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< 1, Jacobi can in fact be a good smoother. If we set the condition A(n+1/2(R,J)=An(R J)l, we obtain w =2/3. We also note 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➠ ✱