1 Model problem 1.1 Poisson Equation in 1D Boundary Value Problem(BVP) (x)=∫(x) (0,1),u(0)=(1)=0,f Describes many simple physical phenomena(e.g) Deformation of an elastic bar Deformation of a string under tension Temperature distribution in a bar The Poisson equation in one dimension is in fact an ordinary differ tion. When dealing with ordinary differential equations we Poisson equation will be used here to illastrate numerical techniques for elliptic PDE's in multi-dimensions. Other techniques specialized for ordinary differen tial equations could be used if we were only interested in the one dimension Note 1 Poisson equation The Poisson equation(in R)is elliptic, per our classification. It is also coercive, nd symmetric(these concepts will be defined more precisely in the Finite Element lectures). These attributes are very important as regards umerical treatment. These properties are reflected in the fact(see first lecture) that the eigenvalues of -V-v are real and positive We denote by cm, more precisely, cm([0, 1]), the set of functions f(a): [0, 1]+ IR with continuous m derivat ives. Thus. cu denotes the set of continuous func Isly, Ck CC for k>❅ ❆❇✬❈❊❉✛❋❍●❏■✼❇▲❑❊❋▼❉✛◆ ❖✼P◗❖ ❘❂❙❯❚◗❱❲❱❳❙❯❨✾❩❊❬❪❭✜❫✠❴❵❚❛❙❯❨❜❚◗❨❝❖✒❞ ❡❣❢❳❤❥✐❧❦✂♠ ♥▲♦❣♣✠q✗r✠s✉t✇✈✂①✤s❵②③♣✗④✎⑤✦t✇♦⑦⑥⑧②⑨④❲⑩❷❶❛♥▲①✆⑤❹❸ ❺❼❻❣❽❾❽❵❿➁➀➃➂❯➄✯➅⑧❿➁➀❣➂ ➆➈➇ ➀❊➉➊❿③➋❵➌ ➇ ➂☎➌➍❻⑧❿⑨➋➎➂✘➄✪❻⑧❿ ➇ ➂✼➄✷➋❵➌➏➅✎➉➑➐➓➒ ➆→➔ ↔❀↕➛➙✞➜✣➝✞➞➠➟❣↕➛➙❪➡➤➢➦➥❲➧➑➙➨➞➩➡✱➫➯➭➠↕✤➫❵➲❲➧✉➙➨➞➩➜➛➢➦➭➃➫❵➲❵↕➛➥❵➳➵➡➸↕▼➥⑦➢ ➆❀➣ ❿ ↕➵➺ ➻➯➺ ➂☎➼ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢➦➥✕↕▼➭➪➢➵➙◗➚➨➞➪➜❀➟⑦➢➦➝ ➆✛➶ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢✱➙◗➚➨➝✞➞➩➥❵➻➸➹❵➥➯➘✉↕➛➝❪➚➨↕▼➥⑦➙◗➞➩➳➵➥ ➆→➴ ➽➬➷↕▼➡➸➫❣↕▼➝✍➢➮➚✞➹❵➝➨↕✤➘✉➞➪➙◗➚➨➝✞➞➠➟❵➹❵➚➨➞➩➳➵➥❂➞➠➥❊➢✱➟➯➢➵➝ ➆❀➱ ✃➃❐❵❒→❮❼❰➮Ï➪Ð✍Ð▼❰➮ÑÒ❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ❊Ï❥Ñ✎❰➦Ñ❣❒➈Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ✎Ï➪Ð→Ï❥Ñ➈Ú✣Õ➎Û✣Ö✓Õ➦Ñ✎❰➮Ü➨Ø➦Ï❥Ñ❣Õ➦Ü☎Ý➑Ø➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß❧❒✍Ó✣Ô➯Õ➦à Ö×Ï⑨❰➮Ñ❣á❝â✬❐❵❒▼Ñ★Ø➎❒✍Õ➮ßãÏ❥Ñ❲ä❍å✼Ï❥Ö➁❐✯❰➮Ü➨Ø➦Ï❥Ñ❣Õ➮Ü☎ÝæØ➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß✦❒✞Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð❊å❯❒✎å✼Ï❥ß❥ß✦Õ➮ßçÐ▼❰❍Ô❲Ð▼❒✲Ö➁❐❵❒ èêéÜ☎Ï❥Ù➤❒❳ë✛Ñ❣❰➦ÖìÕ➦Ö③Ï⑨❰➦Ñ✆Öí❰❹Ï❥Ñ❣Ø➮Ï⑨Û✍Õ➦Öì❒✦Ø➮ÏÞ❀❒✣Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮Ö×Ï⑨❰➮Ñ➃á➤✃❣❐✉Ô❲Ð✍î ❻❽❹ï ❻➃ð î ❻❽❾❽ ➄ñ❻➃ð ð î✼❒✣ÖíÛ➛á✱✃➃❐❵❒ ❮❼❰➦Ï➪Ð✞Ð▼❰➦Ñò❒✍Ó✣Ô➯Õ➦Ö③Ï⑨❰➦Ñ✲å✼Ï❥ß❥ß✠ó✍❒➈Ô❲Ð✣❒✍Ø❹❐➯❒✣Ü✞❒✤Öí❰➸Ï❥ß❥ßãÔ❲Ð☎Ö③Ü✞Õ➮Öí❒✤Ñ➯Ô✉Ù➤❒▼Ü☎Ï⑨Û✞Õ➦ß✒Öí❒✞Û✍❐❲Ñ⑦Ï⑨Ó✣Ô➯❒✣Ð✼Ú✣❰➮Ü❹❒▼ß❥ßãÏéÖ×Ï⑨Û ❮✼ô✦õ➸öÐ➈Ï❥ÑòÙ▲Ô✉ßãÖ×Ï❥à◗Ø➮Ï❥Ù➤❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➯Ð✣á✬÷❪Ö❥❐➯❒✣Ü➈Öí❒✍Û✞❐✉Ñ➯Ï⑨Ó▼Ô➯❒☎Ð✤Ðé❒✍Û✣Ï⑨Õ➦ßãÏ➠ø❾❒✞Ø✦Ú✣❰➦Ü▲❰➮Ü✞Ø➮Ï❥Ñ❣Õ➦Ü☎Ý❂Ø➦ÏÞ✛❒▼Ü➨❒✣Ñ⑦à Ö×Ï⑨Õ➮ß✑❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð✬Û✍❰➦Ô✉ß➠ØÒó✍❒➑Ô❲Ð▼❒✞Ø✎ÏÚ➤å❯❒➑å❯❒✣Ü✞❒✕❰➦Ñ➯ßãÝ✎Ï❥Ñ➯Öí❒✣Ü➨❒✣Ð☎Öì❒✍Ø✲Ï❥Ñ❍Ö➁❐❵❒❂❰➮Ñ➃❒✆Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➃Õ➮ß Û✍Õ✇Ð▼❒❾á ùòú➓û☎ü✎ý þ➈ú❲ÿ✁￾✂￾➵ú☎✄✯ü☎✆✞✝✠✟➓û◗ÿ❛ú☎✄ ➷➲❵↕☛✡⑧➳➵➞➪➙➨➙➨➳➵➥✤↕✌☞➎➹⑦➢➮➚➨➞➩➳➵➥ ❿ ➞➠➥✎✍✏✒✑ ➂ ➞➩➙✗↕▼➭➩➭➩➞➠➫✉➚✞➞➩➜✔✓➦➫❣↕▼➝✠➳➵➹❵➝⑧➜✣➭➪➢➵➙✞➙◗➞✖✕⑦➜➛➢➮➚➨➞➩➳➵➥❧➺✗✍ì➚✠➞➪➙✠➢➦➭➪➙◗➳✛➜▼➳❲↕▼➝✍➜✣➞✙✘➵↕✔✓ ➳➵➝❧➫⑦➳❳➙◗➞➠➚➨➞✙✘➵↕✑➘✉↕✚✕➯➥➯➞ê➚✞↕✔✓➦➢➦➥⑦➘✤➙◗➧❲➡➸➡➸↕✣➚➨➝✞➞➪➜ ❿ ➚✞➲❵↕➛➙➨↕✓➜▼➳➵➥➯➜▼↕▼➫✉➚✍➙✗✛❼➞➩➭➠➭❳➟⑦↕✑➘✉↕✚✕⑦➥❵↕➛➘✤➡➸➳➎➝➨↕✘➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧ ➞➩➥✬➚➨➲➯↕✢✜✠➞➩➥❵➞ê➚✞↕✤✣✘➭➩↕▼➡➸↕▼➥❳➚✓➭➩↕➛➜✣➚➨➹❵➝✞↕➛➙ ➂ ➺ ➷➲❵↕❾➙◗↕✦➢➮➚➨➚➨➝✞➞➠➟➯➹✉➚➨↕❾➙✑➢➵➝➨↕✥✘➵↕➛➝➨➧✱➞➠➡➸➫❣➳➵➝➨➚✞➢➵➥➎➚❼➢➵➙✓➝✞↕▼➻➎➢➵➝✞➘➯➙ ➥❲➹❵➡➸↕▼➝✞➞➩➜➛➢➦➭❳➚➨➝✞↕➛➢➦➚➨➡➸↕▼➥❳➚❾➺ ➷➲❵↕❾➙◗↕❪➫❵➝✞➳➵➫❣↕▼➝➨➚➨➞➩↕➛➙✼➢➵➝➨↕✑➝✞↕✚✦➯↕❾➜☎➚➨↕❾➘❹➞➩➥✱➚➨➲❵↕✑➾⑨➢➎➜☎➚ ❿ ➙➨↕▼↕☛✕➯➝✍➙◗➚⑧➭➩↕➛➜✣➚➨➹❵➝✞↕ ➂ ➚➨➲⑦➢➮➚❼➚➨➲➯↕✤↕▼➞➩➻➵↕▼➥☎✘➮➢➦➭➩➹❵↕❾➙✓➳➵➾ ❺★✧✑✪✩ ➢➦➝✞↕→➝✞↕➛➢➵➭✒➢➦➥➯➘✆➫❣➳➎➙➨➞ê➚✞➞✖✘➎↕➵➺ ùòú➓û☎ü✬✫ ➐✗✭ ￾✚✮✯✟✠✰➎ü✱￾ ✲ò↕✦➘❵↕▼➥❵➳➵➚➨↕✦➟❲➧ ➐✳✭ ✓✉➡➸➳➵➝✞↕❀➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧✴✓ ➐✗✭✱❿✶✵➋❵➌ ➇✚✷ ➂ ✓❳➚➨➲❵↕✤➙➨↕✣➚❼➳➦➾✗➾➁➹❵➥➯➜✣➚➨➞➩➳➵➥➯➙ ➅⑧❿⑨➀❣➂❪➼✸✵➋❵➌ ✍✏✺✛❼➞➠➚➨➲❊➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙☛✻ ➘❵↕▼➝✞➞✖✘➮➢➮➚✞➞✖✘➎↕➛➙➛➺ ➇✚✷✠✹ ➷➲❳➹⑦➙✪✓ ➐➓➒ ➘✉↕▼➥➯➳➦➚➨↕❾➙✑➚➨➲❵↕➈➙➨↕✣➚✜➳➵➾✠➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙✑➾➁➹➯➥➯➜✽✼ ➚➨➞➩➳➵➥⑦➙▼➺✿✾❀➟☎✘❲➞➠➳➎➹➯➙◗➭➩➧✔✓ ➐❁❀❃❂❍➐✗✭ ➾➁➳➎➝★❄❆❅❇✻ò➺ ➇