A general partial derivative can be approximated as bmm=am=∑∑bb 户kmb7n()L(y)0 Therefore, defining the weights L i (a)Lk(y) The method of undetermined coefficient extends to multiple dimensions in a straightfor ward manner if we consider multidimensional Taylor series expan assuming a uniform△zand△g △x)22 U +k△ 28 +G△rk△y) azay I(a) A finite difference approximation of the form can be obtained by inserting the Taylor series expansions for ujk and determining Oik coefficients by the coefficients of the different derivative terms ct differe The finite difference approximations considered here are called divided diffe ences approximations. More sophisticated, and accurate, difference approxima tions are also possible. These approximations are called compact approxima ∑v d a ll knowi at ion to the first deriva erify, using Taylor expansi h that this if fourth order accurate for a sufficiently smooth function, 1. e. p=4
➽➚➾✷➪❦➶✌➪✳➹✩➘✪➴↕➷✕➘✷➹❝➬❷➮❽➘✷➴→➱✜➪✳➹❷➮❽✃✑➘✪➬❖➮❽✃✷➪➏❐✳➘✷➶P❒✎➪➋➘✪➷✕➷✌➹❖❮✑❰✜➮❽Ï✒➘✭➬❖➪➇➱➁➘✼Ð Ñ↕Ò①Ó→Ô✺Õ Ñ✎ÖÒ Ñ✎× ÔÙØ Ñ✎Ò①Ó→Ô♠ÚÕ Ñ✎ÖÒ Ñ✎ÖÔÜÛ ÝÞ ß❖à➍á✚â Þã äà♠á❅å Ñ↕Ò①Ó→Ô Ñ✎ÖÒ Ñ✚× ÔPæß✷çÖ↕è æä ç×✌è↕Õß äPé ê①ë➪❦➹❖➪✳ì▼❮✷➹❷➪✷í✜➱✜➪✘î✚➶✌➮✾➶✕➾❼➬ë➪➄ïð➪✳➮❽➾ë➬❷Ð ñÒ①Ô ß ä Û ÑÒ①Ó→Ô Ñ✎ÖÒ Ñ✎× ÔPæß çÖ↕è æä ç ×✌è✭òò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø ïð➪ ë➘✑✃✷➪ ÑÒðÓ❑Ô ÕÚ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø➍ù ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ê①ë➪PÏ❼➪✘➬ë❮✺➱✙❮✷ì❯ú✌➶✕➱✜➪✳➬❖➪❦➹❖Ï❼➮❽➶✌➪❦➱✙❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✒➪✳❰⑩➬❷➪✳➶✕➱✕Ð➈➬❷❮ÙÏ➈ú✌➴✾➬❖➮❽➷✌➴❽➪✱➱✌➮✾Ï❼➪✳➶✚Ð❝➮❽❮✷➶✕Ð❼➮❽➶✠➘ Ð❝➬❖➹✩➘✪➮❽➾ë➬❝ì▼❮✼➹❖ï①➘✪➹✩➱ÜÏ✒➘✪➶✌➶✕➪✳➹✒➮✾ì➏ïð➪③❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹✒Ï➈ú✌➴✾➬❖➮✫➱✜➮❽Ï❼➪✳➶✕Ð❖➮✾❮✼➶✕➘✪➴ ê➘✑ü✺➴❽❮✷➹✒Ð❝➪❦➹❖➮❽➪❦Ð❼➪✳❰✜➷✕➘✪➶✜ý Ð❖➮✾❮✼➶✕Ð✳þ ê①ëú✕Ð❦í✌➘✷Ð❷Ð❝ú✕Ï➉➮❽➶✌➾✒➘➉ú✌➶✌➮✾ì▼❮✷➹❷Ï ÿ❉Ö③➘✪➶✚➱Pÿ❉×↕í✌ïð➪ ë➘✑✃✷➪ Õ ß ä Û Õ ó✾ô➇õ✄ö ÷✩õ❖ø ✁✄✂ ÿ❉Ö Ñ✎Õ Ñ✎Ö ò ò ò ò ó✾ô õ ö ÷õ ø ✁✆☎ ÿ➈× Ñ✎Õ Ñ✎× ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ❉Ö✎è✞✝ ✟ Ñ✠✝✘Õ Ñ✎Ö✝ ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ➈Ö ☎ÿ❉×✌è Ñ✡✝✳Õ Ñ✚Ö✎Ñ✎× ò ò ò ò ó✾ô õ ö ÷õ ø ✁ ç ☎ÿ❉×✌è✞✝ ✟ Ñ✠✝✳Õ Ñ✎× ✝ ò ò ò ò ó✾ô õ ö ÷õ ø ✁☞☛✌☛✍☛ é ➽ î✕➶✌➮✾➬❖➪➋➱✌➮✏✎↕➪✳➹❷➪✳➶✕❐✳➪➄➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶➁❮✷ì❋➬ë➪➏ì▼❮✷➹❷Ï ÑÒðÓ❑Ô Õ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø Ø ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ❐✳➘✷➶➏❒✚➪r❮✼❒✜➬❷➘✷➮✾➶✕➪❦➱➏❒✺ü❯➮❽➶✕Ð❖➪✳➹❖➬❖➮❽➶✌➾①➬ë➪ ê➘✑ü✺➴❽❮✷➹→Ð❖➪✳➹❷➮✾➪➇Ð❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð↕ì▼❮✷➹→Õß ä ➘✪➶✕➱➏➱✜➪✳➬❖➪✳➹❷Ï❼➮✾➶✕➮✾➶✌➾ ➬ë➪①ñÒ①Ô ß ä ❐✘❮✺➪✘û◆❐✘➮❽➪✳➶⑩➬✩Ð♠❒⑩ü➈➪✌✑⑩ú✕➘✭➬❷➮✾➶✕➾❯➬ë➪❨❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✩Ð❥❮✪ì✕➬ë➪❨➱✜➮✒✎✎➪❦➹❖➪❦➶⑩➬➍➱✌➪✳➹❷➮✾✃✭➘✭➬❷➮✾✃✼➪◗➬❷➪✳➹❷Ï✒Ð✳þ ✓✕✔✗✖✙✘✛✚ ✜✢✔✤✣✕✥✧✦✡★✤✖✪✩✗✫✭✬✮✘✰✯✱✘✰✲✳★✱✘✴✦✱✥✵✥✶✯✱✔✰✷✸✫✹✣✺✦✗✖✻✫✞✔✤✲✵✼ ê①ë➪✒î✚➶✌➮➬❷➪➁➱✜➮✒✎✎➪❦➹❖➪❦➶✕❐✘➪t➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶✕Ð➏❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹❖➪➇➱ ë➪✳➹❷➪◆➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù➱✜➮❽✃✺➮❽➱✜➪➇➱Ü➱✜➮✒✎✎➪❦➹❝ý ➪✳➶✚❐✘➪❦Ð①➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð✳þ✵✽✯❮✷➹❷➪➝Ð❖❮✷➷ë➮❽Ð❝➬❖➮✫❐✳➘✪➬❖➪➇➱❅í✌➘✪➶✕➱t➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪✷í✺➱✜➮✏✎↕➪✳➹❷➪✳➶✚❐✘➪➏➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð➋➘✪➹❷➪◆➘✪➴✫Ð❝❮✯➷✎❮✼Ð❷Ð❖➮✾❒✌➴❽➪✷þ ê①ë➪➇Ð❝➪◆➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð➄➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù❐✘❮✼Ï❼➷✕➘✷❐✘➬➈➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð✮➘✪➶✕➱t➬❷➘✿✾✷➪➝➬ë➪➏ì▼❮✼➹❖Ï ÝÞ❀ ß❖à♠á✎â ❀ ❁Ò Õ ❁ÖÒ ò ò ò ò ô à ô✌❂ Ø ÝÞ ß❖à♠á✎â ñÒ ß Õ ß❄❃ ➘✪➶✚➱t➘➉➷✕➘✷➹❝➬❷➮❽❐✳ú✌➴❽➘✷➹✮ï✁➪❦➴✾➴✗✾✺➶✌❮✭ï✮➶✱❐✘❮✼Ï❼➷✕➘✷❐✘➬✮➘✷➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✭➬❷➮✾❮✼➶◆➬❷❮❉➬ë➪➏î✕➹✩Ð✥➬❨➱✜➪❦➹❖➮❽✃✭➘✭➬❷➮✾✃✼➪➎➮✫Ð ❅ ❆ çÕ✤❇ß Ó❉❈ ✁❋❊Õ●❇ß ✁ Õ✤❇ß✘á ❈ è Û Õ ß Ó❉❈■❍ Õ ß✘á ❈ ✟ÿ❉Ö é ❏➬❑➮✫Ð→➪➇➘✷Ð❖ü❨➬❖❮❯✃✷➪✳➹❷➮✾ì▼ü✷í➇ú✕Ð❝➮❽➶✌➾ ê➘✑ü✺➴✾❮✼➹❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð❅❮✷➹→❮✷➬ë➪❦➹❖ï✮➮✫Ð❖➪✷í❦➬ë➘✪➬❑➬ë➮❽Ð❋➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶ ➮✾ì❋ì▼❮✼ú✌➹❝➬ë ❮✼➹❷➱✌➪✳➹❨➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪❯ì▼❮✼➹❨➘❼Ð❝ú✜û◆❐✳➮✾➪❦➶✼➬❷➴✾üPÐ❝Ï❼❮✺❮✪➬ë ì▼ú✕➶✕❐✄➬❷➮✾❮✼➶→í✌➮✉þ ➪✼þ✗❑ Û ❊ þ ▲
2 Poisson equation in 2D Vu(a, g)=f(a, y) in Q we have seen that one of the criti cal requirements to obtain optimal a-priori error estimates is that the solution be sufficiently regular so that the derivatives appearing in the leading terms of the truncation error eist. In id the regularity of the solution depends exclusively on the regularity of f (see Fourier analysis in Lecture 1). In multidimensions however, the regularity of the boundary plays also a very important role and, for general domains involving corners or non- regular boundary data, the solution may not be very regular. This topic will be dealt with in greater detail in the finite element lectures 2.2 Discretization SLiDE 11 ,j-1 △x=m,△y=m,x1=i△x,v=j△y 2.3 Approximation SLIDE 12
▼ ◆P❖❘◗✍❙❉❙✵❖❘❚❱❯P❲❨❳❬❩❪❭✶◗❫❖❘❚❴◗✍❚❵▼❪❛ ❜■❝✻❞ ❡☞❢✗❣✐❤❦❥✞❧♠❥✻♥✧❤ ♦✡♣●qsr✉t✺✈✰✇ ①❦②④③❫⑤⑦⑥✭⑧❉⑨✻⑩❷❶✧❸❺❹⑦⑥✹⑧✉⑨❻⑩♠❶ ❼❾❽ ❿ ⑤❬❸☞➀ ➁✿❽ ➂ ②③➄➃➆➅③ ➅ ⑧③❘➇ ➅ ③ ➅ ⑩ ③ ⑨ ❹➉➈✛➊✗➋ ➌➉➍➏➎❷➐➒➑➒➍✴➓✍➍➔➍❫→↔➣✭➎❷➐➒➣✢↕✰→✡➍➉↕✻➙➏➣✭➎❷➍➉➛✍➜✙➝s➣➞➝✹➛➔➐➒➟■➜❻➍➔➠✍➡♠➝s➜❻➍✍➢➤➍✍→➥➣➞➓➏➣➦↕✕↕✿➧✍➣➨➐➒➝s→➩↕➭➫✠➣➞➝s➢❄➐➒➟➯➐✰➲❾➫✡➜✙➝✹↕➒➜✙➝ ➍✍➜✙➜❻↕➒➜➳➍✙➓✙➣➵➝s➢➤➐✰➣➦➍❫➓✐➝➸➓➄➣✭➎❷➐➒➣➺➣s➎➥➍❪➓✍↕➒➟➻➡♠➣➵➝✹↕➒→✕➧➭➍✐➓➔➡❫➼↔➛✍➝✹➍❫→➽➣➞➟➻➾④➜❻➍➦➚✿➡♠➟✒➐➒➜❪➓✍↕④➣s➎➥➐➒➣➺➣s➎➥➍✪➪✿➍✍➜✙➝s➑➶➐✰➣➞➝s➑➒➍❫➓ ➐➭➫✱➫❷➍➔➐✰➜✙➝s→●➚❘➝s→❨➣✭➎❷➍➄➟✒➍➔➐✿➪✰➝s→●➚❘➣➨➍❫➜✙➢❘➓✮↕✻➙✮➣✭➎❷➍❦➣➞➜✙➡♠→✠➛➭➐✰➣➞➝✹↕✰→✛➍✍➜✙➜❻↕➒➜❦➍➭➹✱➝➸➓✙➣➦➘➺➴✙→✴➷✌➬➮➣✭➎❷➍➄➜❻➍➦➚✿➡♠➟✒➐➒➜✙➝s➣➵➾ ↕✞➙④➣✭➎❷➍④➓✍↕✰➟➻➡♠➣➞➝✹↕✰→➱➪✿➍➨➫❷➍✍→✡➪➒➓❄➍➭➹✤➛❫➟✏➡✤➓➔➝s➑➒➍✍➟➻➾✃↕➒→❐➣✭➎❷➍➤➜➭➍➵➚✿➡♠➟✒➐✰➜✙➝s➣➞➾➉↕✞➙ ❹❮❒➓✍➍➔➍✢❰❉↕➒➡♠➜✙➝✹➍✍➜➏➐➒→✡➐✰➟➻➾➶➓✙➝➸➓ ➝s→❨Ï✉➍➔➛✍➣➞➡♠➜➭➍➳➷➔Ð✿➘➺➴❫→➉➢❘➡♠➟✏➣➞➝✹➪✰➝s➢➤➍✍→❷➓✙➝✹↕➒→➥➓✮➎➥↕➒Ñ✧➍✍➑➶➍✍➜➔Ò✳➣s➎➥➍✐➜❻➍➦➚✿➡♠➟✒➐➒➜✙➝s➣➵➾➤↕✻➙❦➣✭➎❷➍✢➧➭↕✰➡♠→✡➪✱➐➒➜✙➾➄➫✡➟✒➐✰➾➶➓ ➐➒➟➻➓❫↕Ó➐✛➑➶➍✍➜✙➾❬➝s➢➯➫➥↕➒➜✙➣➨➐➒→➽➣➯➜❻↕➒➟❾➍❨➐➒→✠➪➶Ò✉➙❫↕✰➜✐➚●➍❫→✠➍❫➜➭➐➒➟✧➪✿↕✰➢➤➐✰➝s→❷➓➤➝s→➥➑➒↕➒➟➻➑Ô➝s→●➚✕➛➔↕➒➜✙→✠➍❫➜➔➓❄↕➒➜➤→✡↕✰→➥➲ ➜❻➍➦➚✿➡♠➟✒➐➒➜❘➧➔↕✰➡♠→✡➪✿➐✰➜✙➾✛➪✱➐➒➣➨➐➶Ò■➣s➎➥➍❘➓❫↕✰➟➻➡♠➣➵➝✹↕➒→✕➢➤➐✰➾❨→✡↕✰➣➯➧➭➍❘➑➒➍❫➜✙➾✴➜❻➍➦➚✰➡♠➟❾➐➒➜✍➘➉Õ✠➎✤➝➸➓✢➣➨↕➭➫✠➝✹➛④Ñ✶➝s➟s➟✶➧➔➍ ➪✿➍➔➐✰➟➻➣➺Ñ✶➝s➣✭➎❬➝s→❬➚✰➜➭➍➭➐✰➣➦➍✍➜✢➪✿➍✍➣➨➐➒➝s➟✉➝s→✄➣s➎➥➍✧Ö✧→➥➝s➣➨➍✢➍✍➟✒➍✍➢➤➍✍→➥➣×➟✒➍➔➛❫➣➵➡♠➜❻➍❫➓✍➘ ❜■❝➞❜ ❡❺❥❻Ø●Ù✠Ú➥❢✗❧❷❥✞Û✗Ü✵❧❷❥✞♥✧❤ ♦✡♣●qsr✉t✺✈❷✈ Ý ⑧✛❸ßÞ➥à á✿â❉ã♠ä Ý ⑩❄❸ Þ❷å æ■â❉ã❷ä ⑧➽ç❉❸☞èÝ ⑧ ä ⑩➶é❦❸➱êÝ ⑩ ❜■❝➞ë ì↔í❦í➄Ú❷♥❉î✮❥❻ïðÜ❉❧❷❥✻♥✧❤ ♦✡♣●qsr✉t✺✈✿ñ ò ➁✱ó✮ô❫õ❷ö✰÷➤ø❷ù❾ô➤ú✍ú✌ú ➅ ③❫û ➅ ⑧③ ü ü ü ü ç✭ý é➄þ û✰çâ❉ã ý é➯① ÿ✰û✰ç✭ý é ➇ û✰ç ✁ ã ý é Ý ⑧③ ➅ ③❫û ➅ ⑩ ③ ü ü ü ü ç✹ý é þ ûç✹ý é â❉ã ①❋ÿ➒ûç✹ý é ➇ ûç✹ý é ✁ ã Ý ⑩ ③ ✂➁✿ó Ý ⑧ ä Ý ⑩☎✄❻÷➤ö✿ù✒ù ✆