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