1 Motivation The Poisson problem has a strong formulation a minimization formulation and a weak formulation T weak formulations are more general than the strong formulation in terms of regularity and admissible data SLIDE 2 The minimization/weak formulations are defined by: a space X; a bilinear The minimization/weak formulations identify ESSENTIAL boundary conditions NATURAL boundary conditions ed in a The points of departure for the finite element method are the weak formulation(more generally) the minimization statement (if a is SPD) 2 The dirichlet problem 2.1 Strong Formulation Find u such that in Q on t The boundary condition u=0 is denoted"homogeneous Dirichlet. We consider Dirichlet boundary conditions in Section 4 and Q is a domain in R with boundary r❙ ❚❯✰❱❳❲☎❨✰❩✵❱❬❲❭❯❫❪ ❴❉❵❜❛❞❝❃❡✂❢ ❣✿❤❥✐❧❦✰♠❬♥♣♦rqsqs♥♣t✆✉❧✈s♥①✇❧②③❦❭④⑤✐⑦⑥①q⑧⑥⑨q❶⑩✞✈s♥①t❧❷✓❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t❃❻ ⑥❫④⑨♦③t❧♦③④❼♦③❽❋⑥❾⑩s♦③♥♣t✆❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✮❻❧⑥❾t➁➀✛⑥❼➂✏❦❋⑥♣➃⑨❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✮➄ ❣✿❤❥✐❧❦➅④⑨♦❿t❧♦③④⑨♦❿❽✷⑥❍⑩✞♦❿♥①t❉➆➇➂✏❦❋⑥♣➃➈❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t⑦q✢⑥❾✈✞❦➉④⑨♥♣✈✞❦➅❷♣❦❭t⑦❦❭✈➊⑥❾②✏⑩s✐⑦⑥♣t✺⑩s✐❧❦➋q❶⑩✞✈s♥①t❧❷ ❸❹♥♣✈✞④✓❺❧②r⑥❍⑩✞♦❿♥①t✻♦③t✆⑩✞❦❭✈✞④✢q❥♥❾❸❬✈s❦❋❷♣❺❧②r⑥❾✈✞♦➌⑩❶➍✩⑥♣t⑦➀✛⑥♣➀➎④⑨♦rqsqs♦❿✇⑦②❿❦✰➀❧⑥❾⑩✞⑥❧➄ ❴❉❵❜❛❞❝❃❡➐➏ ❣✿❤❥✐❧❦❥④❼♦③t❧♦③④⑨♦❿❽✷⑥❍⑩s♦③♥♣tP➆➇➂✏❦❋⑥❾➃➑❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t➁q✠⑥♣✈s❦❥➀➎❦❭➒⑦t❧❦❋➀❫✇▲➍P➓✙⑥➑q➔✉➁⑥♣→✍❦❥➣↔❻①⑥↕✇⑦♦❿②③♦❿t⑦❦❋⑥❾✈ ❸❹♥♣✈✞④✧➙❉❻❧⑥⑨②❿♦③t❧❦❋⑥♣✈✏❸❹♥♣✈✞④⑤➛❾➄ ❣✿❤❥✐❧❦➑④❼♦③t❧♦③④⑨♦❿❽✷⑥❍⑩s♦③♥♣tP➆➇➂✏❦❋⑥❾➃⑨❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t➁q❥♦③➀➎❦❋t❜⑩s♦❿❸❹➍ ➜✒➝❧➝➎➜✒➞❤❥➟❶➠↕➡➋✇❉♥♣❺⑦t⑦➀❧⑥❾✈✞➍✻→❭♥♣t⑦➀➎♦❿⑩s♦③♥♣t➁q❭➢ ➤♦❿✈✞♦③→➊✐⑦②❿❦❭⑩❥➥➦✈s❦❭➧⑦❦❋→✍⑩s❦✷➀✻♦③t➉➣↔❻ ➞➠❳❤⑧➨➫➩✸➠↕➡➈✇➁♥①❺❧t⑦➀❧⑥♣✈s➍✆→✍♥♣t➁➀➎♦➌⑩✞♦❿♥①t⑦q❋➢ ➞❦❋❺❧④✢⑥❾t❧t✆➥➭✈✞❦✍➧⑦❦✷→☎⑩s❦✷➀✛♦❿t➅➙❉➯❶➛❾➄ ❴❉❵❜❛❞❝❃❡➐➲ ❣✿❤❥✐❧❦↕➳❧➵❾➸❞➺⑦➻➽➼✦➵➔➾✓➚♣➪➶➳⑦➹❍➘☎➻➷➴➎➘s➪➑❸❹♥①✈❥⑩s✐❧❦✸➬➮➺⑦➸❞➻➶➪✓➪❭➱❿➪✍✃✢➪❭➺⑦➻➑✃⑨➪❭➻❹❐❧➵✷➚✆⑥♣✈s❦①➓ ⑩✞✐❧❦✰➂✒❦✷⑥❾➃✢❸❹♥①✈s④❫❺❧②r⑥❍⑩s♦③♥♣t❮❒❹④⑨♥①✈s❦➑❷♣❦❋t❧❦❭✈➊⑥❾②③②③➍➎❰✍❻ ♥♣✈ ⑩s✐❧❦✰④⑨♦③t❧♦❿④⑨♦③❽❋⑥❾⑩s♦③♥♣t➅q❶⑩➊⑥❍⑩✞❦❭④⑨❦❭t❜⑩✦❒❹♦❿❸✗➙✢♦③q ➝♠➤❰☎➄ Ï ÐÒÑ➉Ó❂ÔÕ❲✍Ö✗❲❭×➮Ñ➅Ø✍Ó❥❱ÚÙ❏Ö❳❯❫Û➅Ø✍Ó↕Ü Ý❥Þ➔ß à✒á➎â❧ã✒ä✸å❁æ⑧ã✗â⑦çéè➫ê❶ë✠á❧ì❶ã✒ä ❴❉❵❜❛❞❝❃❡îí ï♦❿t➁➀✆ð✹qs❺⑦→➊✐✛⑩s✐➁⑥❍⑩ ñ➫ò❼ó ð ô õ ♦❿t✹ö ð ô ÷ ♥♣t➉ø ù❐❧➪✸ú➊➵❾➴➎➺❉➚♣➹❾➘☎û✦ü✞➵❾➺❉➚❾➸❞➻➽➸ý➵❾➺✰ð✛ôþ÷❼➸r➼❥➚①➪✍➺P➵❍➻➶➪➊➚➐ÿ❞❐⑦➵❍✃✢➵✁￾❜➪✍➺P➪➊➵❍➴▲➼✄✂✓➸❞➘☎➸ýü➊❐▲➱❿➪❭➻✆☎✞✝✠✟➉➪⑧ü➊➵❍➺⑦➼☎➸ý➚♣➪❭➘ ✡➪❭➴➎✃✢➹❍➺⑦➺↔ú➊➵❍➴➎➺P➚♣➹❾➘☎û➉ü➊➵❍➺P➚❍➸❞➻➷➸ý➵❍➺⑦➼☞☛✍✌✏✎ ✌✏✑ ➸❞✃✸➳⑦➵➇➼❭➪➊➚✓✒➉➸❞➺✕✔P➪➊ü❭➻➽➸ý➵❾➺✗✖✙✘➫➹❍➺P➚✛➸❞➺▲❐❧➵❾✃⑨➵✚￾❜➪✍➺P➪✞➵❾➴▲➼ ✂✓➸❞➘☎➸ýü✞❐➎➱❿➪✍➻✸ú➊➵❍➴➎➺P➚♣➹❾➘☎û✻ü✞➵❾➺❉➚❾➸❞➻➽➸ý➵❾➺❧➼✦➸❞➺✛✔✶➪➊ü✍➻➷➸ý➵❍➺☞✜✢☎ ➂⑧✐❧❦❋✈s❦ òó✤✣✦✥ó ✥✢✧ó✩★ ✥ ó ✥✢✪ ó ⑥❾t➁➀✛ö✭♦③q➫⑥❼➀❧♥♣④✢⑥❾♦③t✆♦③t➉➟➩ó ➂⑧♦➌⑩✞✐✛✇❉♥♣❺⑦t⑦➀❧⑥❾✈✞➍✻ø➮➄ ✫