Discret ization of the poisson Problem in IR: Theory and Implementation April7&9,2003
✂✁☎✄✝✆✟✞✡✠☞☛✌✁☎✍✏✎✏☛✌✁✒✑✔✓✕✑✗✖✘☛✝✙✚✠✜✛✢✑✔✁☎✄✣✄✌✑✔✓ ✛✤✞✡✑✔✥✧✦✒✠✩★ ✁☎✓✕✪✬✫✮✭✰✯✲✱✳✙✚✠✟✑✴✞✶✵✷✎✸✓✧✹ ✪✺★✼✻✧✦✒✠✟★✽✠✰✓✾☛✿✎❀☛✝✁✒✑✔✓ ❁✧❂✌❃❅❄❇❆❉❈✘❊●❋✌❍✰■✶❏❑❏✶▲
1 Theory 1.1 Goals 1.1.1 A priori LIDE A priori error estimates of 36 nn a in terms of C(@, problem parameters) h [mesh diameter], and u Note 1 A priori theory Clearly, since a priori estimates will be expressed in terms of the unknown exact solution, a, they are not useful in determining in practice whether un is accurate enough. A priori estimates are, however, useful to compare different discretizations(which converge faster in which norms? which is more efficient?) to understand what conditions must be satisfied for rapid convergence (is u smooth enough? ) and to understand if a method has been properly implemented (for a test problem, does wh u at the correct rate?) SLIDE 2 ux=f,(0)=(1)=0 a(u,)=(u),∈X a(a, v) m,0=“md X={v∈m()|v(0)=v(1)=0} Recall that e(u) can in fact be more general-any linear functional in H-(Q) that is, any linear functional which satisfies e(o)l < c ulI(s) for anyUE Ho(Q). For example, e(u)=(0xo, v)=v(ao) is adm is sible a(ulh,)=(),u∈Xh a(u, )= wx Uz dr, e(u)=vda Xh={v∈X|vr∈P1(Th),VTh∈Th} In fact, the theory presented applies equally well to the Neumann problem and at least in R ) the inhomogeneous Dirichlet case
▼ ◆P❖✚◗✏❘✾❙❯❚ ❱☞❲❳❱ ❨❬❩✿❭☞❪❳❫ ❴✶❵❛❴✶❵❛❴ ❜❞❝✌❡❣❢❳❤✐❡❥❢ ❦✶❧♥♠♣♦✣qsr t✤✉❑✈☎✇❛①②✈☎✇✟③⑤④⑥④⑧⑦❣④✟③❅⑨❶⑩⑥❷❹❸❻❺❼⑩⑧③❽⑨❽❾ ❿➁➀ ➂⑦❣➃➅➄➇➆✧➈②❺❼④⑥❷❹⑦❣➃➇⑨➊➉❳❸➊③❽❺❣⑨❳➃➇④⑧③❅⑨⑧➋ ⑦❼➌❯➍✘➎③⑤➏➐❺❣➑☎⑩⑧➒➇➓➔➍➣→✚➎❺❼↔➅↔➅④⑥⑦↕➏➐❷❹❸❻❺❼⑩⑧③✒➒❇➙ ❷➛➄➜⑩⑥③⑤④⑥❸❻⑨➝⑦❼➌✿➞➊➟➡➠➤➢❉↔➅④⑥⑦➂➅➥③⑤❸➦↔➇❺❼④➧❺❼❸➊③✒⑩⑥③⑤④➧⑨❳➨☎➩ ➫ ➎❸➊③❅⑨❳➭✚➆➐❷➯❺❼❸➊③✒⑩⑥③⑤④⑧➒❇➩➇❺❥➄➇➆➜➍✝➲ ➳➸➵➣➺☎➻➽➼ ➾❞➚✌➪②➶❛➹✐➪②➶✩➘❅➴✝➷✺➹✐➪↕➬ ➮➥③❽❺❥④➥➛➱➩❑⑨❳❷➛➄➇➑⑤③✧✃➁✉✶✈☎✇❛①❼✈☎✇❉③❽⑨❳⑩⑧❷➛❸❻❺②⑩⑧③❅⑨❀❐❉❷ ➥➛➥✌➂ ③➁③⑤➏➐↔➅④⑧③❅⑨⑧⑨⑧③❽➆❒❷❹➄❮⑩⑥③⑤④⑥❸❻⑨❀⑦❥➌☞⑩⑧➭➅③✧❰➐Ï✺Ð❼Ï✶①❼Ñ☞Ï ③✒➏➅❺❣➑☎⑩➤⑨⑧⑦➥➃➐⑩⑥❷❹⑦❣➄✡➩✐➍✌➩✐⑩⑧➭➅③➱ ❺❥④⑧③✾➄➅⑦❼⑩➤➃➇⑨⑧③✒➌Ò➃➥ ❷❹➄➔➆➐③✒⑩⑥③⑤④⑥❸Ó❷➛➄➅❷➛➄➅Ô➽✇♣Ï➜✉❑✈⑧✃❥Õ⑤Ö➡✇❛Õ➧×➊❐❉➭➇③✒⑩⑧➭➇③⑤④➤➍→ ❷➛⑨ ❺❥➑❽➑✒➃➅④➧❺②⑩⑥③➁③❽➄➅⑦❥➃➅Ô❣➭✡➲✴tØ✉✶✈☎✇❛①②✈☎✇❀③❽⑨❳⑩⑧❷➛❸➊❺❼⑩⑧③❅⑨➤❺❥④⑧③❣➩✐➭➇⑦②❐✩③❽➈❥③⑤④❅➩✶➃➇⑨❳③⑤➌Ò➃➥ ⑩⑧⑦✚➑✒⑦❣❸Ó↔✐❺❼④⑥③➁➆➐❷❹Ù✶③❽④⑧③❽➄❣⑩ ➆➐❷➯⑨⑧➑⑤④⑧③⑤⑩⑧❷➛Ú❽❺❼⑩⑧❷➛⑦❥➄➇⑨➝➟Ò❐❉➭➇❷➛➑➧➭➊➑✒⑦❣➄♥➈❣③⑤④⑥Ô❥③☞➌❛❺❣⑨❶⑩⑥③⑤④❯❷➛➄✾❐❉➭➅❷➯➑➧➭Ó➄➅⑦❥④⑥❸❻⑨⑥ÛÜ❐❉➭➅❷➯➑➧➭✾❷➯⑨❯❸➊⑦❥④⑥③✩③⑤Ý❻➑⑤❷❹③❽➄♥⑩➧Û↕➨☎➩ ⑩⑧⑦Þ➃➇➄➇➆➐③⑤④➧⑨❳⑩⑥❺❼➄✐➆s❐❉➭✐❺②⑩➜➑⑤⑦❥➄➇➆➐❷❹⑩⑧❷➛⑦❥➄✐⑨❻❸➁➃➇⑨❳⑩ ➂ ③➽⑨⑧❺❼⑩⑧❷➯⑨❶ß✐③❽➆s➌Ò⑦❥④à④➧❺❼↔➇❷➛➆á➑✒⑦❣➄✺➈❥③⑤④⑥Ô❥③❽➄➇➑✒③â➟Ò❷➯⑨à➍ ⑨⑧❸Ó⑦✺⑦❥⑩⑧➭✔③⑤➄➇⑦❥➃➅Ô❣➭✐Û↕➨☎➩↕❺❼➄➇➆➤⑩⑥⑦✏➃➅➄➇➆➐③❽④⑥⑨❳⑩⑥❺❥➄➇➆➤❷✬➌➅❺❉❸➊③⑤⑩⑧➭➅⑦➐➆✗➭➇❺❣⑨ ➂ ③⑤③⑤➄✔↔➇④⑧⑦❣↔✐③❽④➥➛➱ ❷➛❸➊↔➥③❽❸➊③⑤➄♥⑩⑧③❅➆ ➟♣➌Ò⑦❣④✏❺Ó⑩⑧③❅⑨❶⑩❉↔➅④⑥⑦➂➅➥③⑤❸✚➩➇➆➅⑦♥③❅⑨➝➍→✴ã ➍❒❺②⑩➝⑩⑧➭➇③✴➑✒⑦❥④⑥④⑥③❽➑☎⑩➝④➧❺②⑩⑧③↕Û↕➨ ❦✶❧♥♠♣♦✣q➸ä ➍✌❾ ➓❉➍❑å❅å✔æèç✶➢➝➍✌➟➡é❣➨✿æ❬➍✌➟❳➀❅➨✿æ✤é ê ➟Ò➍✌➢❳ë➐➨✿æáì❥➟❛ë✺➨✒➢ í✴ëÜî✧ï ê ➟Òð✴➢⑧ë✺➨✟æ✤ñ✢ò ó ðå ëå❉ô❥õ ➢ ì❥➟❛ë✺➨✿æ ➉ ñsò ó ç➁ë ô❣õ ➋ ïöæø÷❅ëàî✚ùò ➟❇➠❉➨✿ú➧ë❑➟❛é❣➨✩æáë❑➟❶➀❅➨✩æ❬é➐û ü×➧Õ➧✃②ý♣ý✐Ö♣þ➇✃②Ö❑ì❥➟❛ë✺➨➤Õ➧✃②Ï✧✇♣Ï✗ÿ✒✃❣Õ✒Ö✁➧×✄✂❻①②✈⑧×✆☎♥×✒Ï❑×✒✈⑧✃❼ý✞✝ ✃②Ï✠✟➁ý✇♣Ï✶×➧✃②✈✌ÿ➧❰➐Ï✶Õ⑤Ö➡✇❛①❼Ï✶✃❼ý✐✇♣Ï➊ù☛✡ ò ➟➡➠❉➨✌☞ ÖÒþ➅✃②Ö✗✇✎✍✏☞➁✃②Ï✑✟❒ý✇♣Ï✶×➧✃❼✈✸ÿ➧❰➐Ï✶Õ⑤Ö❇✇❛①②Ï✶✃❼ý✰Ñ✝þ➐✇❛Õ⑥þ✒✍⑤✃②Ö❇✇✎✍✔✓✩×✕✍➜ú ì❥➟❛ë✺➨❽ú✁✖➦➞✘✗☎ë✙✗✕✚✜✛✣✢✥✤✠✦❀ÿ✒①❼✈✧✃❼Ï✠✟àë✢î ù òó ➟➡➠❉➨★✧✜✩✝①②✈➁×✣✪✺✃✫✂❀✉✶ý❹×✕☞✡ì❥➟Òë➐➨✰æ✭✬✯✮⑤å✱✰❼➢⑧ë✳✲✿æ✲ë❑➟õ ó ➨✔✇✎✍➁✃✵✴✫✂Ó✇✎✍✣✍☎✇✯⑤ý❹×✱✧ ❦✶❧♥♠♣♦✣q✷✶ ➍➣→➅❾ ê ➟Ò➍➣→➅➢⑧ë✺➨✿æ❬ì❥➟Òë➐➨☎➢ í➁ëàî✧ï❻→ ê ➟Òð✴➢⑧ë✺➨✟æ✤ñ✢ò ó ðå ëå❉ô❥õ ➢ ì❥➟❛ë✺➨✿æ ➉ ñsò ó ç➁ë ô❣õ ➋ ï→ æ ÷❽ë❻î✚ï✮ú✒ë❑ú ✸✫✹➊î✻✺✼ ò ➟✯✽→ ➨☎➢ í✾✽→ î✻✿→ û ❀Ï❻ÿ✒✃❣Õ✒Ö❁☞✏Ö♣þ➇×ÓÖ♣þ➇×➧①②✈✌✟✾✉✶✈⑧×✕✍⑤×✒Ï✐Ö×✏✴✚✃⑥✉❣✉✶ý✇❛×✕✍Ü×✏❂✒❰➇✃❼ý♣ý❃✟✚Ñ✰×✒ý♣ý❯Ö①✧Ö♣þ➇×❅❄➁×⑤❰❆✂➊✃❼Ï➇Ï➜✉✶✈⑧①✵✒ý➛×✕✂ ✃②Ï❇✴ ❈❺②⑩ ➥③❽❺❥⑨❳⑩✏❷➛➄❉✺❊ ò✏❋ ÖÒþ➅×➁✇♣Ï✺þ➇①✫✂❻①●☎♥×⑤Ï✶×➧①②❰✳✍■❍➁✇♣✈☎✇❛Õ⑥þ➐ý❹×⑤Ö❉Õ➧✃✫✍✒×✱✧ ➀
a posteriori er bound various“ measures” of u [exact]-wh [approximate in terms of C(Q, problem parameters) h mesh diameter], and uh A posteriori theory A posteriori error estimates are arguably more useful than a priori esti mates since we know uh. Bear in mind, however, that (i) in most methods for a posteriori error estimation the constants C are not known, and(i)for those methods which do attempt to better quantify the constants C, additional computational effort is required. Nevertheless, a posteriori error analysis is an increasingly important aspect of finite element practice: even when the C are not known precisely, local estimators can provide guidance as to how best to refine a triangulation. We shall restrict at tention in these lectures to the simpler case of a priori estimat es. 1.2 Projection We need several concepts to make the subsequent analysis flow smoothly: pro jection(general) and interpolation(specific to our particular space Xn) 1.2.1 Definition Given Hilbert spaces y and Z cy (Iy,v)y=(y:v)y,Wv∈2 defines the projection of y onto Z, Ily I:Y→Z
❏❇❑✯❏❇❑✎▲ ▼❖◆✙P✑◗❙❘✣❚❱❯✵❲❳P✑❯❱❲ ❨❇❩❭❬❫❪❵❴❜❛ ❝❡❞✠❢❤❣✌✐❦❥✕❧✌♠✯❢❙❧✌♠♦♥q♣●♣✣r❱♣✆♥qs❳t●✉✥✈①✇✫t✣♥qsq② ③✄④ ⑤r✵⑥⑧⑦✠⑨❶⑩✫✇❙♣✣✉❷r✵⑥✠s❹❸❳✈❹♥q✇✵s❳⑥✠♣●♥✱s●❺ r❙❻❽❼❿❾♥★➀❆✇✵➁✌t●➂✠➃➄❼✙➅✻❾✇❙➆⑧➆⑧♣✣r❤➀❆✉❷✈①✇❙t●♥✕➂➈➇ ✉✥⑦❉t✣♥★♣✣✈①s✆r❙❻✁➉❹➊❁➋✄➌✜➆⑧♣✣r⑤⑧➍♥★✈➎➆✠✇❙♣✏✇❙✈❹♥✕t✣♥★♣✏s❳➏✌➐ ➑ ❾✈❹♥✱s❳➒✻⑨❆✉✎✇❙✈❹♥✕t✣♥★♣●➂➈➐✠✇❱⑦✠⑨❉❼✞➅✠➓ ➔✷→✙➣✌↔❶↕ ➙❖➛♦➜✑➝q➞q➟❭➠✫➡✯➜✑➠✫➡➢➣❳➤➥↔✳→✳➦★➧ ❝➨❞⑧❢✫❣✏✐❦❥★❧✌♠✯❢✫❧✌♠❹♥★♣✣♣●r✵♣❹♥qs❳t●✉✥✈❹✇❙t●♥✱s➩✇❙♣✣♥➫✇❙♣✣➭❱⑥✠✇⑤⑧➍✥➯ ✈❹r❱♣✣♥➫⑥✠s❳♥★❻✔⑥➍ t✣➒✠✇❙⑦✭➲➫❞❇❧✌♠✯❢❙❧✌♠➳♥✱s➵t✣✉➺➸ ✈①✇✫t✣♥qs①s●✉❷⑦✠➁★♥❉➻➼♥❜➽✫➾✞❢✫➚➪❼ ➅ ➓➹➶➼♥q✇❱♣➘✉✥⑦➴✈❹✉❷⑦✠⑨➷➐➥➒⑧r✫➻➼♥★⑩✵♥★♣✱➐➬t●➒✠✇❙t➮➊➵♠❦➏➳✉❷⑦➹✈❹r✵s❳t❹✈❹♥★t●➒⑧r❆⑨⑧s ❻✔r❱♣❶➲❹❞✠❢❤❣✌✐❦❥✕❧✌♠✯❢❙❧✌♠✄♥q♣●♣✣r❱♣❅♥✱s➵t✣✉❷✈①✇❙t●✉✥r❱⑦☛t●➒✠♥➩➁✕r✵⑦✠s➵t✏✇❙⑦❭t✣s➘➉➱✇❱♣●♥➫➾✞❢✫✐❐✃✳⑦⑧r✫➻✜⑦➷➐❽✇❙⑦✠⑨❒➊❳♠❫♠❮➏✾❻✔r❱♣ t●➒✠r✵s●♥➢✈❹♥✕t✣➒⑧r❆⑨⑧s➥➻✜➒✠✉✥➁✏➒➩⑨⑧r➘✇✫t●t●♥q✈➳➆⑧t♦t✣r ⑤ ♥✕t❳t✣♥★♣✜❰❭⑥✠✇❙⑦❭t✣✉➺❻➯ t●➒✠♥✄➁★r❱⑦✠s❳t✣✇❱⑦✵t✏s♦➉➳➐❆✇❱⑨✠⑨❆✉➺t✣✉❷r✵⑦✠✇➍ ➁✕r✵✈❹➆⑧⑥❆t✣✇❙t●✉✥r❱⑦✠✇➍ ♥✕Ï✞r❱♣●t➢✉✎s✜♣✣♥q❰❭⑥⑧✉✥♣●♥✱⑨✙➓➼③➢♥★⑩✵♥★♣●t●➒⑧♥➍♥✱s●sq➐Ð➲❅❞⑧❢✫❣✏✐❦❥★❧✌♠✯❢✫❧✌♠✆♥★♣✣♣●r✵♣Ñ✇❙⑦✠✇➍❷➯s●✉✥s✆✉✎s➢✇❱⑦ ✉✥⑦✠➁✕♣✣♥q✇✵s❳✉✥⑦⑧➭➍✥➯ ✉❷✈❹➆❇r❱♣●t✣✇❱⑦✵t❅✇❱s●➆❇♥q➁✌t❅r❱❻✁Ò✠⑦⑧✉❷t●♥①♥➍♥q✈➳♥q⑦❭t✄➆⑧♣✏✇❱➁✕t●✉✎➁✕♥✵②➢♥★⑩✵♥★⑦✷➻✜➒⑧♥★⑦❜t●➒⑧♥Ó➉➎✇❙♣✣♥ ⑦⑧r❱tÔ✃✳⑦⑧r✫➻✜⑦☛➆⑧♣●♥✱➁✕✉✎s❳♥➍❷➯➐ ➍r❆➁★✇➍ ♥qs❳t●✉✥✈①✇✫t●r✵♣✣sÔ➁★✇❙⑦Õ➆✠♣●r✫⑩✳✉✎⑨❆♥❹➭❱⑥⑧✉✎⑨⑧✇❙⑦✑➁✕♥➩✇✵s❐t●r➮➒✠r✫➻ ⑤ ♥qs❳t✾t●r ♣✣♥✕Ò✠⑦⑧♥✆✇■t●♣✣✉✥✇❱⑦⑧➭❱⑥➍✇❙t●✉✥r❱⑦➷➓➷Ö✷♥Ñs●➒✠✇➍✥➍ ♣●♥✱s➵t✣♣●✉✎➁✌t➬✇✫t●t●♥★⑦❭t✣✉❷r✵⑦➘✉✥⑦Ôt✣➒⑧♥qs●♥ ➍♥✱➁✌t✣⑥⑧♣●♥✱sÐt●r✄t●➒⑧♥✜s●✉❷✈❹➆➍♥★♣ ➁★✇✵s❳♥❐r❱❻✆➲✄❞✞❧✌♠✯❢✫❧✌♠➼♥qs❳t●✉✥✈①✇✫t●♥✱s★➓ ×➬Ø❁Ù Ú❜Û✠ÜÐÝ❭Þ❵ß❇à⑧á➵Ü➥â ã❥①➾❇❥✏❥✏ä❉❣★❥✕å✫❥★❧●➲✫æ♦ç✏❢❙➾❇ç✏❥❮❞✞✐✯❣❹✐❦❢❶è❹➲✱➽❱❥Ó✐❫é✠❥❹❣✌ê✠ë✌❣★❥✏ì★ê✠❥✕➾✑✐■➲✫➾✞➲✫æ❃í❤❣✌♠✎❣Ñî✜❢✫➚ï❣✌è❹❢✱❢✫✐✔é✳æ❃í❱ð❐❞❇❧●❢❙ñ ò❥✏ç✕✐➈♠✯❢✫➾ôó✔õ✵❥★➾❇❥★❧●➲✫æö❉➲✫➾❇äÓ♠❫➾✠✐❦❥✕❧➈❞✠❢✫æ✥➲✫✐➈♠✯❢✫➾➹ó❁❣➈❞✠❥✏ç✕♠÷➼ç✾✐❦❢➩❢❙ê❆❧➼❞⑧➲❙❧✌✐❁♠✯ç★ê❆æ❷➲✫❧❐❣➈❞✠➲❱ç✏❥■øÓ➅✫ö❱ù ❏❇❑✎▲➷❑✯❏ ú➟❭ûýüý➡✔➞✱➡✯➜❇ü ❨❇❩❭❬❫❪❵❴✷þ ÿ✉✥⑩❱♥★⑦✁Ñ✉ ➍✥⑤♥q♣❳t■s●➆✠✇✵➁✕♥qs✄✂ ✇❱⑦✠⑨✁☎✝✆✞✂Ó➐ ➊✠✟☛✡ ☞✍✌✏✎✍✑ ✒✔✓ ➌✖✕❆➏✖✗✙✘ ➊✚✡ ☞✛✌✜✎✛✑ ✒ ✗ ➌✢✕✳➏✢✗Ñ➌ ✣✤✕✦✥✧☎ ⑨❆♥★Ò✠⑦⑧♥qs✜t●➒⑧♥✄❞✞❧●❢ò❥✏ç★✐❁♠✯❢❙➾❜r❙❻★✡Ór✵⑦✵t✣r✩☎✜➌✪✟✫✡✞➇ ✟✾②✫✂✭✬✮☎✰✯ ④
The projection Ily minimizes ly-端,Vz∈Z ly-(lly +u)llx =((y-lly)-v,(y-Ily)-uy y-Iyl-2(y-Iy,v)y+|‖g,Vu∈z Note z=Iy+v∈ Z and Ily∈ Z implies∈z, and hence since(Iy,v)y (y, U)r for allv E Z,(y-lly, u)r=0. The above result states that y-llyl2< ly-all for all z# lly. In words, Ily is the best approximation in Z of y in 1.2.3 Geometry SLIDE 7 Geometry of projection y Ily :(y-Iy,v)y=0,Vv∈Z Not surprisingly, if we wish to find the z Ily on the Z aris closest to y, Ily should be perpendicular to the z in the ( r inner prod usual notion of projection in Rn should be self-evident. In the above picture, z is the orthogonal complement of Z in Y: the space of all members of z a) Show that‖ Iy lr≤ ly llr and y-Iyly≤llly, and interpret this (b) Show that II(IIg)= lly
✱✠✲✴✳✵✲✴✳ ✶✸✷✺✹✼✻✾✽✿✷✛❀✪❁ ❂ ❃✿❄❆❅❈❇❊❉ ❋✄●■❍❑❏■▲✪▼❖◆P❍✍◗✜❘✪❙❚▼✔❯❲❱✫❳❲❨❩❙❚❯■❙❬❨❩❙❚❭❪❍❪❫❵❴✏❳✤❛✚❜✠❴❞❝❡❣❢✐❤❵❜❩❥✧❦❑❧ ♠♥●♣♦■q ❴✜❳r❛ts✉❱✫❳✤✈①✇ ② ③✜④ ⑤ ⑥✉⑦✏⑧❵⑨❪⑩✔❶ ❷ ❴❞❝❡①❸❹s✢s❺❳❵❛✚❱✫❳ ❷ ❛❊✇✠❢❪s❺❳❻❛✚❱✫❳ ❷ ❛❼✇ ❷ ❡ ❸✰❴✜❳✤❛✚❱✫❳✵❴✏❝❡❽❛✚❾❣s❿❳❵❛✚❱✫❳➀❢✖✇ ❷ ❡ ② ③✜④ ⑤ ➁✜➂■➃ ⑩✔❶ ✈❩❴✜✇➄❴✏❝❡❣❢✐❤✤✇✦❥✧❦✰➅ ➆❵➇❖➈➊➉✸❜✩❸❹❱✫❳❵✈✞✇➋❥①❦➍➌✺➎✠➏❩❱☛❳➐❥①❦➒➑❆➓✫➔➀→➣➑❿➉✜↔↕✇➐❥①❦✸➙✫➌✺➎➀➏❩➛➜➉✏➎➀➝✉➉❻↔✜➑❆➎✠➝✉➉❩s❿❱☛❳✠❢✢✇ ❷ ❡ ❸ s❺❳➀❢✢✇ ❷ ❡➐➞➇✺➟☛➌❖→❆→➠✇✦❥➐❦❑➙➡s❺❳➢❛❻❱✫❳➀❢✖✇ ❷ ❡ ❸➥➤✼➦❵➧➀➛➜➉↕➌✔➨✉➇✺➩✺➉✫➟✢➉✏↔✜➫➭→➣➈✵↔✜➈➯➌✺➈➯➉✜↔➲➈❆➛➜➌✺➈➳❴✜❳➢❛❻❱✫❳✵❴✏❝❡✙➵ ❴✜❳❻❛①❜➀❴✏❝❡ ➞➇✺➟➸➌✺→❆→➄❜❊➺❸➻❱☛❳✵➦✤➼✜➎❊➽✾➇✺➟✢➏✺↔✉➙➳❱☛❳①➑✴↔r➈❺➛■➉➸➨✉➉✏↔✜➈☛➌✪➔➠➔➀➟✢➇❞➾✔➑❆➓➸➌✺➈➚➑❿➇✺➎①➑❆➎❼❦➪➇➞ ❳①➑❆➎ ➈❺➛■➉❻❴➲➶✔❴ ❡ ➎✠➇✺➟✜➓➸➦ ✱✠✲✴✳✵✲❬➹ ➘➴✽♣✹✠➷①✽✿❀❞✷✛❁ ❂ ❃✿❄❆❅❈❇❽➬ ➮❍❞▼✔❨❩❍✏❘✢▲✪♦✦▼➠➱➡❏■▲✪▼❖◆P❍✍◗✜❘✢❙❬▼➠❯❈✃ ❐➟✜➈❆➛➜➇✢❒✿➇✺➎➀➌✺→➣➑❆➈➚❮➀✃❣s❺❳❻❛❊❱☛❳✠❢✢✇ ❷ ❡ ❸♥➤➜❢ ❤✤✇❲❥➋❦✤❧ ❰✫Ï ➆❵➇❖➈❵↔✜➫➭➟➚➔➀➟✜➑✴↔✉➑❆➎♣❒❖→Ð❮✛➙❻➑➞ ➽✾➉✧➽➳➑✴↔✢➛♥➈➯➇✦Ñ✾➎✠➏✚➈❺➛■➉❲❜Ò❸Ó❱✫❳❹➇❖➎➥➈❺➛■➉✁❦✮➌❞➾✔➑✴↔✧➝❞→❚➇✛↔❞➉✏↔✉➈❻➈➯➇➋❳✠➙ ❳➴❛t❱☛❳Ò↔✢➛■➇❖➫➭→❚➏❽➨✉➉r➔■➉❞➟➚➔➜➉✏➎➀➏✺➑❿➝❞➫➭→❚➌✺➟✁➈➯➇✧➈❺➛■➉❲❦Ó➌❞➾✔➑✴↔➴ÔÕ➑❆➎t➈❺➛■➉➐sP➶❬❢❞➶ ❷ ❡ ➑❆➎➜➎➀➉✏➟❵➔✠➟✪➇❪➏❖➫➜➝✏➈➯➦ ➧➀➛♣➑✴↔❑➌✺➎✠➌❖→❚➇✪❒❖➫➜➉✸➈➯➇❻➇✺➫➭➟☛➫♣↔✜➫➜➌❖→➜➎✠➇❖➈Ö➑❿➇❖➎➐➇➞ ➔✠➟✢➇P×❞➉✪➝❞➈➚➑❿➇✺➎➋➑❆➎❩ØÙ☛Ú➋↔✢➛■➇❖➫➭→❚➏➸➨✉➉Û↔✏➉❞→➞✉Ü ➉❞➩❪➑❿➏✔➉✏➎✼➈➊➦✄➼✏➎ ➈❺➛■➉❵➌✔➨✉➇✺➩✺➉➲➔✠➑❿➝❞➈➚➫➭➟✢➉✜➙➡❦ÛÝÞ➑✴↔↕➈❺➛■➉✄▼✔▲✖❘✪●■▼➠ß✔▼➠❯➜à➠á✼◗❞▼➠❨❩❏■á❬❍❞❨❩❍❞❯✿❘r➇➞ ❦✰➑❆➎❲â❲ã❣➈❺➛■➉❑↔➚➔■➌✔➝✪➉r➇➞ ➌✺→❆→ ➓➸➉✏➓➸➨✉➉✏➟✉↔❵➇➞ âä➇✺➟✜➈❺➛■➇✪❒✔➇❖➎✠➌❖→❈➈➯➇✩➌✺→❆→➡➓❩➉❞➓➸➨✪➉❞➟✉↔✤➇➞ ❦r➦ å✁æÛç★è✺é✔ê✛ë❿ì➠è✧í s❿à❷✁î●■▼✺ïð❘✪●➜à✺❘✧❴❞❱✫❳✵❴ ❡➍ñ ❴✏❳➄❴ ❡ à❖❯➜ò✝❴✜❳➴❛Ò❱✫❳✵❴ ❡óñ ❴✜❳✵❴ ❡✄ô à❖❯✼ò✙❙❬❯✿❘✢❍❞▲✪❏■▲✪❍✏❘❩❘✢●➜❙❬❫ ▲✢❍✍❫✖õ■á❚❘❣ß➠❍❪▼➠❨❩❍✏❘✪▲✢❙✴◗❞à➠á❚á❬♦➠❧ s❺ö ❷➋î●■▼✺ïÒ❘✪●➜à✺❘☛❱rs❿❱☛❳ ❷ ❸➥❱✫❳➀❧ ÷
1.3 The Interpolant 1.3.1 Definition rn∈P1(h),Th∈T} U∈X ∈X, the inte Tn∈Xh;and五h(x;)=w(x;),i=0,,n+1 工()=∑v(x)(m) i=1 132A If∈x,and叫n∈C2(h),VTh∈Th,tho Recall ul (s)=Jo vx da, llaull22(=Jo u2 da, and lell(2=1(9)+1|l12a2
ø➳ùÖú û❊ü✫ý✝þ✔ÿ✁➭ý✄✂✆☎✞✝✠✟☛✡➳ÿ✠ ☞✍✌✏✎✑✌✒☞ ✓✕✔✗✖✙✘✙✚✜✛✢✚✒✣✍✘ ✤✍✥✗✦★✧✑✩✫✪ ✬✮✭✰✯✰✱✳✲★✲ ✴✶✵✸✷✺✹✼✻✾✽✿✴❁❀❂✻❃❀ ❄✳❅❆✽✿❇❈❊❉✳❋✒●❍✵✗■❑❏▼▲◆●✄✵❆✽✿❖✆✵◗P ✤✍✥✗✦★✧✑✩✫❘ ❙❯❚✏❱❳❲❩❨✕❬ ✽✿✴❪❭❴❫❛❵❲✾❜★❝◗❞✭❂❡❣❢◗❤✳✲✐✱❝◗❞❦❥✵ ❬♠❧♦♥❫ ❚✏❧☛♣◗❲✢❧❩q ❥✵ ❬ ✽✿✴✵✆r ♥s❨✉t✈❥✵ ❬ ❋✒✇✉①②■③✷ ❬ ❋✜✇✉①②■❂❏⑤④⑥✷⑧⑦◗❏❩⑨✼⑨❩⑨❂❏❛⑩✶❶❸❷❆⑨ ☞✍✌✏✎✑✌❺❹ ❻❽❼❾❼✙❿✳✣❴➀✄✚✒➁❪➂✆✛✼✚✒✣✍✘➄➃◆➅✙✔✗✣✉❿➇➆ ✤✍✥✗✦★✧✑✩➉➈s➊ ❇➌➋ ❬ ✽➍✴❪❭ ♥❳❨◗t✕❬ ❀ ❄➇❅✾✽➏➎➑➐➒❋✜●✄✵✗■❑❭➍▲◆●❍✵✾✽✕❖✆✵✉❭❴❫❛❵❲❩❨ ❀ ❬⑧➓✿❥✵ ❬ ❀ ➔➣→✰↔✏↕◗➙➜➛ ➝➟➞♥✳➠ ❄✳❅s➡➒➢✼❅✸➤➞♥s➠ ➥ ➡❳❄➇❅ ❀ ❬✮➦ ➦ ❀ ➧ ➨ ❬⑧➓✿❥✵❬ ➨❩➩✆➫ ↔✏↕◗➙➜➛ ➝➐ ➞♥✳➠ ❄❅ ➡➒➢❅ ➤➞♥✳➠ ➥ ➡❳❄❅ ❀ ❬✮➦ ➦ ❀ ➧➭⑨ ✬✮✭✰✯✰✱✳✲★✲✙❀ ✻❍❀ ➐➔➣→❂↔✐↕◗➙ ✷➲➯ ❉ ➳ ✻ ➐➥➣➵✇✑➸ ➨ ✻ ➨ ➐➩✆➫ ↔✐↕◗➙ ✷❸➯ ❉ ➳ ✻ ➐ ➵✇✄➸ ✱❝❃➺ ➨ ✻ ➨ ➐➔❊→❑↔✐↕◗➙ ✷➻❀ ✻❍❀ ➐➔➣→❑↔✐↕◗➙ ❶ ➨ ✻ ➨ ➐➩✆➫ ↔✏↕◗➙❩➼ ✤✍✥✗✦★✧✑✩➉➈✆➈ ➽❴➾➒❲❫♦➚✰❵✕➪s➋⑥➶✆➹♦➪❴➪s➋q ➘
(-工v)lk (-工h2)/lrt LIDE 12 h max w h h, max max lu" The first line follows from Rolle's Theorem(which requires WITh EC(Tn), as is the case here). The second line bounds the K=b integrals by hx the mazimum of the integrand. Note, however that we only require w to be defined in the elements, not at the nodes, so if we place our delta distribution loads at nodes this hypothesis is still satisfied for solutions u of our Poisson problem even Since(ln w'rk is a constant, we are effectively appro imating w by a constant Not surprisingly, this will not work very well if w has jumps(infinite) in Th also, the larger the w, the larger the error, since the more w' will vary way from a constant.(In general, if w has strong singularities, lw-IhWlHI(Q will only converge as some fractional power of h) b Exercise 2 Prove the L2 estimate of Slide 10. Hint: write(w- Lh wlrk ral in terms of (w- Lhw'lrk; then express(w-Lhw'T the h SLIDE 13 Ifw∈x,andw∈H2(9,Th)
➴✍➷✗➬★➮✑➱➉✃❳❐ ❒ ❒ ❒❺❮✜❰❸Ï✿Ð✄Ñs❰❯ÒÔÓ②Õ Ö❃×Ø ❮✒Ù✍Ò ❒ ❒ ❒❴Ú ❒ ❒ ❒ ❒ÜÛ❪Ý Þ × ❮✜❰⑧Ï✿Ð✄Ñs❰❯ÒÔÓ ÓÔÕ Ö❃×ØàßÙ ❒ ❒ ❒ ❒ Ú ❒ ❒ ❒ ❒áÛâÝ Þ × ❰✮Ó Ó ßÙ ❒ ❒ ❒ ❒ ã➄ä⑧å✾æ✳ç Ýsè Ö❃×Ø Õ ❰éÓ Ó②Õ ê ë ì❑í✙î ÛÖ❃×Ø ❮✜❰⑧Ï✿ÐÑ ❰❯ÒÔÓÔÕ ïÖ✍×Ø ßÙ ã➟ðä ä✫ñ❍äòå✾æ✳ç ì❂í❾î✰óáôáôáô óê å✾æ✳ç Ýsè Ö❃×Ø Õ ❰✮Ó Ó➌Õ õ ï ö✁÷ ø❃ù✆ú✙û③ü✰ý❑þ❾ÿ✁✄✂✍ú✆☎✞✝✳ÿ★ÿ✟✝✡✠⑥ý☛☎✰ü☞✝✍✌✏✎✑✝✳ÿ★ÿ✏ú✓✒ý✈ø❃ù✆ú✔✝sü❛ú✞✌✖✕✗✠❾ù✘✚✙✰ù❽ü❛ú✔✛✞✜✢★ü♦ú❑ý ❰àÕ ÖØ✤✣✦✥ î ❮✚✧Ñ Ò✔★✪✩ý✫❺ý þ✜ù✆ú✑✙✩ ý❂ú ù✆ú❩ü❛ú✭✬✍✮✈ø❃ù✆ú❊ý❩ú✔✙✔✝✡✂✰✯➑ÿ✁✄✂✍ú✲✱✔✝✡✜✢✂✳✯✳ý þ✜ù✆ú✵✴ Ú î Ñ ✄✂◗þ②ú✷✶❳ü✩ ÿáý✸✱✺✹ ä✼✻ þ★ù◗ú✸✌✩✺✽✄✌✾✜✢✌ ✝✭☎✶þ★ù◗ú✿✄✂✉þ➌ú✗✶❳ü✩✂✰✯❀✮✦❁✾✝✳þ②ú★ ù❂✝✡✠✠ú✺❃✳ú❂ü★ þ★ù ✩ þ❄✠✠ú❅✝✡✂◗ÿ❆✹ ü❛ú✔✛✞✜✢★ü♦ú ❰Ó Ó þ❇✝❈✱✰ú❅✯❳úû✵✂✍ú✔✯❈✄✂âþ✜ù✆ú ú❩ÿ✐ú✞✌✾ú✺✂◗þý ★ ✂✳✝✳þ ✩ þ❊þ✜ù✆ú❉✂✳✝❊✯❳ú❂ý ★ ý✺✝❋☎●✠✠ú✸❍❃ÿ✩ ✙✰ú■✝✡✜ü❏✯❳ú❩ÿÜþ✩ ✯✡❺ý❑þ❣ü❑✚✱✞✜þ✷✚✝✡✂✫ÿ▲✝✩ ✯✳ý ✩ þ▼✂✳✝❊✯❳ú❂ý ★ þ✜ù✘❺ý✸ù✘✹◆❍✓✝✳þ✜ù✆ú❂ý✔❺ý✤❺ý✈ý❑þ❖★ÿ★ÿ❦ý ✩ þ❖❺ý✜û✁ú✔✯P☎✞✝sü✈ý✞✝sÿ✁✜þ✷✚✝✡✂✆ý❄◗❘✝❙☎■✝✍✜ü❉❚✸✝✡❺ý✰ý✺✝✡✂❯❍✍ü☞✝❱✱❂ÿ✐ú✺✌ ú✺❃✳ú✞✂❲✄✂ þ✜ù✘❺ý❉✙✩ ý❩ú❳✮ ❨✄✂✰✙✰ú ❮Ð✄Ñ➒❰éÒ Ó Õ Ö❃×Ø ❺ý ✩ ✙✔✝✡✂◗ý✰þ✩✂✉þ★ ✠✠ú ✩ ü❛ú❯ú❖❩✮ú✔✙❂þ✷✄❃➇ú❩ÿ✁✹ ✩❍❀❍❃ü☞✝✽ ✄✌✩ þ✷✄✂❬✶ ❰Ó ✱✞✹ ✩ ✙◆✝✍✂✆ý❑þ✩✂◗þ❇✮ ❁❉✝sþ ý❑✜ü✷❍✍ü❑❺ý❑✄✂✘✶sÿ✁✹★ þ★ù✢❺ý❏✠❭★ÿ★ÿ❪✂✳✝sþ✑✠❫✝✳ü☞❴❯❃✳ú❂ü❑✹❵✠✠ú❩ÿ★ÿ❫☎ ❰Ó ù ✩ ý❜❛❑✜✢✌✑❍✉ý❵✕❰Ó Ó ✄✂✢û✵✂❝★þ➌ú✭✬❅✄✂ ✧ ì Ñ✪❞ ✩ ÿáý✺✝ ★ þ✜ù✆ú✕ÿ✩ ü❖✶➒ú❩ü❽þ✜ù✆ú ❰Ó Ó ★ þ✜ù✆ú✕ÿ✩ ü❖✶➒ú❩ü❽þ✜ù✆ú➏ú❩ü❑ü☞✝sü★ ý❑✄✂✰✙♦ú✕þ✜ù✆ú❅✌✤✝✳ü♦ú ❰Ó ✠❭★ÿ★ÿ❜❃✩ ü❑✹ ✩✠ ✩✹▼☎✰ü◆✝✡✌ ✩ ✙◆✝✍✂✆ý❑þ✩✂◗þ❇✮P✕❢❡✞✂■✶✗ú✞✂❃ú❂ü✩ ÿ ★ ☎ ❰ ù ✩ ý ý❑þü☞✝✍✂❬✶✸ý❑✄✂❬✶❀✜ÿ✩ ü❑★þ✷✒ú❂ý ★③Õ ❰ Ï✸Ð✄Ñ➒❰àÕ ❣❜❤✞✐▲❥❝❦ ✠❭★ÿ★ÿ✆✝✍✂◗ÿ✁✹✿✙✔✝✍✂✓❃✳ú❂ü❖✶✗ú ✩ ý➑ý✺✝✡✌✾ú❫☎✰ü✩ ✙❩þ❖✚✝✍✂✩ ÿ❂❍❂✝✍✠✠ú❂ü❏✝✭☎ ä ✮ ✬ ❧✦♠✲♥✪♦✍♣❀qsr✚t❀♦❵✉✦✈❫✇☞①✡②❱③✑④◆⑤❂③❉⑥ï ③❳⑦❙④☞⑧å✾æ④☞③❉①✍⑨✵⑩✘❶▲⑧❸❷✢③ ð❊❹❂❺✸❻ ✄✂✉þ✞❼✵❽✇◆⑧❆④◆③ ❮✜❰❸Ï✿Ð✄Ñ➒❰éÒ❩Õ Ö❃×Ø æ⑦ æ ❷✢③✞❾✓❿✓⑧❆④◆③❉⑧✟❿❱④◆③✺➀❱✇æ❶➁⑧▲❿❵④◆③✺✇å⑦✑①✍⑨ ❮✜❰ Ï✿ÐÑ ❰éÒ Ó Õ Ö❃×Ø✆➂ ④◆⑤❂③✺❿✦③ç✘➃✇☞③❊⑦☞⑦ ❮✒❰❸Ï➏ÐÑ❰❯Ò Ó Õ Ö❃×Ø æ⑦ ⑧✟❿➄④◆⑤❂③●➅î✔➆ ⑦☞③å⑧✟❿❂①❀✇å➇➃✇◆①❬①❀⑨ ❺ ➴✍➷✗➬★➮✑➱➉✃❀➈ ➉⑨ ❰ ✣➄➊❲➋ æ❿✓❷ ❰ ✣ ➅ ï ❮❖➌➎➍◆➏✆Ñ➒Ò ➋ Õ ❰⑧Ï✕ÐÑ ❰àÕ ❣❤ ✐▲❥❝❦ ã ä➐➒➑ ❰ ➑ ❣❪➓✺✐▲❥ ó ➔Ø ❦ ➑ ❰⑧Ï✕ÐÑ ❰ ➑✺→ ➓ ✐▲❥❝❦ ã ä ï ➐ ï ➑ ❰ ➑ ❣➓ ✐✟❥ ó ➔Ø ❦✲➍ ➣
he i(27)=∑|ln N at jumps in the derivative(e.g,, due to a delta distribution or change nductivity) at nodes are fine the function is still in H2over (In fact, the above result is true with just the H seminorm, Norms which have been broken up over elements or subdomains are sometimes The proof of the above is not difficult, but involes the Rayleigh quotient for fourth-order eigenvalue problem. As we have not introduced these con cepts, the demonstration would req uire a major digression at this stage. The reader is referred to ages 45-47 of strang 6 FiT. Note we prefer this second result t that of Slide 10 since the norm on u is "weaker, and consistent with the energy notions that underly the finite element method 1.4rror:卫n 1.4.1 Definition SLIDE 14 Define the energy,or“a”, norm‖llas v2 dx =oh Note: I. is problem-dependent Since a(, )is an SPd bilinear form,(a(v, u))/2 does indeed satisfy all the requirements of a proper norm.( Recall that the H seminorm is in fact a norm Hb(2).) SLIDE 15 Of interest: for u(a)(exact solution) uh(a)(finite element approximation =e(a)=(u-uh(a)(discretization error) find bound for lelll in terms of h, u
↔✸↕❂➙❳➛☞➙ ➜✞➝✾➜✞➞➟▼➠✞➡✟➢➁➤ ➥❊➦✺➧❭➨ ➩ ➫ ➭✞➯☛➲ ➜❑➝❏➜✞➞➟▼➠✞➡▲➳✰➵➦ ➧❭➸ ➩ ➫ ➭✞➯☛➲ ➺ ➳✳➵➦ ➝✑➞➻❳➻✑➼ ➝✑➞➻✲➼ ➝✑➞❫➽❱➾➪➚ ➶❉➹✍➘✗➴➬➷✺➮❬➷✡➱✄✃❐➘❢❒❂➷✡➘❜❮❑❰✢Ï✑Ð❝Ñ✦➱✄✃Ò➘❢❒❂➴✼Ó❱➴✞Ô❑➱✄Õ✡➷✡➘✷➱✄Õs➴×Ö✭➴❊ØÙ➮❱Ø▲Ú➄Ó✍❰✓➴❯➘❇➹Û➷×Ó❀➴✺Ü✁➘❇➷ÝÓ✍➱❸Ñ❑➘❖Ô❑➱✚Þ✺❰✢➘❖➱✚➹✍✃ß➹✍Ô à❒❂➷✡✃✘➮❱➴❯➱✄✃ à➹✡✃✰Ó✡❰à ➘❖➱✄Õ❊➱✄➘❖á✔â✦ã✡ä✤å❂æ✢ç➙❊è ➷✡Ô◆➴Pé✵✃✰➴❯êë➘❢❒❂➴❉ì✔❰✢✃à ➘❖➱✚➹✍✃í➱❸Ñ➄Ñ❑➘✷➱✄Ü✄Ü✸➱✄✃❲î➞ ➹✡Õ✡➴✺Ô ➴✔➷ à❒ï➴✞Ü✟➴✞Ï✤➴✞✃❝➘✗ØðÖ✄ñ✞✃✦ì✞➷ à ➘✚Ú●➘❢❒❂➴✦➷❱Þ◆➹✍Õs➴✦Ô☞➴❑Ñ❑❰✢Ü✁➘P➱❸Ñ➄➘❖Ô❑❰✓➴❅ò❭➱✄➘❢❒✿❮❑❰✘Ñ✔➘P➘❢❒❂➴■î➞ Ñ✺➴✺Ï✾➱✄✃✰➹✡Ô❑Ï✤Ø â ➶❉➹✍Ô❑Ï❉Ñ●ò☛❒✢➱ à❒✦❒✓➷✡Õ✡➴✤Þ✔➴◆➴✺✃➬Þ✺Ô☞➹❊ó❀➴✺✃✼❰❊Ð❲➹✍Õs➴✺Ô❏➴✺Ü▲➴✺Ï■➴✺✃✓➘❖Ñ❉➹✡Ô●Ñ❑❰✓Þ✔Ó❀➹✍Ï■➷✍➱✄✃❂Ñ✾➷✍Ô☞➴✾Ñ✺➹✡Ï✤➴✞➘✷➱✄Ï■➴✞Ñ ó✍✃✳➹✡ò❭✃✼➷sÑ➪ô❇Þ✞Ô☞➹❊ó❀➴✺✃✓õ❄✃✰➹✡Ô❑Ï✾Ñ✞Ø ö✰❒❂➴■Ð✳Ô☞➹❊➹✭ì❯➹❙ì➄➘❢❒❂➴❈➷❀Þ✔➹✡Õ✡➴❯➱❸Ñ❅✃✳➹✍➘✾Ó✡➱÷à❰✢Ü❆➘✚Ú❏Þ✺❰✢➘●➱✄✃❝Õs➹✍Ü✁Õs➴✞Ñ❵➘✄❒✓➴✿ø✑➷✍á✡Ü▲➴✺➱❆➮❊❒Òù✞❰✓➹✍➘❖➱✚➴✺✃✓➘✵ì✞➹✍Ô ➷✾ì✞➹✡❰✢Ô❑➘❢❒✘ú❙➹✡Ô◆Ó❀➴✺Ô➄➴✞➱❆➮❬➴✺✃✓Õ✡➷✡Ü✁❰✓➴●Ð✳Ô◆➹❀Þ✺Ü▲➴✞Ï✤Ø❈û❄Ñ■ò❫➴■❒❂➷✍Õs➴➄✃✰➹✡➘➎➱✄✃✓➘✷Ô☞➹❳Ó✍❰à➴◆Ó❯➘✄❒✓➴❑Ñ✺➴ à➹✡✃ à➴❇Ð✳➘❖Ñ✔Ú ➘❢❒❂➴❉Ó❱➴✞Ï✤➹✡✃✓Ñ❑➘❖Ô☞➷✍➘❖➱✚➹✍✃❯ò❫➹✡❰✢Ü✟Ó✤Ô☞➴✔ù✺❰✢➱✄Ô☞➴●➷■Ï✤➷❇❮✺➹✍Ô❉Ó✡➱❆➮❀Ô☞➴❑Ñ✔Ñ❑➱✚➹✡✃✼➷✡➘❪➘✄❒✢➱❸Ñ✲Ñ❑➘❇➷✺➮❬➴❳Ø✤ö✰❒❂➴●Ô☞➴✔➷❀Ó❀➴✺ÔP➱❸Ñ Ô☞➴❖ì✞➴✞Ô❑Ô◆➴◆Ó❈➘❇➹✾Ð❂➷❳➮❱➴✞Ñ❉ü❝ý✍þ❳ü ÿ✦➹❙ì ✁➘✷Ô☞➷✍✃❬➮✄✂✆☎❫➱✞✝✘Ø❵➶❉➹✡➘❇➴✿ò❫➴❄Ð✰Ô☞➴✚ì✞➴✺Ô❋➘❢❒✘➱❸Ñ■Ñ✺➴à➹✍✃✳Ó❯Ô☞➴❑Ñ❑❰✢Ü✁➘✲➘✗➹ ➘❢❒❂➷✡➘✵➹❙ì ✁Ü❆➱✚Ó❀➴✠✟☛✡❉Ñ❑➱✄✃à➴✲➘❢❒❂➴✲✃✳➹✍Ô❑Ï➇➹✍✃✌☞➬➱❸Ñ❈ô✷ò❫➴✔➷❳ó❱➴✞Ô✔Ú◆õ❄➷✍✃✳Ó à➹✡✃❂Ñ❑➱❸Ñ❑➘❇➴✞✃❝➘☛ò❭➱✄➘❢❒✿➘✄❒✓➴➎➴✺✃✳➴✺Ô❖➮❀á ✃✳➹✍➘❖➱✚➹✍✃❂ÑP➘❢❒❂➷✡➘▼❰✢✃✳Ó❱➴✞Ô❑Ü❆á✤➘✄❒✓➴❫é❫✃✓➱✄➘❇➴❉➴✺Ü▲➴✞Ï✤➴✺✃✓➘▼Ï■➴✺➘✄❒✓➹❳Ó❱Ø ✍✏✎✒✑ ✓✕✔✖✔✖✗✘✔✚✙✛✓✕✜✣✢✤✔✖✥✏✦★✧✩✗✘✔✫✪ ✬✮✭✰✯✱✭✲✬ ✳✵✴✷✶✹✸✹✺✒✻✼✺✲✽✮✸ ✾✮✿✷❀❂❁✚❃❅❄❇❆ ❈➙❊❉å ➙ ä↕❂➙P➙å ➙✺➛●❋☛❍☛■ æ ➛❑❏▼▲❖◆✖■ å❂æ➛▼P❘◗✰◗❙◗ ❚❯◗❙◗✰◗ ã è ◗❙◗✰◗ ❚❯◗✰◗❙◗ ➞ ➸ ▲❲❱✒❚✮❳❨❚❬❩ ❱✒❋❀➙å ➙✺➛ã❪❭❙❭❍❬❩ ➸ ➺➲ ❫ ❚ ➞➻ ➽❱➾ ➸❴◗ ❚❯◗ ➞ ➟❛❵✞➡▲➢✓➧ ❱❢↕❂➙❳➛☞➙❜❩❞❝ ❡✫æ❀ä ➙❣❢✣◗✰◗❙◗❇❤❣◗❙◗❙◗❖✐✟è Ð✰Ô☞➹❀Þ✺Ü▲➴✺Ï✾ú❙Ó❱➴✗Ð✓➴✺✃✳Ó❀➴✺✃✓➘ ❝ ✁➱✄✃ à➴ ▲❲❱❨❤✰❳❥❤ ❩ ➱❸Ñ➬➷✍✃ ✁✖❦✏❧ Þ✞➱✄Ü✁➱✄✃✰➴✔➷✡Ô■ì✞➹✡Ô❑Ï✾Ú ❱✲▲❲❱✒❚✮❳❨❚❬❩❨❩ ➲❨♠ ➞ Ó❀➹❊➴✞Ñ❈➱✄✃✳Ó❱➴◆➴✔ÓÛÑ✺➷✡➘✷➱❸Ñ❢ì✔áï➷✍Ü✄Ü✲➘❢❒❂➴ Ô☞➴✔ù✺❰✢➱✄Ô☞➴✞Ï✤➴✺✃✓➘❖Ñ✲➹✭ì❄➷✸Ð✰Ô☞➹◆Ð✓➴✺Ô✲✃✰➹✡Ô❑Ï✤Ø●Ö❢ø✑➴à➷✡Ü✄Ü✳➘❢❒❂➷✡➘❭➘❢❒❂➴❜î➲ Ñ✺➴✺Ï✾➱✄✃✰➹✡Ô❑Ï ➱❸Ñ✲➱✄✃❄ì✞➷ à ➘▼➷❉✃✰➹✡Ô❑Ï ➹✡Õ✡➴✺Ô✑î➲ ❫ ❱♦♥❛❩ Ø â ✾✮✿✷❀❂❁✚❃❅❄☛♣ qì❄➱✄✃✓➘❇➴✞Ô◆➴❑Ñ❑➘ ❢ ræ ➛ ☞ ❱➾ ❩s❱❢➙❊tã❣✉❑ä èæ❣❭✰✈❂ä ✐æ❀å❩ ☞❯✇ ❱➾❩✠❱✒❉å ✐ä ➙P➙❭➙①P■➙å❱ä✑ã☛②✖②➛æt❬✐✰Pã✡ä ✐æ❱å❩ ③⑤④ ❱➾ ❩✵➸⑥❱☞⑧⑦⑨☞❲✇ ❩❊❱➾❩✠❱ ç ✐✟è ✉ ➛◆➙ä ✐✰⑩ ã✡ä ✐æ❱å ➙❳➛☞➛æ ➛❶❩ ❉å✓ç✵❷✳æ☛✈❂å❝ç ræ ➛❸◗✰◗❙◗ ④ ◗❙◗❙◗❖✐å❅ä ➙✺➛●P■è æ r✏❹✚❳ ☞ ❝ ❺
1.4.2 Orthogonality a(u,u)=C(),vv∈X then e(U),Vv∈Xh a(uh,)=C()],V 1.4.3 General Bound SLIDE 17 or any wh Uh∈ 0: ort hogonality ell h∈X more accurate than in the energy norm So we see that the finite element procedure does as well without knowledge of the ecact solution as you can do with knowledge of the eract solution- so long as we speak of the energy(or" a")norm. The finite element method has trans formed the problem of discretization of a PDe into a problem of appro imation vu∈X
❻✮❼✰❽✱❼❿❾ ➀➂➁❜➃❥➄✹➅✫➆✮➅✫➇✹➈✖➉➋➊✒➃➍➌ ➎✮➏✷➐❂➑✚➒❅➓☛➔ →❖➣❙↔➙↕❊➛ ➜❲➝✒➞✹➟❨➠❬➡✘➢➥➤☛➝✲➠❖➡❊➟➧➦➨➠➫➩✵➭ ➯▼➲➛①↔ ➳✖➵➯ ➜❲➝✲➞✱➟▼➠❖➡➸➢ ➤☛➝✒➠❬➡❶➟➺➦❸➠⑧➩✵➭⑧➻ ➼➾➽➜❲➝✒➞ ➻ ➟▼➠❖➡➸➢ ➤☛➝✒➠❬➡➪➚➪➟➶➦❸➠⑧➩✵➭➻ ➝✒➭⑧➻➘➹➴➭✄➡➷➟ ➬▼➮ ➜✮➝✲➞ ➼ ➞❲➻✖➟❨➠❬➡➱➢ ✃➙➟❐➦❸➠➫➩✵➭⑧➻ ➝➳➣✰❒❙➣❙↔✖➛❥❮☛❰▼➣➯ÐÏ✖➡❶Ñ ❻✮❼✰❽✱❼❙Ò Ó✵Ô❖➇✹Ô✷➁❜➈➙➉ÖÕ×➅✫Ø✹➇✹Ù ➎✮➏✷➐❂➑✚➒❅➓✷Ú Û➮❰Ü❮❪↔Ï⑧Ý➻ ➢Þ➞➻➷ß ➠➻ ➩✕➭➻ ➟ ➠➻ ➩à➭➻ ➜❲➝✒➞ ➼ ÝÜ➻➙➟▼➞ ➼ Ý❛➻✷➡ á â❊ã ä åæåæå ç❣è✮é❲ê☛åæåæå ë ➢Þ➜➷➝▼➝✒➞ ➼ ➞❲➻✷➡ ➼ ➠☛➻✖➟✼➝✒➞ ➼ ➞❯➻❣➡ ➼ ➠☛➻✷➡ ➢✩➜❲➝✒➞ ➼ ➞❯➻✖➟▼➞ ➼ ➞❯➻☛➡ á â❶ã ä åæåæå ì❶åæåæå ë ➼îí ➜❲➝✲➞ ➼ ➞❯➻➙➟❨➠☛➻✷➡ á â❊ã ä ï❶ð❖ñ❶ò✲ó♦ô✼ñ❶õ❶ñ❶ö❇÷➍øæù ó❂ú ß ➜❲➝✒➠☛➻➙➟❨➠☛➻❖➡ á â❶ã ä û ï❛ù ü✤ý ê❬þÿï ✁✂✁✄✁ ☎✆✁✄✁✂✁ ➢ ➣✰↔✞✝ éê✠✟☛✡✏ê ✁✂✁✄✁ ➞ ➼ ÝÜ➻ ✁✂✁✂✁✌☞ ➎✮➏✷➐❂➑✚➒❅➓☛✍ ✎✑✏✓✒✕✔✗✖✙✘✗✚✜✛ ➛✣✢❣➛①↔✵➣✤✝ Ï➮➵✦✥✏✌✧✣✒ ➞✩★ Ï➮➵ ↕➮➵❒✫✪à↔➮➯✭✬↔✮✪✵❮ ÝÜ➻ ➣❙↔ ➭➂➻ ✯➮❰●➛✠❮☛↕①↕➵❰➍❮➯ ➛ ➯▼➲❮❪↔ ➞➻ ✰✏✓✱✳✲✞✧✴✧✵✏✶✧✵✖✸✷✠✹✺✏✶✔✗✖✑✻ Ñ ✼✔✽✒✕✧✺✚✣✧✾✧✿✱❀✲❂❁✗✱❃✱✳✲✞✧❅❄❆✏✰ ✱❇✧❈✧✵❉✂✧✵✻✺✧✵✏✮✱❆❊✶✖✙✔●❋❍✧✾✘✠■✆✖✙✧❈✘✜✔❏✧✵✚✿❁❑✚✺✒✕✧✣❉❀❉▼▲➣➯▼➲➮➵➯ ✥✏✌✔✠✒◆❉✂✧❍✘❏✷✜✧❖✔◗P ✱✳✲✞✧❘✧❍❙❚❁✜❋✵✱✩✚✣✔✗❉✤■✆✱✰ ✔✠✏✦❁✗✚❃✹☛✔✗■❖❋✾❁✗✏✦✘☛✔❯▲➣➯▼➲ ✥✏✶✔✗✒❱❉✄✧✾✘✣✷❚✧❲✔❳P❃✱❀✲❂✧❘✧✙❙❨❁✜❋✵✱❩✚✣✔✗❉✤■✆✱✰ ✔✠✏❖❬❭✚✵✔❪❉✄✔✠✏❚✷ ❁❑✚❘✒✕✧❘✚❫❊❂✧✾❁✥ ✔❳P❴✱✳✲✞✧✴✧✵✏✶✧✵✖✸✷✠✹❖❵❳✔✗✖❜❛➜❆❝❇❞ ✏✌✔✗✖✑✻✺❡✺❢✶✲✞✧❣❄❆✏✰✱❤✧✐✧✣❉✄✧✣✻❥✧✣✏❂✱❣✻❥✧✣✱✳✲✞✔●✘❪✲✞❁✗✚❘✱❫✖✙❁✠✏✞✚✑❦ P✵✔✗✖✑✻✺✧✾✘✴✱❀✲❂✧❆❊✌✖✙✔✜❧✵❉✄✧✣✻♠✔◗P❅✘✰✚✣❋✣✖✙✧✣✱✰✄♥❁✗✱✰ ✔✗✏✓✔◗P❯❁❘♦q♣❴r ✰✏✮✱❇✔❥❁s❊✌✖❍✔☛❧✣❉✄✧✵✻t✔◗P❯❁✾❊☛❊✶✖✙✔✣❙✰✻✺❁✗✱✰ ✔✗✏✌❡ ➎✮➏✷➐❂➑✚➒❅➓☛✉ ✈✧✾✔✗✻✺✧✵✱✸✖✑✹ ➞❯➻➨➢①✇❯②➻ ➞ ✛ ➯▼➲➛❘③✖❰➮✠④ ➛✼↕➯ ➣➮↔ ➮✝ ➝ ↕①❒➮✷➬➛ ➬➯ ③ ➮➣❙↔➯❛➯➮➡ ➞ ➮↔ ➭➻ ➣❙↔ ➯●➲➛ ➜ ↔ ➮❰✯Ñ ➎✮➏✷➐❂➑✚➒⑥⑤✆⑦ ⑧
Miracle a(Iifu, v)=a(u, v),vE Xh NO:a(u,v)=()→a(Ihu,u)=(u),Vv∈Xn Only in the energy inner product can we compute IIn u without knowing u. N3 Note 3 Generality of abstract result We note that our bound that is, that uh= Ihu, is in fact true for any Spd bilinear form a, and any boundary conditions, and any finite element space Xh, and any space dimension For any particular SPD problem(that is, any linear problem for which the bilinear form a in the weak formulation is SPD), the only thing that changes is the definition of the norm; for our particular problem Iell= lehI(Q), though in general that will not be the case Obviously, however, we are not yet quite done; we must understand how llu-wnll depends on h, the smoothness of u, and the parameters of the problem For that we need to introduce the particulars of our finite element approximation 1. 4. 4 Particular bound SLIDE 21 We knot u-hlr1g)≤= Julho(g,) u-D1m()≤|lgea,)[3 e would, of course, prefer to directly use the projection uh for wh, rather tha the interpolant. However, the latter is much easier to work with, and will, in general, yield the correct h dependence. In fact, for our particular pro blem, ah=Lh u(see Exercise 3), but this is a bit of a coincidence
⑨✦⑩❀❶✙❷☛❸✣❹✄❺✓❻❨❼❆❽✌❾❫❿❅➀➁☛➂ ➃➄✑➅➆ ➇✗➈ ➉✙➊❨➋❆➌ ❽✶❾✳➂ ➉◗➊✆➋✑➉◆➍❲➊✿➎➐➏➁➐➑ ➒✞➓✞➔s→❣➣❘↔✆↕✺➙✞↕☛➔✭➛❚➙❂↕✗→ ➂❜➜✣➜❏➜ ➝❯➞❪❼✮❽✌❾➟➂ ➉✙➊❨➋❆➌ ➠ ❾➊✆➋ ➃❑➄✑➅●➆ ➡❇➢✑➤❥➥❤➦✑➧✾➨➫➩❑➧❍➭✸➥ ➯ ❽✶❾➟❿➀➁ ➂ ➃➄✵➅➆ ➇ ➈ ➉◗➊✆➋✕➌➲➠ ❾➊✆➋✑➉◆➍✐➊➳➎❖➏➁➐➵ ➞➙✞➸✂➺➳➻✂➙❈➔❍➼✞➣❘➣✣➙❂➣✣➽❍➾☛➺➳➻✂➙✞➙✞➣❏➽✭➚✞➽✙↕✆↔✆➓✮➪✑➔s➪✣➶☛➙➐→◆➣ ➪✵↕✜➹✴➚❂➓✆➔✙➣ ❿➁ ➂ →✭➻✄➔✙➼✞↕✜➓✆➔s➛❨➙✞↕✗→✭➻✄➙❂➾ ➂✩➜ ➝❅➘ ➴⑥➷➮➬✑➱✦✃ ❐❈➱✠❒❆➱✠❮✗❰✆Ï➟Ð✙➬➟ÑÒ➷✞Ós❰➮Ô✌Õ✜➬✙❮❑❰✶Ö❚➬❘❮✜➱✠Õ❚×✮Ï◗➬ Ø➣❘➙✞↕✠➔❍➣❴➔✙➼❂➶✠➔s↕☛➓✞➽❱➒✌↕☛➓✞➙✮↔ Ù✄Ù✂Ù Ú❨Ù✂Ù✂Ù ➌ ➻✄➙✆Û Ü✶➈✠Ý☛Þ❩➈ Ù✂Ù✄Ù ➂➳ß⑥à➁ Ù✂Ù✄Ù ➉ ➔✙➼✮➶✗➔✐➻✫á✣âã➔✙➼✮➶✗➔ ➂ ➁ ➌ ❿➀➁ ➂ â✩➻✂á✐➻✄➙⑥Û➟➶☛➪✵➔❲➔✙➽❍➓✞➣✺Û✳↕✜➽✐➶✠➙❨➺åä✆æ✕çè➒❂➻✄➸✂➻✄➙❂➣❏➶✠➽❲Û✳↕☛➽❍➹ ❽ â❩➶✠➙❂↔å➶✠➙❨➺ ➒✌↕☛➓✞➙❂↔❂➶✠➽❍➺❯➪✣↕☛➙❂↔✆➻✄➔✙➻✂↕☛➙✮á✣â✠➶☛➙❂↔❲➶✠➙❨➺❅é❂➙✞➻✄➔✙➣◆➣✣➸✂➣✣➹❥➣✣➙❚➔❩á◗➚❂➶✜➪✵➣ ➏ ➁ â✠➶✠➙❂↔✐➶✠➙❨➺❯á✙➚❂➶✜➪✵➣✕↔✆➻✂➹❥➣✣➙❂á✙➻✂↕☛➙ ➜ ê↕☛➽❴➶✠➙❨➺❈➚✮➶✠➽✙➔✙➻✫➪✵➓✞➸✫➶✠➽❴ä✆æ✕çë➚✞➽❍↕☛➒✞➸✂➣✣➹ ❾ ➔❍➼❂➶✗➔❴➻✫á✣â➮➶✠➙❨➺❖➸✂➻✄➙✞➣●➶✠➽❅➚✞➽✙↕✜➒✞➸✂➣✣➹ìÛ✳↕☛➽❯→✭➼✞➻✫➪✾➼✦➔✙➼✞➣ ➒✞➻✂➸✄➻✂➙✞➣●➶✠➽◆Û✳↕☛➽❍➹ ❽ ➻✄➙❈➔✙➼❂➣❯→◆➣❏➶☛➛✴Û✳↕☛➽❍➹✐➓✞➸✫➶✗➔❍➻✄↕✜➙✿➻✫á❱ä✞æ✕ç➋ â❨➔✙➼❂➣❯↕✜➙✞➸✂➺❥➔✙➼✞➻✂➙✞➾✴➔✙➼❂➶✠➔❱➪✾➼✮➶✠➙✞➾✜➣❏á❣➻✂á ➔✙➼❂➣❪↔✆➣✣é❂➙✞➻✄➔✙➻✂↕☛➙✦↕✠Û❩➔❍➼✞➣✐➙✞↕✜➽✙➹➑ Û✳↕☛➽❃↕✜➓✞➽s➚❂➶☛➽◗➔❍➻✂➪✣➓✞➸✫➶✠➽s➚✞➽❍↕☛➒❂➸✄➣❏➹ Ù✂Ù✄Ù Ú✆Ù✄Ù✂Ù ➌ Ù Ú✆Ù í❱î✑ï✄ð❂ñ â❂➔❍➼✞↕☛➓✞➾✜➼ ➻✂➙➐➾✜➣✣➙✞➣❏➽❍➶☛➸✌➔✙➼❂➶✠➔s→✭➻✄➸✂➸ò➙✞↕☛➔❱➒✌➣❴➔✙➼❂➣❲➪✣➶☛á✙➣ ➜ ➞➒❨ó❨➻✄↕✜➓❂á✙➸✄➺✜â✆➼✞↕✗→❣➣❏ó☛➣❏➽❏â❚→❣➣❘➶☛➽✙➣❴➙✞↕☛➔s➺☛➣✵➔sô❚➓✞➻✄➔✙➣❲↔✆↕✜➙✞➣ ➑ →◆➣❴➹✐➓❂á◗➔s➓✞➙❂↔✞➣✣➽✾á❳➔✾➶✠➙❂↔❈➼✞↕✗→ ➻✂➙✆Û Ü➈ Ý☛Þ➈ Ù✂Ù✄Ù ➂➳ß⑥à➁ Ù✂Ù✂Ù ↔✆➣❏➚✮➣❏➙❂↔✞á❪↕☛➙öõ▼âã➔❍➼✞➣✿á✙➹❥↕❚↕☛➔✙➼✞➙❂➣❏á❍á❲↕✠Û ➂ â✩➶☛➙❂↔å➔❍➼✞➣➳➚❂➶☛➽❍➶☛➹✴➣✣➔✙➣❏➽❍á❘↕☛Û❱➔✙➼✞➣❈➚✞➽✙↕✜➒✞➸✂➣✣➹➜✦ê↕☛➽ ➔✙➼✮➶✗➔❲→◆➣❪➙✞➣❏➣❏↔÷➔✙↕❖➻✂➙❚➔✙➽❍↕❨↔✞➓❂➪✵➣✴➔✙➼❂➣✺➚❂➶✠➽✙➔✙➻✫➪✵➓✞➸✫➶✠➽✾á❅↕✠Û❣↕✜➓✞➽❴é❂➙✞➻✄➔✙➣✺➣❏➸✄➣❏➹❥➣✣➙❚➔❲➶✠➚✞➚✞➽❍↕❑ø✆➻✄➹✺➶✠➔✙➻✂↕☛➙ á✙➚❂➶☛➪✣➣ ➜ ù✌ú✄ûãú✄û ü❅ý✆þ❑ÿ✁✄✂✆☎✞✝➟ý✆þ✠✟☛✡☞☎✍✌✞✎ ✏☞✑✓✒✕✔✗✖✙✘✛✚ Ø➣❘➛❚➙❂↕✗→ Ù ➂✿ß✠✜➁ ➂ Ù í✭î✾ï✄ð✮ñ✣✢ õ✤✦✥ ➂ ✥ í★✧✾ï✄ð✪✩ ✫ ➈ ñ ➵ ✬➼❨➓❂á Ù✄Ù✂Ù Ú✆Ù✂Ù✄Ù ➌ ➻✂➙✆Û Ü➈ Ý☛Þ➈ Ù✂Ù✂Ù ➂➳ß⑥à➁ Ù✄Ù✂Ù✭✢ Ù✄Ù✂Ù ➂✿ß✠✜➁ ➂ Ù✂Ù✂Ù ➌ Ù ➂➳ß✮✜➁ ➂ Ù í✭î✑ï✄ð❂ñ✯✢ ➁✰ ✥ ➂ ✥ í✯✧✵ï✄ð✛✩ ✫ ➈ ñ ✱➘ ➝✳✲ ❾ ➶☛á❍á✙➓✞➹❥➻✄➙✞➾ ✥ ➂ ✥ í✯✧✾ï✂ð✛✩ ✫ ➈ ñ é❂➙✞➻✄➔✙➣ ➋ ➜ ✴✦❺✶✵✣✷✹✸✆❹✻✺✹✼✽✷✿✾❅❸❀✷✹✸✆❶❀❁✣❺❂✼❄❃✶❶✙❺❅✾✵❺✣❶✶❆❇✷❈✺✗⑩❀❶✙❺✾❸❉❆❫❹❋❊☛✸✭❁✣❺●❆✕❍❂❺✣❃✌❶■✷❑❏✣❺✾❸❂❆✸⑩❅✷✹▲✺➂ ➁ ✾❂✷✗❶✕à➁ ✼ã❶❍❷✹❆▼❍✞❺✣❶✳❆▼❍✞❷✹▲ ❆▼❍✞❺✿⑩✕▲❄❆❤❺✵❶◆❃❄✷✗❹✄❷❖▲❄❆◗P❙❘☛✷✹✵✕❺❉❚❑❺✣❶❀✼●❆▼❍✞❺✿❹✄❷❖❆✄❆❤❺✵❶❥⑩❯❁❲❱❈✸❂❸❳❍ö❺❍❷✹❁✑⑩➟❺✵❶❲❆◗✷❨✵✣✷✠❶■❩❨✵q⑩✕❆▼❍✓✼❘❷✹▲☞✺❬✵q⑩❀❹❀❹❭✼❅⑩✕▲ ❪ ❺❂▲✶❺✵❶✙❷✠❹❭✼❫❊✗⑩➟❺✵❹✻✺❴❆▼❍✞❺÷❸❳✷✠❶✑❶✙❺✾❸❂❆ õ ✺✜❺❇❃❂❺❉▲☞✺☛❺❉▲✌❸✾❺❵P❜❛❝▲❞✾✵❷✜❸❂❆✄✼❡✾❂✷✗❶❞✷❖✸✆❶❈❃✞❷✠❶❝❆❫⑩➟❸❉✸✆❹✄❷✗❶❲❃✌❶■✷❣❢✵❹✄❺❉❱☛✼ ➂ ➁ ➌ ✜➁ ➂✐❤◆❁✣❺✾❺●❥✽❦❨❺✣❶✙❸✣⑩❯❁✵❺❈❧❂♠❵✼✯❢❉✸♥❆✣❆▼❍❨⑩❯❁❘⑩❯❁❪❷❙❢✵⑩✕❆♦✷❑✾❲❷✦♣❇❸❀✷✗⑩✕▲✶❸✵⑩❅✺✜❺❂▲✶❸✾❺✁P❑q r
We say the above estimate is optimal"in the sense that the pouer of h can not be improved -there exist problems (in fact, almost blems)for which i lell decreases no faster than h.( The term "sharp "is usually reserved for the case in which, for some problem, the bound obtains with strict equality - that is not the case here tho ugh we could tighten things up a bit to ensure sharpnes. Au the above requires essentially no modification for the Neumann problem D Exercise 3 Show that, for our particular problem, uh=Zhu. Hint: Show that alu-Thu, v)=0, VUE Xh, by integrat ion by parts over each element.(We odal superconvergence since it is rather special to -uzz=f, R, and exact quadrat ure. Note 4 Convergence rate and smoot ness First, the above estimate tells us that uh converges to u(at least in the a norm). Second, it tells us that it converges as h. Third, it tells us that u must be sufficiently smooth--finite in the broken H-norm, I IH (Q, T,)-to achieve this convergence It is important to recognize that although we exploit the weak form to look for finite element approximations uh that are only in H( &), we do require additional smoothness on the part of the ecact solution u if we are to obtain apid convergence. Furthermore, as we consider higher order finite elements we will require additional smoothness to achieve the best convergence rates: for ample, for quadratic finite elements,‖l‖≤ch2leg)- a higher power of h, but also a higher norm of u 1.5 Error: H Norm 1.5.1 Reminders lull(9)=luH(2)+lolli?(g) + llellH'(s) measures e and e SLIDE 23
s❨t✈✉❉✇❖①❞②▼③④t✮✇✆⑤❀⑥✹⑦✹t❨t❂✉❀②◆⑧✕⑨❲✇✹②◗t✠⑧❯✉✦⑩❇⑥❀❶☞②◆⑧✕⑨❫✇❖❷◆❸❙⑧✕❹❴②▼③④t❺✉❉t❉❹④✉❉t❙②✕③❄✇✹②❻②✕③❄t❼❶④⑥❖❽✣t❂❾✮⑥❑❿☛➀➂➁❀✇✹❹ ❹☞⑥❖②✯⑤❀t❻⑧✕⑨✳❶☞❾■⑥❖⑦➃t❀➄❺➅➆②▼③④t❉❾■t➇t❳➈❣⑧❯✉❀②☞❶➉❾■⑥✆⑤❉❷✻t❉⑨➇✉❲➊❇⑧✕❹❼❿❂✇✆➁❉②✄➋✯✇✹❷➌⑨❲⑥➃✉❝②✗➍✆➎✻➎➉❶➉❾■⑥✆⑤❉❷✻t❉⑨➇✉❇➏♦❿❂⑥✹❾➐❽✍③♥⑧❅➁❳③ ➑➒➑✻➑ ➓♥➑✻➑➒➑ ➄❣t❀➁❂❾■t❀✇✹✉❂t❂✉❼❹☞⑥✳❿❂✇➃✉❝②◗t❂❾➐②▼③④✇❖❹✠➀✞➔➇➊❉→➉③④t☛②◗t❂❾❝⑨➣⑩✄✉✿③❄✇✹❾◆❶✣❸➐⑧❯✉➐↔✭✉❝↔❄✇✹❷✕❷➌①❺❾■t❝✉❉t❉❾❝⑦➃t❀➄✳❿❂⑥✹❾❼②▼③④t ➁❀✇➃✉❉t✳⑧✕❹❙❽✍③♥⑧❅➁❳③✭➋✭❿❂⑥❖❾↕✉❂⑥❖⑨❫t➙❶➉❾■⑥✆⑤❉❷✻t❉⑨➇➋✞②✕③❄t●⑤❀⑥✹↔♥❹➉➄➇⑥❣⑤❂②◗✇✹⑧✕❹❄✉❡❽✽⑧✕②✕③❲✉❝②✄❾❝⑧❅➁❉②✞t❀➛❉↔❄✇✹❷➌⑧✕②◆①☛➅➜②▼③④✇✹②✞⑧❯✉ ❹☞⑥❖②✣②▼③④t☛➁❳✇✹✉❉t●③④t❉❾■t❂➋➙②✕③❄⑥✹↔✆➝➃③✮❽✣t☛➁❳⑥❖↔♥❷✻➄❲②◆⑧❋➝➃③✭②◗t❂❹❬②▼③✭⑧✕❹✭➝✹✉❻↔❵❶➞✇✈⑤❉⑧✕②★②❇⑥✈t❉❹④✉❝↔♥❾■t❻✉✿③❄✇✹❾◆❶➉❹☞t❂✉❳✉❉➔ ➏ ➟ ❷✕❷✪②▼③④t❈✇✆⑤❀⑥✹⑦✹t☛❾■t❀➛❂↔♥⑧✕❾❳t❝✉❼t❝✉❀✉❉t❉❹❄②◆⑧❅✇✹❷✕❷➌①✈❹➉⑥❲⑨❲⑥✁➄❖⑧➠✯➁❀✇✹②◆⑧❅⑥✹❹❫❿❂⑥✹❾➇②✕③❄t●➡☛t❂↔♥⑨❲✇✹❹➢❹❈❶☞❾■⑥❣⑤❂❷✻t❉⑨❲➔ ➤✈➥●➦➨➧❖➩❣➫❵➭❅➯✆➧❺➲❺➳✭➵❄➸✹➺❴➻❳➵➍➻❵➼✓➽▼➸✆➾★➸✆➚④➾✯➪➍➾■➻■➶❯➹❂➚➎❯➍➾✯➪④➾❳➸✆➘➎➒➴❉➷➼✭➬➉➮❼➱❐✃✪➮✆➬✍❒➙❮⑧✕❹➢②❂❰ ➳✭➵④➸✹➺ ➻■➵➍➻➙Ï➉Ð▼➬❡Ñ❻✃✪➮✹➬✞Ò✿Ó♥Ô➙➱✦Õ❄➼➃Ö✣Ó❲×✠Ø❲➮❄➼✆➘✭Ù➐➶➒Ú✓➻ ➴❉Û➾➍➻■➶➒➸✆Ú☛➘✓Ù➇➪➍➾■➻❳Ü➨➸✹Ý➴ ➾ ➴✁➍➹❀➵ ➴❉➎➒➴❉➷❫➴Ú✓➻❵❒↕Ð❅Þ➴ ß➸❙Ú④➸❖➻ ß➺➴❉➎➒➎ ➸❣Ú❬➻■➵❄➶➒Ü❙à➷➶➒➾➍➹➎➒➴✁á❈â Ú④➸ ß➍✆➎ Ü■➚④➪ ➴ ➾❀➹❂➸✆Ú✭Ý➴ ➾Û✆➴Ú➢➹➴✶â Ü✿➶➒Ú❄➹➴ ➶✻➻✶➶❯Ü✳➾➍➻❳➵➴ ➾ Ü■➪➴ ➹❂➶➍❖➎ ➻■➸❨Ñ♦➬➉ã❵ã❻➱✐ä✞➼♥åæ●ç➃➼ ➍Úß ➴❂è④➍➹❂➻♦é✓➚➍ß➾➍➻■➚❄➾➴ ❒ Ô ê✙ë✛ì ➧✠í îë✭ï✽ð➧❖➩✁ñ✍➧ï➫❣➧✠➩✹ò ì ➧✠òï✣ó ➯✁ôë✗ë➉ì■õ✍ï➧❖➯✁➯ ö➶➒➾❳Ü✿➻✁➼✗➻■➵➴❲➍➘☞➸✹Ý➴❈➴Ü✿➻■➶➷❲➍➻ ➴ ➻ ➴❉➎➒➎ Ü❼➚❄Ü✶➻■➵➍➻➐➬➉➮✮➹❉➸✆Ú✭Ý➴ ➾Û❣➴Ü✳➻■➸✮➬✦Ð ➍➻ ➎➒➴✁➍Ü❑➻➐➶➒Ú✙➻■➵➴ Ï Ú④➸❣➾➷Ô❝❒★➳➴ ➹❉➸✆Úß ➼❄➶❋➻❡➻ ➴❉➎➒➎ Ü❡➚❄Ü↕➻■➵➍➻✳➶✻➻✳➹❂➸❣Ú✓Ý➴ ➾Û✆➴ Ü ➍Ü ➀ ❒★÷↕➵④➶➒➾ß ➼❄➶❋➻❡➻ ➴❉➎➒➎ Ü❡➚❄Ü↕➻■➵➍➻❡➬ ➷➚❄Ü❑➻ ➘ ➴ Ü■➚♥ø✈➹❂➶➴Ú✓➻ ➎Ù❼Ü➷➸✓➸✆➻■➵ âúùÚ④➶❋➻ ➴ ➶✻Ú➇➻■➵➴ ➘④➾❳➸✆û➴Ú➇ü❬ý✯Ú④➸❣➾➷➼❄þ❄ÿ◗þ✁✄✂✆☎✞✝✠✟ ✡☞☛✍✌ â ➻■➸ ➍➹❀➵④➶➴Ý➴ ➻■➵❄➶➒Ü❡➹❉➸✆Ú✭Ý➴ ➾Û❣➴Ú❄➹➴ ➾➍➻ ➴ ❒ å❇➻❡➶❯Ü↕➶➷➪☞➸✆➾■➻ ➍Ú❣➻↕➻❳➸❲➾➴ ➹❂➸ÛÚ④➶✏✎➴ ➻■➵➍➻ ➍✆➎➻❳➵④➸✆➚Û➵✠➺➴❻➴❂è➪➎➸✆➶✻➻↕➻■➵➴ ➺➴✁➍û❲➽▼➸✆➾➷ ➻❳➸ ➎➸✭➸✆û ➽▼➸✆➾ ➠➙❹❄⑧✕②◗t✙t❂❷✻t❉⑨❲t❂❹➢②❫✇❀❶✆❶☞❾❳⑥❉➈❣⑧✕⑨❫✇❖②✄⑧❅⑥❖❹④✉ ➬➮ ➻❳➵➍➻ ➍➾➴ ➸✆Ú➎Ù❴➶➒Ú✦üç Ð✒✑♦Ô❝➼♦➺➴ ß➸ ➾➴é✓➚④➶➒➾➴ ➍ß❄ß➶❋➻❳➶✻➸❣Ú➍❖➎ Ü➷➸✭➸❖➻❳➵④Ú➴ Ü❳Ü❻➸✆Ú✙➻❳➵➴ ➪➍➾■➻❼➸❖➽★➻■➵➴ t■➈✭✇❣➁❂②✳✉❂⑥❖❷➌↔♥②◆⑧❅⑥✹❹ ➬❐➶❋➽♦➺➴❫➍➾➴ ➻❳➸❨➸✆➘♥➻ ➍➶➒Ú ➾➍➪④➶ß ➹❂➸❣Ú✭Ý➴ ➾Û✆➴Ú❄➹➴ ❒ ö➚④➾■➻■➵➴ ➾➷➸❣➾➴ ➼ ➍Ü❼➺➴ ➹❉➸✆Ú❄Ü■➶ß➴ ➾☛➵④➶Û➵➴ ➾❈➸✆➾ß➴ ➾ ùÚ④➶✻➻ ➴❙➴❉➎➒➴❉➷❫➴Ú✓➻❀Ü❉➼ ➺➴ ➺♦➶ ➎➒➎ ➾➴é❣➚❄➶✻➾➴❡➍ß④ß➶❋➻❳➶✻➸❣Ú➍❖➎ Ü➷➸✭➸❖➻❳➵④Ú➴ Ü❳Ü✽➻❳➸ ➍➹❀➵❄➶➴Ý➴ ➻■➵➴ ➘ ➴ Ü✿➻★➹❂➸❣Ú✭Ý➴ ➾Û✆➴Ú❄➹➴ ➾➍➻ ➴ Ü ❰ ➽▼➸✆➾ ➴❂è④➍✆➷➪➎✻➴ ➼✆➽▼➸❣➾★é✓➚➍ß➾➍➻■➶❯➹ ùÚ④➶✻➻ ➴✶➴❉➎➒➴❉➷❫➴Ú✓➻❀Ü❉➼ ➑✻➑➒➑ ➓✭➑➒➑➒➑✔✓✖✕ ➀ý✣þ❝➬✽þ✆✘✗✙☎✞✝✚✌ â ➍ ➵④➶Û➵➴ ➾★➪☞➸✹➺➴ ➾ ➸❖➽ ➀ ➼♥➘④➚♥➻ ➍❖➎ Ü■➸ ➍ ➵④➶Û➵➴ ➾♦Ú④➸❣➾➷ ➸✆➽✍➬✞❒ ✛✢✜✒✣ ✤✦✥✧✥✧★✩✥✫✪✭✬✯✮✦✰✱★✩✥✚✲ ✳✵✴✷✶✸✴✹✳ ✺✼✻✔✽✿✾✹❀❂❁❃✻❅❄❇❆ ❈✵❉❅❊●❋✫❍❏■✧■ →➉③④t üç ❹☞⑥❖❾❝⑨✮❰ þ❂Ó➉þ❉ý▲❑▼☎✞✝✚✌ ➱ ➑ Ó ➑ ý▲❑▼☎✞✝◆✌✸❖ þ❝Ó✛þ❂ýP ✂✆☎✞✝◆✌ ➱ ◗ ç ❘ Óã❚❙❱❯ ý ❖ ◗ ç ❘ Ó ý ❙❱❯❳❲ þ ➓ þ✆❨❑❩☎✞✝✚✌ ➷❫➴❵➍Ü■➚④➾➴ Ü ➓ ➍Úß ➓ ã ❒ ❈✵❉❅❊●❋✫❍❏■✧❬ ❭
