2.2“ Projection” 2.2.1Pl i(a) RR/FE Approximatic fore precisely, what we mean is that uhi is the minimizer of jisi Wj pi) that is, arg min (Z=I wj pi The finite element(FE)approximation is, for this simple problem, a classical Rayleigh-Ritz(RR) approach wnith a particular choice of space and basis h) inimizer over X) er of XI finding the minimizer(uh)ofJ over all functions in Xh. The choice of basis will thus not affect the minimizer(or minimum), though it will affect the particula coefficients. We later prove that JLx, is a paraboloid -as indicated here and that by ertension J over X is an infinite-dimensional paraboloid. We see from this picture that as Xh grows it absorbs more of X, and wh should thus to u as we increase the number of elements; this is indeed the case. Of course J(uh)>J(u), since J(uh)is the minimum of over a subspace(Xh)of X✒✔✓✕✒ ✖✘✗✚✙✜✛✣✢✥✤✧✦✩★✫✪✠✛✑✬✮✭ ✯✱✰✲✯✱✰✍✳ ✴✶✵✕✷✜✸ ✹✩✺✼✻✄✽✿✾❁❀❃❂ ❄✱❅❇❆ ❈✘❉❋❊❍●❏■❑❉✼▲ ▼ ◆P❖ ◗ ❘✿❘✣❙❯❚✜❱✏❲✣❳❨❳❬❩☞❭❫❪❵❴❜❛❞❝❯❡✕❴❜❭❣❢ ❤❥✐❦ ❧❫♠✧♥ ❈✘❉❧♣♦✱❧ ❊✍q✩▲sr t ❅P❆ ❈✉❉❧ ❤✇✈❧ ❆❫①③②✡❆❞④⑥⑤⑧⑦✜⑤⑨④⑥⑤⑧⑩❵❅ ❶❸❷❹❦✐ ❧❫♠✧♥ ✈❧♣♦✿❧❇❺❻❽❼ ❾✚❿➁➀❫➂➄➃✩➀❫➂➆➅❇➇✲➈P➂❇➉✞➊❨➋❞➌✧➍③➎➁➏➐➌✑➂⑥➑➒➂➆➎➁➓➔➇✲➈→➏➣➍✜➎➁➏ ❈✉❉❧ ➇✲➈→➏➣➍✜➂➒➑↔➇✄➓↕➇✄➑↔➇⑧➙❬➂❇➀⑥❿✠➛➜❶ ❊☞➝✐ ❧❫♠✧♥ ✈❧➞♦✿❧ ▲ ➋ ➏➣➍✜➎➁➏➟➇✲➈➆➋✿②✡➠❯➡♣④⑥⑤⑧⑦s❶ ❊➝ ✐ ❧❫♠✧♥ ✈❧✑♦✿❧ ▲❇➢ ➤✘➍✜➂➦➥♣➓↕➇✄➏➧➂➨➂P➉⑧➂❇➑➒➂P➓↕➏→➩✍➫✁➭✧➯✚➎❯➃❃➃✘➀❫❿❇➲✥➇✄➑➒➎➁➏☞➇✍❿➁➓➳➇✲➈➆➋✧➛P❿➁➀➒➏➣➍➵➇✲➈⑥➈❣➇✄➑❞➃✘➉⑧➂✶➃✩➀❫❿✥➸P➉⑧➂❇➑↔➋s➎➺➅❇➉⑧➎❨➈➆➈❣➇✍➅➆➎➁➉ ➻✮➎✡➊➁➉⑧➂❇➇➽➼❬➍✫➾✍➻✮➇✄➏➣➙✏➩➣➻❞➻➟➯❑➎❯➃❃➃✘➀❫❿❬➎❃➅➆➍➺➌➚➇✄➏➣➍➺➎✶➃③➎➁➀❣➏☞➇✍➅P➪✫➉⑧➎✡➀→➅❯➍③❿➁➇✍➅➆➂➒❿❍➛➄➈☞➃③➎❃➅➆➂⑥➎➁➓✘➶✏➸➆➎❨➈❣➇✲➈ ➢ ✹✩✺✼✻✄✽✿✾❁❀❃➹ ➘❅❇➴✥④⑥❅P❆❫➠❯⑤✲➷➄➬✑⑤⑨➷P❆❫➮✜➠❯❅❃➱ ✃➇✄➓✘➅➆➂❐➎➁➓③➊❒➑➒➂P➑➒➸➆➂P➀❏❿❍➛ ■❉ ➅➆➎✡➓✇➸➆➂❮➀❫➂❰➃✩➀❯➂❣➈❇➂❇➓③➏❰➂❯➶➔➎➁➈❏❿➁➪✫➀✏➈❣➪✫➑Ï❿✡Ð❨➂❇➀❮➏✄➍③➂ ♦❧ ➋s➌✑➂✚➎➁➀❫➂ ➥♣➓✘➶➁➇✄➓➵➼s➏✄➍③➂✔➑✆➇✄➓③➇✄➑✆➇⑧➙❵➂❇➀ ❊➣❈ ❉ ▲ ❿✠➛➞❶➔❿➁Ð➁➂❇➀➐➎✡➉✄➉P➛➆➪✫➓✩➅❇➏✕➇✍❿✡➓✜➈➟➇✄➓ ■❉ ➢ ➤✘➍③➂✮➅➆➍✜❿➁➇✍➅➆➂➦❿❍➛➐➸➆➎❨➈❣➇✲➈➟➌➚➇✄➉✄➉ ➏➣➍➵➪➵➈➦➓✘❿➁➏✔➎❍Ñ✮➂➆➅❇➏✑➏➣➍✜➂✶➑✆➇✄➓③➇✄➑✆➇⑧➙❬➂P➀⑥➩❍❿➁➀✶➑✆➇✄➓③➇✄➑✆➪✫➑➦➯❬➋➚➏➣➍✜❿✡➪❃➼❬➍❮➇✄➏➞➌➚➇✄➉✄➉✧➎❍Ñ➦➂❯➅❇➏✑➏➣➍✜➂✔➃✜➎✡➀❣➏✕➇✍➅❇➪✫➉⑧➎✡➀ ➅➆❿❵➂✕ÒÓ➅P➇✍➂❇➓③➏✕➈ ➢ÕÔ➂✏➉⑧➎➁➏❰➂❇➀s➃✩➀❫❿✡Ð❨➂➨➏➣➍✜➎✡➏✮❶✑Ö ×✁ØÙ➇✲➈✏➎⑥➃③➎➁➀❯➎❃➸➆❿➁➉⑧❿✡➇✍➶➔ÚÛ➎❨➈➒➇✄➓✩➶✡➇✍➅❯➎✡➏➧➂➆➶❐➍✜➂❇➀❫➂❮Ú ➎➁➓✘➶❏➏➣➍✜➎➁➏➄➸P➊❐➂❫➲✥➏❰➂P➓③➈❣➇✍❿➁➓➔❶Ü❿✡Ð❨➂❇➀ ■ ➇✲➈➒➎➁➓Ý➇✄➓❬➥♣➓③➇✄➏❰➂❇➾✠➶➁➇✄➑➒➂P➓③➈➆➇✍❿✡➓✩➎✡➉✘➃✜➎✡➀❫➎❃➸➆❿✡➉⑧❿➁➇✍➶➢ÓÔ➂⑥➈P➂➆➂ ➛➆➀❫❿✡➑Þ➏➣➍➵➇✲➈♣➃✘➇✍➅P➏☞➪✫➀❫➂s➏➣➍✜➎✡➏➞➎❨➈ ■❑❉ ➼❃➀❫❿➁➌✁➈➄➇✄➏➞➎❃➸P➈❇❿➁➀❇➸P➈➄➑➒❿➁➀❫➂↔❿❍➛ ■ ➋♣➎➁➓✘➶ ❈✘❉ ➈❫➍③❿➁➪✫➉⑧➶⑥➏➣➍➵➪➵➈➐➼✥❿ ➏❰❿ ❈ ➎❨➈➜➌✑➂↔➇✄➓✩➅❇➀❫➂➆➎❨➈❇➂↔➏➣➍✜➂→➓③➪✫➑➒➸❯➂❇➀→❿❍➛✆➂P➉⑨➂P➑➒➂P➓↕➏✍➈➆ß✑➏➣➍➵➇✲➈➜➇✲➈➜➇✄➓✘➶❃➂➆➂➆➶➨➏✄➍③➂↔➅➆➎❨➈❇➂ ➢➒à➛→➅➆❿➁➪✫➀➆➈❇➂P➋ ❶ ❊➣❈✉❉✥▲➟á ❶ ❊➣❈✉▲ ➋➚➈❣➇✄➓✩➅➆➂➜❶ ❊➣❈✉❉✥▲ ➇✲➈s➏➣➍✜➂→➑✆➇✄➓③➇✄➑✆➪✫➑â❿❍➛✶❶❸❿✡Ð❨➂❇➀→➎⑥➈❣➪③➸P➈☞➃✜➎✥➅❯➂ ❊✍■➒❉✼▲ ❿✠➛ ■❁➢ ã