Miracle a(Iifu, v)=a(u, v),vE Xh NO:a(u,v)=()→a(Ihu,u)=(u),Vv∈Xn Only in the energy inner product can we compute IIn u without knowing u. N3 Note 3 Generality of abstract result We note that our bound that is, that uh= Ihu, is in fact true for any Spd bilinear form a, and any boundary conditions, and any finite element space Xh, and any space dimension For any particular SPD problem(that is, any linear problem for which the bilinear form a in the weak formulation is SPD), the only thing that changes is the definition of the norm; for our particular problem Iell= lehI(Q), though in general that will not be the case Obviously, however, we are not yet quite done; we must understand how llu-wnll depends on h, the smoothness of u, and the parameters of the problem For that we need to introduce the particulars of our finite element approximation 1. 4. 4 Particular bound SLIDE 21 We knot u-hlr1g)≤= Julho(g,) u-D1m()≤|lgea,)[3 e would, of course, prefer to directly use the projection uh for wh, rather tha the interpolant. However, the latter is much easier to work with, and will, in general, yield the correct h dependence. In fact, for our particular pro blem, ah=Lh u(see Exercise 3), but this is a bit of a coincidence.⑨✦⑩❀❶✙❷☛❸✣❹✄❺✓❻❨❼❆❽✌❾❫❿❅➀➁☛➂ ➃➄✑➅➆ ➇✗➈ ➉✙➊❨➋❆➌ ❽✶❾✳➂ ➉◗➊✆➋✑➉◆➍❲➊✿➎➐➏➁➐➑ ➒✞➓✞➔s→❣➣❘↔✆↕✺➙✞↕☛➔✭➛❚➙❂↕✗→ ➂❜➜✣➜❏➜ ➝❯➞❪❼✮❽✌❾➟➂ ➉✙➊❨➋❆➌ ➠ ❾➊✆➋ ➃❑➄✑➅●➆ ➡❇➢✑➤❥➥❤➦✑➧✾➨➫➩❑➧❍➭✸➥ ➯ ❽✶❾➟❿➀➁ ➂ ➃➄✵➅➆ ➇ ➈ ➉◗➊✆➋✕➌➲➠ ❾➊✆➋✑➉◆➍✐➊➳➎❖➏➁➐➵ ➞➙✞➸✂➺➳➻✂➙❈➔❍➼✞➣❘➣✣➙❂➣✣➽❍➾☛➺➳➻✂➙✞➙✞➣❏➽✭➚✞➽✙↕✆↔✆➓✮➪✑➔s➪✣➶☛➙➐→◆➣ ➪✵↕✜➹✴➚❂➓✆➔✙➣ ❿➁ ➂ →✭➻✄➔✙➼✞↕✜➓✆➔s➛❨➙✞↕✗→✭➻✄➙❂➾ ➂✩➜ ➝❅➘ ➴⑥➷➮➬✑➱✦✃ ❐❈➱✠❒❆➱✠❮✗❰✆Ï➟Ð✙➬➟ÑÒ➷✞Ós❰➮Ô✌Õ✜➬✙❮❑❰✶Ö❚➬❘❮✜➱✠Õ❚×✮Ï◗➬ Ø➣❘➙✞↕✠➔❍➣❴➔✙➼❂➶✠➔s↕☛➓✞➽❱➒✌↕☛➓✞➙✮↔ Ù✄Ù✂Ù Ú❨Ù✂Ù✂Ù ➌ ➻✄➙✆Û Ü✶➈✠Ý☛Þ❩➈ Ù✂Ù✄Ù ➂➳ß⑥à➁ Ù✂Ù✄Ù ➉ ➔✙➼✮➶✗➔✐➻✫á✣âã➔✙➼✮➶✗➔ ➂ ➁ ➌ ❿➀➁ ➂ â✩➻✂á✐➻✄➙⑥Û➟➶☛➪✵➔❲➔✙➽❍➓✞➣✺Û✳↕✜➽✐➶✠➙❨➺åä✆æ✕çè➒❂➻✄➸✂➻✄➙❂➣❏➶✠➽❲Û✳↕☛➽❍➹ ❽ â❩➶✠➙❂↔å➶✠➙❨➺ ➒✌↕☛➓✞➙❂↔❂➶✠➽❍➺❯➪✣↕☛➙❂↔✆➻✄➔✙➻✂↕☛➙✮á✣â✠➶☛➙❂↔❲➶✠➙❨➺❅é❂➙✞➻✄➔✙➣◆➣✣➸✂➣✣➹❥➣✣➙❚➔❩á◗➚❂➶✜➪✵➣ ➏ ➁ â✠➶✠➙❂↔✐➶✠➙❨➺❯á✙➚❂➶✜➪✵➣✕↔✆➻✂➹❥➣✣➙❂á✙➻✂↕☛➙ ➜ ê↕☛➽❴➶✠➙❨➺❈➚✮➶✠➽✙➔✙➻✫➪✵➓✞➸✫➶✠➽❴ä✆æ✕çë➚✞➽❍↕☛➒✞➸✂➣✣➹ ❾ ➔❍➼❂➶✗➔❴➻✫á✣â➮➶✠➙❨➺❖➸✂➻✄➙✞➣●➶✠➽❅➚✞➽✙↕✜➒✞➸✂➣✣➹ìÛ✳↕☛➽❯→✭➼✞➻✫➪✾➼✦➔✙➼✞➣ ➒✞➻✂➸✄➻✂➙✞➣●➶✠➽◆Û✳↕☛➽❍➹ ❽ ➻✄➙❈➔✙➼❂➣❯→◆➣❏➶☛➛✴Û✳↕☛➽❍➹✐➓✞➸✫➶✗➔❍➻✄↕✜➙✿➻✫á❱ä✞æ✕ç➋ â❨➔✙➼❂➣❯↕✜➙✞➸✂➺❥➔✙➼✞➻✂➙✞➾✴➔✙➼❂➶✠➔❱➪✾➼✮➶✠➙✞➾✜➣❏á❣➻✂á ➔✙➼❂➣❪↔✆➣✣é❂➙✞➻✄➔✙➻✂↕☛➙✦↕✠Û❩➔❍➼✞➣✐➙✞↕✜➽✙➹➑ Û✳↕☛➽❃↕✜➓✞➽s➚❂➶☛➽◗➔❍➻✂➪✣➓✞➸✫➶✠➽s➚✞➽❍↕☛➒❂➸✄➣❏➹ Ù✂Ù✄Ù Ú✆Ù✄Ù✂Ù ➌ Ù Ú✆Ù í❱î✑ï✄ð❂ñ â❂➔❍➼✞↕☛➓✞➾✜➼ ➻✂➙➐➾✜➣✣➙✞➣❏➽❍➶☛➸✌➔✙➼❂➶✠➔s→✭➻✄➸✂➸ò➙✞↕☛➔❱➒✌➣❴➔✙➼❂➣❲➪✣➶☛á✙➣ ➜ ➞➒❨ó❨➻✄↕✜➓❂á✙➸✄➺✜â✆➼✞↕✗→❣➣❏ó☛➣❏➽❏â❚→❣➣❘➶☛➽✙➣❴➙✞↕☛➔s➺☛➣✵➔sô❚➓✞➻✄➔✙➣❲↔✆↕✜➙✞➣ ➑ →◆➣❴➹✐➓❂á◗➔s➓✞➙❂↔✞➣✣➽✾á❳➔✾➶✠➙❂↔❈➼✞↕✗→ ➻✂➙✆Û Ü➈ Ý☛Þ➈ Ù✂Ù✄Ù ➂➳ß⑥à➁ Ù✂Ù✂Ù ↔✆➣❏➚✮➣❏➙❂↔✞á❪↕☛➙öõ▼âã➔❍➼✞➣✿á✙➹❥↕❚↕☛➔✙➼✞➙❂➣❏á❍á❲↕✠Û ➂ â✩➶☛➙❂↔å➔❍➼✞➣➳➚❂➶☛➽❍➶☛➹✴➣✣➔✙➣❏➽❍á❘↕☛Û❱➔✙➼✞➣❈➚✞➽✙↕✜➒✞➸✂➣✣➹➜✦ê↕☛➽ ➔✙➼✮➶✗➔❲→◆➣❪➙✞➣❏➣❏↔÷➔✙↕❖➻✂➙❚➔✙➽❍↕❨↔✞➓❂➪✵➣✴➔✙➼❂➣✺➚❂➶✠➽✙➔✙➻✫➪✵➓✞➸✫➶✠➽✾á❅↕✠Û❣↕✜➓✞➽❴é❂➙✞➻✄➔✙➣✺➣❏➸✄➣❏➹❥➣✣➙❚➔❲➶✠➚✞➚✞➽❍↕❑ø✆➻✄➹✺➶✠➔✙➻✂↕☛➙ á✙➚❂➶☛➪✣➣ ➜ ù✌ú✄ûãú✄û ü❅ý✆þ❑ÿ✁￾✄✂✆☎✞✝➟ý✆þ✠✟☛✡☞☎✍✌✞✎ ✏☞✑✓✒✕✔✗✖✙✘✛✚ Ø➣❘➛❚➙❂↕✗→ Ù ➂✿ß✠✜➁ ➂ Ù í✭î✾ï✄ð✮ñ✣✢ õ✤✦✥ ➂ ✥ í★✧✾ï✄ð✪✩ ✫ ➈ ñ ➵ ✬➼❨➓❂á Ù✄Ù✂Ù Ú✆Ù✂Ù✄Ù ➌ ➻✂➙✆Û Ü➈ Ý☛Þ➈ Ù✂Ù✂Ù ➂➳ß⑥à➁ Ù✄Ù✂Ù✭✢ Ù✄Ù✂Ù ➂✿ß✠✜➁ ➂ Ù✂Ù✂Ù ➌ Ù ➂➳ß✮✜➁ ➂ Ù í✭î✑ï✄ð❂ñ✯✢ ➁✰ ✥ ➂ ✥ í✯✧✵ï✄ð✛✩ ✫ ➈ ñ ✱➘ ➝✳✲ ❾ ➶☛á❍á✙➓✞➹❥➻✄➙✞➾ ✥ ➂ ✥ í✯✧✾ï✂ð✛✩ ✫ ➈ ñ é❂➙✞➻✄➔✙➣ ➋ ➜ ✴✦❺✶✵✣✷✹✸✆❹✻✺✹✼✽✷✿✾❅❸❀✷✹✸✆❶❀❁✣❺❂✼❄❃✶❶✙❺❅✾✵❺✣❶✶❆❇✷❈✺✗⑩❀❶✙❺✾❸❉❆❫❹❋❊☛✸✭❁✣❺●❆✕❍❂❺✣❃✌❶■✷❑❏✣❺✾❸❂❆✸⑩❅✷✹▲✺➂ ➁ ✾❂✷✗❶✕à➁ ✼ã❶❍❷✹❆▼❍✞❺✣❶✳❆▼❍✞❷✹▲ ❆▼❍✞❺✿⑩✕▲❄❆❤❺✵❶◆❃❄✷✗❹✄❷❖▲❄❆◗P❙❘☛✷✹✵✕❺❉❚❑❺✣❶❀✼●❆▼❍✞❺✿❹✄❷❖❆✄❆❤❺✵❶❥⑩❯❁❲❱❈✸❂❸❳❍ö❺❍❷✹❁✑⑩➟❺✵❶❲❆◗✷❨✵✣✷✠❶■❩❨✵q⑩✕❆▼❍✓✼❘❷✹▲☞✺❬✵q⑩❀❹❀❹❭✼❅⑩✕▲ ❪ ❺❂▲✶❺✵❶✙❷✠❹❭✼❫❊✗⑩➟❺✵❹✻✺❴❆▼❍✞❺÷❸❳✷✠❶✑❶✙❺✾❸❂❆ õ ✺✜❺❇❃❂❺❉▲☞✺☛❺❉▲✌❸✾❺❵P❜❛❝▲❞✾✵❷✜❸❂❆✄✼❡✾❂✷✗❶❞✷❖✸✆❶❈❃✞❷✠❶❝❆❫⑩➟❸❉✸✆❹✄❷✗❶❲❃✌❶■✷❣❢✵❹✄❺❉❱☛✼ ➂ ➁ ➌ ✜➁ ➂✐❤◆❁✣❺✾❺●❥✽❦❨❺✣❶✙❸✣⑩❯❁✵❺❈❧❂♠❵✼✯❢❉✸♥❆✣❆▼❍❨⑩❯❁❘⑩❯❁❪❷❙❢✵⑩✕❆♦✷❑✾❲❷✦♣❇❸❀✷✗⑩✕▲✶❸✵⑩❅✺✜❺❂▲✶❸✾❺✁P❑q r