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 Definitiî☎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