1 Model problem 1.1 Formulations 1.1.1 Strong formulation LIDE Find a such that for Q a polygonal domain Generalizat ion We look here at a particularly simple but nevertheless illustrative problem First, we require our domain to be polygonal. More general domains demand at least for accurate treatment parametric(non-affine) mappings that result in elements with curved edges. Although the implementation of such elements is not particularly difficult, it requires more machinery that we can develop in this short series of lectures Second, we consider the homogeneous Dirichlet problem. In one space mension, the difference between homogeneous and inhomogeneous Dirichlet con- ditions was quite small as regards formulation, analysis, and implemention. In yo space dimensions, unless the inhomogeneous Dirichlet data ecewise olynomial, in our case here piecewise linear, we will not longer be able to ex- actly represent ud within our finite element approximation space. In practice one typically replaces u with its interpolant, but this clearly leads to new com plications as regards the theory(a new "variational crime") Treatment of Neumann and robin conditions is a relatively straight-forward extension of what we present here. It is particularly in higher space dimensions that the convenience of natural boundary conditions is most beneficial, since calculation of normals and gradients at the boundary can be complicated(and n many cases, ambiguous ). As regards inhomogeneous Neumann conditio the same issues that arise in higher space dimensions for the inhomogeneous Dirichlet problem also arise for the Neumann problem Third, we restrict ourselves to two space dimensions. In fact, three dimen sions is quite similar, though obviously the implementation(in particular mesh generation)is more complex, and computational cost potentially much higher 1.1.2 Minimization/weak formulation SLIDE 2 Find or find u∈ X such that❃ ❄❅❇❆❉❈❋❊❍●☛■❏❅▲❑❉❊✤❈❋▼ ◆❏❖P◆ ◗❙❘❯❚❀❱☎❲❋❳P❨✌❩❭❬P❘❫❪❋❴ ❵❜❛❝❵❜❛❝❵ ❞✚❡❣❢✐❤❜❥✌❦♠❧✌❤♥❢✐♦q♣✌rts✼❡✲✉t❤❀❥ ✈❜✇✸①③②✖④✒⑤ ⑥✾⑦⑨⑧❀⑩❷❶❹❸❻❺♥❼❾❽✣❿❻❽❀➀✐❿ ➁❋➂▲➃ ❶ ➄ ➅ ⑦✪⑧❹➆ ❶ ➄ ➇ ➈➉⑧✣➊ ➋➈✞➌❋➆➍➀➏➎❭➐➒➑➔➓❣→✸➐✐➣❜↔➒➑✰↕➉➐✐➙➛↔✐➜③➣✰➝ ➞➏➟ ➠q➡➤➢✕➥➧➦ ➨❷➥➒➩❫➥➒➫➯➭❭➲❝➳❝➵✸➭➤➢P➳➸➡➺➩ ➻q➼➏➽✪➈➺➈✞➾➚❽♥➼❣➌➪➼➏➀✐❿➶➀❇➹♥➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽✪➴❷❸❻⑦⑨➷➬➹❭➽✪➼✠➮❭❺✼❿➱⑧❭➼❣✃➉➼❣➌❻❿❻❽❭➼✲➽⑨➼❐❸❻❸❒⑦⑨➽✪➽⑨❺❀❸➸❿➪➌➪➀➒❿❻⑦✪✃✞➼✠➹❭➌➪➈✞➮♥➽⑨➼✲➷✣➝ ⑥✾⑦⑨➌❾❸P❿✲❮✚❰✦➼Ï➌➪➼✲Ð✸❺❭⑦✪➌➪➼Ï➈➉❺❭➌➱⑩❭➈✞➷➛➀➒⑦✪⑧➧❿➪➈❷➮❜➼➬➹❀➈➉➽⑨➴➺Ñ➉➈✞⑧♥➀✞➽t➝➏Ò❹➈✞➌➪➼ÏÑ➉➼❣⑧❭➼✲➌➪➀✞➽✾⑩✼➈➉➷➛➀➒⑦✪⑧♥❸✶⑩✼➼❣➷➛➀➒⑧❀⑩ Ó ➀➒❿❒➽⑨➼❐➀✞❸P❿ ➋➈✞➌➶➀✞❼❣❼❣❺❭➌❾➀✐❿❻➼➱❿➪➌❻➼❐➀✐❿❻➷➬➼✲⑧➉❿ Ó ⑦➘❸P➈➉➹♥➀➒➌❾➀➒➷➬➼✤❿➪➌❻⑦➘❼▲ÔÕ⑧❭➈➉⑧✼Ö×➀➒Ø➛⑧❭➼❐ÙÚ➷➛➀✞➹❭➹❭⑦✪⑧❭Ñ➉❸Ú❿➪❽♥➀✐❿ ➌➪➼✲❸❻❺❭➽Û❿❇⑦✪⑧Ü➼❣➽✪➼❣➷➬➼✲⑧➉❿❾❸➛❰❙⑦⑨❿❻❽✂❼✤❺♥➌❻✃➉➼✲⑩✒➼✲⑩✼Ñ➉➼✲❸✲➝➍Ý❒➽⑨❿❻❽♥➈✞❺❭Ñ➉❽✒❿➪❽❭➼❹⑦✪➷➬➹❭➽⑨➼✲➷➬➼❣⑧✸❿➪➀➒❿❻⑦✪➈✞⑧Þ➈➋ ❸❻❺♥❼❾❽ ➼❣➽✪➼❣➷➬➼✲⑧➉❿❾❸➏⑦➘❸Ï⑧❭➈➒❿▲➹♥➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽✪➴q⑩✼⑦⑨Ø❇❼✤❺❭➽⑨❿✲❮❫⑦Û❿▲➌❻➼❐Ð➉❺♥⑦⑨➌➪➼✲❸➏➷▲➈➉➌❻➼➚➷➬➀➉❼❾❽❭⑦✪⑧❭➼❣➌➪➴q❿❻❽♥➀➒❿Ï❰✦➼❷❼❣➀✞⑧ ⑩✼➼✲✃✞➼❣➽✪➈✞➹✣⑦✪⑧❷❿➪❽❭⑦➘❸❒❸P❽♥➈✞➌❻❿Ú❸❻➼❣➌➪⑦⑨➼❐❸Ú➈➋ ➽⑨➼❐❼✕❿➪❺❭➌❻➼❐❸❣➝ ß➼❐❼✤➈➉⑧♥⑩➤❮✾❰à➼➚❼❣➈✞⑧♥❸❻⑦➘⑩✼➼❣➌✶❿➪❽❭➼❇❽❭➈➉➷▲➈➉Ñ✞➼✲⑧❭➼❣➈➉❺♥❸✜á❋⑦✪➌➪⑦✪❼❾❽❭➽✪➼✤❿Ï➹❭➌❻➈➉➮❭➽✪➼❣➷✎➝➚âã⑧♠➈➉⑧❭➼❇❸P➹❀➀✞❼✤➼❇⑩❭⑦ÛÖ ➷➬➼❣⑧♥❸❻⑦✪➈✞⑧✚❮❐❿❻❽♥➼✦⑩✼⑦⑨ä❁➼❣➌➪➼❣⑧♥❼❣➼❫➮❜➼✤❿➸❰✦➼❣➼❣⑧➏❽❭➈✞➷➬➈✞Ñ➉➼❣⑧❭➼✲➈✞❺♥❸✖➀➒⑧♥⑩✜⑦✪⑧❭❽❭➈➉➷➬➈✞Ñ✞➼✲⑧❭➼❣➈➉❺♥❸✖á➶⑦✪➌❻⑦➘❼❾❽❭➽✪➼✤❿✾❼❣➈✞⑧✼Ö ⑩✼⑦⑨❿❻⑦✪➈✞⑧♥❸❋❰✦➀➉❸❒Ð✸❺❭⑦⑨❿❻➼➏❸❻➷➛➀➒➽✪➽✌➀✞❸❒➌❻➼✲Ñ➉➀➒➌❾⑩❭❸ ➋➈➉➌❻➷Ï❺❭➽✪➀➒❿❻⑦✪➈✞⑧✚❮❀➀➒⑧♥➀✞➽⑨➴✼❸❻⑦✪❸✲❮♥➀➒⑧❀⑩✎⑦⑨➷➬➹❭➽✪➼❣➷➬➼❣⑧✸❿➪⑦⑨➈➉⑧✚➝Úâã⑧ ❿➸❰✦➈▲❸❻➹♥➀✞❼❣➼➶⑩✼⑦✪➷➬➼❣⑧♥❸❻⑦✪➈✞⑧♥❸✲❮✼❺❭⑧❭➽✪➼✲❸➪❸à❿❻❽❭➼✶⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸àá➶⑦⑨➌➪⑦➘❼❾❽❭➽⑨➼❣❿Ú⑩♥➀✐❿➪➀Ï❶❁åÜ⑦✪❸Ú➹♥⑦⑨➼❐❼✤➼❣❰❙⑦➘❸❻➼ ➹❜➈✞➽✪➴✸⑧♥➈✞➷➬⑦✪➀✞➽t❮❀⑦⑨⑧➧➈✞❺❭➌❋❼✲➀✞❸❻➼✜❽❭➼❣➌➪➼✠➹❭⑦✪➼✲❼❣➼❣❰❙⑦➘❸P➼➏➽⑨⑦✪⑧❭➼✲➀✞➌✲❮❀❰à➼➏❰❙⑦⑨➽✪➽✖⑧❭➈✞❿❋➽✪➈✞⑧❭Ñ➉➼❣➌❒➮❜➼Ï➀✞➮❭➽⑨➼✠❿❻➈❇➼❣æ✸Ö ➀✞❼✤❿❻➽✪➴❷➌➪➼❣➹♥➌❻➼❐❸P➼✲⑧➉❿❒❶❁åç❰❙⑦Û❿➪❽❭⑦✪⑧❹➈✞❺❭➌❒è♥⑧❭⑦⑨❿❻➼➏➼✲➽⑨➼✲➷➬➼❣⑧✸❿➶➀➒➹♥➹❭➌❻➈➯æ✼⑦✪➷➛➀✐❿❻⑦✪➈✞⑧❉❸❻➹♥➀✞❼❣➼✞➝Úâã⑧❉➹❭➌❾➀✞❼✤❿❻⑦➘❼✤➼✞❮ ➈✞⑧♥➼✎❿➸➴✸➹♥⑦✪❼✲➀➒➽✪➽⑨➴✒➌❻➼✲➹❭➽✪➀➉❼✤➼❐❸➛❶❜åé❰❙⑦⑨❿❻❽Ü⑦⑨❿➪❸➚⑦⑨⑧✸❿❻➼✲➌❻➹❜➈✞➽➘➀➒⑧✸❿❐❮✦➮♥❺✼❿❇❿❻❽♥⑦✪❸❷❼✤➽✪➼✲➀➒➌➪➽✪➴❍➽✪➼✲➀✞⑩♥❸➬❿❻➈♠⑧❭➼✲❰ ❼✤➈➉➷➬➹❭➽⑨⑦➘❼❣➀➒❿❻⑦✪➈✞⑧♥❸❙➀➉❸Ú➌❻➼✲Ñ➉➀✞➌➪⑩❭❸❫❿❻❽♥➼✶❿❻❽❭➼✲➈✞➌➪➴➧Ô❝➀▲⑧❭➼✲❰ëê❻✃➯➀✞➌❻⑦➘➀✐❿➪⑦⑨➈➉⑧♥➀➒➽➤❼❣➌❻⑦✪➷➬➼✲ì➉Ù✤➝ í✌➌❻➼❐➀✐❿➪➷▲➼✲⑧✸❿❫➈➋ ➞❒➼✲❺❭➷➛➀➒⑧❭⑧❇➀✞⑧♥⑩➛î❙➈➉➮❭⑦⑨⑧❇❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸❏⑦➘❸❫➀✜➌➪➼❣➽➘➀✐❿➪⑦⑨✃➉➼❣➽✪➴Ï❸P❿❻➌❾➀➒⑦✪Ñ✞❽✸❿PÖ ➋➈✞➌➪❰✦➀✞➌➪⑩ ➼✤æ➺❿➪➼❣⑧♥❸❻⑦⑨➈➉⑧❇➈➋ ❰❙❽♥➀✐❿✦❰à➼➱➹❭➌➪➼✲❸❻➼❣⑧✸❿à❽❭➼✲➌❻➼➉➝✌â×❿Ú⑦➘❸à➹❀➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽⑨➴▲⑦✪⑧➚❽❭⑦✪Ñ✞❽❭➼✲➌✦❸❻➹♥➀➉❼✤➼➶⑩❭⑦⑨➷➬➼❣⑧❀❸P⑦✪➈✞⑧♥❸ ❿❻❽❀➀✐❿▲❿❻❽❭➼✣❼✤➈✞⑧➺✃➉➼❣⑧❭⑦✪➼❣⑧♥❼❣➼❇➈➋ ⑧❀➀✐❿❻❺♥➌➪➀✞➽à➮❜➈✞❺❭⑧❀⑩❭➀➒➌➪➴q❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸➏⑦➘❸Ï➷▲➈✸❸➸❿▲➮❀➼✲⑧❭➼✤è❜❼✤⑦➘➀➒➽✴❮❫❸P⑦✪⑧♥❼❣➼ ❼❣➀✞➽✪❼❣❺❭➽➘➀✐❿❻⑦✪➈✞⑧✣➈➋ ⑧❭➈➉➌❻➷➛➀➒➽➘❸❙➀➒⑧❀⑩❷Ñ➉➌➪➀➉⑩✼⑦✪➼❣⑧✸❿➪❸❙➀➒❿❙❿❻❽❭➼✜➮❜➈✞❺❭⑧❀⑩❭➀➒➌➪➴➚❼❣➀✞⑧✎➮❀➼✠❼❣➈✞➷➬➹❭➽✪⑦✪❼✲➀✐❿❻➼❐⑩ïÔ❝➀➒⑧❀⑩ ⑦✪⑧ð➷➛➀➒⑧➺➴q❼✲➀✞❸❻➼✲❸✲❮✾➀✞➷➏➮❭⑦✪Ñ✞❺♥➈✞❺♥❸❾Ù✕➝❹Ý➶❸➏➌➪➼❣Ñ✸➀➒➌❾⑩❭❸✜⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸✠➞❒➼✲❺❭➷➛➀➒⑧❭⑧❍❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸❣❮ ❿❻❽♥➼✣❸➪➀➒➷➬➼➚⑦➘❸➪❸P❺❭➼❐❸✠❿➪❽♥➀✐❿❇➀✞➌❻⑦➘❸P➼➚⑦⑨⑧✒❽❭⑦⑨Ñ➉❽❭➼❣➌➬❸❻➹♥➀✞❼❣➼❷⑩✼⑦✪➷➬➼❣⑧♥❸❻⑦⑨➈➉⑧♥❸ ➋➈✞➌Ï❿❻❽❭➼✣⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸ á➶⑦⑨➌➪⑦✪❼❾❽♥➽⑨➼❣❿❒➹❭➌❻➈➉➮❭➽✪➼❣➷ñ➀✞➽✪❸❻➈▲➀✞➌❻⑦➘❸❻➼ ➋➈➉➌Ú❿❻❽❭➼✠➞❋➼❣❺❭➷➛➀✞⑧❭⑧✎➹❭➌❻➈➉➮❭➽✪➼❣➷✎➝ íÚ❽❭⑦✪➌❾⑩➤❮❁❰à➼Ï➌❻➼❐❸➸❿➪➌❻⑦➘❼✕❿➱➈✞❺❭➌❾❸P➼✲➽⑨✃➉➼✲❸❒❿❻➈❇❿➸❰✦➈❷❸❻➹♥➀➉❼✤➼Ï⑩❭⑦⑨➷➬➼❣⑧❀❸P⑦✪➈✞⑧♥❸✲➝➱âã⑧ ➋➀✞❼✤❿✲❮❁❿❻❽♥➌❻➼✲➼Ï⑩✼⑦✪➷➬➼❣⑧✼Ö ❸❻⑦⑨➈➉⑧♥❸✦⑦✪❸ÚÐ✸❺❭⑦⑨❿❻➼✜❸P⑦✪➷➬⑦⑨➽➘➀➒➌❐❮✸❿❻❽❭➈➉❺❭Ñ✞❽❷➈✞➮➺✃➺⑦⑨➈➉❺♥❸❻➽⑨➴▲❿❻❽♥➼➱⑦✪➷➬➹❭➽✪➼❣➷➬➼❣⑧✸❿➪➀➒❿❻⑦✪➈✞⑧ïÔÕ⑦✪⑧❷➹❀➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌✦➷➬➼✲❸❻❽ Ñ✞➼✲⑧❭➼❣➌❾➀✐❿➪⑦⑨➈➉⑧❀Ù❫⑦✪❸❙➷➬➈➉➌❻➼✜❼✤➈➉➷➬➹❭➽⑨➼❣æ➤❮❭➀➒⑧♥⑩✣❼✤➈➉➷▲➹♥❺✼❿➪➀➒❿❻⑦✪➈✞⑧♥➀✞➽✚❼✤➈➉❸P❿❙➹❀➈✞❿❻➼✲⑧➉❿➪⑦✪➀✞➽⑨➽✪➴➚➷Ï❺♥❼❾❽✣❽❭⑦⑨Ñ➉❽❭➼❣➌❐➝ ❵❜❛❝❵❜❛➘ò ó☛✉❝❥✾✉❝♦ð✉tô✞s✼❡❐✉❝❤❜❥✰õ❭ö☛÷➺s✼ø♠❧✖❤❀❢✐♦ù♣✾r❝s✼❡❐✉❝❤❜❥✌ú ✈❜✇✸①③②✖④qû ⑥✾⑦⑨⑧❀⑩❷❶✎➄Ü➀➒➌➪Ñ➱➷▲⑦✪⑧ ü✾ý✞þëÿ➃✁￾ Ô✄✂✆☎✝✂➶Ù ➁✟✞ Ô✄✂➱Ù ✠ ✡☞☛ ✌ ✍✏✎ü✒✑ ✓ ➈✞➌Úè❀⑧♥⑩✣❶✕✔✗✖✑❸❻❺♥❼❾❽➚❿➪❽♥➀✐❿ ➟