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 23s❨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 ❒ ❈✵❉❅❊●❋✫❍❏■✧❬ ❭