Discretization of the poisson Problem in r Formulation april 2, 2003
✂✁☎✄✝✆✟✞✡✠☞☛✌✁☎✍✏✎✏☛✌✁✒✑✔✓✕✑✗✖✘☛✝✙✚✠✜✛✢✑✔✁☎✄✣✄✌✑✔✓ ✛✤✞✡✑✔✥✧✦✒✠✩★ ✁☎✓✫✪✭✬✯✮✱✰✳✲✴✑✵✞✣★✷✶✧✦✸✎✏☛✝✁✒✑✔✓ ✹✻✺✽✼✸✾❀✿✏❁✝❂✱❁❄❃❅❃❄❆
1 Model problems 1.1 Dirichlet 1.1.1 Strong Form SLIDE ain:9=(0,1) Find a h that in Q (1) 1.1.2 Minimization statement Define X≡H(9) Find he w2 d-f This follows from the previous lecture, noting that dA is now dz, and vw 1.1.3 Weak Formulation sLIdE 3 Find u∈ X such that 6J(u)=0 u∈X f vdx Vu∈X Again, this follows f er lecture with Vu. Vu now given by ur Ux
❇ ❈❉❋❊❍●❏■▲❑◆▼☞❉P❖❍■✒●❏◗❙❘ ❚☞❯❱❚ ❲❨❳❱❩❬❳❱❭❫❪❏❴❛❵✡❜ ❝❄❞❡❝❄❞❡❝ ❢✡❣✐❤❦❥❄❧✝♠♦♥✝❥❬❤❦♣ q❄rts✈✉✣✇②① ③⑤④⑦⑥⑨⑧❶⑩❸❷✡❹ ❺✢❻❽❼❡❾➀❿➂➁✸➃➅➄ ➆✌⑩➇❷❫➈➊➉➌➋❱➍❬➎➐➏✻➑➒➏❫⑧❦➑ ➓ ➉❅➔✸➔→❻ ➣ ⑩❸❷↔❺ ➉✽❼❡❾↕➃➙❻ ➉✌❼❱➁✸➃➙❻ ❾ ❿ ➛④⑦➜➞➝↕⑩➇➟↕➠✐❷➡➣✌➄ ❝❄❞❡❝❄❞➤➢ ➥◆➦❡❧✌➦❡♣✘➦❀➧⑦➨➩❣✸➦❡❥❄❧✢❢✡❣➂➨➩❣✸➫t♣▲➫t❧❅❣ q❄rts✈✉✣✇➯➭ ③⑤➠✒➲❬❷➀➠ ➳➸➵➻➺➽➼➾ ❼❀❺➞➃➅➄ ➆✌⑩➇❷❫➈ ➉✻❻✤⑧❶➜➚➝✘⑥➪⑩➇❷ ➶✌➹⑦➘➷➴ ❼❡➬⑤➃ ➮➏➀➠➂➜➒➠ ➴ ❼➱➬⑤➃✟❻ ➁✃➻❐ ➼ ➾ ➬❏❒➔❏❮⑦❰ ➓ ❐ ➼ ➾ ➣❄➬ ❮⑦❰♦Ï Ð❅ÑÓÒ➤Ô✗Õ✒Ö❦×✈×➇Ö❶Ø✽Ô➅Õ➐Ù➒Ö❶Ú✜Û✈Ñ❬Ü✔Ý❅Ù➒Ü✐Þ➂Ò❡Ö❶ßÓÔ✻×➇Ü➐à✐Û❀ß➩Ù➒Ü✒á✗â❄Ö❦ÛãÒ✈âtä➽Û➱Ñ➀å❶Û ❮tæ Ò➤Ô✴â❄Ö❶Ø ❮↕❰ áçå❦â❅è✚éç➬êÒ➤Ô â❄Ö❶Ø♦➬➞➔❫ë ❝❄❞❡❝❄❞❸ì íî➫t➨➀ï②♥✣❥❫❤❦♣➽ð✌ñ❡➨➀❣➂➦❡❥❄❧ q❄rts✈✉✣✇➯ò ➆✌⑩➇❷❫➈➊➉❍ó✚➳ ➋❱➍❫➎➐➏✻➑➚➏❬⑧❦➑ ô ➴↕õ ❼➱➉ö➃÷❻✤❾✻❿ øçù❋ó✧➳ ú ❐ ➼ ➾ ➉❄➔ûù❶➔ ❮↕❰ ❻ ❐ ➼ ➾ ➣✵ù ❮⑦❰ ❿ ø✵ù✴ó✧➳ Ï üä↕å❶Ò✈â➀á÷Û➱ÑÓÒ➤Ô✱Õ✒Ö❦×✈×❸Ö❦Ø✽Ô✱Õ➐Ù➒Ö❶ÚýÖ❶ß➩Ù✵Ü➚å❶Ù☎×þÒ❡Ü✐Ù✵×➇Ü➐à✒Ûãß➩Ù➒Ü✵Ø☞Ò✈Û✈Ñ✻éÿ➉✁➂éÿù â❄Ö❦Ø✘ä⑦Ò✈Þ Ü✐â✄✂✆☎✔➉➔ ù➔ ë ➁
1.1. 4 Notation a(,U) U Minim u=arg min 5 a(u, w)-e() u∈X:a(u,u)=(u),vv∈X 1.1.5 Generalization For any(u)∈H-(9) fndu∈H(9) such that u=arg min 2a(u, w)-e(a),or a(u,v)=C(),VU∈H(2); for example, e(o)=0zo, D)=v(ao) is admissible As indicated earlier, the delta distribution is not admis sible if& Cir be motivated by considering the green,'s function 1. 1. 6 Regularity SLIDE 6 Ife∈H-(9) lal (2)≤C If∈2(9),(v)=/fndr ll()≤Co‖ flea(S Recall ulPH( te 1 If e(o)=So fudr, with fE L2(Q2), we immediately obtain from I|lH1(2)< IellH-1(s)that lalH(Q)S ClIfz (o), since the H- norm is al ways bounded by the L2 norm(there is "more"in the denominator). But from the strong form uz= f we can see that uH2()≤‖fz2()· It thus follows that Co in the
✝✟✞✠✝✟✞☛✡ ☞✍✌✏✎✒✑✓✎✒✔✠✌✟✕ ✖✟✗✙✘✛✚✢✜✤✣ ✥✧✦✆★✪✩✓✦ ✫✭✬✯✮✱✰✳✲✵✴✷✶ ✸✺✹ ✻ ✮✽✼✾✲✿✼❁❀❃❂ ❄ ✬✯✲✵✴❅✶ ✸✹ ✻❇❆ ✲❈❀❃❂❊❉ ❋✍●❍✩✓●❍■❏●❍❑✒▲✿▼◆●❍❖P✩❘◗ ❙ ✶❚▲✿❯❲❱❊■❳●☛✩ ❨❬❩P❭❫❪❴ ✫✟✬✠✮❵✰✳✮❛✴❝❜ ❄ ✬✯✮✧✴ ❞❡✦✒▲✿❢✭◗ ❙✍❣✐❤ ◗✾✫✭✬❙ ✰✳✲✵✴❥✶ ❄ ✬✠✲❦✴❛✰♠❧❵✲ ❣♥❤ ✝✟✞✠✝✟✞♣♦ q♥r❦✕❬r✙st✑✪✉✠✔✠✈❃✑✵✎✒✔✇✌✏✕ ✖✟✗✙✘✛✚✢✜❡① ②✓❖❃❯❁▲✿✩❦③ ❄ ✬✠✲❦✴ ❣✍④⑥⑤ ✹ ✬✇⑦✽✴✆⑧ ★✪✩✪⑨ ❙⑩❣✐④✹✻ ✬✇⑦✽✴✾❶◆❷✪❸❺❹♥▼◆❹✪▲✿▼ ❙ ✶❚▲✿❯❲❱ ■❏●❍✩ ❨❬❩P❻✽❼❽❿❾☛➀✏➁ ➂✹ ✫✭✬✯✮❵✰◆✮✧✴❝❜ ❄ ✬✯✮❛✴➄➃➅❖P❯ ✫✭✬❙ ✰◆✲❦✴➆✶ ❄ ✬✯✲✵✴➇✰ ❧✱✲ ❣✍④✹✻ ✬➈⑦✽✴❛➃ ➉❖P❯✽✦✆➊✓▲P■❏➋✪➌☛✦❃⑧ ❄ ✬✯✲✵✴❥✶➎➍✇➏➐✼ ❽ ✰✳✲✵➑➆✶➒✲✟✬✠❂✻ ✴♠●❍❶❁▲❃⑨✵■❳●❍❶❲❶◆●☛➓✓➌❍✦P➔ →❛➣↕↔✛➙✟➛❿↔✠➜❺➝✿➞➠➟❺➛✍➟❺➝❿➡➇➢➤↔✠➟➐➡❺➥✽➞✛➦✪➟➧➛P➟➐➢➤➞➨➝✍➛❿↔♣➣➇➞✇➡➇↔✠➩➐➫✵➞➈↔✠➭❿➙✄↔♣➣✱➙✭➭❿➞❈➝P➛✿➯↕↔♣➣❺➣➇↔✠➩➐➢☛➟➧↔➲ ⑦➵➳➺➸➻➂ ➥❁➝t➣❏➜❺➝❿➙ ➩❺➟❵➯➧➭❿➞➈↔✛➼❿➝❿➞➨➟❲➛➽➩✆➾♥➜❲➭✿➙✓➣➇↔✠➛P➟➐➡➇↔✛➙✙➚➽➞✯➦✓➟➶➪✾➡◆➟❺➟✆➙➘➹➣❝➲❺➫✵➙✟➜➐➞✇↔✠➭✿➙✟➴ ✝✟✞✠✝✟✞❍➷ ➬➽r❦➮✏➱❬✉✠✑✵s❿✔✠✎❲✃ ✖✟✗✙✘✛✚✢✜❡❐ ➸➉❬❄ ❣✐④⑤ ✹ ✬✇⑦✽✴✆⑧ ❒ ❙ ❒ ❻❼ ❾☛➀✏➁❰❮ÐÏ ❒ ❄ ❒ ❻❈Ñ ❼ ❾❍➀✪➁ ❉ ➸➉❬❄ ❣✐Ò➂ ✬✇⑦✽✴➇⑧ ❄ ✬✠✲❦✴➆✶Ó✸✹ ✻Ô❆ ✲✧❀❃❂ ❒ ❙ ❒ ❻♠Õ ❾☛➀✏➁♠❮✺Ï✻ ❒ ❆ ❒➐Ö Õ ❾☛➀✏➁ ❉ × ❪ Ø➟❺➜❺➝❿➢✛➢ ❒ ✲ ❒ ➂ ❻♠Õ ❾☛➀✏➁ ✶ÚÙ ✲✭Ù ➂ ❻❰Õ ❾☛➀✏➁➘Û ❒ ✲ ❒ ➂ ❻❼ ❾☛➀✪➁ ✶❚Ü✻ ✹ ✲ ➂✼Ý✼ Û ✲ ➂✼ Û ✲ ➂ ❀❃❂ ➴ Þ❡ß➘à➇á✤â ã⑩á❿äæå✏ç✇è✵étê◆à✯ë ➸➉✭❄ ✬✯✲✵✴❥✶ Ü ✻ ✹ ❆ ✲æ❀P❂ì⑧❃í✽●☛▼◆❹ ❆ ❣♥Ò➂ ✬➈⑦✽✴➇⑧✙í❰✦✽●❍■❳■❏✦Ý⑨✵●♣▲❿▼◆✦✒➌☛③❏❖P➓✓▼❲▲✿●❍✩ ➉❯◆❖❃■ ❒ ❙ ❒ ❻❼ ❾☛➀✪➁♠î Ï ❒ ❄ ❒ ❻Ñ ❼ ❾☛➀✪➁ ▼◆❹✪▲✿▼ ❒ ❙ ❒ ❻❼ ❾☛➀✪➁❰❮ïÏ ❒ ❆ ❒✆Ö Õ ❾☛➀✪➁ ⑧❃❶✳●❍✩✪❸✆✦❥▼◆❹✓✦ ④⑤ ✹ ✩✪❖P❯❲■Ó●♣❶æ▲✿➌❍í♠▲t③✵❶✢➓✏❖❃❷✓✩✪⑨✵✦Ý⑨ ➓❦③ð▼◆❹✪✦ Ò➂ ✩✓❖❃❯◆■ñ✬✯▼◆❹✓✦✒❯◆✦✽●♣❶ðò✳■❳❖❃❯◆✦Ýó❁●❍✩↕▼❲❹✓✦❁⑨✵✦✒✩✓❖P■❳●❍✩✪▲❿▼❲❖P❯➇✴➇➔ìô♠❷✵▼ ➉❯❲❖P■➎▼◆❹✓✦❁❶✳▼◆❯❲❖P✩✓❱ ➉❖❃❯◆■ ❜ ❙✼✒✼ ✶ ❆ í♠✦❳❸➐▲✿✩õ❶✳✦✒✦❵▼◆❹✏▲❿▼✁Ù ❙ Ù ❻♠Õ ❾☛➀✪➁✧❮ ❒ ❆ ❒✆Ö Õ ❾❍➀✪➁ ➔➄➸➠▼❛▼◆❹❦❷✪❶ ➉❖P➌❍➌☛❖❿í❁❶❈▼◆❹✪▲✿▼ Ï✻ ●☛✩✤▼◆❹✓✦ ❴
bove slide is(1+C2)1/2. This can also be shown by explicit construction of u (see Lecture 2 from earlier in the course) The fact that u is regular when f is regular(and in IR2, the domain is suitably regular) has very important implications as regards the convergence rate of the of a priori and a poste st es 1.2.1 Strong form SLIDE 7 2=(0,1) Find a such tha u(0) (1) for given∫, 1.2.2 Minimization statement SLIDE 8 {U∈H1(9)|v(0)=0} arg f wdr-g w(1 This follows from the previous lecture, noting that rn g v da is here ju 9v(1). We can also show this explicitly by integrating by parts to find dJ,(u) ∫0v{-x-∫}dr+v(1){uz(1)-g}=0,V∈X 1.2.3 Weak Statement Find u∈ X such that yu∈X +gv(1),VU∈X
ö✿÷✟ø❿ùPú❛û✳ü❍ý♣þ✵ú➄ý❍û➄ÿ✁✄✂✆☎✞✝✠✟☛✡☛☞✌✝✎✍✄✏✒✑✓ý♣û✒✓➐ö✎✔♥ö✿ü♣û✳ø❏÷✟ú➄û☛✑✓ø✖✕✗✔➽÷✙✘➧ú✛✚✢✜✓ü☛ý✣✓✆ý✥✤✦✓✆ø✧✔✪û✁✤✌★☛✩✪✓✫✤◆ý❍ø✬✔➶ø✎✭✯✮ ÿ✠û◆ú➐ú✱✰✢ú✲✓✳✤☛✩✴★❲ú✱✵✱✭✶★❲ø✬✷ ú✒ö✎★❲ü❍ý☛ú✲★♠ý✸✔✹✤☛✑✓ú✱✓➐ø✬✩✴★❺û◆ú✺✟✫✍ ✏✒✑✓ú✻✭✠ö✬✓✳✤✼✤☛✑✪ö✎✤✯✮↕ý❍û✽★◆ú✲✾✬✩✓ü♣ö✎★✼✕✗✑✪ú✛✔❀✿↕ý❍û✯★❲ú✛✾✬✩✪ü❍ö✬★➆ÿ✠ö✎✔✏þðý✥✔❂❁❃✦✝✬❄✺✤☛✑✓ú♠þ✵ø✬✷➧ö✿ý✸✔ðý♣ûìû☛✩✓ý✥✤❲ö✿÷✪ü✥✘ ★❲ú✛✾✬✩✪ü❍ö✬★❅✟✄✑✪öPû❰ùPú✲★☛✘❳ý✸✷❆✜✟ø✬★☛✤❲ö✬✔✙✤♠ý✸✷❇✜✓ü☛ý✣✓➐ö✎✤◆ý❍ø✬✔✪û✾ö❃û❈★❲ú✛✾✙ö✎★❺þ✓û❉✤✌✑✓ú❂✓✆ø✧✔✙ù❃ú✛★✌✾Pú✲✔❊✓✆ú✗★❺ö✖✤❲ú✧ø✬✭✼✤☛✑✓ú ❋✔✓ý✥✤◆ú➶ú➐ü❍ú✛✷❳ú✛✔✙✤✱✷❳ú✳✤✌✑✓ø✵þ❡ö✬✔✪þ●✤☛✑✓ú❍✓✆ø✬✔✏û✁✤✌★☛✩❊✓✳✤◆ý❍ø✬✔❡ø✎✭✱■❆❏▲❑✫▼❖◆✖❑✫▼ðö✎✔✪þP■❇❏✴◆✖◗❅❘❚❙✛❑✫▼❖◆✖❑✫▼❛ú✛★✌★◆ø✧★ ú✒û❯✤◆ý✸✷➧ö✖✤❲ú✒û✲✍ ❱❉❲❨❳ ❩❭❬❫❪✽❴❛❵❝❜❡❞❛❞❣❢ ❤❥✐✣❦✼✐❖❤ ❧✼♠✛♥✖♦❥♣✯q✆r✯♦❊♥✖s t❥✉✙✈①✇✽②✆③ ④ø✬✷➧ö✿ý✸✔✼⑤ ⑥⑧⑦➎ÿ❖⑨✴⑩✲✺✟❶✍ ❷ý✥✔✏þ❸✮❫û❯✩❊✓❅✑❍✤☛✑✏ö✖✤ ❹ ✮❥❺✺❺ ⑦ ✿ ý✥✔❻⑥❼⑩ ✮æÿ❖⑨✙✟❽⑦ ⑨❾⑩ ✮▲❺✪ÿ✁✠✟❽⑦ ❿➀⑩ ✭✯ø✬★✗✾❃ý☛ù❃ú✛✔➁✿❥⑩❯❿➂✍ ❤❥✐✣❦✼✐✣❦ ➃➅➄❖♣➆➄❖s➇➄❨➈✬➉✢♠✺➄❖♦❥♣⑧❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍➌ ④ú❋✔✓ú ➎➐➏❼➑✧➒➔➓❾→●✡❿ÿ➣⑥✗✟↕↔➙➒✟ÿ❨⑨✧✟✄⑦❫⑨❈➛❾✍ ❷ý✥✔✏þ ✮❍⑦❚ö✎★✌✾➇✷❳ý✥✔ ➜➆➝✬➞❝➟ ÿ❖➠❛✟ ✕✗✑✓ú✲★◆ú ➟ ÿ❖➠❛✟✻⑦ ✵ ➡✡ ➢ ➠❺✦➤✧➥ ✝ ❹ ➡✡ ➢ ✿❀➠ ➤✬➥ ❹ ❿❛➠✱ÿ✁✠✟❡➦ ➧▲➨▼✣◗➔➩✳◆✖➫①➫✸◆✖➭❡◗❍➩❅❑☛◆✎➯➲❘➨ ❙➳❏❥❑☛❙✛➵✺▼❖◆✖➸➙◗✆➫✥❙❅➺✳❘➣➸✢❑☛❙✳➻❍➼▲◆✖❘➣▼①➼✙➽❫❘➨ ■✖❘❆➾✲➚✧➪➶❿✹➒ ➤✙➹ ▼✣◗ ➨ ❙✛❑☛❙➴➘✫➸➙◗❅❘ ❿✒➒✟ÿ❯✺✟✛➷➴➬➳❙❀➺❅■✖➼➀■✎➫➮◗✛◆❇◗➨ ◆✎➭⑧❘➨▼✣◗❂❙☛➱✳❏▲➫✃▼❖➺✳▼①❘➣➫✃❐➴❒✛❐❇▼①➼❊❘❚❙➣➽✬❑☛■✎❘❨▼①➼➙➽➔❒✳❐❛❏❊■✖❑✫❘❨◗❶❘❚◆✒❮✄➼▲❰✞Ï ➟✧Ð ÿ✶✮❭✟✄⑦ ➾ ✡ ➢ ➒●Ñ ❹ ✮▲❺✲❺ ❹ ✿✯Ò ➤✧➥ ✂➇➒✟ÿ❯✺✟✫Ñ✲✮▲❺✓ÿ✁✠✟ ❹ ❿❥Ò❛⑦P⑨✴➻▲Ó❀➒➔➓❍➎⑧➷ ❤❥✐✣❦✼✐✸Ô ÕÖ➊✙➉✴×Ø❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍Ù ❷ý✥✔✏þ❸✮➳➓❾➎ û❯✩✪✓❅✑❍✤✌✑✪ö✖✤ Ï ➟Ð ÿ✶✮❭✟✄⑦❫⑨❍⑩ Ó❀➒➔➓✹➎ Ú ➡✡ ➢ ✮❥❺✒➒✎❺ ➤✧➥ ⑦ ➡✡ ➢ ✿❀➒ ➤✧➥ ✂✆❿❣➒✭ÿ✁✺✟❶⑩ Ó✱➒➴➓✹➎Û➦ Ü
1.2.4 Notation SLIDE 1 ∫vdx+gv(1) imitation u=arg min 5a(w, w)-e(w) Weak u∈X:a(u,)=(v),v∈X Note 2 Neumann and delta distributions(Optional) We note that, in R, our Neumann condition looks exactly like a delt distribution forcing at the boundary, a l. This is fine, since we know the delta distribution is an admissible(bounded) linear functional, that is, is in the pace(Q), for this one-dimensional problem We know that in the interior a delta distribution imposes(weakly) a jump in the derivative(see Note 10 of last lecture). On the boundary, it imposes (weakly) the value of the derivative itself -since there is no "other side "to the 2 Rayleigh-Ritz Approach 2.1 Approximation 2.1.1 Mesh SLIDE 11 Note our default problem is the Dirichlet problem; we shall explicitly indicate Neumann when we wish to consider that problem (primarily e erercise =0 h =UT k=1..,K=n+1: elements b n+1: nodes Triangulations Th The above decomposition is known as a triangulation, Th, even though in IR our elements are not really triangles(though they are simplices--which are
Ý❥Þ✣ß✼Þ✥à á➳â✪ã✲ä✴ã✲å❖â❥æ ç❥è✙é①ê✽ëíì✎î ï❛ð✳ñ❊ò✴ð ó▲ô✶õ✱ö☛÷➙øúù û⑧ü ý õ✗þ✗÷✎þ✗ÿ✁ ✂ ô❖÷➙ø❽ù ûü ý ✄ ÷❶ÿ✁✆☎✞✝❣÷▲ô✠✟✠ø☛✡ ☞☛✌ ✍✏✎✸ò✑✎✓✒✔✎✓✕✗✖✙✘✚✎✓✛✬ò✢✜ ✣ ù✤✖✁✥✚✦✧✒✔✎✸ò ★✪✩✁✫ ✟ ✌ ó▲ô✶õ❀ö❯õ❛ø✭✬ ✂ ô❖õ❛ø ✮➍ð✗✖✙✯✰✜ ✣✏✱✳✲ ✜✒ó▲ô✣ ö❯÷✢ø✻ù ✂ ô❖÷➙ø❶ö✵✴✱÷ ✱✶✲ ✷✹✸✻✺✽✼✶✾ ✷✹✼✑✿❁❀❃❂❅❄✪❄✤❂❅❄❇❆✤❆❈✼✙❉❊✺❋❂●❆✻❍❏■❑✺❊▲▼❍✚◆✪✿❈✺❊❍✠✸❖❄✪■◗P❙❘☛❚❯✺❊❍✠✸❖❄❱❂❅❉❳❲ ✮➍ð↕ò✑✛✙✘✌ð◗✘❋❨❩✖❬✘❪❭❫✎✸ò❵❴❛ ü ❭❫✛❝❜✑✥ ☞ð❞❜✑✒❡✖✬ò✴ò❣❢❞✛✬ò❩❤✑✎✐✘❋✎❥✛✧ò❵❦✓✛❑✛❝✯❅❧➴ð❞♠❅✖❝❢✽✘❋❦❥♥✤❦✓✎❥✯✧ð✹✖✧❤✢ð❞❦❥✘❋✖ ❤❅✎♦❧✠✘❋✥✚✎✓♣✑❜❅✘❋✎❥✛✧ò●q❳✛✁✥❙❢r✎✸ò✑✦◗✖✙✘s✘❋❨✴ðt♣✉✛✁❜✴ò❁❤✑✖✙✥❋♥✁❭✈ ù✇✟✁①◗②③❨❩✎✓❧④✎✓❧❀ñ❊ò✴ð❝❭✭❧❊✎✸ò❩❢✛ðt⑤ðt✯➙ò✑✛❬⑤⑥✘✚❨✴ð ❤✢ð✗❦✐✘❙✖s❤❅✎♦❧✠✘❋✥✚✎✓♣✑❜❅✘❋✎❥✛✧òt✎♦❧⑦✖✬òt✖✁❤✑✒✔✎♦❧❋❧❊✎✓♣✑❦✸ð ô❳♣✉✛✁❜❊ò❩❤✢ð✗❤✪ø✭❦✓✎✥ò❊ð✗✖✙✥⑦q❳❜✴ò❩❢r✘✚✎✓✛✬ò❩✖✁❦⑧❭❑✘❋❨❩✖❬✘✵✎✓❧✗❭❖✎✓❧⑦✎✥òt✘✚❨✴ð ❧✚⑨❩✖✁❢✛ð❶⑩●❷ ü ô❹❸✗ø✽❭❅q❳✛❝✥③✘✚❨✑✎♦❧❺✛✧ò✴ðr❻❼❤❅✎✓✒❆ð✲ò❩❧✚✎❥✛✧ò❩✖✙❦✰⑨✑✥❋✛✁♣✑❦✸ð❞✒✳① ✮➍ð④✯➙ò✑✛❬⑤❽✘❋❨❩✖❬✘❾✎✸ò❿✘❋❨✴ð④✎✸ò❑✘☛ð✗✥✚✎✓✛✁✥☛✖➀❤✢ð✗❦✐✘❙✖t❤❅✎♦❧❊✘✚✥❋✎❥♣✑❜✑✘✚✎✓✛✬ò◗✎✓✒✆⑨❁✛❑❧❯ð❪❧❇ô❳⑤ð✗✖✙✯❖❦✓♥✴ø❯✖❾➁✠❜❩✒✔⑨ ✎✸ò❃✘❋❨✴ð✏❤✢ð❞✥❋✎❥➂❬✖✙✘✚✎✓➂✬ð↕ô❏❧☛ð✛ð ☞✛✙✘✌ð✞✟✗➃➄✛✙q➅❦♦✖✁❧❊✘❡❦✥ð❪❢✽✘✚❜❩✥☛ð✠ø✽①➇➆❛ò❃✘✚❨✴ð✳♣✉✛✁❜✴ò❁❤✑✖✙✥❋♥✁❭⑦✎✐✘❡✎✓✒✔⑨✉✛❝❧☛ð✗❧ ô❳⑤ð✗✖✁✯❖❦❥♥✴ø❈✘✚❨❊ð❯➂❬✖✙❦✓❜✴ð❺✛✙q✰✘✚❨❊ð➅❤✢ð❞✥❋✎✓➂▼✖✙✘✚✎✓➂✬ð❺✎❥✘❋❧☛ð❞❦❥q✉➈➉❧❊✎✸ò❩❢✛ð❺✘✚❨✴ð✗✥☛ð❺✎♦❧✄ò❩✛✏➊❊✛✙✘❋❨✴ð❞✥✭❧✚✎♦❤✢ð✗➋☛✘❋✛❾✘✚❨✴ð ➁✠❜✑✒✆⑨✢① ➌ ➍➏➎❯➐➀➑r➒➅➓❞➔④→t➣❁➍↔➓✽↕✪➙➜➛✇➝✏➝✳➞✈➟s➎❾➠❇→ ➡③➢❊➤ ➥➇➦➅➦❫➧✑➨❈➩❺➫✚➭⑥➯❈➲✑➫❊➨❇➳ ß✼Þ❖Ý❥Þ❖Ý ➵➺➸❑➻❪➼ ç❥è✙é①ê✽ëíì✴ì ➽s➾✙➚➶➪✳➾✙➹❅➘➀➴✁➪⑧➷r➬❬➹❅➮➱➚✵✃✰➘✚➾✁❐❞➮❥➪❞❒❰❮♦ÏÐ➚ÒÑ❩➪❡ÓÔ❮Ò➘✽❮❏Õ❙Ñ❖➮✓➪r➚✵✃✰➘✚➾✁❐❞➮❥➪❞❒sÖs×❇➪tÏ✚Ñ✑➬✙➮Ò➮③➪❋Ør✃✉➮➱❮❏Õ❞❮Ò➚❹➮➱Ù➄❮ÒÚ✰➴❬❮❏Õ❙➬❬➚❼➪ ➽s➪❞➹❅❒❡➬❬Ú❩Ú◗×❈Ñ✑➪❞Ú◗×❇➪s×✈❮♦Ï❊Ñ❿➚❼➾ÐÕ❙➾❬Ú✑Ï✽❮❏➴❝➪r➘Ô➚❳Ñ✑➬✙➚✢✃✰➘✚➾✁❐❞➮❥➪❞❒ÜÛ✐✃✰➘✽❮Ò❒✆➬✙➘✽❮Ò➮➱Ùt❮ÒÚ➄➚ÒÑ❩➪s➪❋Ø❖➪r➘✚Õ❞❮♦Ï❞➪✽Ï➶Ý✁Þ ❸Øù àßá â✽ã ü äâå äâå ö✇æ ù❽✟✬ö✗✡❞✡✗✡✳ö✚ç➐ù✤èÐ☎é✟✁✜Ô➪❞➮❥➪❞❒✆➪❞Ú❩➚⑧Ï ❁ê✦ö↔ë➆ù❵➃✴ö❞✡✗✡❞✡✳ö✚è❡☎é✟✁✜❶Ú✰➾✗➴❝➪✽Ï ☞❫ì ✷✹✸✻✺✽✼✏í î❈▲❪❍❹❂❅❄❈ï❱✿❁❉⑧❂✻✺✚❍❼✸❅❄❈■✆ðå ②③❨✴ð✶✖✙♣✉✛❬➂✬ð❡❤✢ð❪❢r✛✁✒✆⑨✉✛❝❧✚✎✐✘❋✎❥✛✧ò●✎✓❧s✯➙ò✑✛❬⑤✗òñ✖✁❧Ô✖ñ➚⑧➘✽❮❏➬✙Ú❑ò✁➹❅➮❥➬❬➚❹❮❏➾❬Ú✪❭✭ðå ❭❡ð✗➂✬ð✛ò✞✘❋❨✑✛✁❜❩✦✁❨●✎✸ò ❴❛ ü ✛✁❜❩✥➆ð✗❦✥ð✗✒❇ð✛ò❑✘❋❧✭✖✁✥☛ð✒ò✑✛✁✘✈✥☛ð❪✖✙❦✓❦❥♥❾✘❋✥✚✎♦✖✎ò❩✦✁❦✸ð✗❧✒ôÒ✘❋❨✑✛✁❜✑✦❝❨✔✘✚❨✴ð✗♥④✖✙✥✌ð③❧❊✎✓✒✆⑨✑❦❥✎♦❢✳ð❪❧✪➈➜⑤❺❨✑✎♦❢❙❨✆✖✙✥✌ð ó
())/0=),6()/+2g= edrhugdr)/2M),oxdx) o: Elex Enwk'< uz=x), a(z qurgdiea= uuv y= adwvopu)z(o j): ac=2 Ghwceo(udo: 2M( Hw(uy 0: w cloks rE o Chu Tnr(i b)iwgiffadu cu wa rdo h ouno pk- which wite pexy a ca)=dre halfan a)+x=gub)whwxppdpxiv z=ioy u)udre Wwv ay co)(igad )o)a)i o u v( which +wutu u)= ww axi uu) dizi dud oo rt uw a io podu o du wp/ wise i): aC=,w co)Cua)ug wih a(aua)Ca -di ) ua-io)( th wih s x W (ay hax- oud (aqua)cwo: -dix),ua-io)( i( bsfki-snaiforx i: dorio cm. I/cmay oOd t hi(, ou)gug: u, uow d c s xp ww(hale zewry( a(uywhi(+, w+u Q2hudw z)oud Wazy+ guci, wut u)=rg =dix), uea=io)( i) which Tn rle daco pC却=0ODe=4P=h(09=)h)O)1)=0g(c,, Crdiou( appdoxi x=io)() =v(O: i( a,, dOr=ug )o==io)2 hzQvta whudwTh e th i) gica=u(+=kwHwu)io) oOd zle ufn u= Q-OS“”BXvX leigh, Z LXA= Apex Fror f c N k=1 pr o xeCl WifwEojc年卟p敌rkhM1 k tiff pAn kWMItiecee ikF-linefr Mr v2 on Efcmfle cfn fl Xd∈0 a ∈Xh piecewise linear U(0)=0
ô✚õ❞ö✁÷✆õ✗ø❝ù❙ô❱ú❥ø❡ûü❫ý✙þ✙ù❋ÿ✚ú✁✙ø✑ö✄✂❥õ❪ô❱ú❥ø✆ûü✆☎✁þ✄✙ø✞✝sù❋õrù❋ÿ✟✡✠✑õ☛✝❅ÿ☞☛ú❥ø✆ûü✆✌✎✍✑✏❱û❼ø✆ö✁õ❞ø❩õ❞ÿ☞✒✂❹þ✡☛ù✚ÿ❋ú✓✁ø✑ö✡✔✕✂✓✒✖ ù✚ú✓✗✁ø✙✘✛✚❡ÿ✚õ✢✜❳õ❞ÿ❙ô③ù✣✗✆ù✟✠✑õ✥✤✦✗✄✂✧✂✓õ☛✤rù✚ú✓✗✁ø★✗✒✜✆✩✢✪✧✩✦✫✬✩✢✭✕✮✰✯✲✱❏ô✚õ❞ö✁÷✆õ✗ø❝ù❙ô❞þ✑ù✚ÿ❋ú✓✁ø✑ö✡✂✓õ✗ô✗þ✛✳✴✔✕✡✝❅ÿ❋ú✓✂✓✙ù✚õ❞ÿ☞✒✂♦ô✗þ ✏✢✏☛✏✟✍✶✵✸✷✚ þ⑦ù✣✠✑õ✹✔❩ø✑ú✧✗❝ø✺✗✒✜✸✻✼✠❩ú✓✤☞✠➇ÿ❋õ☛✤✦✗❝ø❩ô❊ù✚ú❥ù✣✔❅ù❋õ✗ô✔ù✣✠❩õ✹✗✁ÿ❋ú❥ö❝ú❥ø✕✡✂✼✝✽✗✁÷✬✙ú✓ø✿✾❀✏❂❁✆✗✙ù❋õ✳ù✣✠✑õ õ✢✂✓õ❞÷✆õ✗ø❝ù❙ô❃✙ÿ❋õ❄✗✡❅✉õ❞ø✢þ✢ô✣✗tú✓ø✙✜❆✄✤✽ù ✾❇✱Òù✟✠✑õ✬✤✦✂✓✗❝ô✣✔✑ÿ❋õ❄✗✒✜❈✾✼✍➅ú♦ô➅ù✟✠✑õ✆ô❉✔✑÷❊✗✡✜✭ù✣✠❩õ★❋✢✪✧●❍✯✑■✽❏✣✩❑✗✡✜ ù✣✠❩õ❀✵✚ ✷ ✏▼▲➅ô✵ú❥ø❖◆❩ø✑ú❥ù✚õP✝❅ú✧◗✰õ❞ÿ❋õ❞ø✕✤❞õ✗ô✗þ✽✻⑦õ❃✒✂♦ô❉✗❑✠✕❍❘❝õ✥✭❙●✎❚✡✩✦✯✸❯❱✻✼✠✑ú✓✤☞✠❖✻❺ú✧✂✓✂❲❅✛✂✁❍❳❨❩✤❞õ❞ø❑ù✚ÿ☞✒✂ ÿ✟✗✡✂✓õ❬❯❱❭✕✔❅ù➅ú❥ù❯ú✓ô✵ù✟✠✑õ❾õ✢✂✓õ❞÷✆õ❞ø❑ù❙ô③ù✣✠✕✙ù✆✝❅õ✦◆❩ø❩õ❶ù✣✠✑õ❄✡❅✛❅✑ÿ✟✗❍❪❖ú✓÷✬❬ù❋ú✧✗❝ø❫✏ û❼ø❿ö❝õ❞ø✑õ✗ÿ✟✡✂⑧þ❙✻✵õÔ÷✬❍❳❴✤✢✗✁ø❩ô✚ú✓✝✑õ❞ÿ➅ø✕✗✁ø✽✖❵✔✑ø✑ú✧✜❛✗✁ÿ❋÷ ÷✔õ❪ô❉✠❩õ✗ô❯ú✓ø✙✻✼✠✑ú✁✤☞✠✏ù✟✠✑õ④õ☛✂❥õ✗÷✆õ❞ø❑ù❋ô✸✙ÿ❋õ ✗✒✜❜✝❅ú✧◗✉õ✗ÿ✚õ✗ø❝ù❝✂✓õ❞ø✑ö✁ù✣✠❩ô✗þ✒✗❝ÿP❞✣✝❅ú✁✙÷✆õrù❋õ❞ÿ❙ô❞þ ❡❃❢✞✷❣✏✪û❼ø④ù✣✠✑ú♦ô▼✤✢❝ô❊õ⑦ù✣✠❩õ❬❢❑✻✼✠✑ú✓✤☞✠❤✒❅✕❅❁õ✎✙ÿ❙ô❈ú✓ø❤✘✛✚☛ú✓ô ù✣✠❩õ☛÷✬✒❪❖ú✓÷✥✔❩÷✐✝❅ú✁✙÷✆õrù❋õ❞ÿ✼✗❥❘✁õ✗ÿ❈✡✂✧✂✻õ☛✂❥õ✗÷✆õ❞ø❑ù❋ô☛✏❱û➶ù❯ú✓ô③ú✓÷❤❅✞✗❝ÿ❊ù☞✙ø❑ù⑦ù✟✗✆ÿ✚õ✗÷✔õ✗÷✥❭✉õ❞ÿ✵ù✣✠✞❬ù✆✻✵õ ✻❺ú✓✂✧✂❈ú❥ø★✜❆✄✤✽ù❬❭✉õ✥✤✦✗❝ø✕✤rõ✗ÿ✚ø❩õ☛✝❖✻❺ú✐ù✟✠❦❡ô❊õ✎✳✴✔✑õ❞ø✕✤❞õP✗✒✜❱ù✚ÿ❋ú✁✙ø✑ö✄✔✛✂✓✙ù✚ú✓✗✁ø❩ô✼✘✛✚❨✻❺ú✐ù✟✠❦❢❖❧♥♠✛✏❈♦➄õ ô✟❍❳❡ù✣✠✕✙ù❬✗✡✔❩ÿ❯ô❊õ✎✳✄✔❩õ❞ø✕✤❞õP✗✒✜❱ù✚ÿ❋ú✓✁ø✑ö✡✔✕✂✓✙ù✚ú✓✗✁ø❩ô✵ú♦ô❩♣✢■✕q❍✯✑rts✉■✽✭✕r✈✦●✒❏✑✫ ú✧✜❱ù✣✠✑õÔÿ☞❬ù❋ú✧✗❨❢✕✇❝① ②✄③✒❢✕✇❝④❉⑤ ✗❥❘✁õ✗ÿ❈✘✛✚✔ú♦ô❈❭✞✗✄✔✑ø✕✝❅õ✎✝❨✜❳ÿ✟✗✁÷❇❭✉õ✢✂✓✗❥✻⑥❝ô⑦❢❖❧⑧♠✛⑨✽✻✵õ☛ô✣✠✕✡✂✧✂❲✒✂✓✻⑦❍❳❖ô❈✁ô❋ô✣✔✑÷✆õ➅ù✟✠✑ú✓ô✵ù✟✗❩❭✉õ☛ù✣✠✑õ ✤✢❝ô❊õ✄✏❱û❼ø✲✠✑ú✓ö✡✠✑õ✗ÿ❈✝✑ú❥÷✆õ❞ø❁ô❊ú✓✗✁ø❩ô❈✻✵õ✸✻❺ú✓✂✓✂❫✒✂♦ô❉✗❑✝❅õ✦◆❩ø❩õ❀❖❏✣✩❵⑩✒■✽✪✓q❥❏✑rt✮❷❶❑✠❣❳❣❅❙✗✙ù✣✠❩õ✗ô✚ú✓ô✵ÿ❋õ✢✂✁❬ù✚õ✎✝✆ù✣✗ ù✣✠❩õÔô❉✠✕✡❅❁õ❃✗✡✜❈ù✟✠✑õ❾õ✢✂✓õ❞÷✆õ❞ø❑ù❙ô✢✏ ❸✠✑õ✗ÿ✚õtú✓ô❩✙ø✛✗✁ù✣✠✑õ✗ÿ❄✻❈❍❳◗ù✣✗✙✝❅õ❪ô✣✤❞ÿ✚ú✓❭✉õÐõ✢✂✓õ❞÷✆õ❞ø❑ù❙ô✥✙ø✞✝✹ù✚ÿ❋ú✓✁ø✑ö✡✔✛✂✁❬ù❋ú✧✗❝ø❩ô❶ú✓ø❹✻✼✠✑ú✁✤☞✠❺✵❲✚ ô✣✠✕✒✂✓✂✭ÿ❋õ✦✜❳õ❞ÿÔù✣✗✙✙ø❣❳❻❅✕✙ÿ✚ù✚ú✁✤✦✔✛✂✁✙ÿ❾÷✆õ✗÷✥❭✉õ❞ÿ❄✗✡✜⑦✘✚ ❯ ù✣✠✞❬ùÔú♦ô❞þ❈ù✣✠✑õtõ❞ø❣✔✑÷✆õ❞ÿ☞❬ù❋ú✧✗❝ø❼✁ø✕✝❺❽ ô✣✔✛❅❁õ✗ÿ❋ô✟✤rÿ❋ú✓❅❅ù❀✡❭✞✗❥❘❝õsú♦ô❬✂✓õ✦✜Òù❾ú✓÷❑❅✕✂❥ú✁✤rú❥ù☛✏ ❸✠✑ú♦ô☛ú✓ô❀✗✡✜Òù✚õ❞ø➄÷❤✗✁ÿ❋õ❩✤✦✗❝ø❣❘✁õ❞ø❩ú❥õ✗ø❝ù✸✜❛✗✁ÿ❃✝❅õ✗ô✟✤rÿ❋ú✓❭✑ú❥ø❩ö ❘❥✙ÿ❋ú✧✗✄✔❩ô⑦✒❅✛❅❩ÿ✣✗❍❪❅ú✓÷❤✙ù✚ú✓✗✁ø❩ô☛✏✪û❼ø➀ù❋õ❞ÿ❋÷❡ô⑦✗✒✜✪ù✣✠✑ú♦ô✆✒❭✕❭✑ÿ✚õ☛❘❖ú✓✙ù✚õ☛✝tø✛✗✙ù☞❬ù❋ú✧✗❝ø✢þ✽✻⑦õP✠✞❍❘✁õ❫ù✣✠✕✙ù ✾✺❾ ❿ ➀❥➁✡➂✄➃☛➁ ✵✚ ✻✼✠✑õ✗ÿ✚õ❃✵✚❤➄ ✘✚ ú❥ø✞✝❅ú✓✤☛❬ù❋õ✗ô✵ù✣✗✆ù☞✒➅❝õ❫ù✟✠✑õP✔✑ø✑ú✓✗✁ø❴✗❥❘✁õ✗ÿ✼✒✂✓✂✻õ✢✂✓õ❞÷✆õ❞ø❑ù❋ô☛✏ ➆❫➇❆➈❙➇✁➆ ➉❫➊➌➋✛➍✡➎✹➏✚❤➐ ➏ ➑❙➒✴➓t➔➣→↕↔✡➙ ➏✚ ❾➜➛✞➝ ➄ ➏ ➞ ➞ ➞ ➝❲➟ ➀❜➠➁ ➄ û➡ ý ✱❛✵✚ ✷ ✍✑➢ ❽✬❾➥➤✡➢☛➦✢➦☛➦✦➢✣➧✺➨ ➩✩☞❋☞q❥✪t✪➫✮t➭✕q❥✮➌➝❲➟ ➀❜➠➁ ✫✬✩☞q❥✭✛✯✼➝✳ÿ❋õ✗ô❊ù✚ÿ❋ú✓✤rù✚õ☛✝➀ù✟✗❖✵✸✷✚➲➯❴➳➭❣■❣✯❄✮❛➭✛✩❤q✡➵☞●❥➸❥✩❩✯✢q❥❶❍✯❄✮❛➭✛q✒✮✼q❩➝➺rt✭ ➏✚ ✫❑■❣✯☞✮✥➵☞✩★rt✭ ➏ ❾➼➻➄ý ➽ ✱✰✾✼✍☞➾❤q❥✭❙❚❺✫❩■❣✯✑✮✥➵☞✩❩➚❙r❆✩☞❋☞✩✢➪❝r✁✯✦✩✢s✉✪➶rt✭❜✩✟q✒❏❦➹àû➡ ý ✱t➘❤✍☞➾✆➘ ➐ ✾✸➾✥r✁✯ ✮❛➭✛✩❩✯❷➚✛q✄❋☞✩✬●➴✈❩✪➶rt✭❙✩☞q❥❏✆➚✛●✒✪➶❶❥✭❙●✒✫❩r❆q✒✪➷✯❨●❥➸❥✩✦❏❬➘➬➹➮●✒✭❹✩✟q✄❋✟➭➺✩✢✪✧✩✢✫❤✩✢✭✕✮ ➯❻➱✩✬❋☞q✒✭➺q✒✪➷✯✢●❴➪❝❏✑rt✮❵✩ ➏✚ ❾❐✃✎➝ ➄ ➏ ➟✦➝❜➟ ➀➁ ➄ û➡ ý ✱❆✵✚ ✍✑➢✆❒❄✵✚✬➄ ✘✚✛❮ ➯ ❁✆❰ Ï
Note 4 Continuity of v in X It is clear that if v E Xh, then since Xh CX(X ember of Xn is a member of X because Xh Z vEX+ggg u(Mz u(ma M aaamembers of v in t h(Q2(anr hence Xh CX anish at l z Manr la nWt Xn CX also tells us that v must be co X uous Tthe (r istributionalreriatiwe of the function r epicter abo"e is piecet ise constant on each element, anr hence are integrable, as re Wuirer by t h(s thot e"er, if t e har almps in v bett een elements, the reri ati e t oulr generate relta r istributions at the nor es, t hich are Xoq in i r(Q-(see rote b of the last lecture thus Xo gin t h(s rIt is important e that t e ro not re Wire that our u be in Ch(Q, that is, ha"e continuous Tst reriaties-this is much more r iy cult to implement numerical We X - these are knot n as Xo Conforming approlimations, as opposer to tI conforming appro imations t e consirer herent Y, set of members pHE Y, y a nQ.Si is a basis for y if anr only if ∈Y,miWe= HE F such that p a=HpHi rim(ension(Y M n 回n回 It follot s from our re nition of a basis that any set o members pH- members such that HpHa Mt Hr my a nQ. SM t ill ser e basis it is al ily choice of basis is not at all uniMue, the rimension of Y, rim(y, w uniwi pr simplicity t e t ill use the basis concept primarily in the contel t of miter imensional spaces such as Xnubut in mite r imensional space t h(@ can also be rescriber in these terms mr ote t
Ð❼Ñ❲Ò✑Ó❴Ô Õ✥Ñ❣Ö❝Ò✣×❆Ö▼Ø✞×❉Ò❆ÙÚÑ✛Û⑦Ü✙×✰Ö↕Ý❨Þ ß❵à✆á✁â✼ã✦ä✓å☛æ✡çèà✟é✕æ❥à✆á✧ê➌Ü❨ë★ÝÞ✛ì à✣é✕å✢í✹â❉á✓í✕ã✢å❀ÝÞ❤î Ýðï❆ÝÞ á✓â✼æ❖ñ✑ò✕ó✦ñ✰ô✕õ✡ö☞÷❄ø✒ê➫Ý➼ùúæ✒í❣û üåü❩ýå☛çþø✡ê✞Ý❨Þ✸á✓â➌æ üåü✥ý å✢ç❝ø✡ê✕Ý ý å☛ã☛æ✒ÿ✕â✣å❈Ý✬Þ✁✄✂☛Ü✲ë❴Ý✆☎✞✝✟✝✠✝☛✡✌☞➣Ü❙ï✎✍✏☞✑✿Ü❙ï✓✒✔☞✑✕✍✼ù õ✞✖✗✖ üåü❩ýå☛ç✟â❝ø✡ê✞Ü❄á✓í✙✘✛✚✜ ï✣✢✤☞✼ï❆æ✡í✦✥❩é✛å✢í✕ã✢å✼Ý❨Þ î Ý✧☞✩★❥æ✡í✛á✓â✣é✬æ❥à✫✪✬✕✍Pæ✒í✦✥✁✪✬✆✒✏✭✯✮❈ÿ✽à Ý❨Þ î Ý æ✒ä✁â❉ø❩à✣å☛ä✧ä✁â❈ÿ✕â❈à✣é✞æ❥à⑦Ü üÿ✕â❉à ý å❨ö✱✰✞✲✦✳✣✴✗✲✕ò✦✰✒ò❣ñ✶✵þà✣é✛å✬ï✎✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞æ✒ä✗☞✑✥✽å☛ç✣á✷★❥æ❥à✟á☛★✄å ø✒ê❜à✟é✛å✆ê❛ÿ✛í✞ã✑à✣á✓ø✡í✸✥✛å✠✹✛á✁ã✑à✟å✟✥❨æýø✞★✄å✼á✓â✑✹✛á✧å✎ã✦å✟✺✼á✓â✣å❬ã✦ø✄í✕â❉à✟æ✒í✴à▼ø✄í✬å☛æ✡ã☞é❤å☛ä✧åüå☛í✴à ì æ✒í✦✥❤é✛å☛í✕ã✦å â✼✻✄ÿ✞æ✒ç✟å✼á✧í✴à✣å✟✽✡ç☞æýä✓å ì æ✡â❝ç✟å✟✻✴ÿ✛á✓ç✣å✔✥ ýû✾✘✛✚✒ï✎✢✤☞❀✿✴é✛ø✞✺❈å✠★✄å✢ç ì áê❁✺❈å✆é✕æ✏✥❃❂➴ÿü✹✕â▼á✓í✬Ü ý å✦à❄✺❈å✢å☛í å✢ä✓åüå☛í✄à☞â ì à✣é✛å❅✥✽å☛ç✣á✷★❥æ❥à✣á✷★✡å❆✺❈ø✡ÿ✛ä❇✥❈✽✡å✢í✕å✢ç☞æ❥à✣å❅✥✽å☛äà☞æ✾✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞â✆æ❥à⑦à✣é✕åPí✛ø❉✥✽å☛â ì ✺✼é✛á✁ã☞é æ✒ç✟å❊✲❋✰●✳❃á✧í■❍❑❏✡ï✎✢✤☞❨ï❆â✣å✢å▼▲❬ø✒à✣å❈◆★ø✒ê❈à✣é✛å❨ä✁æ✡â❉à✥ä✓å☛ã✦à✣ÿ✛ç✟å✔☞✸ù æ★ê❛ÿ✛í✕ã✦à✣á✓ø✡í❖✺✼á✧à✣éP❂➴ÿü✹✕â✥á✓â à✣é❣ÿ✕â◗✲❋✰●✳❝á✧í❈✘✛✚❥ï✣✢✤☞❀✭❝ß❵àèá✁âèáü✹❙ø✡ç✣à✟æ✒í✴à▼à✟ø✥í✛ø✡à✣å❬à✣é✕æ✒à❑✺❈å❘✥✽ø✥í✕ø✒àèç✟å✟✻✴ÿ✛á✓ç✟å✼à✣é✕æ✒àèø✄ÿ✛ç➲Ü ýå á✓í▼❙❚✚❥ï✎✢✤☞ ì à✟é✕æ❥àèá✓â ì é✞æ✌★✡å✼ã✦ø✄í✄à✟á✧í❣ÿ✛ø✄ÿ✕â❱❯✕ç☞â❉à✑✥✽å✢ç✟á☛★❥æ✒à✣á✷★✡å☛â➫ùðà✣é✛á✁â▼á✁â üÿ✕ã☞é üø✡ç✟å❲✥✽á❨❳❨ã✢ÿ✛äà à✣ø❤áü✹✛ä✧åüå✢í✴à❬í❣ÿüå✢ç✟á✓ã☛æ✒ä✓ä✧û✏✭ ❩❼å❻ç✟åüæ✡ç❭❬↕à✣é✕æ✒à❖à✟é✛å✢ç✟å❻æ✡ç✣å❪❯✕í✛á✧à✣å❼å✢ä✓åüå✢í✴à✹æ●✹❫✹✕ç✣ø✌❴✽áüæ✒à✣á✓ø✡í✕â❨á✧í❵✺✼é✛á✓ã☞é⑥ÝÞ❜❛î Ý❱ù à✣é✛å✎â❉å✹æ✡ç✣åP❬❣í✛ø✞✺✼í↕æ✄âP✲❋✰●✲❙ã✢ø✡í✽ê❛ø✄çüá✓í❫✽❺æ●✹❫✹✛ç✟ø✌❴✽áüæ✒à✣á✓ø✡í✕â ì æ✡â❤ø✏✹❫✹❙ø✄â✣å✟✥ à✣ø❼à✣é✛å ã✦ø✄í✽ê❛ø✡çüá✧í✦✽❑æ✶✹❫✹✛ç✟ø✌❴❣áüæ❥à✟á✧ø✄í✕â❝✺èåPã✦ø✄í✕â✣á✷✥✽å☛ç✼é✛å✢ç✟å✶✭ ❞❢❡❤❣❋❡✷✐ ❥❧❦❫♠✔♥❤♠ ♦❋♣rq✗s✩t✈✉✶✇ ①÷②✲❜÷②③✟õ✞✖⑤④✄÷❤⑥✑✲✦✴✗✳✣✴❤✰✞✲❋✵ ✽✄á☛★✄å✢í★æ❩ä✓á✓í✛å☛æ✡ç✆â✓✹✕æ✄ã✦å⑧⑦ ì æ❑â✣å✦à✆ø✡ê üåü❩ýå☛ç✟â❵⑨✌⑩Pë✬⑦ ì✾❶ ❷✒✏❸✠❹✟❹✠❹②❸✱❺ì á✓â✥õ ýæ✡â✣á✓âèê❛ø✡ç✆⑦ á✧ê❝æ✒í✦✥❴ø✡í✕ä✧û❨á✧ê ❻ ⑨✲ëP⑦ ì❽❼ ÿ✛í✛á❇✻✴ÿ✛å❿❾⑩ ë❖ß➀➮â✣ÿ✕ã☞é❴à✣é✕æ✒à ⑨✙➂➁➃ ⑩❭➄ ✚ ❾❁⑩✫⑨✌⑩✙✿ ✥✽áüï❛å✢í✞â❉á✓ø✡í❽☞❃ï✎⑦✁☞➅❵❺➆✭ ▲❃➇ ▲❘➈ ➉❲✒ ➉❑➊ Ð❼Ñ❲Ò✑Ó➌➋ ➍➲×✰Ö▼Ó✶➎❉➏P➐þÓ✔➑❬Ó✒Ö➅➐þÓ✒Ö➅➒✄Ó❈➎✽Ö✑➐➓➐❜×✎➔✺Ó✒Ö⑤→✎×✉Ñ✽Ö ß❵à➫ê❛ø✡ä✓ä✧ø✞✺✆â❫ê❛ç✣øü ø✄ÿ✛ç✩✥✽å✠❯✕í✛á✧à✣á✓ø✡í❄ø✡ê✕æ ýæ✡â✣á✁â❜à✣é✕æ✒àþæ✡í✴û❃â✣å✦à➣ø✡ê❽❺♥ä✓á✓í✛å☛æ✡ç✣ä✓û❬á✓í✦✥✽å✟✹✞å☛í✦✥✽å✢í✴à üåü❩ýå☛ç✟â➣⑨⑩ ù üåü❩ýå☛ç✟â✆â✣ÿ✕ã☞é❖à✣é✕æ✒à ➃➁ ⑩❭➄ ✚ ❾⑩ ⑨⑩ ↔✍➙↕➛❾⑩ ↔✍❫❸ ❶ ✆✒✶❸✟❹✠❹✟❹②❸✼❺ ù➜✺✼á✓ä✓ä❀â✣å✢ç✼★✡å✹æ✡â❖æ ýæ✡â✣á✁â✠✭➥ß❵à❖á✁â✲æ✡ä✓â✣ø❺ç✟å☛æ✶✥✛á✧ä✓û✈✥✽åüø✡í✕â❉à✣ç☞æ❥à✟å✟✥↕à✣é✕æ✒à ì æ✡äà✟é✛ø✡ÿ✦✽✡é✺ø✡ÿ✛ç ã☞é✛ø✄á✓ã✢å❴ø✡ê ýæ✡â✣á✓â❤á✁â✬í✛ø✡à✲æ❥à❨æ✒ä✓ä✆ÿ✛í✛á❇✻✴ÿ✛å ì à✟é✛å➌✥✽áüå☛í✕â✣á✧ø✄í ø✡ê❃⑦ ì ✥✛áüï❤⑦❧☞ ì ✴✁ñ✹ÿ✛í✛á❇✻✴ÿ✛å✶✭ ➝✛ø✄çPâ❉áü✹✛ä✧á✁ã✦á✧à➴û❪✺❈å❧✺✼á✧ä✓ä▼ÿ✕â✣å❑à✣é✛å ýæ✄â❉á✁âPã✦ø✄í✕ã✦å✟✹✽à❃✹✕ç✣áüæ✒ç✟á✧ä✓û✹á✧í❼à✟é✛å✬ã✦ø✡í✴à✟å②❴❣àPø✒ê✑❯✞í✛áà✟å②➞ ✥✽áüå✢í✕â✣á✧ø✄í✕æ✒ä⑦â❭✹✕æ✡ã✢å☛â✥â✣ÿ✕ã☞é æ✡â❄ÝÞ ✿ ýÿ✽à❑á✓í❉❯✕í✕áà✟å✬✥✽áüå✢í✕â✣á✓ø✡í✕æ✡ä❈â❭✹✕æ✡ã✢å☛â❩â❉ÿ✕ã☞é↕æ✡â❩Ý➟ ✘✛✚✜ ï✎✢✤☞✼ã☛æ✒í★æ✒ä✁â❉ø ý å➙✥✽å☛â✟ã✦ç✟áýå✔✥❖á✓í❴à✣é✛å✎â❉å❃à✟å✢çüâ✠✭✑▲❬ø✒à✟å❅✺èå❄ã☛æ✒í❴å②❴❉✹✛ç✟å☛â✟â✼æ✬â✓✹✕æ✄ã✦å❅⑦ á✓í ➈
terms of any basis as Y= span yj, 3=1,., M, meaning that any member of y can be represented as a linear combination of the Orthogonality If our space Y is a Hilbert space with inner product(, Y, we can introduce the notion of orthogonality: two members yi E y and y2 E Y are orthogonal if An orthogonal basis is thus a basis for which the y, are mutually orthogona (yi, 9j)Y ≠J.I, furt hermore,(v;y)Y=‖yi=1, the basis D Exercise 1 Consider the Hilbert space R2 "equipped clidean inner product, ([1, y1],[2, y2])=2132+u132, and hence norm ,yll Is(1, 1),(1,0)a basis for R2? an orthogonal basis? (b)If(1, -1)/v2 is one of our basis vectors, find a second vector such that we have an or thonormal basis Exercise 2 Consider Y= P2([-1, 1]=span (1, r) equipped with the 2 inner product, (3, x)r= y z dz(here y and z are two members of Y (a) Replace x with another basis vector (in fact, polynomial) such that b) Appropriately with what famous French mathematician) are you generating by the above Gram-Schmidt"process Nodal basis for Xh dim(Xn)
➠❭➡✟➢❭➤✙➥✤➦✶➧➅➨●➩➭➫✸➯✦➨✏➥✓➲❇➥❲➨✶➥❘➳➸➵❵➥✓➺✦➨✶➩❧➻✟➼✌➽●➾❑➚✾➵⑧➪✶➾✠➶✟➶✠➶②➾✱➹➸➘✏➴❫➤✾➡✔➨●➩❫➲✷➩❫➷✾➠❭➬❽➨✞➠◗➨✶➩r➫▼➤❧➡✟➤➙➯❋➡✠➢ ➦●➧✫➳➱➮✠➨✶➩❈➯❋➡❅➢❭➡✟➺❫➢✼➡✟➥❭➡✠➩r➠❭➡✔✃▼➨✶➥✤➨✾❐✷➲✷➩❫➡✟➨✶➢➣➮✠➦✶➤✁➯❫➲☛➩❽➨✞➠❭➲✷➦✶➩✬➦●➧✯➠❭➬✦➡❅➼✌➽●❒ ❮✧❰❁Ï❀Ð❪Ñ Ò❚Ó➭Ï❭Ô➅❰➭Õ✩❰❉Ö❱×❉Ø❤Ù❭ÏÛÚ Ü➧❢➦✏Ý❫➢❑➥❭➺✦➨✶➮✠➡❲➳❷➲❇➥❑➨❚Þ❘➲☛❐✷➯❋➡✠➢❭➠❝➥❭➺✦➨✶➮✠➡✤ß✤➲❨➠✼➬✸➲☛➩✦➩❫➡✠➢❑➺❫➢❭➦❉✃❉Ý❽➮❀➠❃à✓á☛➾✟á â❭ã❲➴rß❝➡❘➮✠➨✶➩✙➲☛➩r➠❭➢✼➦❉✃❉Ý✦➮✠➡ ➠❭➬✦➡❅➩❫➦●➠✼➲☛➦✏➩❈➦✶➧✯➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐☛➲☛➠❄➫åä❢➠❄ß❝➦❧➤✾➡✟➤➙➯❋➡✠➢✱➥➣➼ræ◗ç❊➳❿➨✶➩✦✃✬➼✶è❆çP➳❿➨✶➢❭➡✸é●ê❀ë✗ì✦é❭íré✞îåï✞ð❱➲❨➧ àÛ➼ræ✞➾✓➼✶è✔â ã ➵↔ñ✬➶ ò➩ó➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐✩➯✦➨✏➥✓➲❇➥❆➲✷➥❃➠✼➬➭Ý✦➥❅➨✬➯✦➨✶➥❭➲✷➥◗➧Û➦✏➢❅ß✤➬❫➲✷➮✱➬✛➠❭➬✦➡❧➼➽ ➨●➢✼➡❧➤➙Ý❉➠✼Ý✦➨●❐✷❐☛➫✛➦✶➢❭➠❭➬❫➦✏➷✶➦✏➩✦➨●❐✣➴ àÛ➼✏ô✓➾❭➼✌➽✟â ã ➵õñ❫➴❘öø÷➵➛➚❽❒ Ü➧❄➴❘➧ÛÝ✦➢✓➠✼➬❫➡✠➢✼➤✾➦✶➢✼➡✶➴➙à❤➼✶ô✓➾❭➼✶ô✎â ã ➵➟ù②➼✶ô❭ù èã ➵ú➪✏➴❲➠❭➬✦➡✛➯✦➨✏➥✓➲❇➥❈➲✷➥ é✞ê❀ëÛì❫é●î❋é●ê❀û✾ï●ðü❒ ý➓þ◗ÿÐ●Ó✁✌Ù✄✂✶Ð✆☎✞✝➦✶➩❽➥✓➲❇✃❉➡✠➢❧➠✼➬❫➡➌Þ❲➲✷❐☛➯❋➡✠➢❭➠▼➥❭➺✦➨✶➮✠➡ Ü✟è✡✠ ➡☞☛rÝ❫➲✷➺❫➺❽➡✔✃✍✌✛ß✤➲❨➠✼➬øÝ❽➥✓Ý✦➨✶❐✏✎➅Ý✒✑ ➮②❐✷➲❇✃❉➡✟➨✶➩❃➲✷➩❫➩❫➡✟➢❢➺❫➢✼➦❉✃❉Ý✦➮❀➠✔➴❉à✔✓✕åæ●➾✓➼ræ✗✖ ➾☞✓✕❽è✏➾✓➼✏è✘✖Ûâ➅➵✙✕❁æ✘✕❋è✛✚◗➼ræ❀➼✶è●➴✌➨●➩✦✃❃➬❫➡✟➩✦➮②➡✑➩❫➦✶➢✼➤❿ù✁✓✕⑤➾✓➼✜✖üù✤➵ à✢✕è ✚■➼ è â æ✗✣✼è ❒✥✤❘➦●➠❭➡❆➠✼➬❫➡❅➺✦➨✶➲☛➢✦✓✕✩➾✓➼✜✖✩➢✼➡②➧Û➡✟➢✼➥❝➠✼➦✙➨✙➥✓➲✷➩❫➷✶❐✷➡❆➤✾➡✠➤➙➯❋➡✠➢❚à❤➺❽➦✏➲☛➩r➠❀â➣➲☛➩ Ü✟è ❒ à❤➨râ Ü➥❅à❄➪✶➾✟➪✔â②➾✟à❄➪✏➾❭ñrâ❑➨❧➯✦➨✏➥✓➲❇➥❝➧Û➦✶➢ Ü✟è✛✧ ➨✶➩P➦✶➢❭➠❭➬❫➦✏➷✶➦✶➩❽➨●❐❋➯✦➨✶➥❭➲✷➥ ✧ àÛ➯❋â Ü➧❆à✓➪✶➾✩★❅➪✌â✗✪✬✫✭❈➲✷➥❚➦✶➩❫➡✾➦●➧➣➦✶Ý✦➢❃➯❽➨✶➥❭➲✷➥✯✮✏➡✟➮❀➠✼➦✶➢✱➥✠➴✱✰❽➩✦✃ó➨❊➥✓➡✔➮②➦✶➩❽✃✲✮✏➡✟➮❀➠✼➦✶➢❅➥❭Ý✦➮✱➬❪➠✼➬✦➨✞➠ ß❑➡❆➬✦➨✳✮✏➡❃➨✶➩✬➦✏➢✓➠✼➬❫➦✶➩✦➦✶➢✼➤✾➨✶❐❽➯❽➨✶➥❭➲✷➥✟❒ ý✸þ◗ÿÐ●Ó✬✔Ù✄✂✶Ð✵✴✶✝➦✏➩✦➥❭➲✷✃❉➡✟➢❑➳⑧➵ Ü✷ è✶à✗✓✸★❅➪✏➾✠➪✹✖Ûâ✫➵✕➥❭➺✦➨✶➩❆➻✏➪✶➾✗✕✩➾✗✕è ➘❃➡☞☛rÝ❫➲✷➺❫➺❋➡✟✃✸ß✤➲☛➠❭➬▼➠❭➬❫➡ ✺è ➲✷➩❫➩❫➡✟➢❅➺❫➢✼➦➭✃❫Ý✦➮❀➠✔➴❝à❤➼❋➾✼✻râ ã ➵✾✽ æ✿ æ ➼❀✻✦❁✁✕✕à❤➬❫➡✠➢✼➡✙➼❪➨●➩❽✃❂✻❊➨✶➢❭➡❧➠❄ß❝➦P➤❧➡✟➤➙➯❋➡✠➢✱➥❅➦●➧✤➳✙➴ ➠❭➬❽➨✞➠❲➲❇➥✠➴➭➠❄ß❝➦✾➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐❇➥✱â❀❒ à❤➨râ ✟➡✠➺❫❐❇➨✶➮✠➡❃✕è ß✤➲❨➠✼➬✧➨✶➩❫➦●➠✼➬❫➡✠➢❆➯✦➨✏➥✓➲❇➥❄✮✶➡✔➮❀➠❭➦✏➢✙àÛ➲✷➩❪➧❤➨✏➮❀➠✟➴✩➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐✗â❃➥✓Ý❽➮✱➬❪➠✼➬✦➨✞➠❅ß❝➡ ➩❫➦✞ß↔➬✦➨✳✮✏➡❃➨✶➩❈➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐å➯✦➨✶➥❭➲✷➥✟❒ àÛ➯❋â ò➺❫➺❫➢✼➦✶➺✦➢❭➲❇➨✞➠✼➡✠❐✷➫❖➩❫➦✶➢✼➤✙➨●❐✷➲❆❅✟➡✟✃❁➴❑ß✤➬✦➨✞➠✸➺❋➦✶❐✷➫➭➩❫➦✶➤✾➲❇➨●❐❲➥❭➫➭➥✓➠❭➡✟➤➂à✗➠✼➬✦➨✞➠❈➲✷➥✟➴❝➨✏➥❭➥❭➦❉➮②➲❇➨✞➠✼➡✟✃ ß✤➲❨➠✼➬➙ß✤➬✦➨●➠✯➧❤➨✶➤✾➦✶Ý✦➥❈❇❫➢✼➡✠➩❽➮✱➬➙➤✙➨✞➠✼➬❫➡✠➤✙➨✞➠✼➲✷➮✠➲✷➨✶➩❽â⑤➨●➢✼➡✑➫✶➦✏Ý➙➷✏➡✠➩❫➡✟➢✼➨●➠❭➲✷➩❫➷❘➯➭➫❆➠❭➬✦➡➣➨✶➯❽➦❉✮✏➡ ✠✼❊➢✱➨●➤❋✑❍●❫➮✱➬❫➤✾➲✷✃➭➠■✌➙➺✦➢❭➦❉➮②➡✔➥❭➥✟❒ ❏▲❑✜▼❖◆◗P❙❘❉❚ ❯é✛❱✶ï●ð❉➯✦➨✶➥❭➲❇➥❝➧Û➦✶➢❳❲❩❨❉ä ❬ ➽✶➾✑➚✁➵❷➪✶➾✠➶✟➶✠➶②➾✗❭❊➵↔✃❉➲✷➤❊à✢❲❩❨✏â ❪
pP: nonzero only on TU N7|N8 We can convince ourselves independent of any basis that dim(r=n. First ve note that on any element TA, utK =a+6 z; since we have K boundary conditions(at a =0, face continuity conditi (ai),i=l,.., n), for a total of n+2 constraints. Th dim(Y)=2+2-(m+2)=n Note 8 Interpretation of basis The nodal in nodal basis refers to the fact that the basis coefficients are not st Fourier-like"coefficients, but also have a"physical-space"significance: if is a member of Xh, we know from the definition of a basis that an vi Pi(aj)=uj,j n,since the pi are zero at al an be uniquely defined by the condit Xh, pi (ai)=dij,i=l,., n; here dij is the Kronecker-delta symbol )Note that there is no“y0”or“n+1” in the basis since we must have u(0)=v(1)=0 fo Thus vi=v(ai), the value of u at a =ai, the ith node; and 2i=l vi pPi(a connects"the values of v at the nodes with linear segments on each element It is then patently clear that we can represent any piecewise-linear continuous function v that vanishes at r=0 and l by the choice vi u(ai),i Furthermore, the vi are unique no choice except vi u(i) will work. It thus follows that the pj are indeed a basis There are many other possible choices for basis - we explore a particularly useful one in a later exercise. However, the nodal represent ation remains the most common, first because of the convenient inter pretation as nodal values, and econd because of the matrix sparsity induced by the minimal over lap between
❫❈❴✞❵✍❛✬❵✍❜✩❝☞❞✗❛❋❛✁❵❢❡❆❣❤❛✁❵ ✐ ❴❥❧❦ ✐ ❴❆♠♦♥ ❥ ♣✯q ♣sr t❂✉✇✈✘①③② ④⑤✉✇⑥⑧⑦⑨✈✔⑩❶⑦◗❷✞❸✒❹✛❷⑨⑥❢❺✙①❉⑦⑨✈ ❻❝❽❼☞❾❿❵❩❼✹❛✬❵➁➀✜➂➃❵❢❼✩❝✏❛✁➄✍❞■➅✗❝✩❡➃➀✁❝☞➅❈➂➃❵❢➆✒❝☞➇⑧❝☞❵❢➆✒❝☞❵✬➈➉❛❿➊➋❾✁❵✜❣❋➌❢❾✬➅✔➂➍➅⑨➈✼➎❢❾❉➈➏➆✍➂❆➐➒➑❶➓❃➔✥→↔➣⑨↕♦➙♦➂➃❞✼➅✔➈☞➛ ➜❝➝❵✍❛❿➈✼❝➞➈✗➎❢❾❿➈⑤❛✁❵③❾❿❵➁❣➟❝✩❡➃❝✩➐➞❝✩❵✜➈⑤✐❽➠❥ ➛♦➡✇➢ ➤✱➥➦ →✾➧✒➠❄➨➫➩✹➠✛➭◗➯✥➅✗➂➃❵❢❼✹❝ ➜❝➝➎❢❾✳➀✬❝➞➲➳→➵➣➸➨➻➺ ❝✩❡➃❝✩➐➞❝☞❵✬➈■➅✩➛s➈✗➎✍➂➍➅✶➼✁➂➃➀✁❝✛➅❩➄❢➅➾➽❉➣➚➨➪➽➶➆✍❝✩➼✁❞✼❝✩❝✛➅➘➹➴❛❿➊❖➹➷➊✢❞✼❝✩❝☞➆✍❛✁➐➸↕➮➬s❛➜❝☞➀✁❝✩❞ ➜❝➚❾❿❡➍➅✔❛✞➎❢❾✳➀✁❝➚➽ ➌▲❛✁➄✍❵❢➆❢❾❿❞✼❣❂❼✩❛✁❵❢➆✒➂❆➈✗➂➃❛✁❵⑧➅✵➑✄❾❿➈❃➭✙→✾➱❢➛⑨➭✡→✃➺✛➔⑤❾❿❵❢➆➚➣✙➂➃❵✜➈✗❝☞❞✔➊✄❾✬❼✹❝✵❼✩❛✁❵✜➈✗➂➃❵➁➄✍➂✸➈➘❣➚❼✹❛✬❵❢➆✒➂❆➈✗➂➃❛✁❵❢➅ ➑✢➡✇➢ ➤⑧❐➦ ➑✢➭▲❴❍➔➞→❒➡✱➢ ➤❐❆❮➁❰ ➦ ➑✢➭▲❴➷➔✹➛ÐÏ❋→Ñ➺✁Ò☞Ó✩Ó✩Ó✩Ò✔➣◗➔✘➛Ð➊✢❛✁❞➞❾➒➈✼❛❿➈■❾❿❡➉❛❿➊s➣✲➨Ô➽➒❼✩❛✁❵❢➅✔➈✗❞■❾❿➂➃❵✜➈✼➅☞↕➶Õ➉➎➁➄❢➅✩➛ ➆✒➂➃➐➾➑✄➓❃➔➏→↔➽❉➣❩➨❙➽❄Ö×➑✄➣❩➨❙➽✁➔✥→✙➣⑨↕ t❂✉✇✈✘①➾Ø Ù✹⑦⑨✈✘①❿❹✩Ú⑨❹✬①✍✈✗❸➋✈✔⑩❍✉✒⑦Û✉✍Ü✦Ý✱❸✒Þ☞⑩❶Þ Õ➉➎✍❝➞ß✱à☞á✬â❉ã◗➂➃❵✵❵❢❛➁➆❢❾❿❡✱➌❢❾✁➅✗➂➍➅➏❞✼❝✹➊✢❝☞❞✼➅Ð➈✗❛❋➈✼➎✍❝s➊✄❾✬❼✘➈➉➈✗➎⑧❾❉➈➉➈✗➎✍❝❄➌❢❾✬➅✔➂➍➅➉❼✹❛➁❝✹ä❩❼✹➂➃❝✩❵✜➈■➅❧❾✁❞✗❝❽❵✍❛❿➈ å ➄❢➅✔➈✵æ✗➙❢❛✁➄✍❞✼➂➃❝✩❞✗➹➷❡➃➂❆ç✬❝☞è❋❼✹❛➁❝✹ä❩❼✩➂❆❝☞❵✬➈■➅✩➛⑧➌✍➄✒➈❽❾✁❡➃➅✗❛➝➎❢❾✳➀✬❝❀❾➚æ✔➇✍➎➁❣✒➅✔➂➍❼✩❾✁❡✸➹❍➅✗➇❢❾✁❼✩❝☞è❋➅✔➂➃➼✁❵✍➂❆é⑧❼☞❾❿❵❢❼✩❝✁ê✥➂❆➊ ➡❩➂➍➅❳❾➞➐➞❝✩➐⑤➌▲❝✩❞❳❛✁➊♦ë❥ ➛ ➜❝✯ç➁❵✍❛➜ ➊✢❞✗❛✬➐ì➈✗➎✍❝❀➆✍❝✹é❢❵✍➂❆➈✗➂➃❛✁❵➸❛✁➊❈❾❋➌❢❾✬➅✔➂➍➅❧➈✗➎❢❾❿➈ ➡❋→îíï ❴❆ð♦♥ ➡❴ ❫❴ ➑✢➭▲➔✦➯ ➎✍❛➜❝✩➀✬❝✩❞✛➛✜➡✱➑✢➭✍ñ✳➔✥→Ôòí ❴➃ðó♥ ➡❴ ❫❴ ➑✢➭✒ñ❉➔✥→✙➡✛ñ✬➛✁ô➞→➪➺✁Ò✩Ó☞Ó✩Ó✹Ò✗➣⑨➛✍➅✔➂➃❵❢❼✩❝✯➈✗➎✍❝❀❫❴ ❾✁❞✗❝❄❜✩❝☞❞✗❛➝❾❉➈s❾✁❡❆❡ ❵✍❛✒➆✒❝✛➅✥❝✹õ✍❼✩❝✩➇✒➈✥➭▲❴✔↕❳➑✢ö❍❵⑧➆✒❝✩❝✛➆✇➛✁➈✼➎✍❝s❫❈❴◗❼☞❾❿❵➞➌▲❝✏➄✍❵✍➂➍÷✜➄✍❝✩❡➃❣➸á✬ø✄ùÐß▲ø■á❋➌➁❣⑤➈✼➎✍❝s❼✩❛✁❵❢➆✒➂❆➈✗➂➃❛✁❵⑧➅✥❫❈❴⑨ú ë ❥ ➛♦❫❈❴✗➑✄➭ñ ➔✦→üû☞❴ñ ➛óÏ❄→ý➺✁Ò☞Ó✩Ó☞Ó✹Ò✔➣⑨➯♦➎✍❝✩❞✼❝❩û✩❴ñ ➂➃➅❄➈✼➎✍❝✵þ✯❞✼❛✁❵✍❝✛❼■ç✁❝✩❞✗➹❍➆✒❝✩❡❆➈✼❾❩➅✗❣➁➐⑤➌▲❛✁❡➷↕ ➔ ♣❛❿➈✗❝ ➈✗➎⑧❾❉➈✥➈✼➎✍❝✩❞✼❝✏➂➍➅⑨❵✍❛➒æ✼❫óÿ✛è✯❛✁❞❀æ✼❫ í♠ó♥✘è❀➂❆❵➞➈✼➎✍❝✏➌❢❾✬➅✔➂➍➅✥➅✗➂❆❵❢❼✩❝ ➜❝❳➐⑤➄❢➅✔➈Ð➎❢❾✳➀✬❝❳➡▲➑❶➱✬➔✥→↔➡▲➑✔➺✛➔✥→➫➱ ➊✢❛✁❞❳➡✵ú✶ë❥ ↕ Õ➉➎➁➄❢➅❽➡❿❴➏→Û➡✱➑✢➭▲❴❍➔✘➛✁✄✂❢ø✆☎✳â❿ã✞✝❢ø❩à✠✟s➡❙â✡⑨➭➟→Û➭▲❴✔➛▲➈✗➎❢❝❃Ï☞☛✍✌❩❵❢❛➁➆✍❝✁➯➋❾❿❵⑧➆ ò í ❴➃ðó♥ ➡❿❴▲❫❈❴✗➑✄➭▲➔ æ✗❼✩❛✁❵✍❵❢❝☞❼✘➈■➅✗è➝➈✗➎✍❝❃➀✳❾✁❡❆➄❢❝☞➅❽❛❿➊✥➡➸❾❉➈s➈✼➎✍❝❋❵✍❛✒➆✒❝☞➅ ➜➂✸➈✼➎➒❡➃➂❆❵✍❝✛❾❿❞✯➅✔❝☞➼✁➐➞❝✩❵✜➈■➅✏❛✁❵✲❝☞❾✁❼■➎➾❝☞❡❆❝☞➐➞❝✩❵✜➈☞↕ ö➴➈❽➂➍➅s➈✗➎✍❝☞❵✲➇❢❾❿➈✗❝✩❵✜➈✼❡❆❣➾❼✹❡➃❝☞❾✁❞✏➈✼➎❢❾❉➈ ➜❝❋❼✩❾❿❵✲❞✗❝☞➇✍❞✗❝✛➅✔❝☞❵✜➈❽❾❿❵➁❣✶➇✍➂❆❝✛❼✹❝➜➂➃➅✗❝✹➹➴❡❆➂➃❵✍❝✛❾❿❞s❼✩❛✁❵✜➈✗➂➃❵➁➄✍❛✁➄❢➅ ➊✢➄✍❵❢❼✹➈✗➂➃❛✁❵×➡➚➈✗➎❢❾❿➈❋➀❉❾✁❵✍➂➃➅✗➎✍❝✛➅❋❾❿➈➝➭✆→❒➱❂❾❿❵❢➆❙➭✆→ ➺✶➌➁❣➚➈✗➎✍❝➾❼■➎✍❛✬➂➃❼✩❝❤➡❿❴⑤→ ➡▲➑✄➭⑧❴➴➔✘➛➏Ï➞→ ➺✁Ò☞Ó✩Ó☞Ó✹Ò✔➣⑨↕❂➙✍➄✍❞✗➈✗➎✍❝☞❞✗➐➞❛✬❞✗❝✬➛ó➈✼➎✍❝✶➡❴ ❾✁❞✗❝❩➄❢❵✍➂➃÷✜➄✍❝✏✎ ❵✍❛➟❼■➎✍❛✬➂➃❼✩❝❩❝✹õ✍❼✩❝✩➇✒➈❋➡❴ → ➡✱➑✢➭❴ ➔ ➜➂➃❡❆❡ ➜❛✁❞✼ç▲↕♦ö➴➈❳➈✗➎➁➄❢➅❳➊✢❛✁❡➃❡❆❛➜➅❧➈✼➎❢❾❉➈❳➈✼➎✍❝❀❫◗ñ✯❾❿❞✼❝✯➂❆❵⑧➆✒❝✩❝✛➆✶❾❋➌❢❾✬➅✔➂➍➅☞↕ Õ➉➎✍❝☞❞✗❝✦❾✁❞✗❝❽➐➞❾✁❵➁❣❩❛❿➈✼➎✍❝✩❞➉➇▲❛✬➅✼➅✔➂➃➌✍❡➃❝❄❼■➎✍❛✬➂➃❼✩❝☞➅➏➊✢❛✁❞➉➌❢❾✬➅✔➂➍➅✑✎ ➜❝✯❝✩õ➁➇❢❡❆❛✬❞✗❝❄❾❃➇⑧❾❿❞✗➈✗➂➍❼✹➄✍❡➍❾❿❞✼❡❆❣ ➄❢➅✗❝✹➊✢➄✍❡➏❛✁❵✍❝➝➂➃❵➶❾➸❡➍❾❉➈✗❝☞❞✦❝✹õ✒❝✩❞■❼✹➂➍➅✗❝✁↕❤➬s❛➜❝☞➀✁❝☞❞☞➛✇➈✼➎✍❝❩❵✍❛✒➆✍❾✁❡✥❞✼❝✩➇✍❞✼❝☞➅✗❝✩❵✜➈✼❾❿➈✗➂➃❛✁❵➟❞✗❝☞➐➞❾✁➂❆❵⑧➅✯➈✗➎✍❝ ➐➞❛✬➅✔➈♦❼✩❛✁➐➞➐➞❛✁❵➋➛☞é❢❞■➅➘➈♦➌⑧❝✛❼✩❾❿➄⑧➅✔❝✥❛✁➊✒➈✗➎✍❝❧❼✹❛✬❵✜➀✬❝✩❵✍➂➃❝✩❵✜➈➋➂➃❵✜➈✗❝✩❞✼➇✍❞✼❝✹➈■❾❉➈✗➂➃❛✁❵❀❾✬➅➋❵✍❛✒➆✍❾✁❡✬➀❉❾✁❡❆➄✍❝✛➅✩➛❉❾✁❵❢➆ ➅✗❝☞❼✹❛✬❵❢➆❤➌⑧❝✛❼✩❾❿➄⑧➅✔❝✯❛❿➊♦➈✗➎❢❝✦➐➞❾❿➈✗❞✼➂✸õ✶➅✔➇❢❾✁❞✼➅✗➂❆➈➘❣➝➂❆❵❢➆✍➄❢❼✹❝✛➆❤➌➁❣➝➈✼➎✍❝✦➐➞➂❆❵❢➂❆➐➝❾❿❡◗❛❉➀✁❝✩❞✼❡➍❾❿➇❩➌▲❝✹➈➜❝✩❝☞❵ ➈✗➎❢❝❀❫ñ ↕ r
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
