(u,)=(u),v∈X SLIdE 3 where ∈H(9)vr=0}≡Hb(2 Vw. Vo de Recall that f need not be in L(Q2): we can consider any e EH-(Q2),for cample a "line source"e(u)= 1 on Te, 0 elsewhere, where Te is some line Q.(Note, he that the delta distributi H-1(2)frg∈R2.) 1.2 Regularity SLIDE 4 If L( Q)and Q is convex Allez2(a)≤C∫ portant for convergence rate tullia =fal vul2+u2 dA, and that e(o ul Note also that in IR the H- norm includes the square of the cross derivatives regula In one space dimension it was sufficient that f be suitably smooth to ensure that u would be in H-( Q). In two space dimensions that is no longer true, due to the potential conflict between what the boundary data tells the derivatives to do and what the equation tells the derivatives to do. however, in the case in which the domain Q is convex, then fEL2(9)is sufficient to ensure that u is in H-(Q)✘✚✙✄✛✢✜✤✣✏✥✧✦✩★✪✙✫✣✬✥☞✜✮✭✯✣✱✰✳✲✵✴ ✶✯✷✁✸✺✹✼✻✾✽ ✿❁❀❃❂❅❄✝❂ ✲❆✦❈❇❉✣✱✰✗❊✟❋●✙■❍❏✥✧❑☞✣✚❑ ▲▼✦❖◆❃P❘◗❙❊✟❋❚ ✙■❍❏✥❯✜ ✘✯✙✫❱✆✜✤✣✏✥❲✦ ❳✬❨✆❩❬❱❖❭❉❩❬✣❫❪✁❴ ❵❃❛❝❜❞✜ ★✪✙✫✣✬✥❡✦ ❢❣❳❨✆❤ ✣❫❪✁❴✆✐ ❥✯❦✪❧♥♠♥♦✏♣❉♦✾q r❂tst✉✇✈✺✈❯①✄❀❃✉✇① ❤❞② ❂t❂④③ ②✯⑤ ①▼⑥④❂✾⑦②⑨⑧❝⑩ ✙■❍❏✥❉❶❷✿❝❂❸st✉② s⑤✇②❃❹ ⑦✫③❺❂☞❄❻✉②♥❼ ★❷✰❞❊❸❽ ❋ ✙❾❍❏✥t❿❘➀⑤ ❄ ❂④➁✁✉●➂➄➃✚✈➅❂➆✉➈➇■✈➉⑦② ❂ ❹❅⑤✇➊❄④s④❂✬➋➌★✪✙✄✣✏✥✳✦➎➍ ⑤✇②➐➏➒➑ ❿❫◆❖❂❅✈❹ ❂❅✿❁❀❃❂❅❄✝❂❣❿➓✿❁❀❃❂❅❄✝❂ ➏➒➑ ⑦❹✳❹❅⑤➂➔❂→✈➉⑦②❂ ⑤➀➣➇↔②⑦✺①↕❂✕❂④➁❺①↕❂②①☞➋✳⑦② ❍➛➙➝➜✺➞⑤ ①✮❂❣❿❯❀ ⑤✿❝❂❅➟➠❂❅❄t❿➔①✺❀♥✉✇①✆①✄❀❃❂→③❺❂☞✈➡①↕✉➐③●⑦❹ ①■❄❣⑦✫⑥ ➊①■⑦ ⑤●② ⑦❹✳②✚⑤ ①✆⑦② ❊❻❽ ❋ ✙■❍❏✥❝➀⑤ ❄❫❍❙✰✗➢➤⑩ ➙ ➥ ➦➨➧■➩ ➫⑨➭➒➯❝➲❫➳➸➵➻➺♥➼✮➽✪➾ ✶✯✷✁✸✺✹✼✻➆➚ ➢✮♠➶➪✪♣❉♠❃♣❅➹t➘●➴❾➷➒➬❣✛➨➬☞➮❏➱④✃❨❒❐❝❮❷❰ ➬t★❺➬ ➮➄Ï✁➱t✃❨♥❐tÐ ➢↕Ñ ❤ ✰ ⑧❝⑩ ✙■❍❏✥➛✉② ③➶❍❷ÒÔÓ➄Õ☞❦❺♠✁Ö❺♣☞×✒➷ ➬☞✛➻➬❅➮ÙØt✃ ❨♥❐Ú❮✩❰ ➬ ❤ ➬☞Û❃Øt✃ ❨♥❐ ✴ Ü❯Ý ÒßÞ➓à✯❦✪➹✝á④➘✪♠✁áÚÑ✄❦❺➹➌s⑤✇②➟✇❂❅❄■â❺❂②st❂➌❄✝✉✇①✮❂ Ð r❂tst✉✇✈✺✈✼①✺❀♥✉✇①❏➬❣✛➨➬ ⑩ ➮➱ ✃ ❨❒❐ ✦äã❨ ❑ ❩❬✛➨❑ ⑩Ùå ✛⑩ ❪❺❴✆❿Ù✉②③✱①✄❀❃✉✇① ➬t★❺➬ ➮Ï✁➱ ✃❨♥❐ ✦ Ó✝❧❃à æèç ➮❏➱❣✃ ❨❒❐ ★✪✙✫✣✬✥ ➬☞✣✚➬❅➮➱ ✃❨♥❐ q ➞⑤ ①↕❂é✉●✈❹❅⑤ ①✺❀♥✉✇①❏⑦② ➢➤⑩ ①✄❀❃❂ê❊⑩ ②✯⑤ ❄❣➂❆⑦②s❅✈➊③✪❂ ❹ ①✄❀❃❂ ❹❅ë❅➊✉✇❄✝❂ ⑤➀➌①✄❀❃❂✱s☞❄⑤✇❹t❹ ③✪❂❅❄❣⑦✺➟➠✉●①■⑦✺➟✇❂ ❹ ✉❹ ✿❝❂❅✈✺✈➅➙ ì✾í✒î❣ï✗ð ñ✕ï✇ò➨ó♥ô❾õ✏ö➠÷✤î✫ø⑨÷■ù ➢➤⑩ ➢✮♠✳❦✪♠❃♣❯Ó✤à♥➘❺Õ☞♣❯♦✏Ò➅Þ➔♣❉♠♥Ó✤Òß❦✪♠✱Ò➡áûúÚ➘❺ÓÙÓ✝❧✏ü▼Õ☞Òß♣❅♠✁áÚá✝ý♥➘●á ❤ ❥✯♣êÓ✤❧❃Ò➅á④➘✪❥❃➴ßþ➔Ó✝Þ➔❦✬❦●á✝ý✱á✝❦❬♣❅♠♥Ó✝❧❃➹④♣ á✝ý❒➘✇á➄✛✕úÚ❦✪❧❃➴Ô♦▼❥✯♣➛Ò➅♠✕❊⑩ ✙■❍❏✥ Ð ➢✮♠✳á➸úÚ❦➔Ó✤à♥➘❺Õ☞♣➛♦✏ÒßÞ➔♣❅♠♥Ó✝Ò➅❦❺♠♥ÓÚá✝ý❒➘✇á❏ÒßÓ➄♠❃❦➔➴➅❦❺♠❃➪✪♣❉➹Ùá④➹✝❧♥♣✪➷❃♦✏❧❃♣ á✝❦✳á✝ý♥♣➓à✯❦●á④♣❅♠✁á✝ÒÔ➘●➴➨Õ❅❦✪♠✏ÿ♥ÒÔÕ❣áê❥✯♣☞á➸úÚ♣❅♣❅♠✟ú❏ý♥➘✇áêá④ý❃♣❬❥✯❦✪❧♥♠♥♦❃➘●➹④þ→♦❃➘●á④➘éá④♣❅➴ß➴ßÓ❯á④ý❃♣➔♦✏♣❅➹④Ò➅Ö✇➘●á✝ÒßÖ✪♣❉Ó á✝❦➔♦✏❦➓➘●♠♥♦éú❏ý❒➘✇áÚá✝ý❃♣❯♣✁￾✁❧♥➘●á✝Òß❦✪♠éá④♣❅➴ß➴ßÓÙá✝ý♥♣❯♦✏♣❉➹✝ÒßÖ✇➘✇á④Ò➅Ö❺♣❉Ó❝á✝❦❬♦❃❦ Ð✄✂❦✇úÚ♣❅Ö❺♣❅➹è➷✪Òß♠▼á✝ý♥♣❯Õ❉➘✪Ó✝♣❘Òß♠ ú❏ý❃ÒÔÕtý➶á✝ý❃♣❬♦✏❦✪Þé➘●Òß♠→❍äÒßÓ❫Õ☞❦✪♠✬Ö❺♣☞×✒➷✏á✝ý♥♣❅♠ ❤ ✰ ⑧Ù⑩ ✙■❍❏✥✆⑦❹ Ó✤❧✏ü▼Õ❅Ò➅♣❉♠❺á❘á✝❦▼♣❉♠♥Ó✤❧♥➹✝♣êá④ý♥➘✇á❘✛➆ÒßÓ Òß♠➶❊⑩ ✙■❍❏✥ Ð Ý