≤(1+2)a-五叫ll( 8)=(S,Th) The error in the H norm converges at the same rate as the error in the energy norm.(This must be the case since the two norms are equivalent 1.6 Error: L2 Norm 1.6.1 Reminde In the L--like norm, ue found for finite differences that the error converged as h; given the similarity of An for finite elements to the corresponding system matric for finite differences, we expect h- behaviour here as well. Note the h dependence(as opposed to h dependence)of the h norm is not surprising 1.6.2 Particular Result SLIDE 27 A General Result is possible, but not as transparent as in the other cases, so we directly to the particalar res ul The [2 error satisfies lle(g)≤Chl(a) C hall for C independent of h and✳✵✴✶✳✸✷✺✹✑✻✽✼✿✾❁❀ ❂❄❃☎❅❇❆❈✦❉ ❊●❋✶❍ ■✍❏✓❑✓▲▼❏ ✳❖◆✁P✢◗❙❘✍✳✵✷❚✹✑✻✽✼✬✾ ❯ ❂❄❃☎❅❇❆❈ ❉❱✳❖◆❲P❨❳❘ ◆❩✳✸✷✹ ✻✽✼✿✾ ❯ ❬❭ ❂✝❃☎❅❪❆❈❫❉❵❴❛ ✳✵◆❩✳✸✷☎❜✵✻✽✼✝❝ ❞ ❏ ✾☞❡ ❢✡❣✐❤✖❤✸❥❖❥✙❦✰❥✦❧♥♠♣♦q❣✐❤✺r✢st♠✡❦✉❥❖✈①✇✑❦✉♠✬②③❤✸❥⑤④⑥❤❖⑦⑨⑧✉♦✤♦q❣✐❤✦⑦✸⑧✰✈✮❤✫❥✒⑧✉♦⑩❤✖⑧③⑦☞♦q❣✐❤❶❤✵❥❖❥✙❦✰❥✦❧♥♠❨♦q❣✐❤✖❤✸♠✍❤✸❥⑤④✓❷ ♠✍❦✰❥❖✈✮❸t❹✸❢✡❣❺❧❻⑦✖✈❽❼❺⑦❖♦☎❾✑❤✖♦q❣✐❤❽✇✑⑧③⑦✸❤⑨⑦❖❧♥♠✡✇✒❤❶♦q❣✐❤❶♦⑤❿➀❦➁♠✍❦✉❥❖✈t⑦❶⑧✉❥✙❤❽❤✑➂✵❼✶❧♥②✉⑧✉➃●❤✵♠✬♦➄❸ ➅ ➆▼➇⑤➈ ➉❨➊✐➊✐➋✤➊➍➌➏➎⑨➐✢➑❵➋✤➊✿➒ ➓✍➔●→➍➔✣➓ ➣↕↔❺➙➜➛✣➝➟➞➠↔⑥➡ ➢✍➤⑥➥♥➦➍➧➩➨✐➫ ❢✡❣✐❤t➭➀➯❽♠✍❦✰❥❖✈♣➲ ✳✵➳✡✳✸➵✐❜❖✻●✼✿✾▼❀➸❂✬➺ s ➻ ➳ ➯➀➼✓➽❉ s✙➾ ➯➪➚ ✳✸✴❺✳✸➵✐❜✒✻✽✼✬✾➀➶✮➹➴➘✲➷✩➬✿➮✙➹➱➷❚✴✲✃ ❐❖♠✏♦♥❣✿❤❙➭➯✓❒ ➃❮❧✽❰✲❤✖♠✡❦✉❥❖✈tÏ✤❿➀❤➀Ð✵❦✰❼✶♠✍Ñ❚Ð✵❦✰❥✤Ò✤♠✿❧♥♦⑩❤❶Ñ✰❧Ó☞❤✸❥✙❤✵♠✡✇✑❤❖⑦⑨♦q❣✐⑧✉♦Ô♦q❣✐❤❶❤✸❥❖❥✙❦✰❥❶✇✒❦✰✈t②✉❤✸❥⑤④✲❤✑Ñ↕⑧③⑦ ❴ ➯✉Õ ④✰❧♥②✉❤✸♠Ö♦q❣✐❤➁⑦❖❧♥✈t❧♥➃✽⑧✰❥❖❧♥♦⑤❷×❦ØÐtÙ❘ Ð✵❦✉❥✦Ò➀♠✿❧♥♦⑩❤↕❤✸➃✽❤✸✈✮❤✸♠✿♦⑤⑦❲♦➄❦➏♦♥❣✿❤❨✇✒❦✰❥❖❥✙❤✵⑦⑤Ú✿❦✉♠✡Ñ✉❧♥♠❺④➏⑦❖❷③⑦❖♦⑩❤✵✈ ✈❲⑧✉♦Û❥❖❧ÝÜtÐ✵❦✰❥✦Ò➀♠✿❧♥♦⑩❤↕Ñ✰❧Ó☞❤✵❥✒❤✵♠✡✇✒❤✵⑦✑Ï❫❿➀❤❨❤✒Ü✵Ú✐❤✑✇✸♦ ❴ ➯ ❾✑❤✑❣✐⑧✉②➱❧✣❦✉❼✶❥❲❣✐❤✸❥✙❤♣⑧③⑦❲❿➀❤✸➃♥➃✽❸✏Þt❦✉♦⑩❤↕♦♥❣✿❤ ❴ Ñ✓❤⑩Ú✿❤✵♠✡Ñ✓❤✸♠✍✇✑❤➁❹Ø⑧✉⑦✖❦✑Ú✓Ú✿❦③⑦✸❤✑Ñ➁♦⑩❦ ❴ ➯ Ñ✲❤➄Ú✿❤✵♠✡Ñ✓❤✸♠✍✇✑❤Ø➅➁❦✩Ð✫♦♥❣✿❤❙r✢s❲♠✍❦✰❥❖✈①❧❻⑦⑨♠✍❦✰♦▼⑦❖❼✶❥ÛÚ✍❥❖❧❻⑦❖❧♥♠⑥④àß ❧♥♦Ô✈✮❤✑⑧③⑦❖❼✶❥✙❤✵⑦✫♦♥❣✿❤t❤✸❥❖❥✙❦✉❥⑨❧♥♠×⑧✮⑦❖♦Û❥✙❦✉♠❺④✲❤✸❥⑨♠✍❦✰❥❖✈✮❸ ➓✍➔●→➍➔❻á â✦ã✶➡③ä➴➛⑤å✓æ➟ç✣ã✶➡à➣✁↔❺è➴æ➟çqä ➢✍➤⑥➥♥➦➍➧➩➨✬é ê①ë❤✵♠✡❤✵❥✙⑧✰➃✐ì☞❤✵⑦✑❼✶➃Ý♦▼❧❻⑦✤Ú✿❦③⑦✑⑦❖❧✣❾✵➃✽❤✵Ï☎❾✸❼✶♦▼♠✍❦✉♦➀⑧✉⑦✦♦Û❥✙⑧✉♠✿⑦ÛÚ✐⑧✰❥✙❤✵♠✬♦➀⑧③⑦☞❧♥♠♣♦q❣✐❤❶❦✉♦q❣✐❤✸❥✫✇✑⑧③⑦✸❤✵⑦✑Ï➟⑦✸❦❽❿➀❤ ④⑥❦✁Ñ✉❧♥❥✒❤✒✇✸♦Û➃❮❷➁♦⑩❦❲♦q❣✐❤❚Ú✐⑧✰❥❖♦⑤❧✣✇✸❼✶➃✽⑧✉❥⑨❥✒❤❖⑦❖❼✶➃❮♦⑩❸ í✺î➹ ➭➯ ➹➴➮✙➮✒ï✓➮❚➷✒➘✉ð✒❊●➷✩ñ✿➹➴➷ ✳✸✴✶✳✵➵ ❜ ✻●✼✿✾ ❯ ò ❴ ✳✸✴❺✳✸✷✹ ✻✽✼✿✾ ❯ ò ❴ ➯ ✳❖◆▼✳✵✷Ô❜❖✻✽✼❄❝ ❞ ❏ ✾☞ó ❍qï✓➮ ò ❊●❋✿ô✶➹✸õ✍➹✸❋✬ô✶➹✸❋⑥ð❙ï✰❍ ❴ ➘✓❋✿ô✁◆➟✃ ö☞÷ ❃ ❭