equire that a be“ lipschitzian.” We recall functie K such that Jw(a)-w(y)I< Kle-yI for all r, y of interest. a domain Q is Lipschitzian if the boundary f at any point admits a locally Lipschitzian repre- sentation - it can t be too wiggly or singular. Note also that, unless otherw. indicated, we will be speaking of open domains Q (e.g, Q=(0, 1),which de not include 0 and 1); the closure of such a domain will be denoted Q(e.g 2.2 Minimization Principle The finite element method is not based on the strong form, but rather a min ization statement or, more generally, a weak formulation. We must thus develop and understand these formulations before proceeding with the finite ele 2.2.1 Statement SLIDE 5 Find he X=u sufficiently smooth X here is a linear space, the precis e definition of which will be given shortly; we shall also make "sufficiently smooth"precise during the course of this lecture. los Note 1 Notation We explain here some of the notation that we will be using. First arg min The lol e)of";C subset or subspace of”; means“ for all;彐 means" there exists”;|(ands.t)✬✮✭✰✯✲✱✳✭✴✱✳✵✁✶✸✷✺✹✼✻✽✱✾✵✁✱✓✿✳❀❂❁❃✵✁✱✾❄❆❅❇✶✸❄❉❈❋❊✓✱❍●❆■✄❁❏▲❑✍▼✚❅❂❁❃❄❃◆✍❁❖✶✙✭✴P✞◗❙❘❚✱✾✵✁✱✓▼✓✶✙✷❃✷❯❄❃❅❱✶✙❄❲✶❨❳✓❀❂✭▲▼✍❄❩❁❖❬✸✭✠❬❭❳ ❪❑✍✶✙❫✓❴❵❬✙✭✴✱❜❛✙✶✙✵✮❁❖✶❝❊✍✷❞✱✳✹❢❡✼✹✤❑✍✶✙❄❣❁❤❑❃✐❯✱✮❑❥✶❦■❧❁❏✢❑✍▼✚❅❂❁❃❄❃◆♠▼✓❬✸✭▲♥✙❁❃❄❣❁❖❬✙✭♦❁❳♣❄❃❅❱✱✳✵✚✱❦✱✚q❝❁❤❑✮❄❩❑❜✶❵▼✚❬✸✭❇❑✮❄r✶✙✭❱❄ st❑✮❀❱▼✚❅✉❄❆❅❇✶✸❄✇✈ ❡②①❆③✴④❯⑤✗❡②①❖⑥❂④⑦✈⑨⑧⑩s❶✈ ③☞⑤✗⑥⑨✈❂❳✳❬✸✵❜✶✸✷❃✷⑨③❸❷✁⑥❶❬❭❳✩❁❃✭✢❄✆✱✍✵✁✱✳❑✮❄✆P❥❹❺♥❻❬✙❼❜✶✙❁❃✭✉❈❽❁❤❑ ■✄❁❏▲❑✍▼✚❅❂❁❃❄❃◆✍❁❖✶✙✭✕❁❳✼❄❆❅❇✱❨❊✓❬✙❀❂✭✴♥❝✶✙✵✮❫✩❾❍✶✙❄❿✶✙✭✢❫✇❏❇❬✸❁❃✭❱❄➀✶❝♥✸❼✩❁❃❄❩❑②✶❥✷❞❬✏▼✓✶✙✷❃✷➁❫♣■✄❁❏✢❑✍▼✓❅➂❁❃❄❆◆✍❁❖✶✙✭✕✵✁✱r❏▲✵✁✱✍➃ ❑✍✱✳✭✢❄✆✶✸❄❩❁❖❬✸✭➅➄➆❁❃❄❿▼✓✶✙✭➈➇❄❿❊✚✱✩❄r❬⑦❬❦✻❧❁➉✯✏✯✸✷➁❫♠❬✙✵➊❑✮❁❃✭✲✯❝❀❂✷❞✶✸✵✍P❉➋②❬✸❄✆✱❨✶✙✷➁❑✳❬✾❄❆❅❇✶✸❄❖✹✽❀❂✭✢✷❞✱✳❑✚❑②❬✸❄❃❅❱✱✳✵✮✻❧❁❤❑✍✱ ❁❃✭▲♥✸❁❖▼✓✶✙❄r✱✚♥✙✹➀✻✽✱♣✻❧❁❃✷❃✷✽❊✓✱✩❑❣❏❇✱✓✶⑦➌✸❁❃✭✲✯✉❬✞❳✤➍❝➎▲➏✍➐➑♥❝❬✸❼♣✶✸❁❃✭❇❑❲❈ ❪✱✏P✺✯❻P❞✹❢❈➓➒➔①❩→❇❷⑦➣✏④✓✹↔✻↕❅➂❁❖▼✓❅✗♥❻❬⑦✱✳❑ ✭▲❬✸❄②❁❃✭✴▼✳✷➁❀❱♥❻✱✕➙➛✶✙✭✴♥➛➜✮❴✏➝✾❄❃❅❱✱✉▼✍✷❞❬➞❑✮❀❂✵✁✱➅❬✞❳☞❑✮❀❱▼✓❅➑✶♦♥❝❬✙❼❜✶✙❁❃✭➓✻❧❁❃✷❃✷➊❊✚✱✉♥❝✱✍✭▲❬✸❄✆✱✓♥ ❈ ❪✱✏P✺✯✲P❞✹ ❈✠➒⑩➟→❇❷⑦➣✳➠➡❴✸P ➢↔➤❩➢ ➥➧➦✁➨❉➦❭➩➫➦❭➭➈➯↕➲❇➦❭➳✽➨➸➵❵➺✢➦❭➨❉➻➈➦❭➼✤➽❭➾ ➚✴❅❇✱✇✐➪✭✢❁❃❄✆✱✾✱✳✷➡✱✳❼❜✱✳✭✢❄✤❼❜✱✍❄❃❅❱❬⑦♥❵❁❤❑❜✭▲❬✙❄✼❊✓✶➞❑✍✱✓♥❵❬✸✭♦❄❆❅❇✱❥❑✮❄❩✵✁❬✸✭✲✯②❳✳❬✙✵✮❼✩✹✼❊✍❀❂❄❉✵✁✶✸❄❃❅❱✱✳✵❦✶❵❼❨❁❃✭❱➃ ❁❃❼❨❁❞◆⑦✶✸❄❩❁❖❬✸✭➶❑✓❄r✶✙❄r✱✍❼♣✱✍✭❱❄❜❬✙✵✓✹❥❼❜❬✙✵✁✱♠✯✲✱✳✭✴✱✳✵✚✶✙✷❃✷➁❫➞✹☞✶➛✻✽✱✓✶⑦➌✾❳✳❬✙✵✮❼❨❀❂✷❞✶✸❄❩❁❖❬✸✭▲P➹❘♠✱✉❼❨❀➂❑✮❄✩❄❆❅➂❀➂❑ ♥❝✱✍❛✙✱✳✷❞❬✓❏✗✶✸✭▲♥♣❀❂✭✴♥❝✱✍✵✓❑✮❄✆✶✸✭▲♥♣❄❆❅❇✱✳❑✍✱➪❳✳❬✙✵✮❼❨❀❂✷❞✶✸❄❩❁❖❬✸✭❇❑②❊✓✱❖❳✳❬✸✵✁✱↔❏✴✵✁❬✏▼✚✱✓✱✓♥✙❁❃✭➂✯❥✻❧❁❃❄❆❅☞❄❆❅❇✱❢✐➪✭✢❁❃❄✆✱❲✱✍✷❞✱✍➃ ❼❜✱✳✭✢❄✽❼❜✱✳❄❆❅❇❬✏♥❝P ➘⑨➴❤➘⑨➴❖➷ ➬⑨➮⑦➱❇➮⑦✃➂❐✗✃➂❒✴➮ ❮▲❰✲Ï❃Ð❸Ñ➅Ò ÓÕÔ❞➐✢Ö × ➒➑Ø✸Ù✚Ú➛Û♣Ô➡➐ ÜÕÝ❝Þàß ①❖❡✤④ á❿â➏⑦Ù✁➏ ã②➣ ä❋➒➓å⑦æ❺ç❭è❇é❜ê✍Ô❞➏⑦➐✲ë✁ì➡í✾ç✁Û❨➍➂➍❝ëâ ✈➂æ✴✈ î❥➒➑→❇ï✽❷ äð❅❱✱✳✵✚✱➊❁❤❑➊✶✩✷➁❁❃✭✴✱✚✶✸✵❉❑❣❏❱✶❝▼✓✱✳✹✄❄❆❅❇✱✽❏✴✵✁✱✓▼✳❁❤❑✍✱❲♥❻✱❖✐✽✭❱❁❃❄❣❁❖❬✙✭❵❬❭❳❉✻↕❅❂❁❖▼✚❅☞✻❧❁❃✷❃✷⑨❊✓✱➀✯❝❁❃❛➞✱✍✭♠❑✁❅❇❬✸✵✮❄❩✷➁❫➞➝❧✻✽✱ ❑✁❅❇✶✸✷❃✷Õ✶✙✷➁❑✳❬❦❼❜✶⑦➌❻✱✗●❩❑✮❀✳ñ❶▼✍❁❖✱✳✭✢❄❩✷➉❫♣❑✓❼❜❬✏❬✙❄❆❅❸◗↔❏▲✵✁✱✓▼✍❁❤❑✳✱♣♥✸❀❂✵✮❁❃✭✲✯❦❄❃❅❱✱✩▼✓❬✙❀❂✵✓❑✍✱✩❬✞❳✼❄❆❅➂❁❤❑✼✷❞✱✓▼✍❄❩❀❂✵✁✱✏P Ø✸➐✢Ö ß ①❖❡✤④❢➒ ➣ ò ó➂ôöõ❡ö÷ õ ❡ ø ù✳ú û Ü➈üý✍þÜ➈üÿ✁￾✂ ⑤ ó➂ô☎✄ ❡ ￾✂✝✆ ãò ✞✠✟☛✡✌☞✎✍ ✞✠✟✏✡✒✑☛✡✔✓✕✟✗✖ ✘➅➏❜➏✚✙❂➎❇ì❤Ø✸Ô➡➐ â➏⑦Ù✁➏❜ç✁➍❝Û♣➏❨➍✜✛✽ëâ➏❜➐❇➍❝ë✚Ø✙ë✚Ô❞➍❻➐✛ëâØ✙ë á➏ áÔ➡ì➡ì✣✢✢➏❜è❱ç✁Ô➡➐❇Ú✥✤❨Ó↕Ô➡Ù✓ç✞ë❲Ø✸Ù✚Ú➪Û♣Ô➡➐ Û♣➏⑦Ø❝➐❱ç✧✦✞ëâ➏❥Ø✸Ù✚Ú❝è❱Û❨➏⑦➐✲ë✇ëâØ✙ë❲Û♣Ô➡➐❇Ô➡Û❨Ô✩★✍➏✏ç✫✪ ✬♠ëâØ✙ë②Ô❤ç✭✪⑨ëâ➏❜Û❨Ô➡➐❇Ô➡Û♣Ô✮★➂✱✳✵✕①❖Ø❻ç✼➍❝➎❇➎▲➍❻ç✁➏⑦Ö❵ë✁➍ ëâ➏❥Û♣Ô❞➐❇Ô➡Û♠❀❂❼❥④✌✤✰✯â➏❦ç❭í➂Û✱✢✢➍❻ì✳✲❶Û♣➏✏Ø✸➐❱ç✧✦✁Ô❞➐➅ëâ➏❦ç✁➏✳ë❦①❆➍❻Ù②ç✁➎❱Ø❝ê✍➏✏④✤➍✜✛✴✬✥✵✳✶ Û♣➏⑦Ø❝➐❱ç✧✦✚Ø ç✁è✷✢❱ç✁➏✳ë➊➍❝Ù➊ç❭è✷✢✢ç❭➎❱Ø❻ê✳➏❨➍✜✛✴✬✥✵✏✸➅Û♣➏✏Ø✸➐❱ç✹✦✕✛❆➍❻Ù✇Ø❝ì❞ì✺✬✷✵✼✻❵Û♣➏⑦Ø❝➐❱ç✹✦✞ëâ➏✍Ù✚➏②➏✚✙❂Ô❤ç✞ë✓ç✴✬✥✵✽✈✄①❖Ø❝➐❱Ö❵ç✭✤ ë✽✤ ④ ò