1.4.2 Orthogonality a(u,u)=C(),vv∈X then e(U),Vv∈Xh a(uh,)=C()],V 1.4.3 General Bound SLIDE 17 or any wh Uh∈ 0: ort hogonality ell h∈X more accurate than in the energy norm So we see that the finite element procedure does as well without knowledge of the ecact solution as you can do with knowledge of the eract solution- so long as we speak of the energy(or" a")norm. The finite element method has trans formed the problem of discretization of a PDe into a problem of appro imation vu∈X❻✮❼✰❽✱❼❿❾ ➀➂➁❜➃❥➄✹➅✫➆✮➅✫➇✹➈✖➉➋➊✒➃➍➌ ➎✮➏✷➐❂➑✚➒❅➓☛➔ →❖➣❙↔➙↕❊➛ ➜❲➝✒➞✹➟❨➠❬➡✘➢➥➤☛➝✲➠❖➡❊➟➧➦➨➠➫➩✵➭ ➯▼➲➛①↔ ➳✖➵➯ ➜❲➝✲➞✱➟▼➠❖➡➸➢ ➤☛➝✒➠❬➡❶➟➺➦❸➠⑧➩✵➭⑧➻ ➼➾➽➜❲➝✒➞ ➻ ➟▼➠❖➡➸➢ ➤☛➝✒➠❬➡➪➚➪➟➶➦❸➠⑧➩✵➭➻ ➝✒➭⑧➻➘➹➴➭✄➡➷➟ ➬▼➮ ➜✮➝✲➞ ➼ ➞❲➻✖➟❨➠❬➡➱➢ ✃➙➟❐➦❸➠➫➩✵➭⑧➻ ➝➳➣✰❒❙➣❙↔✖➛❥❮☛❰▼➣➯ÐÏ✖➡❶Ñ ❻✮❼✰❽✱❼❙Ò Ó✵Ô❖➇✹Ô✷➁❜➈➙➉ÖÕ×➅✫Ø✹➇✹Ù ➎✮➏✷➐❂➑✚➒❅➓✷Ú Û➮❰Ü❮❪↔Ï⑧Ý➻ ➢Þ➞➻➷ß ➠➻ ➩✕➭➻ ➟ ➠➻ ➩à➭➻ ➜❲➝✒➞ ➼ ÝÜ➻➙➟▼➞ ➼ Ý❛➻✷➡ á â❊ã ä åæåæå ç❣è✮é❲ê☛åæåæå ë ➢Þ➜➷➝▼➝✒➞ ➼ ➞❲➻✷➡ ➼ ➠☛➻✖➟✼➝✒➞ ➼ ➞❯➻❣➡ ➼ ➠☛➻✷➡ ➢✩➜❲➝✒➞ ➼ ➞❯➻✖➟▼➞ ➼ ➞❯➻☛➡ á â❶ã ä åæåæå ì❶åæåæå ë ➼îí ➜❲➝✲➞ ➼ ➞❯➻➙➟❨➠☛➻✷➡ á â❊ã ä ï❶ð❖ñ❶ò✲ó♦ô✼ñ❶õ❶ñ❶ö❇÷➍øæù ó❂ú ß ➜❲➝✒➠☛➻➙➟❨➠☛➻❖➡ á â❶ã ä û ï❛ù ü✤ý ê❬þÿï ￾ ✁✂✁✄✁ ☎✆✁✄✁✂✁ ➢ ➣✰↔✞✝ éê✠✟☛✡✏ê ✁✂✁✄✁ ➞ ➼ ÝÜ➻ ✁✂✁✂✁✌☞ ➎✮➏✷➐❂➑✚➒❅➓☛✍ ✎✑✏✓✒✕✔✗✖✙✘✗✚✜✛ ➛✣✢❣➛①↔✵➣✤✝ Ï➮➵✦✥✏✌✧✣✒ ➞✩★ Ï➮➵ ↕➮➵❒✫✪à↔➮➯✭✬↔✮✪✵❮ ÝÜ➻ ➣❙↔ ➭➂➻ ✯➮❰●➛✠❮☛↕①↕➵❰➍❮➯ ➛ ➯▼➲❮❪↔ ➞➻ ✰✏✓✱✳✲✞✧✴✧✵✏✶✧✵✖✸✷✠✹✺✏✶✔✗✖✑✻ Ñ ✼✔✽✒✕✧✺✚✣✧✾✧✿✱❀✲❂❁✗✱❃✱✳✲✞✧❅❄❆✏✰ ✱❇✧❈✧✵❉✂✧✵✻✺✧✵✏✮✱❆❊✶✖✙✔●❋❍✧✾✘✠■✆✖✙✧❈✘✜✔❏✧✵✚✿❁❑✚✺✒✕✧✣❉❀❉▼▲➣➯▼➲➮➵➯ ✥✏✌✔✠✒◆❉✂✧❍✘❏✷✜✧❖✔◗P ✱✳✲✞✧❘✧❍❙❚❁✜❋✵✱✩✚✣✔✗❉✤■✆✱✰ ✔✠✏✦❁✗✚❃✹☛✔✗■❖❋✾❁✗✏✦✘☛✔❯▲➣➯▼➲ ✥✏✶✔✗✒❱❉✄✧✾✘✣✷❚✧❲✔❳P❃✱❀✲❂✧❘✧✙❙❨❁✜❋✵✱❩✚✣✔✗❉✤■✆✱✰ ✔✠✏❖❬❭✚✵✔❪❉✄✔✠✏❚✷ ❁❑✚❘✒✕✧❘✚❫❊❂✧✾❁✥ ✔❳P❴✱✳✲✞✧✴✧✵✏✶✧✵✖✸✷✠✹❖❵❳✔✗✖❜❛➜❆❝❇❞ ✏✌✔✗✖✑✻✺❡✺❢✶✲✞✧❣❄❆✏✰✱❤✧✐✧✣❉✄✧✣✻❥✧✣✏❂✱❣✻❥✧✣✱✳✲✞✔●✘❪✲✞❁✗✚❘✱❫✖✙❁✠✏✞✚✑❦ P✵✔✗✖✑✻✺✧✾✘✴✱❀✲❂✧❆❊✌✖✙✔✜❧✵❉✄✧✣✻♠✔◗P❅✘✰✚✣❋✣✖✙✧✣✱✰✄♥❁✗✱✰ ✔✗✏✓✔◗P❯❁❘♦q♣❴r ✰✏✮✱❇✔❥❁s❊✌✖❍✔☛❧✣❉✄✧✵✻t✔◗P❯❁✾❊☛❊✶✖✙✔✣❙✰✻✺❁✗✱✰ ✔✗✏✌❡ ➎✮➏✷➐❂➑✚➒❅➓☛✉ ✈✧✾✔✗✻✺✧✵✱✸✖✑✹ ￾ ➞❯➻➨➢①✇❯②➻ ➞ ✛ ➯▼➲➛❘③✖❰➮✠④ ➛✼↕➯ ➣➮↔ ➮✝ ➝ ↕①❒➮✷➬➛ ➬➯ ③ ➮➣❙↔➯❛➯➮➡ ➞ ➮↔ ➭➻ ➣❙↔ ➯●➲➛ ➜ ↔ ➮❰✯Ñ ➎✮➏✷➐❂➑✚➒⑥⑤✆⑦ ⑧