result is demonstrated most easily in indicial notation. In particular, we note auau dA dA ax ni ds uVu·ndS一 sIn ce we ar J(u+y=J(0)+2/ u VudA, VeX J(u)>J(u),V∈X,u≠u a is the minimizer of j(w) What PDEs admit such a inimization statement? PDEs associated with oper ators that are SPD (symmetric positive definite). We define this more precisely, and indicate how the FEM (inite element method) proceeds in the absence of this property, in a future lecture. For now, we focus on the simplest case almost all of which turns out to be directly relevant to the more general case We could also derive the result above by applying the general machinery of the calculus of variations. In this sense, we may view -Vu=f Euler o Euler- Lagrange equations as sociated with minimization of the functional J b Exercise 1 Consider the problem -xz=1,0<a<l, u(0)=u(1) 0, with solution (1-a). Show by explicit calculation that 8J(u) Jo urUz -U dr=0 for all (smooth) v such that v(0)=v(1)=0.B 2.3 Weak Formulation 2.3.1 Statement SLIDE 11 Find a∈ X such that
ã❬ä✕å♦æ✓çéè➙ê❛å❭ë❩ä✄ì✷í➂î➑å❡è❬ã❬ï❆è♦ä✕ëðì✷í✿å➊è➙ä✧ï➂å♦ê➏ç×ñ➡ê➏î❚ê×î❲ë❩ê❛ò✥ê❛ï❆ç❋î✓í➂è❬ï❫è❬ê➏í✿î✁ó❄ôqî❚õ❲ï➂ã➊è❬ê×ò✄æ✓ç×ï➂ã✕ö➔÷tä❴î❲í❆è♦ä è♦ø➑ï❫è ù úüû✎ý û➑þ✓ÿ û✁ û✎þ✓ÿ ✂☎✄ ✆ ù ú ✝ û û✎þ❩ÿ ✞ ý û✁ û✎þ✓ÿ ✟✡✠ ý û☞☛✌ û✎þ❩ÿ✕û✎þ❩ÿ ✍ ✂☎✄ ✆ ù ✎ ý û✁ û➑þ✓ÿ✑✏✒ ÿ ✂✔✓ ✠ ù ú ý û☛ û✎þ❩ÿ✧û✎þ✓ÿ ✂☎✄ ✆ ù ✎ ý✖✕✗✙✘ ✏✒ ✂✔✓ ✠ ù ú ý✖✕☛ ✂☎✄✛✚ ÷❺ø✓ä✕ã♦ä❋÷tä✘ø❲ï✢✜✿ä✛æ➑å➊ä✧ë✤✣➙ï❆æ❲å❬å✦✥★✧✒ø✓ä✄í✿ã♦ä✕ì è♦í❉ò✥í➂î✔✜✿ä✄ã♦è✎è❬ø✓ä✩✜✿í➂ç×æ✓ì✷ä❋ê➏î✻è❬ä✦✪➂ã❙ï❆ç❫ê×î✻è♦í❣ï❺å♦æ✓ã✬✫✶ï➂ò✄ä è♦ä✕ã♦ì➡ó✮✭❣í❆è❬ä❄÷tä❄ï➂ë❩í✿õ❩è❣è❬ø✓ä❴ò✄í➂î✔✜➂ä✕î✻è♦ê×í➂î➡í✯✫◆å♦æ✓ì✷ì✲ï❫è♦ê×í➂î➈í✰✜✿ä✄ã✆ã❬ä✄õ✎ä✕ï❆è♦ä✧ë➈ê➏î➑ë❩ê×ò✄ä✕å✕ö➑ø❲ä✄ã❬ä ✫✱ã❬í➂ì✲✱↔è♦í✙✳☛å♦ê➏î➑ò✥ä❭÷tä➙ï➂ã♦ä↔ê×î■ô✴ ☛ ó ✵✁✶☎✷✹✸✻✺✽✼✯✾ ✿❁❀✡❂✌ý ❃ ❄❆❅ ❇ ❈❊❉ ✆ ✿❁❀ ❉ ❂ ✱ ✳ ù ú ✕❄ý✗✘★✕❴ý ✂☎✄ ❃ ❄❆❅ ❇ ❋❍●❏■✦❑★▲ ▼❖◆P◆❘◗❚❙❯● ✚❲❱ ý✙❳❩❨ ❬ ✿❁❀P❭❉❫❪ ✿❴❀❉ ✚ ❱ ❭ ❳❵❨ ✚ ❭❜❛✆ ❝ ê×å❡❞❣❢✐❤sì✷ê➏î✓ê×ì✷ê❦❥✕ä✄ã✆í❧✫ ✿❁❀❣❭❉ ♠ ✱ ♥❢✐♦✰❞q♣❁r❚s✉t❡♦❧✈✯✇✗①✹❞②t❆③⑤④⑥❢❵♦⑦✇✡①✹⑧⑤①✹✇✡①❦⑨⑩♦✰❞❶①P❷✰⑧❵t❆❞❖♦✰❞❖❤✌✇❸❤✌⑧❹❞✦❺✗♣❴r❻s✉t❡♦✢t⑥t✦❷★④✦①P♦✰❞❖❤⑥✈✡❼❴①✹❞✹❢❵❷⑥❽✐❤✦❾❆❿ ♦✰❞❖❷✰❾⑥t❏❞❣❢✐♦✯❞②♦✯❾✬❤✮➀✖♣❴r➂➁❶t❆➃✰✇✗✇❸❤✌❞❶❾❆①P④❁❽✐❷✰t❆①✹❞➄①✹➅✰❤❡✈❧❤➄➆❁⑧❹①✹❞➇❤➉➈❧➊ ♥❤➋✈➌❤P➆✩⑧✁❤➋❞❣❢✔①➍t❏✇❸❷✰❾✬❤❴❽✁❾✬❤⑥④✦①➍t✦❤✌➎➏➃✢➐ ♦✰⑧☞✈✑①✹⑧☞✈✰①P④⑥♦✰❞❖❤❸❢✐❷✯❼➑❞✹❢⑤❤✡➒②s✉➓➔➁→➆✩⑧⑤①✹❞❖❤➣❤✦➎❦❤✦✇↔❤✦⑧⑤❞❚✇↔❤✦❞❣❢✐❷⑩✈⑥➈↔❽✁❾↕❷★④⑥❤⑥❤⑥✈✢t❸①✹⑧➙❞❣❢✐❤❵♦❧➛✌t✦❤✌⑧☞④⑥❤➣❷➝➜ ❞❣❢✔①➍t❡❽☞❾✬❷↕❽⑤❤✌❾❆❞❶➃✢➐✤①✹⑧➞♦✡➜⑥③✖❞➄③✖❾✬❤✙➎❦❤⑥④✦❞➄③✖❾✬❤⑩➊➟➒✻❷✯❾✙⑧☞❷✰❼②➐❡❼✩❤❻➜✌❷★④✦③✔t❩❷✯⑧➠❞❣❢✐❤➣t❆①✹✇✮❽✁➎❦❤✌t❆❞⑦④⑥♦✢t✦❤❍➡ ♦✰➎➢✇↔❷✰t⑥❞❫♦✰➎✹➎②❷➉➜❡❼q❢✖①P④↕❢✑❞❶③✖❾❆⑧✐t❻❷✯③✖❞✉❞❖❷➤➛⑥❤✡✈✰①✹❾✬❤⑥④✦❞➄➎➏➃✙❾✬❤✌➎➥❤✌➅✰♦✰⑧❹❞✉❞❖❷❸❞❣❢✐❤✤✇↔❷✯❾✬❤➋➦☎❤✦⑧✁❤✦❾✬♦✰➎②④↕♦✰t✦❤★➊ ♥❤✡④⑥❷✯③✖➎❦✈❵♦✰➎→t✦❷❍✈❧❤✦❾❆①✹➅✰❤✡❞✹❢⑤❤✡❾✬❤❆t❆③✖➎➏❞❏♦➌➛↕❷✯➅✢❤❸➛✦➃❩♦⑥❽❧❽☞➎➏➃✰①✹⑧☎➦❵❞❣❢✐❤❡➦☎❤✦⑧✁❤✦❾✬♦✰➎❘✇↔♦➌④↕❢✖①✹⑧✁❤✦❾❆➃❵❷➝➜✤❞❣❢✐❤ ④⑥♦✰➎❦④✦③✖➎➏③✔t➣❷➝➜❸➅✢♦✯❾❆①P♦✰❞❶①P❷✰⑧⑤t✌➊➨➧❆⑧➠❞❣❢✔①➍t❸t✦❤✌⑧⑤t✦❤❆➐❚❼✩❤✙✇❸♦✰➃➩➅★①P❤✦❼ ✠ ✕✗ ✆➭➫ ♦✢t➤❞❣❢✐❤❸s❲③✖➎➥❤✌❾➣❷✯❾ s❲③✖➎❦❤✦❾❆❿➄➯q♦✦➦❧❾✬♦✰⑧✔➦➌❤✡❤⑥➲✌③⑤♦✯❞➄①P❷✯⑧✐t⑦♦✢t⑥t✦❷★④✦①P♦✰❞❖❤⑥✈➣❼❴①✹❞✹❢❍✇✗①✹⑧❹①✹✇✗①❦⑨⑩♦✰❞❶①P❷✰⑧➟❷➉➜✤❞✹❢⑤❤✩➜⑥③✖⑧☞④✌❞❶①P❷✰⑧☞♦✰➎ ✿ ➊ ➳➸➵➋➺②➻✰➼➌➽✢➾P➚❧➻➙➪❯➶í➂î❲å♦ê×ë✓ä✄ã↔è♦ø✓ä✷õ✓ã❬í❧➹❲ç➏ä✕ì ✠ ✁➘⑩➘ ✆ ✱✿ö✻➴✑➷ þ ➷✛✱➂ö ❀ ➴ ❉ ✆ ❀ ✱ ❉ ✆ ➴✓ö❋÷❺ê➏è♦ø⑧å♦í➂ç×æ❩è♦ê×í➂î ✆➮➬ ☛ þ ❀ ✱ ✠ þ❉ ó➙➱✯ø✓í❫÷✛➹✯ñ✳ä✌✃❩õ✓ç×ê×ò✄êéè✲ò✄ï➂ç×ò✄æ✓ç❛ï❫è♦ê×í➂î➌è❬ø❲ï❫è❸❐ ✿ ◗ ❀❉ ✆ ❒ ➬ ● ☞➘❆ý✯➘ ✠ ý ✂þ ✆ ➴✗✫✱í✿ã❣ï❆ç×ç ❀ å♦ì✷í✻í➂è♦ø❉ ý å➊æ❲ò❙ø①è♦ø➑ï❫è ý ❀ ➴ ❉ ✆ ý ❀ ✱ ❉ ✆ ➴✓ó ❮Ï❰➄Ð ÑÓÒ✻Ô❘Õ➭Ö❫×❁Ø⑤ÙÛÚ✮Ü➉Ô❘Ý✐Þ➉×✉ß àâá➥ã✻áPä åâæ★ç✐æ★è✔éêè✔ë☞æ ✵✁✶☎✷✹✸✻✺✽✼✐✼ ìê➏î➑ë ❯❳➣❨ å➊æ➑ò❙ø❍è❬ø❲ï❫è í
0,Vu∈X vu:VUdA=/fudA,vu∈X see Slide 9 for proof. This equation has a great deal of structure which we cannot obviously see this explicit statement. We thus digress to some more general mathematical definitions so that we can present a more succinct restatement. Note that the eak formulation of a pDe, in which we introduce a test function u to absorb" some of the derivatives, will always exist (indeed is more general than the strong statement)even when no minimization principle is available that is, even when the problem is not SPD. The weak formulation is thus the most general point of departure for the finite element method Note 5 Du Bois-Reymond lemma In fact, we have already derived the weak statement: we know from Side 9 that if u satisfies-V2u=f in 9, ulr=0, then SJ,(u=0,vEX; the latter is simply (defined to be)the weak statemen We might ask whether we can go "the other way, that is, show that if E X satisfies SJ,(u)=0,VvE X, then u satisfies-V2u=f in Q. Yes: B lo Vu Vuda=/ vunds- UVu dA, ut-Vu-f dA= Vu∈X V-u-f does not equal zero at some point; we can then take v nonzero ed about this point, which contradicts dJ,(U)=0,VUE X that-V-u=f in Q; this is known(in certain circles)as the Du Bois-Rc 2.3.2 Definiti
î☎ï➌ð✖ñ❣òôó❁õ÷ö❫øúù✤û➤ü❵ý þ ÿ✁ ✂ò☎✄ ✂û✝✆✟✞ õ ÿ✠☛✡ û✝✆✟✞➟ø ù⑦û➤ü❵ý ☞ ✌✎✍✏✍✒✑✁✓✕✔✕✖✠✍✒✗✙✘✛✚✢✜✤✣✥✜✎✚✁✚✦✘★✧ ✩✝✪ ✫✭✬✁✮✰✯✲✱✴✳✶✵✸✷✺✹✻✮✽✼✺✾✿✬✥✷❀✯❁✷❃❂✦❄✎✱✴✷❀✹❆❅✦✱✴✷❀❇❈✼❊❉❋✯✴✹●❄❍✵✸■✏✹✻✵✠❄✎✱❑❏▲✬✠✮✽■▼✬◆❏❖✱✲■✴✷❀✾P✾◗✼❀✹❆✼✦❘✏❙❚✮✽✼✺✵✁✯✴❇❱❯❲✯✶✱✴✱❳✮❨✾ ✹✛✬✁✮✰✯✲✱✎❩✶❬✭❇❭✮✽■✶✮❨✹✙✯❍✹❪✷✺✹❪✱✏❫❴✱✶✾P✹❪❵❜❛❳✱✲✹❨✬✠✵✁✯❁❅✺✮❱❂✺❄▼✱❍✯✴✯❁✹❪✼❝✯✏✼❀❫❴✱❃❫❴✼❀❄✎✱❋❂✟✱✶✾✭✱✶❄✎✷✺❇❞❫❴✷❀✹✛✬✥✱✏❫❡✷✺✹✻✮✽■✴✷❀❇ ❅✦✱✻❢❣✾P✮❨✹✻✮✽✼✺✾✥✯❡✯✏✼❳✹✛✬✥✷❀✹✝❏❖✱☎■✴✷❀✾❁❬◗❄✎✱✶✯✏✱✶✾P✹❤✷❁❫❴✼❀❄▼✱❡✯❍✵✸■✴■✶✮❨✾✭■✶✹❞❄▼✱❍✯❍✹✐✷❀✹✐✱✶❫❴✱✶✾P✹❪❵☎❥❦✼❀✹✐✱❴✹✛✬✥✷✺✹❈✹✛✬✥✱ ❏❖✱✴✷❚❧♠❉✶✼✺❄❍❫✙✵✠❇♥✷✺✹✻✮✽✼✺✾✲✼★❉❤✷♣♦❣q❤rts✉✮❨✾❳❏▲✬✁✮✽■✴✬❑❏❖✱❈✮❨✾P✹✻❄▼✼❚❅✺✵✸■▼✱✈✷❦✹✐✱❍✯❍✹✠❉✴✵✠✾◗■✏✹✻✮✽✼✺✾ û ✹✐✼✿✇❪✷✢❘❍✯✏✼✺❄✏❘✟① ✯✏✼❀❫❴✱♣✼★❉✤✹✛✬✥✱❤❅✦✱✏❄❍✮❨❙②✷✺✹✻✮❨❙❀✱✶✯▼s③❏③✮❨❇❨❇✭✷❀❇❭❏❖✷✺❯②✯❤✱✎❩✢✮✰✯❍✹♠④●✮❨✾✭❅✦✱✴✱✴❅✒✮✰✯❞❫❴✼❀❄▼✱t❂✟✱✶✾✭✱✶❄▼✷❀❇✸✹✛✬✥✷✺✾☎✹✛✬✥✱❈✯❍✹✻❄▼✼❀✾✟❂ ✯❍✹❪✷✺✹❪✱✏❫❴✱✶✾P✹❱⑤⑥✱✏❙❀✱✶✾⑦❏▲✬✥✱✏✾◆✾◗✼⑥❫❆✮❨✾✸✮❨❫❆✮♥⑧⑨✷❀✹●✮✽✼❀✾❲❬✭❄❍✮❨✾◗■✏✮❬◗❇♥✱❃✮✰✯❳✷✺❙②✷✺✮❨❇♥✷✦❘✏❇♥✱❲⑩❶✹✛✬✥✷✺✹✒✮✰✯✴s✙✱✶❙❀✱✶✾ ❏▲✬✥✱✏✾❷✹✛✬✥✱✈❬◗❄▼✼✦❘✏❇♥✱✶❫❸✮✰✯❴✾◗✼✺✹❞❹✥♦③q✙❵✒✫✭✬✸✱☎❏❖✱✴✷⑨❧✒❉✶✼✺❄❍❫✙✵✠❇♥✷✺✹✻✮✽✼✺✾✿✮✰✯❴✹✛✬✁✵✁✯❡✹❨✬✸✱❺❫❴✼②✯❍✹✤❂✢✱✏✾◗✱✏❄✎✷✺❇ ❬✸✼❀✮❨✾✸✹✤✼★❉✙❅✦✱✐❬✥✷✺❄❍✹✻✵✠❄✎✱t❉✶✼❀❄✈✹✛✬✥✱t❢❣✾P✮❨✹❪✱✙✱✶❇✕✱✶❫❴✱✶✾P✹t❫❴✱✏✹❨✬✸✼❚❅✢❵ ❻⑥❼❾❽❍❿❃➀ ➁⑥➂❲➃☎❼✁➄✻➅⑨➆❊➇❁❿⑨➈▲➉⑦❼✠➊❣➋⑥➌✐❿✺➉➍➉❷➎ ➏✐➐ ✘✽➑✢➒❍➓❚➔P→t✍✈➣P➑②↔✦✍✈➑✺✓✕✜✎✍⑨➑✦✖✠↕❑✖✠✍✏✜▼✔♥↔✢✍❚✖❑➓✎➣✸✍✈→t✍⑨➑✺➙❑✌❊➓▼➑✺➓✎✍❚➛❆✍➐➓❚➜❣→♠✍✈➙➐✚❀→➝✘✛✜▼✚✦➛➞✑✁✓✕✔✕✖✠✍✙✗ ➓✎➣P➑❀➓✤✔❱✘ ò ✌✎➑✺➓✎✔✰✌★➟✸✍⑨✌❤➠ ✂✙➡ ò❩õ ✡ ✔➐❁➢ ➔ ò③➤ ➥➤õ÷ö ➔✁➓✎➣✥✍➐ î☎ïð ñPò☞ó✩õ➞ö ➔ ù❡û➤ü❩ý➍☞ ➓✎➣✥✍♣✓✰➑❀➓✎➓✎✍✏✜ ✔✰✌✤✌❊✔✕➛❡✣✥✓♥↕ ñ✖✠✍✏➟➐✍⑨✖☎➓▼✚❴➦P✍ ó ➓✎➣✥✍✈→♠✍❚➑✦➙☎✌❊➓▼➑❀➓▼✍✏➛❡✍➐➓⑨✧ ➧✍✲➛❡✔✕➨✦➣✟➓❑➑✦✌✎➙☛→✤➣✥✍✶➓▼➣✥✍✏✜❑→t✍✲➒❚➑➐ ➨✢✚➫➩★➓✎➣✸✍✲✚✺➓▼➣✥✍✏✜☎→♠➑②↕✢➔ ➭❷➓✎➣✸➑✺➓❑✔✰✌✏➔❞✌✎➣✥✚❀→➯➓✎➣✸➑✺➓❑✔♥✘ ò➨ü❯ý ✌▼➑❀➓✎✔✰✌❊➟✸✍❚✌ î☎ïð ñ❣òôó❲õ ö ➔ ù⑦û❍ü✑ý ➔✥➓✎➣✸✍➐ ò ✌✎➑✺➓✎✔✰✌★➟✸✍⑨✌✈➠ ✂❆➡ ò❯õ ✡ ✔➐⑥➢ ✧➳➲❣✍❚✌❚➜➳➵♠↕ ✔➐➓▼✍✏➨✢✜▼➑✺➓✎✔✕✚➐ ➦✁↕❑✣✸➑✺✜✎➓▼✌♠→♠✍✈➙➐✚❀→◆➓✎➣✸➑✺➓ ÿ ✂ò☎✄ ✂û❈✆✟✞ õ ÿ ➥➸û ö ✂ò❺✄P➺➻ ✆✟➼ ➠ ÿ û✂➡ ò❦✆✟✞ ø ➑➐✖❺➓▼➣✟➽P✌ ÿ✁ ✂ò☎✄ ✂û ➠ ✡ û❈✆✟✞ õ ÿ✁ ûP➾ ➠ ✂➡ ò ➠ ✡▲➚ ✆✟✞ õ÷ö⑤ø ù✤û✙ü❩ý➶➪ ✩❞✚❀→☛➑✦✌▼✌❊➽✸➛❆✍➳➓▼➣✸➑❀➓❞➠ ✂❆➡ ò ➠ ✡ ✖✠✚✁✍❚✌ ➐✚✺➓❣✍⑨➹✢➽P➑✺✓✠➘✏✍❚✜✎✚✈➑✺➓❖✌❊✚✢➛❆✍➳✣◗✚✦✔➐➓ ☞ →t✍✤➒✏➑➐ ➓✎➣✥✍➐ ➓✴➑✺➙✢✍ û ➐✚➐➘✏✍❚✜✎✚❳✓✕✚✠➒✏➑✺✓✕✔✕➘✏✍❚✖❲➑✦➦P✚✢➽✠➓✈➓✎➣✥✔✰✌✈✣◗✚✦✔➐➓❚➔✉→✤➣✥✔✕➒✴➣❝➒✏✚➐➓✎✜✴➑✦✖✠✔✰➒❍➓✴✌ î☎ïð ñ❣òôó✤õ ö ➔ ù❸û➟ü✽ý ✧ ➧✍✝➓▼➣✁➽✸✌✤➒✶✚➐➒✶✓✕➽✸✖✠✍❤➓✎➣P➑❀➓✈➠ ✂❆➡ ò➣õ ✡ ✔➐❳➢ ☞ ➓▼➣✥✔✕✌➳✔✰✌➳➙➐✚❀→➐ ñ ✔➐ ➒✶✍❚✜❊➓✴➑✺✔➐ ➒✏✔♥✜✴➒✶✓✕✍❚✌ ó ➑✦✌t➓✎➣✥✍ ➴➽❳➵♠✚✢✔✕✌❊➷❪➬❞✍✏↕✁➛❡✚➐✖☎✓♥✍❚➛❆➛❴➑✸✧ ➮✃➱✕❐❒➱✰➮ ❮❋❰✟Ï✉Ð✉Ñ✛Ò⑨Ñ✽Ó◗Ð▲Ô Õ◗Ö✟×❨Ø❒Ù✿Ú✦Û Ü
Linear space. A set Y is a linear (or vector ) space vu1,v2∈Y, va∈R,u∈Y, aU∈Y Linear forms, L(u) L:、Y.,→、R,( form or functional) L(av +U2)=al(on)+l(u2)(linear) Va∈R,V Bilinear forms, B(w, v) B(w, t)linear form in w for fixed D B(U, u)linear form in v for fixed w(bilinear) Note that B:YxZ,R indicates that b has two inputs (arguments the first from the space y, the second from the space Z; the output is a real number. SLIDE 15 B(U,U)=B(U, w) SPI B(,w)>0 ≠0SPD 2.3.3 Restatement SLIDE 16 a(w, v=
ÝßÞ❨à✭á✴â❀ã♣ä●å✥â✢æ✴á✏çPè❴é ê➫ë✎ì✶í èïî ë✤ð❡ñ î✕òì❚ð✦ó✈ô✛õ✦ó✤ö✢ì❚÷✶í✎õ✦ó❍ø♠ë❊ùPð✦÷✶ì î♥ú û✒ü✢ý❀þ✎ü✺ÿ✁ èþ ü✢ý✄✂❝ü✺ÿ☎ è û✝✆✞✠✟✡✒þ û❦ü☛ èþ ✆▲ü☞ è ✌ ✍✏✎✒✑✔✓✖✕✘✗✚✙ ÝßÞ❨à◗á✴â✺ã✜✛✣✢❀ã✥✤✙ä✢ç✧✦ ôü ø é ✦❈é è ★✪✩✣✫✪✬ ✭ ✮✰✯✲✱✣✳ ✴ ✟✡ ★✪✩✣✫✪✬ ✵ ✱✣✳✶✯✰✱✣✳ ô✛✣✢❀ã✥✤ õ✦ó ✛✸✷✠à✭æ✣✹●Þ✶✢❀à✭â✻✺ ø ✦ ô✆▲ü✢ý✜✂❝ü✺ÿ ø✜✼ ✆✦ ôü✢ý ø ✂ ✦ ôü✺ÿ ø ô ✺❱Þ❨à◗á✴â❀ã ø û✝✆✞✠✟✡❲þ û✒ü✢ý✺þ❊ü✺ÿ✁ è✾✽ ✍✏✎✒✑✔✓✖✕✘✗✻✿ ❀❞Þ✔✺❭Þ❨à◗á✴â❀ã❁✛✣✢❀ã✥✤✙ä✶ç✧❂ ô✶❃ þ❊ü ø é ❂☎é❞è❅❄❇❆ ✴ ✟✡ ô✛✣✢❀ã✥✤ø❉❈ ❂ ô❊❃ þ ü ø♠ñ î✕òì❚ð✦ó úõ✦ó●❋ î♥ò ❃ úõ✢ó■❍❑❏✠ì✰▲ ü ç ❂ ô❃ þ✎ü ø♠ñ î✕òì❚ð✦ó úõ✦ó●❋ î♥ò ü úõ✢ó■❍❑❏✠ì✰▲ ❃ ô●▼ Þ✔✺❱Þ❨à◗á✴â❀ã ø ✌ ◆❖✢P✹❪á❉✹❊◗✥âP✹❘❂❺é✤è❙❄❚❆ ✴ ✟✡ Þ❨à✏❯✺Þ✽æ✴â✻✹✐á❍ä❉✹✔◗✸â✻✹❘❂❱◗✥â❀ä❲✹❨❳✄✢❦Þ❨à✺å✏✷❩✹✻ä❖❬★â❀ã❪❭✚✷❩✤❴á✶à❫✹✽ä❵❴❜❛❝✹❊◗✥á❡❞❖ã✴ä✸✹ ✛✴ã❢✢P✤❣✹❊◗✥á✒ä●å✥â✢æ▼á❤è❤❛✜✹❊◗✥á✒ä✏á✴æ●✢✺à✏❯✐✛✴ã❢✢✻✤❥✹❊◗✥á✒ä✻å✸â✦æ✴á✁❆✁❦✜✹❊◗✥á❧✢✻✷❩✹❱å✏✷❩✹❖Þ✰ä✙â❴ã✎á✴â✻✺❒à✧✷❩✤▼á✏ã✲♠ ✍✏✎✒✑✔✓✖✕✘✗✚♥ ♦❩♣✜q❱rî ñ î♥òì⑨ð✺ó úõ✢ó❢❋❴ë ç❑❂ ô❊❃ þ❊ü ø é ❂☎é❞è❅❄❋è ✴ ✟✡ î ës▼ Þ✔✺❭Þ❨à✭á▼â✺ã ❈ ❂ ô❊❃ þ✎ü øt✼ ❂ ô üPþ ❃✝ø ♦✈✉❁✇ ❈ ❂ ô❊❃ þ ❃✝ø②①④③ ç û ❃ è➞ç ❃❅⑤✼⑥③✾⑦♣⑧q ✌ ⑨❘⑩❷❶✖⑩❷❶ ❸❺❹❼❻✲❽✪❾❩❽✰❹❼❿➀❹❼➁➂❽ ✍✏✎✒✑✔✓✖✕✘✗✚➃ ➄ì✏í ➅ ô❊❃ þ✎ü ø❁✼➇➆❼➈⑥➉✝❃⑥➊✰➉ü❲➋✒➌❝þ û ❃ þ✎ü☞➎➍ ➏
SPD bili fvdA,u∈X b Exercise 2 prove that a is indeed an spd bilinear form over X hint. ou must use the boundary conditions. (Note a is SPD because the underlying Minimization prine gm要 k sta (u,0)=(u),VU∈X (a) Show that if J:Y+R is defined by J(w)=sa(w, w)-t(w)for any SPD bilinear for form e over Y, then the mi ay, g “ anti-variation” to find J) (b) Take Y= R, and thus show, by appropriate choice of a and e, that the minimizer u E y of J()=5wGw-wTF- for any SPD matrix G∈R ndF∈R ∈X ince a involves only first derivatives {u∈H2(9) 0}≡H0() H()={1/v2dA,/v2d
➐P➑➓➒❩➔⑧→✾➣✣↔✔↕➙↔✔➛✏➜✸➝✻➞✄➟✣➠P➞✥➡ ➢❁➤ ➐P➑❫➥ ➦✚➧✶➨❼➩✜➫❙➭❼➯⑥➲s➨❉➳➸➵❖➺➼➻s➨☞➽➎➾ ➐❺↕➙↔✔➛✏➜✸➝✻➞✄➟✣➠P➞✥➡➪➚ ➶➘➹❲➴❡➷✻➬➸➮❜➱✶✃✚➷➘❐❮❒✜❰●Ï✻Ð✚Ñ❖Ò❢Ó➐Ò❖Ô❇Õ❷ÖsÕ➑✧➥Ñ✲Ñ➥❮➐P➑✞×❒❁ØÚÙ❑Õ❷ÛÜÕ➑Ñ➐❰✁Ý❊Ï✚❰●ÞßÏ✻Ð✚Ñ✰❰ ➾ ➚❚às↔✔➛✧á✣â ãÏ➸ä☞Þ✝ä✧ÖåÒæä❫ÖçÑ❉Ò❢Ó✧Ñ❉Ù✏Ï✚ä➑✧➥❑➐❰ã☞èÏ➑✧➥ÕÜÒ❢Õ❷Ï➑Ö ➚ ➧✶éÏPÒ●Ñ✁Ô☛Õ❷Ö ×❒❁ØêÙ✏Ñè➐ä✧Ö❢Ñ✐Ò●Ó❑Ñ☎ä➑✧➥Ñ✰❰❢ÛãÕ➑✧ë Ï✚ì✏Ñ✲❰➐Ò●Ï✚❰■ÕíÖ ×❒✄Ø ➚ ➩ î ï✒ð✔ñ✖ò✘ó✒ô õÕ➑Õ❷Þ❧Õ❷ö➐Ò❢Õ❷Ï➑ ❒✜❰●Õ➑èÕ❷ì❑Û❷Ñ â÷ ➫ ➐❰ë Þ❧Õ➑ ø✈ù✚úüû➤ Ô ➧❊ý❖➺çý❲➩✜þÿ➦✚➧✶ý❲➩ ✁✄✂ ☎ ✆✞✝ø✠✟ ✡ ☛Ñ➐✌☞➎×Ò ➐Ò❢Ñ✲Þ❤Ñ➑Ò â ÷ ➽❺➾ ✍ Ô ➧÷ ➺❢➨❼➩✜➫⑥➦✚➧❊➨❩➩ ✁✄✂ ☎ ✎✑✏ ✆✓✒✔✝✖✕✟✘✗✠✙ ➺ ➻☎➨❺➽❺➾ ✡ ➢✛✚ ➶✠➹❲➴❡➷✻➬➸➮❜➱✶✃✚➷✢✜ ➧ ➐ ➩ ×Ó❑Ï✤✣✞Ò●Ó➐Ò✜ÕÜÝ✦✥ â★✧✪✩✬✫✭ Õ❷Ö ➥Ñ✓✮➑Ñ➥ Ùã ✥ ➧❊ý❉➩❁➫ ✯ ✰ Ô ➧❊ýs➺❢ý❲➩✒þ✝➦✚➧❊ý❲➩ Ý❊Ï➸❰ ➝✻➛✲✱❖×❒✄Ø Ù❑ÕÜÛ❷Õ➑Ñ➐❰✁Ý❊Ï✚❰●Þ➪Ô ➐✚➑✧➥ ÛÜÕ➑Ñ➐❰✁Ý❊Ï✚❰●Þ ➦ Ï✻Ð✚Ñ✲❰ ✧ ✍ Ò❢Ó✧Ñ➑ Ò❢Ó❑Ñ☞Þ❧Õ➑Õ❷Þ❤ÕÜö✰Ñ✲❰ ÷ Ö ➐Ò●Õ❷Ö✳✮✧Ñ✰Ö Ô ➧÷ ➺❢➨❼➩s➫❅➦✚➧❊➨❩➩ ✍ ➻❚➨ ➽ ✧❚➚ ➧ ✫➑ Ò❢Ó❑ÕíÖ✴✣➐ã ✍ ëÕ❷Ð✚Ñ➑➘➐ ✣tÑ➐✵☞ ÖçÒ ➐Ò●Ñ✲Þ❤Ñ➑Ò ✍ Ï➑Ñ è➐✚➑ ✶➐P➑Ò●Õ✸✷❵Ð➐❰●Õ➐Ò❢Õ❷Ï➑✺✹ Ò●Ï✻✮➑✧➥ ✥ ➚ ➩ ➧Ù ➩ ✼➐✵☞Ñ ✧ ➫ ✫✭✾✽ ✍ ➐✚➑✧➥ Ò❢Ó❼ä✧Ö☛Ö❢Ó❑Ï✤✣✍ Ùã ➐ì❑ì❑❰●Ï✚ì❑❰●Õ➐Ò❢Ñ èÓ❑Ï➸ÕèÑ➎Ï✚Ý❉Ô ➐P➑✧➥ ➦ ✍ Ò●Ó➐Ò Ò❢Ó❑Ñ❧Þ❤Õ➑Õ❷Þ❤ÕÜö✰Ñ✲❰ ÷ ➽ ✧ Ï✚Ý✿✥ ➧❊ý❲➩✁➫ ✯ ✰ ý ❀❂❁ý þ ý ❀✦❃✑❄ Ý❊Ï✚❰ ➐P➑ã ×❒✄Ø❅Þ➐Ò❢❰●Õ✸❅ ❁ ➽ ✫✭✽✲❆❇✽ ➐P➑❫➥ ❃ ➽ ✫✭✽ ❄ Ö ➐Ò●Õ❷Ö✳✮✧Ñ✰Ö ❁ ÷ ➫ ❃ ➚ ❈❊❉●❋❍❉✖■ ❏▲❑✤▼✲◆✛❖P❑❘◗❊◆❂❙❯❚❱❖✞❲✵❳ ÷ ➽✠➾ î ï✒ð✔ñ✖ò✘ó✵❨ ×Õ➑èÑsÔ☛Õ➑Ð✚Ï➸ÛÜÐ➸Ñ✰Ö ➠✻➛❫↕❩✱✿❬✜➞❪❭✥á❴❫✚➜✲➞✥↔❛❵❜➝Pá❪↔❛❵✻➜✄❭ ➾ß➫ ✍ ❜ ➨☛➽❘❝✯ ➧❡❞②➩❣❢❩➨❤❢ ✐ ➫❦❥❇❧ ♠ ❝ ✯ ✙ ➧♥❞②➩ â ❝ ✯ ➧❡❞②➩ ♠ ❜ ➨♦❢❘➭❑➯ ➨ ✰ ➳✒➵✞➺❲➭❼➯☎➨ ✰♣ ➳✒➵✞➺❲➭❼➯ ➨ ✰ q ➳➸➵ ✮➑Õ➙Ò●Ñ ❧ r s
(u, ma-J vu Vu+uedA AllYl(s) Vu2+w2 dA N6 E4 Important theoretical and numerical implications Important spaces, inner products, and norms Hilbert and Banach Spaces A Hilbert space is a linear space Y with which we associate an inner product Dr this is simply an SPD bilinear form which then induces a norm, lolly =(u, /2. In fact, what we have just described is an inner product space: a hilbert space is a complete inner product space; by completeness we mean that any Cauchy sequence(n∈ Y such that lyn-ymly→0asm,m→ oo) converges to a member of y A Hilbert space is a special case of a Banach space Z, which is a(complete) ormed linear space. The norm z associated with a Banach space is not, in her l, induced from any bilinear form, but must still satisfy certain conditions (the conditions we intuitively associate with any measure of "length") ∈2,u≠0 lawllz al llwlla va∈R,Vu∈ l+叫ll≤‖llx+lxyu∈z,vu∈z the last being the triangle inequality(the shortest distance between two points It can readily be shown that a norm induced by an inner product automat ically satisfies the above conditions. The triangle inequality is proven with the help of the Cauchy-Schwarz inequality, which states that for an inner product (u,y)y≤ Jully lully We give the proof here 0< (u,)y (,) l =(x-2)y l lolly
t✈✉✴✇✳①❇②④③⑥⑤⑧⑦✖⑨✺⑩ ❶ ❷✄❸ ❹ ❺❻❼❻✓❽❿❾✲➀✓❾✘➁✄➂❼➃✓➄➆➅ ➇ ➈⑨➊➉✉➌➋ ➉ ①▲➍➎✉✿①➐➏P➑✑➒ ➓⑧✉➔➓⑧③⑤ ⑦✖⑨✺⑩ ❶ ❷✄❸ ❹ ❻❼➁→❾✈➣ ➇ ↔➈⑨➙↕ ➉ ✉ ↕ ➛ ➍➜✉➛ ➏❱➑✾➝▲➞→➟ ➛➡➠ ➢✾➤ ➥➧➦ ➨✄➩✾➫✺➭✤➯✄➲❿➳✤➵✺➲✦➸→➺✺➻✓➼✵➽➾➻⑧➸➾➚●➪❼➶✌➹➘➳✌➵➷➴➊➬✞➮❯➱✻➻✓➽➾➚●➪❼➶✌➹❐✃❛➩✿➫❤❒❩✃✈❮❪➳✤➲♥✃✈➭✤➵✺❰⑧Ï ÐÒÑ✠Ó✄Ô♦Õ Ö✓×ÒØ✾Ñ❇ÙPÓ➾Ú❇Û★ÓÝÜ✄Ø✛Ú➷Þ✵Ô✌Ü❱ß❂à❡Û✦Û✛Ô✌ÙáØ➧Ù✵Ñ❍â★ã❊ÞPÓ→Ü✵ß★Ú❇Û✛â➡Û➧Ñ❇Ùä×➎Ü å❣✃❛❒●æ➾ç✓➯✄➲⑥➳✤➵➷➴áè✾➳✌➵➷➳❱❮➾é✢ê✌➫✺➳✵❮❪ç⑧❰ ëíì➚✖➹●î➷➻✓➽→➸ðï→ñ✺➶✵➪✓➻⑥➚●ï➘➶✻❒❩✃❛➵➷ç❪➳✤➯✾❰♥➫❯➳❱❮➾çÝòôó⑥➚✖➸→➺õó⑥➺✺➚●➪❪➺õóð➻✾➶✵ï➾ï→➼✞➪✓➚●➶✌➸→➻⑥➶✌➬✢✃❛➵✲➵➷ç✓➯ö➫➷➯➾➭❼➴✌÷✺❮⑧➲ t④➋●✇✓➋ø②✳ùûú ➸➾➺❯➚●ï➊➚●ï➔ï✳➚●➱✻ñ❯➹●üÒ➶✵➬➜ý❯þ✛ÿ î✺➚✖➹●➚✖➬✺➻❼➶✌➽ ✁➼✵➽➾➱ ú ó⑥➺❯➚●➪❪➺ ➸➾➺❯➻✓➬➜➚●➬✄✂❇➮✲➪⑧➻❼ï➔➶ ➵➷➭✌➯✄➩✆☎ ➓✄✉á➓ ù✞✝ t✘✉✴✇→✉➐② ➞→➟ ➛✠✟☛✡➬ ✁➶❱➪✄➸☞☎★ó⑥➺✲➶✤➸➔óð➻➙➺✺➶✍✌❱➻✏✎④➮✺ï✳➸✑✂❯➻❼ï➾➪⑧➽➾➚✖î➷➻☞✂➜➚ ïá➶✌➬➎➚●➬❯➬❯➻❼➽➊ñ❯➽➾➼✒✂❇➮✺➪✄➸ ï→ñ✺➶✵➪✓➻✔✓▲➶ ì➚✖➹●î✲➻❼➽✳➸✴ï→ñ✺➶✵➪✓➻➔➚ ï❣➶ ❮➾➭✌➩✿➫❤❒✖ç✓➲➆ç➙➚✖➬❯➬✺➻✓➽❣ñ❯➽➾➼✒✂❇➮✺➪✄➸✴ï→ñ✺➶✵➪✓➻ ➒ î✞ü ➪✓➼✵➱✻ñ❯➹●➻⑧➸→➻❼➬❯➻❼ï➾ï▲óð➻ ➱✻➻❼➶✵➬➙➸→➺✲➶✤➸✿➶✌➬✞ü✖✕ð➶✌➮✲➪❪➺Pü➙ï→➻☞✗P➮❯➻❼➬✺➪⑧➻ t✙✘✠✚✜✛ ò✑ï✳➮✺➪❪➺ ➸→➺✺➶✌➸ ➓✢✘✔✚✏✣✤✘✔✥➔➓ ù✧✦✩★ ➶✵ï✫✪ ✇✭✬ ✦ ✮② ➪⑧➼✵➬✯✌❱➻✓➽✱✰✵➻❼ï✛➸→➼õ➶✻➱✻➻✓➱➔î✲➻❼➽⑥➼✁ ò ✟ ëûì➚●➹●î✲➻❼➽✳➸⑥ï✳ñ✲➶✵➪⑧➻Ý➚ ïð➶áï→ñ✲➻✔➪⑧➚➶✌➹❤➪✓➶✵ï→➻Ý➼✁ ➶✳✲❴➶✌➬✺➶❱➪❪➺➙ï✳ñ✺➶❱➪⑧➻✵✴✏☎✞ó⑥➺❯➚ ➪❪➺ ➚ ï❴➶ t ➪⑧➼❱➱áñ✺➹✖➻✓➸→➻ ② ➵➷➭✌➯✄➩✻ç❪➴➙➹✖➚●➬❯➻❼➶✵➽➘ï→ñ✺➶✵➪✓➻ ✟✷✶➺❯➻Ý➬❯➼✵➽➾➱ ➓ö➋✔➓☞✸ ➶✵ï➾ï✳➼❇➪⑧➚➶✤➸➾➻☞✂✻ó⑥➚✖➸→➺❘➶✹✲❴➶✌➬✺➶❱➪❪➺ ï→ñ✺➶✵➪✓➻✾➚ ï✛➬❯➼✵➸☞☎✞➚●➬ ✰✵➻❼➬❯➻✓➽❪➶✌➹✺☎✤➚●➬✄✂❇➮✺➪✓➻☞✂ ✁➽→➼❱➱ ➶✵➬Pü➊î❯➚✖➹●➚●➬❯➻❼➶✵➽ ✁➼✵➽➾➱✻☎✵î❯➮❇➸✛➱➔➮✺ï④➸➧ï✳➸→➚●➹✖➹✲ï→➶✌➸→➚ ï✁üá➪⑧➻✓➽→➸➾➶✵➚✖➬✻➪⑧➼❱➬✄✂❇➚✖➸→➚●➼✵➬✺ï t ➸➾➺❯➻✴➪⑧➼❱➬✄✂❇➚✖➸→➚●➼✵➬✺ï❴óð➻❣➚●➬❱➸➾➮❯➚✖➸→➚✼✌✵➻✓➹●ü ➶❱ï→ï→➼❇➪⑧➚➶✤➸→➻Ýó⑥➚✖➸→➺ ➶✌➬✞üõ➱✻➻❼➶❱ï✳➮✺➽→➻▲➼✁✾✽ ➹●➻✓➬✿✰✵➸→➺✄❀ ② ✓ ➓✄✉➔➓ ✸ ❁ ★ ❂ ✉❃✛ ✴✇ð✉✞❄➇ ★ ✇ ➓❆❅✦✉➔➓ ✸ ➇ ↕ ❅ ↕ ➓✄✉á➓ ✸ ✇ ❂ ❅✧✛ ✡❇ ✇ ❂ ✉❈✛ ✴ ✇ ➓✄✉ ➍➎①❤➓☞✸ ❉ ➓⑧✉➔➓❊✸❘➍❦➓⑧①❤➓☞✸ ❂ ✉❃✛ ✴✇ ❂ ①✜✛ ✴ ✇ ➸→➺✺➻❣➹●➶❱ï④➸❴î✲➻❼➚✖➬✄✰✻➸→➺❯➻Ý➸→➽➾➚➶✌➬✿✰❱➹✖➻Ý➚✖➬✺➻☞✗P➮✺➶✌➹●➚✖➸④ü t ➸➾➺❯➻✴ï✳➺✺➼✵➽→➸→➻❼ï✳➸❋✂❇➚●ï✳➸➾➶✵➬✺➪⑧➻▲î➷➻⑧➸④óð➻✓➻❼➬ ➸④óð➼➔ñ➷➼✵➚●➬P➸➾ï ✟❊✟☞✟ ② ✟ ✡ ➸➐➪❼➶✌➬❘➽→➻✔➶✔✂❇➚●➹●ü➙î✲➻✴ï→➺❯➼✤ó⑥➬❘➸→➺✲➶✤➸✾➶á➬❯➼✵➽➾➱ ➚●➬✄✂❇➮✲➪⑧➻☞✂❘î✞ü ➶✵➬ ➚●➬❯➬❯➻❼➽✿ñ❯➽➾➼✒✂❇➮✺➪✄➸✾➶✵➮❇➸→➼❱➱õ➶✤➸✭● ➚ ➪✓➶✌➹●➹●ü ï➾➶✤➸→➚ ï✭❍✺➻❼ï❴➸➾➺❯➻➊➶✵î✲➼■✌❱➻▲➪✓➼✵➬✄✂❯➚✸➸➾➚✖➼❱➬✺ï ✟❏✶➺❯➻▲➸➾➽→➚➶✌➬✿✰❱➹✖➻❣➚●➬❯➻❑✗❱➮✲➶✌➹●➚✸➸④ü ➚●ï✾ñ❯➽→➼■✌❱➻✓➬❘ó⑥➚✸➸➾➺ ➸→➺❯➻ ➺❯➻❼➹✖ñ ➼✁ ➸➾➺❯➻▲✕ð➶✵➮✺➪❪➺✞ü▼●❿ý❇➪❪➺✞ó❴➶✌➽✱◆▲➚✖➬❯➻❑✗P➮✺➶✌➹●➚✸➸④ü✠☎❤ó⑥➺❯➚●➪❪➺Òï④➸❪➶✤➸➾➻❼ï✾➸→➺✺➶✌➸ ✁➼❱➽Ý➶✵➬♦➚●➬❯➬❯➻✓➽Ýñ❯➽➾➼✒✂❇➮✺➪✄➸ t④➋●✇✓➋ø②✳ù ☎ t✈✉✴✇✳①❇②✳ù❖❉û➓⑧✉➔➓✓ù➜➓⑧①❤➓❼ù ➠ PÒ➻✏✰✵➚✼✌✵➻Ý➸→➺❯➻❣ñ✺➽→➼✞➼✁ ➺❯➻✓➽➾➻✔✓ ★ ❉❘◗◗ ◗ ◗ ✉❙✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ① ❚◗◗ ◗ ◗ ➛ ù ➇ ↔✉❙✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ①➷✇✳✉❖✣ t✘✉✴✇→①✞② ù ➓✄①✠➓ ➛ù ①➝ ù ➇ ➓✄✉➔➓ ➛ù ✣❱❯ t✘✉➊✇✳①❇② ➛ù ➓✄①✠➓ ➛ù ➍ t✘✉✴✇→①✞② ➛ù ➓✄①✠➓ ➛ù ➇ ➓✄✉➔➓ ➛ù ✣ t✈✉✴✇✳①❇② ➛ù ➓✄①✠➓ ➛ù ➒ ❲