✦î❣ù⑧ç q
there is less work per iteration; and because fewer iterations are required, but the number of iterations needed to olve An un=fh still O(n2) Since some smooth components of the error will still remain SLIDE 3 2. If after a few iterations, the error is smooth, we could solve for the error on a coarser mesh, e. g A2h e2h=r2h Smooth functions can be represented on coarser grids Coarse grid solutions are cheaper This idea is in fact the central idea af multigrid techniques. In order to turn this idea into a practical algorithm, several ingredients will be required 2.2S SLIDE 4 If the high frequency components of the error decay faster than the low frequency components, we say that the iterative method is a smoother 2.2.1 Jacobi SLIDE 5 Is Jacobi a smoother? →NO
❊●❋■❍✁❏❑❍✶▲◆▼✰❖✴❍☞▼✦▼✰P✥◗✿❏❑❘✩❙■❍✁❏✰▲☛❊❂❍☞❏✞❚✿❊❯▲❱◗✿❲■❳✫❚✿❲❩❨❭❬✦❍✦❪✦❚✿❫✲▼✁❍✹❴☞❍☞P✥❍✁❏✰▲☛❊❂❍☞❏❑❚❵❊❛▲❱◗❵❲■▼❜❚❵❏❑❍✗❏❑❍✦❝☞❫❞▲☛❏❑❍✦❨✿❡ ▲❱❢☛❍❈❢✹❣❤▲◆▼✐▼❦❥❧❚❵❖☛❖✴❍☞❏✁❢ ♠★♥❞♦✰♣✁♣✼♣ ♦❑q■r✩s✲♥■t✶♠✾r✼✉✇✈❵①✷②✯♦✞r✁✉✦③✿♦❑②④✈⑤s✾⑥✍s■r✁r❈⑦❞r✼⑦❭♦❑✈ ⑥⑧✈⑩⑨✴❶⑩r✐❷❇❸✍❹✹❸✶❺✙❻❸ ⑥❼♦✞②✴⑨④⑨❾❽❧❿❣✛➀➂➁✔➃ ➄ ❺ ➅ ➆❵➇ ➅ ➈▲☛❲❩❪✦❍✐▼✁◗✿❥➉❍✗▼✦❥➉◗❈◗✿❊●❋➊❪✦◗✿❥✪❙■◗❵❲➋❍✁❲★❊❛▼✩◗⑧❴✩❊●❋■❍❜❍☞❏❦❏❑◗❵❏✩P➌▲☛❖☛❖✛▼❦❊❯▲☛❖☛❖✛❏❑❍✁❥❧❚❵▲☛❲➋❢ ➍➋➎➐➏☛➑➓➒→➔ ➣ ♣✍↔✁↕✗③✿①☛♦✞r✁✉✪③❜①●r✼➙✙②✴♦❑r✼✉✞③❵♦❑②④✈⑤s★⑥✼➛✲♦❑q★r ❍☞❏❦❏✞◗✿❏✩▲◆▼✐▼❦❥❧◗❈◗✿❊●❋➛★➙✮r✰➜✁✈⑤♥■⑨◆⑦✸⑥⑧✈⑩⑨✴❶⑩r✔①●✈⑩✉✍♦❑q■r✗r✁✉✞✉❑✈⑩✉ ✈⑩s➝③ ❪✦◗❈❚✿❏✦▼✁❍✁❏✩❥❧❍☞▼❑❋➛❞r⑩♣ ➞➟❷➀ ❸✫➠ ➀ ❸ ❺✏➡ ➀ ❸ ♣ ➢➥➤◗❈◗❈❨➉▲❱❨⑤❍✦❚➦❬✦❍✞❪✦❚❵❫✲▼☞❍❈➧ ➨❆➩t❧✈✲✈❵♦✞q❭①●♥★s★➜❦♦✞②✴✈⑩s★⑥✇➜✁③⑤s❭♠➋r✩✉✞r✁➫■✉✞r✼⑥❑r✁s➐♦❑r❈⑦➟✈⑩s➦➜✁✈⑩③❵✉✦⑥❑r✁✉✍➞⑤✉✞②④⑦★⑥✁➭ ➨✙➯✈⑩③⑤✉✞⑥❑r❇➞⑤✉✞②④⑦➦⑥⑧✈⑩⑨✴♥■♦❑②④✈⑤s★⑥✍③❵✉✞r✩➜✦q■r✼③⑤➫✾r✼✉✼♣ ➲❋✲▲◆▼✄▲❱❨⑩❍✦❚➊▲◆▼➟▲☛❲✸❴☞❚⑤❪✁❊✐❊☛❋★❍➦❪✦❍✁❲★❊❯❏❑❚✿❖✮▲❱❨⑤❍✦❚→◗❼❴➟❥❜❫❞❖✆❊❯▲➤❏❦▲❱❨➳❊❂❍✞❪✦❋✲❲✾▲❱❝☞❫★❍☞▼✁❢➳➵☞❲➸◗✿❏✞❨⑤❍✁❏✄❊❂◗➝❊❯❫❞❏❦❲ ❊●❋✲▲◆▼✩▲❱❨⑤❍✦❚✄▲☛❲★❊❂◗✄❚❇❙❩❏❑❚⑤❪✁❊❯▲❱❪✞❚❵❖✷❚❵❖➤◗❵❏❦▲☛❊☛❋❞❥✗❡✮▼✁❍☞➺✿❍☞❏✞❚✿❖✛▲☛❲➤❏❑❍✦❨❵▲❱❍☞❲✾❊❱▼✩P➌▲☛❖☛❖✷❬✦❍✗❏❑❍✦❝☞❫❞▲☛❏✞❍✞❨⑩❢ ➻✍➼❛➻ ➽✍➾➪➚✍➚➌➶■➹✇➘➓➴ ➍➋➎➐➏☛➑➓➒➳➷ ➬①❞♦❑q★r ❋❞▲➤❋✗❴✦❏❑❍✦❝☞❫★❍✁❲➋❪✁➮ ➜☞✈⑩t❧➫✾✈⑩s■r✁s➐♦✞⑥❾✈❵①❞♦✞q■r✥r✼✉❑✉✞✈⑤✉❾⑦❞r✼➜✼③➂➱✪①❱③⑩⑥❼♦✞r✁✉❾♦❑q★③⑤s✩♦❑q■r ❖✴◗✿P✃❴✦❏❑❍✦❝☞❫★❍✁❲➋❪✁➮ ➜☞✈⑩t❧➫✾✈⑩s■r✁s➐♦✞⑥✼➛❞➙✹r✐⑥✞③➂➱➉♦✞q★③✿♦✍♦❑q★r✩②✯♦✞r✁✉✦③✿♦❑②④❶⑤r✐t❧r☞♦✞q■✈❞⑦➦②④⑥✇③ ▼❦❥➉◗✼◗❵❊☛❋★❍☞❏♣ ❐✛❒◆❐✛❒❱❮ ❰✛Ï■Ð⑩Ñ★ÒÔÓ ➍➋➎➐➏☛➑➓➒→Õ -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n=19 λ(RJ) mode k v 2 (mode k=2) v 15 (mode k=15) 01.02468 LOW MODES HIGH MODES 0 2 4 6 8 10 12 14 16 18 20 ➬⑥✫Ö➐③⑤➜☞✈⑩♠■②✛③❧⑥⑧t❧✈✲✈❵♦✞q■r✁✉❦× ♣✁♣✼♣✦ØÚÙ➟Û ➣
We see that P(R3)=A(R3)l= a (R3)l and, since n is the highest frequency mode, it is clear that jacobi is not a smoother 2.2.2 Under-Relaxed Jacobi SLIDE 6 RwJ=wRJ+(1-w)I 入(R)=u入(F3)+(1-)=1-(1-(F3) k=1 e obs erve that for w l the method becomes unstable (does not converge) since for some smoother is a slow down in co nvergence of the low frequency mode o be a good to be paid for Jacobi to IDE 7 Iterations required to reduce an error mode by a factor of 100 The graph shows the number of iterations required to reduce the the amplitude of each error mode by a factor of 100. We see that the standard Jacobi(w=1) algorithm, requires many iterations to eliminate the highest frequency mode On the other hand, the under-relazed Jacobi scheme, eliminates the high modes very quickly, but on the other hand, the low frequency modes take longer, that with standard Jacobi, to dis appear. We shall see that this slow down in the convergence of the low frequency modes is not really a problem and that, by using hes we will be able to speed up the oo nvergence of these modes 2.2.3 Gauss-Seidel
Ü➝Ý✰Þ✁Ý✦Ý❜ß●à■á✿ßÔâ❩ã❱ä✔å❵æ✹çéè ê✾ëìã❱ä✔å✿æ✼è❞çéè ê❩í⑤ã❱ä✔å➂æ✼è➌á✿î➋ï✿ð✮Þ❦ñ☛î❩ò✞Ý✔ó✙ñ◆Þ✰ß☛à★Ý✰à✲ñ✯ô➂à■Ý☞Þ✦ß❾õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú➉ûï⑩Ý❦ð✥ñ☛ß✮ñ◆Þ✗ò☞ü✴Ý✦á❵ö✰ß☛à★á✿ß✍ý❞á⑤òû⑤þ ñ➌ñ◆Þ✰îû ß✹á❧Þú➉û✼û ß☛à★Ý☞ö✁ÿ ✂✁✄✂✁✄ ☎✝✆✟✞✡✠☞☛✍✌✏✎✑✠✓✒✕✔✗✖✂✠✓✞✙✘✚✔✜✛✣✢✥✤✡✦ ✧✩★☞✪✬✫✮✭✰✯ ä✲✱✲å❇ç✴✳✥ä✔å✶✵✠ã✸✷✺✹✻✳✮æ✂✼ -1 -0.5 0 0.5 1 mode k λ(RωJ) ω =1.1 (UNSTABLE) ω =1/2 ω =2/3 ω =1 ê✩✽❞ã❱ä✾✱✲å➂æ✥ç✿✳✮ê✥✽■ã❱ä✔å✿æ✂✵❆ã❀✷✲✹✻✳✮æ✢ç❁✷✺✹✻✳✪ã✸✷✺✹ ê✩✽❞ã❱ä✔å❵æ⑧æ❃❂ ❄ ç❅✷✣❂❇❆❈❆❇❆❈❂⑧ó Ü➝Ý û⑩þ Þ✁Ý✁ö❊❉➂Ý➦ß●à■á❵ßÔõû ö❋✳❍●■✷⑤ð❇ý❞á⑤òû⑤þ ñ❇ò✦á❵î✻ñ☛î➦õ☞á⑤ò✁ß þÝ✸á➦ôû❈ûï➳Þú➉û✼û ß☛à★Ý☞ö✁ÿ❑❏❱õ▼▲✥Ý✄Þ✁Ý☞ß ß●à■Ý➦òû î➋ï❵ñ☛ß❛ñ û î✏è ê✚◆④ëP❖❾í✸◗❙❘✏❚⑩ã❛ä✱✲å æ✁è➓ç è ê✾ë✛ã❱ä✱✲å æ✼è ð❃▲✥Ý û⑩þ ß❂á✿ñ☛î❯✳ ç❲❱❨❳P❩✾ÿ❆Ü✸Ý➦á✿üÞ û îû ßÝ ß●à■á✿ß❩õû ö❬✳✿❭❪✷✄ß●à■Ý úÝ☞ß●à ûï þÝ✦òû✿úÝ☞Þ✐ø❞î★Þ❦ßá þ ü✴Ý✑❫❼ïûÝ❦Þ✰î û ß✫òûî✜❉✿Ý✁ö❛ô⑩Ý❀❴➉Þ❦ñ☛î➋ò✦Ý✮õû ö✰Þ û❵úÝ ❄ ð✔è ê✥✽★ã❱ä✾✱✲å➂æ✼è✮❭❲✷➐ÿ✑❏☞î Þ û✿úÝ❧Þ✁Ý✁î■Þ✁Ý❦ð✔ß☛à★Ý❃❵➋ö❦ñ❱ò✦Ý✄ßû✃þ Ý❃❵★á✿ñ❱ï✶õû ö➟ý❞á⑤òû⑤þ ñ✫ßû➊þ Ý✄á✄ô û❈ûï Þú❧û❈û ß●à■Ý✁ö✩ñ◆Þ✗á❧Þ❦üû▲✙ïû▲➌î➳ñ☛î→òû î✜❉✿Ý☞ö❛ô➐Ý✁î➋ò✦Ý ûõ✩ß●à■Ý✗üû▲➊õ✦ö❑Ý✦÷✁ø★Ý☞î❩ò☞ù ú➉ûï⑤Ý☞Þ☞ÿ ✧✩★☞✪✬✫✮✭❜❛ ❝❡❞✏❢❈❣✐❤❥❞❧❦♥♠✣♦✥♣q❣❧❢✍r❨s✜❦t❣✏❢❇✉✑❞❧♠✈❣✏❢❇✉✗s✜✇❈❢❋❤P♦①❢❈❣✏❣❧♠❨❣③②✈♠✗✉✗❢❋④✓⑤✑❤⑦⑥❙❤✣✇⑧❞❧♠❨❣✺♠P⑥q✷❇⑨✣⑨ 0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 n=19 Number of iterations mode k ω = 1 ω=2/3 ⑩❩à■Ý✶ô❵ö❑á✐❵■à✃Þ❑à û▲ÔÞ❧ß☛à★Ý❧î★øú➉þ Ý☞ö ûõ❜ñ☛ßÝ✁ö❑á❵ß❛ñ û î■Þ✶ö✞Ý✞÷✁ø❞ñ☛ö❑Ý✦ï➦ßû ö❑Ý✦ï✿ø★ò✦Ý❧ß●à■Ý❧ß●à■Ý➟áú❵❩ü✆ñ☛ß❯ø★ï⑤Ý ûõ✩Ý✦á⑤ò✦à➊Ý✁ö❦öû ö ú➉ûï⑤Ý þ ù➟á✫õ☞á⑤ò✁ßû ö ûõ❷❶❨❸✣❸⑤ÿ❭Ü➝Ý✐Þ✁Ý✦Ý✩ß●à■á❵ß✢ß☛à★Ý✐Þ❦ß❂á✿î➋ï⑩á✿ö❑ï➟ý✲á⑩òû⑤þ ñ✢ã✬✳ ç❅✷❈æ á✿üô û ö❦ñ☛ß☛àúð✗ö✞Ý✞÷✁ø❞ñ☛ö❑Ý☞Þ úá✿î✾ù✃ñ☛ß❂Ý☞ö✞á✿ß❯ñ ûî■Þ➦ßû Ý✁ü✆ñúñ☛î➋á❵ßÝ✸ß●à■Ý❭à✲ñ✯ô➂à■Ý☞Þ✦ß✹õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú❧ûï⑤Ý☞Þ✁ÿ ❹î➝ß☛à★Ý û ß●à■Ý✁ö✪à■á❵î➋ï➂ð✢ß●à■Ý✰ø❞î➋ï⑩Ý☞ö❊❺❂ö❑Ý✁ü✴á❇❻➐Ý✦ï➉ý❞á⑩òû⑩þ ñ✷Þ✁ò✦à■ÝúÝ☞ð✮Ý☞ü✯ñúñ☛î❩á✿ß❂Ý❦Þ✩ß●à■Ý✪à❞ñ✯ô❈à ú❧ûï⑤Ý☞Þ ❉✿Ý☞ö❦ù✸÷☞ø❞ñ❱ò✐❼✿ü✯ù➂ð þ ø❞ß ûî ß☛à★Ý û ß●à■Ý✁ö✐à★á✿î➋ï✿ð✫ß☛à★Ý✗üû▲ õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú➉ûï⑩Ý❦Þ✶ßá✍❼⑤Ý❜üû î➐ô➐Ý☞ö✦ð✍ß●à■á✿î ▲➌ñ☛ß●à➸Þ❦ßá❵î➋ï⑩á✿ö❑ï ý✲á⑩òû⑤þ ñ◆ð❜ßû ï✿ñ◆Þ✁á✐❵✣❵★Ý✞á❵ö✁ÿ Ü✸Ý➊Þ❑à■á✿ü☛ü✇Þ☞Ý✦Ý➳ß☛à★á✿ß✶ß☛à❞ñ◆Þ➦Þ❦üû▲ ï û▲➌î✠ñ☛î❆ß●à■Ý òûî✥❉➂Ý✁ö❛ô➐Ý☞î❩ò✞Ý ûõ✹ß●à■Ý✍üû▲➟õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú❧ûï⑤Ý☞Þ✹ñ◆Þ✍îû ß✛ö❑Ý✦á✿ü☛ü✯ù✗á③❵➋öû⑩þ ü④Ýú á✿î➋ï✩ß☛à★á✿ß❛ð þ ù✐ø✲Þ❦ñ☛î➐ô òûá❵ö✦Þ✁Ý☞ö úÝ❦Þ❑à★Ý❦Þ✦ð❽▲✥Ý❾▲➌ñ☛ü☛ü þ Ý❜á þ ü✴Ý✶ßû Þ❿❵★Ý✦Ý✞ï➟ø✍❵→ß●à■Ý❜òûî✜❉✿Ý✁ö❛ô⑩Ý✁î➋ò✦Ý ûõ✩ß●à■Ý☞Þ✁Ý ú➉ûï⑤Ý☞Þ☞ÿ ✂✁✄✂✁♥➀ ➁✝✔➃➂✡➄✍➄⑧✌⑧➅✂✠✓✦❙✞✡✠✓✒ ✧✩★☞✪✬✫✮✭✰➆ ➇✺❢✍✇❈❤P➈♥➈❿➉ ❩
Is g Since the eigenvectors of RGs and a do not coincide, there is little we can say about the smoothing properties of Gauss-Seidel by looking at the eigenvalues of the iteration matri Iterations required to reduce an A error mode by a factor of 100 By looking at the number of iterations required to reduce the amplitude of each mode, of the A matrit, by a factor of 100, we can determ ine the smoothing operties of the Gauss-Seidel scheme. It turns out that based on the above graph, the high frequency modes do in fact decay at a much faster rate than the 2.3 Restriction SLIDE 10 We shall require procedures for transferring information between grids. The process of transferring a vector from a fine to a coarse mesh is called restriction Given wh we obtain w 2h by restriction 2=功hh
n=19 mode k v 2 (mode k=2) v 15 (mode k=15) λ(RGS) 0 2 4 6 8 10 12 14 16 18 20 -1 -0.8 -0.6 -0.4 0 0.2 0.4 0.6 0.8 1 ➊❀➋❬➌➎➍P➏✥➋❧➋✸➐➒➑✗➓❈➔✄→✗➓❈➣✂➍↕↔✣➙✓➙✗→✝➋✸➛✈➙✓➙P➜✏➝➃➓❈➞❊➟ ➠❈➠❈➠ ➡✥➢✬➤➦➥✐➧❷➨➫➩➃➧✈➧❈➢➯➭❨➧❈➤✜➲❥➧✐➥⑧➨➒➳❥➵✐➸↕➳✸➺✲➻❃➼✂➽❯➾P➤✩➚⑦➪➶➚❨➳➹➤✩➳❥➨✺➥✐➳❥➢✬➤➦➥⑧➢❙➚❨➧❊➘q➨✬➩✜➧⑧➵✏➧❾➢✄➸❷➴➯➢✬➨✕➨❿➴t➧⑦➷❽➧▼➥✐➾❥➤❑➸❈➾P➬ ➾✣➮✐➳P➱✗➨✶➨➫➩➃➧❾➸❊✃▼➳❇➳P➨✬➩✗➢✬➤☞➭❾❐✩➵✏➳✏❐✜➧⑧➵❊➨❿➢❙➧❊➸✈➳❀➺❮❒❬➾P➱✓➸✏➸❊❰❡➡✚➧⑧➢❙➚❨➧⑧➴Ï➮⑧➬✑➴t➳❇➳✍Ð❥➢✬➤✓➭❑➾❥➨q➨➫➩➃➧✈➧❈➢➯➭❨➧❈➤✜➲❥➾❥➴Ñ➱✜➧⑧➸↕➳✸➺ ➨➫➩➃➧❾➢✬➨❡➧❈➵❧➾❥➨❿➢❙➳❥➤❯✃✈➾P➨✕➵❊➢➯Ò✓Ó Ô✩Õ☞Ö✬×✮Ø✰Ù ➊❡➜✏➓❈➞✐➍❥➜✏➔t➙❨Ú✜➋✶➞❧➓✍Û❨➏✜➔t➞✏➓❇→✑➜❧➙✈➞✏➓❇→✗➏✜Ü❈➓❷➍PÚ❑Ý ➧❈➵❊➵❧➳❥➵❋✃▼➳❇➚❨➧✺Þ✓ß ➍⑦à❙➍✣Ü⑧➜❧➙❨➞✺➙Pàqá❇â✣â 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 120 140 160 180 200 n=19 Number of iterations mode k ➠❇➠❈➠✐➌❋➑✑➔✄➋✺➍ ➭☞➳❇➳✍➚✈➸❊✃▼➳❇➳P➨✬➩✜➧⑧➵ ➠ ã➬❮➴t➳✍➳❇ÐP➢✬➤☞➭❑➾P➨✶➨➫➩➃➧⑦➤✥➱✗✃✈➮✐➧❈➵❾➳❀➺❷➢✬➨➒➧❈➵❧➾❥➨❿➢❙➳❥➤✜➸❾➵❧➧✐ä⑧➱✗➢✬➵✏➧✏➚✑➨➒➳❮➵❧➧✐➚P➱✜➥✏➧⑦➨➫➩➃➧✈➾❥✃✲❐✩➴➯➢✬➨✕➱✜➚❨➧▼➳❀➺⑦➧✐➾❨➥✏➩ ✃▼➳❇➚❨➧❊➘▼➳❀➺①➨➫➩➃➧ Ý ✃✈➾P➨✕➵❊➢➯Ò✣➘↕➮❈➬å➾▼➺⑧➾✣➥❈➨➒➳❥➵✻➳❀➺❑æ❨ç✣ç❥➘↕➷❽➧❯➥✏➾P➤è➚✣➧❈➨➒➧⑧➵❊✃↕➢✬➤✩➧✻➨✬➩✜➧①➸❊✃▼➳❇➳P➨✬➩✗➢✬➤☞➭ ❐✩➵✏➳✏❐✜➧⑧➵❊➨❿➢❙➧❊➸❯➳✸➺é➨➫➩➃➧å❒✺➾P➱✓➸✐➸✐❰❡➡✚➧⑧➢❙➚❨➧⑧➴✲➸❈➥✏➩✜➧⑧✃▼➧❇Óëê❊➨❾➨❿➱✗➵❊➤➃➸❑➳❥➱✗➨↕➨➫➩➃➾❥➨▼➮✐➾ì➸❈➧✐➚å➳P➤è➨➫➩➃➧✰➾✣➮✐➳❥➲❥➧ ➭✣➵❧➾✏❐✜➩☞➘í➨➫➩➃➧✾➩✗➢➯➭✍➩❷➺✐➵❧➧✐ä⑧➱✜➧❈➤✩➥❈➬✈✃✈➳✍➚✣➧⑧➸❾➚✣➳✈➢✬➤⑦➺⑧➾✣➥❈➨q➚❨➧✏➥✐➾P➬▼➾P➨✶➾✈✃↕➱✜➥✐➩➎➺⑧➾ì➸❊➨➒➧⑧➵➎➵❧➾P➨❡➧❷➨✬➩✜➾❥➤❑➨➫➩➃➧ ➴t➳P➷❑➺✐➵❧➧✐ä⑧➱✜➧❈➤✩➥❈➬✑➳❥➤➦➧❊➸❈Ó îqï✕ð ñ✿ò✮ó❨ô✗õ✜ö✸÷✩ô➃ö❀ø③ù Ô✩Õ☞Ö✬×✮ØûúPü ý➧❑➸❧➩✜➾❥➴✬➴❃➵❧➧✐ä⑧➱✗➢✬➵❧➧▼❐✩➵✏➳❇➥✐➧✐➚❥➱✗➵❧➧⑧➸❾➺⑧➳❥➵❯➨✕➵✏➾❥➤➃➸➫➺⑧➧❈➵❊➵❊➢✬➤☞➭û➢✬➤❇➺⑧➳P➵❊✃✈➾P➨✕➢❙➳P➤þ➮✏➧❈➨❿➷❽➧✏➧❈➤✙➭✣➵❊➢❙➚ì➸❈Ó ÿ➦➩➃➧ ❐✩➵✏➳❇➥✐➧⑧➸✏➸✲➳✸➺✺➨✕➵✏➾❥➤➃➸➫➺⑧➧❈➵❊➵❊➢✬➤☞➭↕➾❷➲❥➧✐➥⑧➨➒➳❥➵✂➺✐➵✏➳❥✃ ➾✂✁❽➤✩➧✺➨➒➳❾➾↕➥✐➳❇➾P➵✐➸❈➧❬✃▼➧❊➸❧➩❮➢✄➸✲➥✏➾P➴✬➴t➧✐➚❷➵✏➧❊➸❊➨❿➵❊➢❙➥⑧➨❿➢❙➳❥➤➦Ó ➌❃➔☎✄✣➓❈Ú✝✆✟✞✡✠✶➓➎➙Þ➜✐➍P➔♥Ú☛✆✌☞✍✞ Þ☞ß✟✎✑✏✓✒✕✔✖✎✑✗✙✘✚✔✕✗✙✛✢✜ ✆✝☞✍✞✤✣✦✥✞ ☞✧✞ ✆✝✞ ✥ ✞ ☞✧✞✩★ ➞✏➓❇➋✸➜❧➞✏➔♥Ü⑧➜❧➔♥➙✣Ú✝➙✫✪✩➓❈➞✐➍❥➜❧➙❨➞✭✬❙➛▼➍❥➜❧➞✏➔✯✮✩✰⑧➠ ✱
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)
×➘Ø☎Ù❯Ø◆Ú Û✟Ü◆Ý◆Þ✿ß✕Ý✙àâá ã❻ä✓å●æ❯ç➊è✫é 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 2nd Eigenvector (n=19) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 18th Eigenvector (n=19) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 2nd Eigenvector (n=9) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 2nd Eigenvector (n=9) ê❽ë❃ì❃í➃î✟ïñð◆ò✟ó↔ô✺õ✍ö✚÷❡ë✫ø✝ù✫ú✍û❈ì✝ü☛ý❻í✕þ✖ë✫ÿ✡í✁❊ð◆ò✟ó③î✩õ➛ÿ✡ë❃ì❃í❊ë✫ø✝ù✫ú✍û❈ì✝÷❳ü✄✂ ☎✝✆✟✞✠✞☛✡☞✞✍✌✏✎✒✑✔✓✖✕✗✞✙✘✚✎✛✕✚✜☛✌✏✎✢✜✤✣✦✥★✧✠✘✏✑✪✩✬✫✭✎✛✜☛✑✮✣✰✯✱✆✲✜☛✌✍✆✳✆✴✧✵✘✖✘✍✜✬✥✶✣✴✜✷✸✌✍✧✪✣✴✎✹✆✲✜✬✥✵✆✺✓✍✕✗✞✍✻✙✫✴✞✏✣✼✌✏✜☛✞✙✘✽✌✍✧✪✣ ✆✟✧✮✾✵✞✿✾✪✞✏✕✚❀❁✣✼✞❂✥✦✧✪✎✢✜✤✾✵✞❃✞✛✡☞✞✍✌✏✎☛✘✍❄❅✘✚✜✤✣✝✌❆✞❇✆✭✜✬✥❈✆✠❉❃✑✁❊❋✞✚✘●✑✪✣↕ü❍❉●✧✮❀✖✧❆■❋■✟✞✍✧✮✕✿✧✵✘✒✩❏✑✪✯❑❉●✑❈❊✶✞✙✘✿✑✪✣ ÷✫ü❍✧✪✣✝❊✒✆✟✞✏✣✼✌✍✞✿✞✗▲✪✆✭✜☛▼✙✜✤✎✸✘✚✩❏✑✪✯◆✌✍✑✮✣✴✾✪✞✙✕✛✥✦✞✏✣✼✌✍✞✁❖ ×➘Ø☎Ù❯Ø❈× P❘◗✱❙✳❙③Þ✭❚✵❯ ã❻ä✓å●æ❯ç➊è✪❱ ❲◆❳ø✴❨❏❩❬❨✯ë✪❭✦ÿ✡ë❃ì❃í❈❩û✯ø↕ü☛þ✁❪✚ø✝ý✢í❊ú✍í✏❫✩ú✍í✁❁í✖ø✦❴✍í✕ì❬❭➛í✁❨❏❨❜û☎ø↕÷✫ü❘✂ ❲❛❵ë✪❭➜ÿ✡ë❃ì❃í✁➁ë❳ø↕ü✝ý✢í✺þ❄ë✫ÿ✡í❇❜✩û☎ù✶❜✿í✖ú❩ÿ✡ë❃ì❃í✁➁û☎ø↕÷✫ü❘✂ ❝✒✞✙✣✝✌✍✞❞✆✴✧✪✾❈✜✤✣✦✥✺✧❡✓✙✧✪✘✍✎❢✞✏✕❣✌❆✑✮✣✴✾✪✞✙✕✢✥❋✞✏✣✼✌✍✞❣✕✗✧✪✎❢✞✁❖ ❤❥✐❧❦ ♠♦♥✴♣❅q✔♣❅r☞s✉t✇✈✟①✔♣❅r ã❻ä✓å●æ❯ç➊è✶② ☎✝✆✟✞③■✼✕✗✑❈✌✍✞✚✘✍✘④✑✔✓✹✎✢✕✗✧✪✣✴✘❧✓✙✞✙✕✚✕✚✜✤✣✲✥❬✧●✾✪✞✍✌✙✎❢✑✪✕❣▼✍✞✏✎✛✯❅✞✍✞✏✣✠✧③✷❅✣✼✞❣✧✮✣✼❊⑤✧❬✌✍✑✁✧✮✕✍✘✙✞❣❉❃✞✚✘✗✆✖✜⑥✘⑧⑦✮✣✼✑✪✯⑨✣ ✧✵✘✸■✝✕✗✑✪✩❏✑✮✣✦✥✦✧✪✎✢✜☛✑✪✣✝❖ ⑩û❷❶✫í✖ø✖❸❁❹✍❺●❭➡í❊ë✫ý✭❴✍❪✚û☎ø❬❸❬❺♠ý✦❩❬❻❚✪❼❻Ü☛❼❻àâá❻Þ✭❽✺Ý☛❼❻à ❸✖❺❣❾✰❿❹✍❺ ❺ ❸❬❹✍❺ ❿ ❹❆❺ ❺◆➀ ❫✩ú❁ë❋❨✯ë❳ø✩ù❋❪✮❴❁û☎ë✫ø✟ë✶❫❻í✖ú✍❪✪❴❁ë❳ú✭ð◆ÿ❃❪✪❴❁ú✍û❏➁✩õ✙✂ ➂✤ô ➃
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➄➆➅✦➇✴➈❷➉✁➊✮➋❷➋❏➅❋➌✭➍➆➎✹➏✴➐✗➎●➑✤➒✴➓❢➔✙→✢➣✴↔✪↕❏➙✮➓✛➑☛↔✮➒✝➛ ➜✇➝➟➞❋➠✏➡✁➡✏➡✙➠✦➢✲➤➦➥ ➧ ➨❡➩✶➫ ➧✗➭ ➝ ➨➧ ➩✶➫ ➭ ➨❡➩✶➫ ➧✗➭❷➯ ➥ ➝ ➥➧❣➲➨➧ ➩✶➫ ➭➵➳ ➨➧ ➩✶➫ ➭❏➯ ➥✏➸ ➺✼➻✦➼✤➽❘➾❍➚✶➪ ➶➘➹➦➴✚➷♦➬ ➮❃➱✭✃✔➷✪❐❈❒✄❮✛❰❛Ï⑨➹✭❐❈ÐÒÑ➵✃✛➱➦➴✗❮❢➹✭❰ Ó❂Ô✇Õ✗Ö✟➎❇×❆➎✁➐ØÕ✗×❆➈❷➉✙Õ✗➈❷Ù✶Ú✖➊✮Ú✴Û✖➇✟×❆Ù✶➋❷Ù✶Ú✟Ü✦➊✪Õ❆➈❏Ù❋Ú⑤Ý❃➊✪Õ✗×❆➈⑥➉✙➎✁➐③➐❆➊✪Õ❆➈❷➐ØÔ❧➅ Þ➧ ➩ ➩ ➝◆ß ➲ Þ➩ ➧ ➩ ➸✔à Ô❧Ù✶×á➊☞➉✏Ù✶Ú✴➐ØÕ❆➊✶Ú✦Õ✇ß✪➌❈Õ✗Ö✴➎➆Ý●➎✙Õ❆Ö✟Ù✭Û④➈⑥➐á➐✗➊✶➈❷Û✹Õ❆Ù☞Ö✴➊✵â❋➎⑨Õ✗Ö✴➎➆ã✟ä✭å✪æ✛ä✭ç✁æ✛è➵é✇ä✟ê➦ë✇å✪è✴ë❅ì✦å✵ç✍í✉➛⑨➄❥Ö✟➈⑥➐❘➈⑥➐✱➊ ➇✟×❆Ù✶➇✼➎✏×✗Õ✔➅❞Õ✗Ö✴➊✮Õ✇Ô☛➊❋➉✙➈❷➋❏➈❏Õ❆➊✮Õ✗➎❈➐⑨➐ØÙ❋Ý✿➎✸Ù✮Ô➵Õ✗Ö✟➎✸Õ❆Ö✟➎✏Ù❋×✗➎✏Õ✗➈⑥➉✏➊✮➋✭➉✏Ù✶Ú✲â✶➎✁×✗Ü❋➎✏Ú✴➉✏➎✉➇✟×❆Ù✦Ù✶Ô☛➐✱Ô❧Ù✶×áÝ❣➏✟➋❏Õ✗➈❷Ü✶×❆➈⑥Û Ý●➎✙Õ❆Ö✟Ù✭Û✟➐✏➛❃î☞Ù✪➍➆➎✁â✶➎✏×❈➌✝Ù✮Õ❆Ö✟➎✏×④➐✗➈❏Ý●➇✟➋❷➎✏×④➉✍Ö✴Ù✶➈⑥➉✙➎✁➐✁➌➦➋❷➈❏ï❋➎✒Õ❆Ö✟➎❃Ù✶Ú✴➎✁➐✹➉✙Ù❋Ú✴➐Ø➈⑥Û✭➎✁×✗➎❈Û♦➊✮ð✼Ù✪â✶➎❋➌✄➉✏➊✶Ú ➊✮➋⑥➐✗Ù✒➍✸Ù✶×❆ï❃➍➆➎✁➋❏➋❘➈❏Ú✺➇✟×✍➊✶➉✙Õ✗➈⑥➉✙➎✶➛ ñÙ✶Ú➵➐Ø➈⑥Û✭➎✏×❈➌✼Ô❧Ù✶×❇➈❷➋❏➋❷➏✴➐ØÕ✗×✍➊✪Õ❆➈❏Ù❋Ú♦➇✟➏✟×❆➇✼Ù❋➐✗➎✁➐✁➌➵Õ❆Ö✴➊✪Õ✹Õ❆Ö✟➎✿ò✴Ú✟➎●Ü✶×❆➈❷Û➘ó✠Ö✴➊✶➐✹ô✳➝öõ❁➇➵Ù❋➈❏Ú✦Õ❆➐✁➌✄➊✮Ú➵Û Õ✗Ö✴➎④➉✙Ù❋➊✶×❆➐✗➎❞Ü✶×❆➈❷Û✺÷✶ó✺Ö➵➊✶➐❥ô♦➝✰ø✿➇✼Ù✶➈❷Ú✦Õ❆➐✁➛❅➄❥Ö✟➎❇➇✟×❆Ù✶➋❷Ù✶Ú✟Ü✦➊✪Õ❆➈❏Ù❋Ú❁Ù✶➇✼➎✏×✍➊✪Õ❆Ù✶×③➉✏➊✶Ú✺ð➵➎❇➍③×❆➈✬Õ✗Õ✗➎✁Ú ➊✶➐ Þ➧ ➩ ➩úù➧ ➩ ➝ ➞ ÷ ûü ü ü ü ü ü ü ü ý ➞ ÷ ➞þ➞ ÷ ➞ÿ➞ ÷ ➞ ✂✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ û ý ➨➥ ➨➧ ➨✆☎ ✄ ➧ ➩ ➝ ûü ü ü ü ü ü ü ü ý ➨➥ ➨➧ ➨✝☎ ➨✆✞ ➨✝✟ ➨✝✠ ➨✝✡ ✂✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ➝ ù➩ ➄❥Ö✟➎✹➉✙Ù❋×✗×❆➎✁➐✗➇➵Ù❋Ú✴Û✭➈❷Ú✟Ü❣×❆➎✁➐ØÕ✗×❆➈⑥➉✚Õ✗➈❷Ù✶Ú❬Ù✶➇✼➎✏×✍➊✪Õ❆Ù✶×❅➈⑥➐❥➏✴➐✗➏✴➊✮➋❷➋❷➅❃×✗➎✏Ô❧➎✏×❆×✗➎❈Û●Õ✗Ù❃➊✶➐❥➊❣Ô❧➏✟➋❷➋➦➍➆➎✁➈❏Ü❋Ö❋Õ❆➈❏Ú✴Ü Ù✶➇✼➎✏×✍➊✪Õ❆Ù✶×③➊✮Ú➵Û❁Ö✴➊❋➐✸Õ✗Ö✟➎✹Ô❧Ù❋×✗Ý Þ➩ ➧ ➩❡ù➩ ➝ ➞ ☛ û ý ➞ ÷ ➞ ➞ ÷ ➞ ➞ ÷ ➞ ✄ ûü ü ü ü ü ü ü ü ý ➨➥ ➨➧ ➨✆☎ ➨☞✞ ➨✆✟ ➨✆✠ ➨✆✡ ✂✁✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ➩ ➝ û ý ➨➥ ➨➧ ➨✝☎ ✄ ➧ ➩ ➝ ù➧ ➩ ✌❋➏✴➐✔Õ❅Ô❧Ù✶×➆➉✏Ù✶Ý●➇✟➋❷➎✙Õ❆➎✏Ú✟➎❈➐✗➐⑨➍✸➎❡Ü❋➈❏â❋➎❡Ö✟➎✏×❆➎③Õ❆Ö✟➎✎✍✹➊✮➋❷➎✏×❆ï✲➈❏Ú❃➇✴×✗Ù❋➋❏Ù❋Ú✟Ü❋➊✮Õ✗➈❷Ù✶Ú●➊✮Ú✴Û✿Ô❧➏✟➋❷➋✼➍➆➎✁➈❏Ü❋Ö❋Õ❆➈❏Ú✴Ü ×❆➎✁➐ØÕ✗×❆➈❷➉✙Õ✗➈❷Ù✶Ú➘Ù✶➇✼➎✏×✍➊✪Õ✗Ù❋×❆➐❞➈❷Ú➘Õ✔➍➆Ù Û✭➈❷Ý●➎✏Ú✴➐✗➈❷Ù✶Ú✴➐✁➛⑤Ó❂Ô ù➧ ➩ ➝öÞ➩ ➧ ➩ ù➩ ➌✄Õ❆Ö✟➎❁➉✙Ù❋Ý●➇➵Ù❋Ú✟➎✏Ú✦Õ❆➐✹Ù✶Ô õ
w2h are given by (W2h iS +w2h is 1S) (ugh iS +Uzh i, S 1) wn 2is 1, 2S 1=4( iS+u2h is 1. $+w2h, S 1+Ugh is 1, S 1) an: ffr wn=in w2h 1,ef pf nenls ff w2h are Wwn 2il 1,2S 1 +Wn 2il 1,291+ Wn 2is 1, 2S 1+Wn 2is 1, 1 4r(m221+2m i, 2Sf, 2g5 rm-terb Rld +iRn ErrAir LIEE1IwlE, yl E It rtnlo y D notynvyI"E“.tv,“1DIyf1rwtE“mnfm2 D∑ fiEUiyIE“∑ my n lEy1 r vnt mmrt wI IVI V IE I lI, rDfrDevD E1o TI i y Ito i ftu,“iDVo"Ewnl:ogv,m,“1DW“ E unll r:2AEJD“ym1i olerpf ralif n inlrf i k es, ig, fre-k-en y errf r LD“gmD.rf, y DburiD 3 ipoD,“1Dw=s“ E unll r:ox1rwtt. ym >h JEUy“EoJD,“1 Ul1:∑ftit,i护jgyL∑m
✏✒✑✔✓✖✕✘✗✚✙✜✛✣✢✥✤✣✙✧✦✩★✫✪ ✬✓✭✑✯✮✱✰ ✑✳✲✵✴ ✬✑✔✓✶✮✷✰ ✲ ✬✸✓✭✑✯✮✥✹✻✺✼✰ ✑✳✲ ✴ ✽✾❀✿✬✝✑✔✓✭✮✷✰ ✲☞❁❂✬✆✑✔✓✶✮❃✹✶✺✔✰ ✲❅❄ ✬✸✓✭✑✯✮✱✰ ✑❆✲✯✹✻✺ ✴ ✽✾❀✿✬✝✑✔✓✭✮✷✰ ✲☞❁❂✬✆✑✔✓✶✮✷✰ ✲✯✹✶✺✧❄ ✬✓✭✑✚✮❃✹✻✺✼✰ ✑❆✲✯✹✻✺❇✴ ✽ ❈ ✿✬✑✔✓✭✮✷✰ ✲ ❁❂✬✑✔✓✶✮❃✹✶✺✔✰ ✲ ❁❂✬✑✔✓✶✮✷✰ ✲✯✹✶✺ ❁❂✬✑✔✓✭✮✥✹✻✺✔✰ ✲✯✹✶✺ ❄ ❉❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊❋ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ● ✽■❍❑❏✔▲◆▼❖❍◗P✩❘ ✽ ✾ ✕✘✦❚❙❱❯✷❲❳✗✝✏✓ ✴❩❨✑✔✓ ✓ ✏✑✔✓❭❬✚❪✙❴❫❵❲❳❛❭❜❝❲✣✦❞✙✧✦❬✚❡ ❲✣❯✶✏✑✔✓ ✕✣✗✯✙ ✬✝✑✔✓✭✮✷✰ ✲ ✴ ✽ ✽❅❢❤❣✬✸✓✶✑✯✮✱✐❥✺✼✰ ✑❆✲❵✐❥✺❦❁❂✬✸✓✭✑✯✮✱✐❥✺✼✰ ✑❆✲✯✹✻✺❦❁❂✬✸✓✻✑✚✮✥✹✻✺✔✰ ✑❆✲❵✐❥✺❧❁♠✬✸✓✶✑✚✮❃✹✻✺✼✰ ✑❆✲✯✹✻✺ ❁ ✾ ✿✬✓✶✑✚✮✷✰ ✑❆✲❵✐❥✺ ❁♠✬✓✭✑✚✮✷✰ ✑❆✲✯✹✻✺ ❁❂✬✓✭✑✯✮✱✐❥✺✼✰ ✑✳✲ ❁❂✬✓✻✑✚✮✥✹✻✺✔✰ ✑❆✲ ❄♥❁ ❈✬✓✶✑✯✮✱✰ ✑❆✲✧♦ ▲ ✽♣❍❑❏✔▲◆▼❖❍◗Pq❘ ✽ ✾ r s✆t✈✉ ✇❳①❧②④③♥⑤⑦⑥■⑧❧⑨❆⑩✶②⑦❶❆⑧❦①❸❷❤⑤⑦⑤⑦⑧❹⑤ ❺❝❻❽❼❿❾➁➀➃➂❽➄ ➅❱➆⑦➇✧➈➊➉✘➋❿➌✯➇✔➍❵➎❀➋❿➈➐➏✧➇✔➍❵➎◆➋✱➑➒➈➊➋➔➓➣→✫➓✧➇✔➉↕↔✭➎✷➆⑦➇✜➌✯➇❵➓✼➎✈➌✼➋✱➍✧➎✈➋✱➑✘➈➙➑❆➛♣➜➞➝❃➑➒➟❑➠❖➑❅➉✣➇✜➋❿➈❚➎✈➌✚➑➐➉✘→❞➍✚➇❵➓➣➈➡➑❭➇✧➌✼➌✯➑✘➌✧➢ ➤➡➆⑦➜✘➎♥➋➔➓✔↔✻➜➣➝✥➑➒➟➥➠❖➑❅➉✣➇❹➦➡↔♥➑✘➈➨➧✘➌✼➋✱➉✝➩♥↔➁➫✔➇✚➍✔➑✘➠❖➇❵➓❧➠✖➑➐➉❳➇❹➦✩➑➒➈➨➧✣➌✼➋✱➉ ✾ ➩✶➢✜➭☞➈❖➎✷➆⑦➇✸➑✘➎❿➆❞➇❵➌❹➆⑦➜✘➈❝➉➒↔ ➎✷➆⑦➇✝➯❝➌✯➑✘➝❃➑➒➈✫➧❳➜✘➎✈➋✱➑✘➈➲➑✔➯⑦➇✧➌✯➜➒➎✳➑➒➌➨➋❿➈❞➎◆➌✯➑➐➉✘→❞➍✔➇✼➓❴➇❵➌✼➌✚➑➒➌✔➓■➟✻➆✫➋✱➍✔➆➥➍✚➑✘➈❞➎✳➜➒➋❿➈➊➆✫➋✂➧↕➆❴➛✔➌✯➇✔➳✧→❞➇❵➈➡➍❵➋✱➇❵➓✧➢ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 (n=19) grid h grid 2h 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 0.5 1 Interpolation error ➵✦❬ ✙✧✗✚❜❝❲✣➸➔✕❬ ✢✥❲✣✦✩✢✥✦❬ ✗✯❲④❙④➺❚❫❵✙❡✆❪✢❃✛❪ ❯✷✗✚✙➐➻❽➺⑦✙➐✦❞❫❵✪q✙✧✗✚✗✯❲❳✗ ❡➐➼ ➅➊➇■➑✣➫❵➓❵➇✧➌✼➽➒➇❹➛✔➌✯➑✘➠➾➎❿➆❞➇✭➚✻➧✣→④➌✯➇✜➎✷➆⑦➜➒➎✈↔✭➎❿➆❞➇♣➠❖➑❅➉✣➇➣➦ ✴❩➪ ➑➒➈✒➧✣➌✼➋✱➉ ✾ ➩♥↔❞➯➡➌✯➑➐➉✘→❞➍✔➇✼➓■➜❆➛✔➎✳➇❵➌❦➯❝➌✯➑✘➶ ➝❃➑✘➈❽➧❽➜➒➎◆➋✱➑➒➈⑦↔❀➎✷➆⑦➇♣➠✖➑❅➉✣➇✸➦ ✴➹➪ ➑✘➈✒➧✘➌✼➋✱➉■➩✩➯❝➝➘→✫➓■➜➞➓✔➠✖➜➒➝❿➝➘↔❹➫✧→④➎❀➋❿➠➣➯⑦➑✘➌✼➎➴➜✘➈❞➎✈↔➁➆④➋✂➧❅➆■➛✔➌✯➇✔➳❵→❞➇✧➈❝➍✧➷ ➇✧➌✼➌✯➑➒➌✧➢ ➬
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
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