彐a>0 such that (,0)a|lgg),Vv∈X 22 dx (Recall this Poincare-Friedricks inequality follows from the fact that v E X satisfy v(0)=v(1)=0. For our problem,a=2works, "as can be shown (say) by considering the Rayleigh quotient. 彐B(=1)>0 such that a(u,y)≤B‖ln(g)‖l(g) (Recall this is derived from the Cauchy-Schuvarz inequality The error e=u-uh satisfies ela()≤(x< degradation error in HI projection of u on Xh n general ah is not the H projection of u on Xh b Exercise 4 For what particular problem(give the strong form)is uh H projection of u on Xn?L Note 5 Proof of H norm general bound(Optional) To begin, we note that for any wn E Xh a( (u )(orthogonality Bla-wn H()lah -wnH(Q2)(continuity)
❪✩❫✍❴✁❵❜❛✁❝●❞✍❝●❡❣❢✼❫✐❤❦❥♠❧♦♥✏♣✁♥rq▼s t✈✉❳✇✖①③②✐④◆⑤✙⑥✼⑦❜⑥✚⑧❇⑦ ❥♠❧⑩⑨✵♣✐⑨❶q❸❷ ✉❏❹ ⑨ ❹✆❺❻❨❼❩❽✞❾✚❿ ♣➁➀➂⑨➄➃✦➅ ➆✵➇➉➈ ➊ ⑨ ❺➋❚➌❱➍ ❷ ✉ ➆✚➇➎➈ ➊ ⑨ ❺➋❨➌➏➍➑➐ ➇➎➈ ➊ ⑨ ❺ ➌❱➍◆➒➓➒→➔ ➣⑩↔❴✙❛✙↕❇➙●➙➛❡●➜❶❝✷➝➟➞❨❫➠❝●➡✵❛✙↕❇❵➤➢❴✁➥✒➦✸❵▼❝✹❴❩➧➠❵▼❝✹❛❩➨❇➝➩❝●➡✵❴✙➫✆➭◆↕➠➙➯❝●❡❣❢➲❤✆❫➠➙●➙✞❫❇➳➤➝➵❤✙❵❜❫❇➸➺❡⑩➜✧❴③❤✆↕❱❛✆❡➻❡●➜◆↕❇❡➼⑨➽➃➾➅ ➝✁↕❇❡❣❝✷➝⑩❤✙❢✼⑨♠❧ ① q➑➚➪⑨♠❧♦➶➹q➑➚ ①✵➘ ➦✫❫➠❵➲❫❇➭❶❵➂➴✵❵❜❫❱➷✆➙✏❴✆➸➮➬ ✉ ➚ ➈❺✃➱➳❐❫➠❵❜➨➹➝✙➬❩❒✭↕❇➝➲❛✙↕❇➡❮➷✙❴✼➝❜➜✧❫❇➳✢➡ ➣➝✁↕❇❢✙❰➻➷✆❢➻❛✙❫➠➡✧➝▼❝✹➧➏❴✁❵▼❝●➡❅Ï➲❡⑩➜✧❴ ↔↕➠❢❇➙✞❴✆❝ÐÏ➹➜Ñ➫✆➭◆❫➠❡✒❝✹❴✁➡◆❡ ➘ ❰ ❪✩❫❇➡✚❡✒❝●➡✚➭❶❝●❡✒❢➻❫✐❤✭❥♠❧♦♥✏♣✁♥rq▼s t➓Ò ❧Ó➚Ô➶➹q ✇➎①③②❜④◆⑤✙⑥✼⑦❜⑥◆⑧➠⑦ ❥✵❧✹Õ➂♣✐⑨❶q▲Ö Ò❦❹ Õ ❹ ❻❼ ❽✞❾✚❿ ❹ ⑨ ❹ ❻❼ ❽✞❾◆❿ ➔ ➣⑩↔❴✙❛✙↕❇➙●➙✸❡⑩➜✔❝✷➝➼❝✷➝×➧➏❴✁❵▼❝●❞❇❴❩➧➓❤✙❵❜❫➠➸Ø❡⑩➜✧❴➄❪✩↕❇➭◆❛✙➜✔❢❇➥ÓÙ♠❛✙➜✔➳❐↕➠❵❜Ú➑❝●➡✵❴✙➫✁➭◆↕❇➙➯❝●❡❣❢➘ ❰ Û✵Ü✷Ý✸Ü✷Þ ß➲à✔á❂à❅â➹ã◆ä✄å✼à❅æ☞ç❂ä⑩è é✵ê❅ë●ì✫í❏î✔ï ð⑥✧ñ➼ñ✍ò❜ò❩ó➏ò▲ô ➚❮õ➄ö❏õ♠÷ ②❩⑧❇⑦❩ø✏②✐ù◆ñ✍② ❹✆ô❶❹ ❻❨❼❩❽✞❾✚❿ Ö ❧✐➶ ➐ Ò ✉ q ú û▼ü ý þ➹ÿ✁✄✂✆☎✙þ✝☎✟✞✡✠☞☛✄✌ ø✎✍✑✏ ✒✔✓✖✕✘✗ ❹ õ➄ö❏Õ❨÷ ❹ ❻❨❼▼❽✞❾◆❿ ú û▼ü ý ÿ✁✂✡✂✡☛✄✂✙✠☞✌✛✚✢✜✔✣✝✂✡☛✥✤✹ÿ✁✦✧✞✡✠☞☛✄✌★☛✥✩✫✪✛☛✄✌✛✬✮✭ ✯ ø✎✍✱✰❱ñ✲✍✧ñ✍ò❩⑧✖✳ õ♠÷ ø✷② ➡✵❫➠❡ ⑦❩⑥✧ñ✢✴➈★✵ò❩ó✷✶♦ñ☞⑤▼⑦❩ø✞ó✸✍➻ó✷✏ õ ó✖✍ ➅➄÷✺✹ ✻✽✼ ✾✛✿ ❀❂❁★❃✘❄❆❅✸❇✝❈❊❉✖❄✱❋❂●ó❱ò✮❍❨⑥◆⑧❇⑦ ✵⑧➏ò✐⑦❩ø✏⑤✁④✺✳✷⑧➠ò ✵ò❩ó✖■✺✳✏ñ✲❏ ❧ ✰➏ø✎❑➏ñ➛⑦❜⑥◆ñ➂②♦⑦❩ò❜ó✸✍✺✰▲✏⑩ó➏ò✟❏q➛❝✷➝➼õ✠÷ ⑦❜⑥✧ñ ✴Ñ➈ ✵ò❩ó✷✶♦ñ☞⑤▼⑦❩ø✞ó✸✍➻ó✷✏ õ ó✖✍ ➅➄÷✸▼ ◆P❖❘◗ ❄❚❙ ❯❱❅❖✙❖✺❲★❖✺❲ ✴Ñ➈❨❳❖❅✝❩❭❬❪❄❳❄✷❅❆❫✑❴❛❵ ❖❘❜❳❞❝❢❡✥❣✛❤◗ ❈❖❳❫✑❴❥✐ ðó❦■✚ñ❧✰➏ø✎✍✙♠✺❍✘ñ✛✍◆ó➠⑦❜ñ✈⑦❩⑥◆⑧❇⑦✮✏⑩ó❱ò❚⑧✷✍♦♥ Õ❚÷③➃✦➅➄÷ ♠ ✉Ñ❹ õ♠÷➼öÑÕ❚÷ ❹ ❺❻❨❼✙❽✞❾✚❿ Ö ❥♠❧⑩õ✠÷✈ö❏Õ❨÷✚♣❜õ♠÷➛ö❏Õ❚÷❅q ❧ ⑤✁ó✔ñ✁ò✙⑤✆ø✎❑❅ø✞⑦✁♥q ➚ ❥♠❧⑩õ✠÷✈ö❏Õ❨÷ ➐ ❧✹õ➄ö❏õ♠÷❅q▼♣❜õ♠÷✈ö❏Õ❚÷❱q ❧ó❱ò✐⑦❩⑥✧ó✖✰❱ó✖✍◆⑧✖✳✞ø✞⑦✁♥q ➚ ❥♠❧⑩õ➄ö❏Õ❚÷✚♣✐õ✠÷➛öÑÕ❚÷✔q Ö Ò❦❹ õ➵ö❏Õ÷♠❹ ❻❼ ❽✞❾✚❿ ❹ õ÷ öÑÕ÷✵❹ ❻❼ ❽✞❾✚❿ ❧ ⑤✁ó✖✍❅⑦❜ø✎✍✔④✧ø✞⑦✁♥q ➶ ①
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♣rq❨s✟t✈✉❆s ✇ ①❘②✛③P④✮② ✇ ⑤⑦⑥✄⑧⑩⑨✈❶❸❷❺❹❻ ✇ ①❼③P④❽② ✇ ⑤⑦⑥✄⑧⑩⑨✫❶❿❾ ➀➂➁s✮srt✈➃✲➄ ✇ ①❼③P① ② ✇ ⑤⑥ ⑧⑩⑨✈❶➆➅ ✇ ①❼③P④②✛➇ ④② ③➈① ② ✇ ⑤⑥ ⑧⑩⑨✈❶ ❷ ✇ ①❼③P④② ✇ ⑤⑥ ⑧⑩⑨✈❶ ➇ ✇ ① ② ③➈④② ✇ ⑤⑥ ⑧✎⑨✈❶ ➉ sr➊✟➋➌✉✷➄✺➍✸➎⑩➃❛➋✎➄✺➃❧➏➁✉✖➎⑩➋⑩s✁➐✺➑ ❷ ➒❘➓ ➇ ❻★➔ ❹ ✇ ①→③➈④② ✇ ⑤✮⑥✥⑧⑩⑨✫❶ ➣ ④②❦↔❂↕➙②✱➛ ✉✷➄✫➜➙s✟t➁♣ ✇ ①➙③P①➝② ✇ ⑤⑥ ⑧✎⑨✈❶❸❷ ➒ ➓ ➇ ❻★➔ ❹ ➋✎➄✑➞ ➟➡➠✖➢✖➤➥➠ ✇ ①➙③➈④❽② ✇ ⑤⑥ ⑧✎⑨✈❶★➦ ✉✖♣⑦➜✑➃➧♣➨➋✎➊✟➃❧➜❘➩➥➫❽q✖sr➃✛➋⑩➄➭srt✈➋✎♣⑦➯✺➊✟q♦q✷➞➳➲➂➃✛➋⑩➄➭➞❊✉✖➵➸s✢➺✷➻❊➺❼➼➝➽❆➾✽➚♦➪✲➶✛➪✄➹❆➘▲➘❦➶✲➾➷➴✄➹❨➬➮s✟t✺➃❛➯✺➊rq♦q✖➞❪✉✷➯✑➱ ➯✺➎✎➋⑩➃➧♣✔srq❱✉✷➄♦➐❱➎✎➋⑩➄✈➃❧✉✷➊✘➯✺➊✟q✖✃✈➎⑩➃❧❐❒➞❥q✖➊✘➲✮t✺➋➌➵✥t❮s✟t✺➃✮✃✺➋✎➎⑩➋✎➄✺➃➧✉✷➊✔➞❥q✖➊✟❐Ï❰❱q✖➞✈srt✈➃➂➲➂➃❧✉✖Ð❛➞❥q✖➊✟❐➁➎➌✉❆s✟➋⑩q✸➄ ➋➌♣✢Ñ✟➽➧➶✲➴rÑ➸➻ÓÒ❆➶ ➉ ✉✖➄✈➜➭➵✲q✖➄Ôsr➋✎➄➁q➁♣✟➑➸➩ Õ➄➈➞❊✉✖➵➸s❧Ö➝➞❥q✖➊❛q➁➊✛➵➁➊✟➊r➃❧➄✸s❱➵✲✉✸♣➨➃✸Ö➝➲✮t✺➋✎➵✥t➈➋✎♣❛✉✖➎✎♣rq➭♣r➐♦❐▲❐❦➃✲sr➊✟➋✎➵▲✉✷➄✈➜❚srt➁♣❛t✈✉✸♣★✉➭❐❦➋⑩➄✈➋×➱ ❐❦➋⑩Ø➧✉❆s✟➋⑩q✸➄P♣➨s✟✉✷sr➃✲❐❦➃❧➄✸s➧Ö➳➲➂➃❼➵✲✉✷➄P➋✎❐▲➯✈➊rq❆Ù✸➃❮s✟t✺➋✎♣✢➊r➃➧♣➁➎⑩s❧Ú❛➋✎➄Psrt✺➃❼➃❧➄✺➃✲➊✟➍✖➐Û➄✺q✸➊r❐Ü➲❸➃❦Ð♦➄✺q❆➲ srt✫✉❆s ❰ ➉①➙③➈①❘② ➦ ①❼③P①➝② ➑ ➅ ➋⑩➄✺➞ ➟➝➠✷➢✖➤✘➠ ❰ ➉①→③P④❽② ➦ ①➙③➈④❽② ➑ ➛ srt➁♣✮➞❥➊rq✸❐Ý➵➸q♦➃✲➊✥➵➸➋✎Ù♦➋×s✁➐➙✉✖➄✈➜❂➵➸q✸➄✸s✟➋⑩➄➁➋⑩s✁➐ ❻ ✇ ①➙③➈①❘② ✇✲Þ⑤⑥ ⑧⑩⑨✫❶ ❷ ➋✎➄✑➞ ➟➠ ➢✖➤➠ ❹ ✇ ①❼③➈④❽② ✇➸Þ⑤⑥ ⑧⑩⑨✫❶ q✖➊ ✇ ①➙③P①❘② ✇ ⑤✮⑥✄⑧⑩⑨✈❶❸❷àß❹ ❻ ➋⑩➄✺➞ ➟✔➢✖➤➠ ✇ ①→③➈④❽② ✇ ⑤✮⑥✥⑧⑩⑨✫❶❿➦ ➲✮t✺➋➌➵✥t❚➋➌♣★♣rt✈✉✷➊✟➯➡➃✲➊✮s✟t✈✉✷➄❚q➁➊❿➯✈➊r➃❧ÙÔ➋✎q➁♣❽➊✟➃❧♣➁➎×s❛♣➨➋✎➄✈➵✲➃ ❹ ➵✲✉✖➄❚➄✺q✖s★✃➡➃❮➎✎➃❧♣✟♣✮srt✫✉✷➄ ❻ ➉ s✥✉✷Ð✸➃ á ➅ ④ ➋⑩➄✱srt✈➃✢➵➸q✖➄Ôs✟➋⑩➄➁➋⑩s✁➐→➵➸q✸➄✈➜✑➋⑩sr➋✎q✖➄✫➑➸➩ ➫❿q✷sr➃❼➋⑩➞⑦➲❸➃➙➵✲q✖❐❦➯✈✉✷➊✟➃❨srt✺➃➙✉✖✃✫q❆Ù✸➃▲➯✈➊rq♦q✷➞❊♣✛s✟qÛ♣➨➋✎❐❦➋⑩➎➌✉✷➊❛â✈➄✈➋×s✟➃→➜✑➋⑩ã➡➃❧➊r➃❧➄✈➵➸➃❦➯✺➊✟q♦q✷➞❊♣❧Ö➳➲➂➃ ♣r➃✲➃✛srt✈✉✷s❽➵➸q♦➃✲➊✥➵➸➋✎Ù♦➋×s✁➐❼➯✺➎➌✉✝➐✑♣➂srt✺➃❛➊✟q✖➎✎➃❛q✷➞✔♣➨s✟✉✖✃✺➋✎➎⑩➋⑩s✁➐✖Ö✺✉✖➄✈➜→q✸➊➨s✟t✺q✖➍✸q✖➄✈✉✖➎⑩➋⑩s✁➐▲srt✺➃❱➊✟q✖➎✎➃✛q✷➞✔➵✲q✖➄✑➱ ♣r➋✎♣➨sr➃❧➄✈➵➸➐✸➩✽ä➳q✸➍✖➃➸s✟t✺➃✲➊➂s✟t✺➃✲➐➭➋⑩❐❦➯✺➎✎➐➭➵✲q✖➄♦Ù✖➃❧➊r➍✸➃✲➄✈➵✲➃✖➩ å➡æ➌ç✙æ✎è é★ê✑ë✝ì❧í➷î✖ï✔ð❊ê✑ë✱ñ→ò♦ó❧ï✔ð❥ì ô➡õÔöÓ÷➳øPù✺ú û➃❱ÐÔ➄✈q❆➲ ✇ ①❼③❂ü✙②✖① ✇ ⑤⑥ ⑧⑩⑨✈❶❸❷þýÿ✁✂ ✇ ① ✇ ⑤☎✄✥⑧✎⑨✝✆ ✞➠ ❶ ➩✽ä⑦t➁♣ ✟✡✠ ➶ ýÿ ➻➌➪❨➪✄➻Ó➘☞☛✍✌➹✏✎Û➪✑✌✎➽✒☛✓☛➝➹✏✔✥➽❆➚✑➼➡➺✖✕➸➽❆➴ ➉✁➓ ➇ ② ✄ ✗✄ ➑✙✘✙✚ Þ ✛✠➻❊Ñ✠ ✎✷➴✄➻➌➪✲➶✄➪✁✔✥➶✥Ñ✒✎✷➚♦➪✲➶❦➾✠ ➶✖✜✢✘ ➼➡➽✷➴✄➘ ✠ ✎✝➪❮Ñ✥➽❆➼✫➾➷➴✄➻✣✔✲➚✑➾➷➻❊➽✷➼✺➪✤✕✥➴r➽✷➘✥✔✥➽❆➾✠ ➾✠ ➶✦✜✢✘❦➪➸➶✲➘▲➻Ó➼➡➽❆➴✄➘✧✎❆➼➝➺✖★Þ ➼➡➽✷➴✄➘ ➽✩✕✫✪✬✌➻❊➺✸➶✮✭✰✯✲✱ ➓✸➓