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 ➘