1.2.4 Notation SLIDE 1 ∫vdx+gv(1) imitation u=arg min 5a(w, w)-e(w) Weak u∈X:a(u,)=(v),v∈X Note 2 Neumann and delta distributions(Optional) We note that, in R, our Neumann condition looks exactly like a delt distribution forcing at the boundary, a l. This is fine, since we know the delta distribution is an admissible(bounded) linear functional, that is, is in the pace(Q), for this one-dimensional problem We know that in the interior a delta distribution imposes(weakly) a jump in the derivative(see Note 10 of last lecture). On the boundary, it imposes (weakly) the value of the derivative itself -since there is no "other side "to the 2 Rayleigh-Ritz Approach 2.1 Approximation 2.1.1 Mesh SLIDE 11 Note our default problem is the Dirichlet problem; we shall explicitly indicate Neumann when we wish to consider that problem (primarily e erercise =0 h =UT k=1..,K=n+1: elements b n+1: nodes Triangulations Th The above decomposition is known as a triangulation, Th, even though in IR our elements are not really triangles(though they are simplices--which areÝ❥Þ✣ß✼Þ✥à á➳â✪ã✲ä✴ã✲å❖â❥æ ç❥è✙é①ê✽ëíì✎î ï❛ð✳ñ❊ò✴ð ó▲ô✶õ✱ö☛÷➙øúù û⑧ü ý õ✗þ✗÷✎þ✗ÿ✁￾ ✂ ô❖÷➙ø❽ù ûü ý ✄ ÷❶ÿ✁￾✆☎✞✝❣÷▲ô✠✟✠ø☛✡ ☞☛✌ ✍✏✎✸ò✑✎✓✒✔✎✓✕✗✖✙✘✚✎✓✛✬ò✢✜ ✣ ù✤✖✁✥✚✦✧✒✔✎✸ò ★✪✩✁✫ ✟ ✌ ó▲ô✶õ❀ö❯õ❛ø✭✬ ✂ ô❖õ❛ø ✮➍ð✗✖✙✯✰✜ ✣✏✱✳✲ ✜✒ó▲ô✣ ö❯÷✢ø✻ù ✂ ô❖÷➙ø❶ö✵✴✱÷ ✱✶✲ ✷✹✸✻✺✽✼✶✾ ✷✹✼✑✿❁❀❃❂❅❄✪❄✤❂❅❄❇❆✤❆❈✼✙❉❊✺❋❂●❆✻❍❏■❑✺❊▲▼❍✚◆✪✿❈✺❊❍✠✸❖❄✪■◗P❙❘☛❚❯✺❊❍✠✸❖❄❱❂❅❉❳❲ ✮➍ð↕ò✑✛✙✘✌ð◗✘❋❨❩✖❬✘❪❭❫✎✸ò❵❴❛ ü ❭❫✛❝❜✑✥ ☞ð❞❜✑✒❡✖✬ò✴ò❣❢❞✛✬ò❩❤✑✎✐✘❋✎❥✛✧ò❵❦✓✛❑✛❝✯❅❧➴ð❞♠❅✖❝❢✽✘❋❦❥♥✤❦✓✎❥✯✧ð✹✖✧❤✢ð❞❦❥✘❋✖ ❤❅✎♦❧✠✘❋✥✚✎✓♣✑❜❅✘❋✎❥✛✧ò●q❳✛✁✥❙❢r✎✸ò✑✦◗✖✙✘s✘❋❨✴ðt♣✉✛✁❜✴ò❁❤✑✖✙✥❋♥✁❭✈￾ ù✇✟✁①◗②③❨❩✎✓❧④✎✓❧❀ñ❊ò✴ð❝❭✭❧❊✎✸ò❩❢✛ðt⑤ðt✯➙ò✑✛❬⑤⑥✘✚❨✴ð ❤✢ð✗❦✐✘❙✖s❤❅✎♦❧✠✘❋✥✚✎✓♣✑❜❅✘❋✎❥✛✧òt✎♦❧⑦✖✬òt✖✁❤✑✒✔✎♦❧❋❧❊✎✓♣✑❦✸ð ô❳♣✉✛✁❜❊ò❩❤✢ð✗❤✪ø✭❦✓✎✥ò❊ð✗✖✙✥⑦q❳❜✴ò❩❢r✘✚✎✓✛✬ò❩✖✁❦⑧❭❑✘❋❨❩✖❬✘✵✎✓❧✗❭❖✎✓❧⑦✎✥òt✘✚❨✴ð ❧✚⑨❩✖✁❢✛ð❶⑩●❷ ü ô❹❸✗ø✽❭❅q❳✛❝✥③✘✚❨✑✎♦❧❺✛✧ò✴ðr❻❼❤❅✎✓✒❆ð✲ò❩❧✚✎❥✛✧ò❩✖✙❦✰⑨✑✥❋✛✁♣✑❦✸ð❞✒✳① ✮➍ð④✯➙ò✑✛❬⑤❽✘❋❨❩✖❬✘❾✎✸ò❿✘❋❨✴ð④✎✸ò❑✘☛ð✗✥✚✎✓✛✁✥☛✖➀❤✢ð✗❦✐✘❙✖t❤❅✎♦❧❊✘✚✥❋✎❥♣✑❜✑✘✚✎✓✛✬ò◗✎✓✒✆⑨❁✛❑❧❯ð❪❧❇ô❳⑤ð✗✖✙✯❖❦✓♥✴ø❯✖❾➁✠❜❩✒✔⑨ ✎✸ò❃✘❋❨✴ð✏❤✢ð❞✥❋✎❥➂❬✖✙✘✚✎✓➂✬ð↕ô❏❧☛ð✛ð ☞✛✙✘✌ð✞✟✗➃➄✛✙q➅❦♦✖✁❧❊✘❡❦✥ð❪❢✽✘✚❜❩✥☛ð✠ø✽①➇➆❛ò❃✘✚❨✴ð✳♣✉✛✁❜✴ò❁❤✑✖✙✥❋♥✁❭⑦✎✐✘❡✎✓✒✔⑨✉✛❝❧☛ð✗❧ ô❳⑤ð✗✖✁✯❖❦❥♥✴ø❈✘✚❨❊ð❯➂❬✖✙❦✓❜✴ð❺✛✙q✰✘✚❨❊ð➅❤✢ð❞✥❋✎✓➂▼✖✙✘✚✎✓➂✬ð❺✎❥✘❋❧☛ð❞❦❥q✉➈➉❧❊✎✸ò❩❢✛ð❺✘✚❨✴ð✗✥☛ð❺✎♦❧✄ò❩✛✏➊❊✛✙✘❋❨✴ð❞✥✭❧✚✎♦❤✢ð✗➋☛✘❋✛❾✘✚❨✴ð ➁✠❜✑✒✆⑨✢① ➌ ➍➏➎❯➐➀➑r➒➅➓❞➔④→t➣❁➍↔➓✽↕✪➙➜➛✇➝✏➝✳➞✈➟s➎❾➠❇→ ➡③➢❊➤ ➥➇➦➅➦❫➧✑➨❈➩❺➫✚➭⑥➯❈➲✑➫❊➨❇➳ ß✼Þ❖Ý❥Þ❖Ý ➵➺➸❑➻❪➼ ç❥è✙é①ê✽ëíì✴ì ➽s➾✙➚➶➪✳➾✙➹❅➘➀➴✁➪⑧➷r➬❬➹❅➮➱➚✵✃✰➘✚➾✁❐❞➮❥➪❞❒❰❮♦ÏÐ➚ÒÑ❩➪❡ÓÔ❮Ò➘✽❮❏Õ❙Ñ❖➮✓➪r➚✵✃✰➘✚➾✁❐❞➮❥➪❞❒sÖs×❇➪tÏ✚Ñ✑➬✙➮Ò➮③➪❋Ør✃✉➮➱❮❏Õ❞❮Ò➚❹➮➱Ù➄❮ÒÚ✰➴❬❮❏Õ❙➬❬➚❼➪ ➽s➪❞➹❅❒❡➬❬Ú❩Ú◗×❈Ñ✑➪❞Ú◗×❇➪s×✈❮♦Ï❊Ñ❿➚❼➾ÐÕ❙➾❬Ú✑Ï✽❮❏➴❝➪r➘Ô➚❳Ñ✑➬✙➚✢✃✰➘✚➾✁❐❞➮❥➪❞❒ÜÛ✐✃✰➘✽❮Ò❒✆➬✙➘✽❮Ò➮➱Ùt❮ÒÚ➄➚ÒÑ❩➪s➪❋Ø❖➪r➘✚Õ❞❮♦Ï❞➪✽Ï➶Ý✁Þ ❸Øù àßá â✽ã ü äâå äâå ö✇æ ù❽✟✬ö✗✡❞✡✗✡✳ö✚ç➐ù✤èÐ☎é✟✁✜Ô➪❞➮❥➪❞❒✆➪❞Ú❩➚⑧Ï ￾❁ê✦ö↔ë➆ù❵➃✴ö❞✡✗✡❞✡✳ö✚è❡☎é✟✁✜❶Ú✰➾✗➴❝➪✽Ï ☞❫ì ✷✹✸✻✺✽✼✏í î❈▲❪❍❹❂❅❄❈ï❱✿❁❉⑧❂✻✺✚❍❼✸❅❄❈■✆ðå ②③❨✴ð✶✖✙♣✉✛❬➂✬ð❡❤✢ð❪❢r✛✁✒✆⑨✉✛❝❧✚✎✐✘❋✎❥✛✧ò●✎✓❧s✯➙ò✑✛❬⑤✗òñ✖✁❧Ô✖ñ➚⑧➘✽❮❏➬✙Ú❑ò✁➹❅➮❥➬❬➚❹❮❏➾❬Ú✪❭✭ðå ❭❡ð✗➂✬ð✛ò✞✘❋❨✑✛✁❜❩✦✁❨●✎✸ò ❴❛ ü ✛✁❜❩✥➆ð✗❦✥ð✗✒❇ð✛ò❑✘❋❧✭✖✁✥☛ð✒ò✑✛✁✘✈✥☛ð❪✖✙❦✓❦❥♥❾✘❋✥✚✎♦✖✎ò❩✦✁❦✸ð✗❧✒ôÒ✘❋❨✑✛✁❜✑✦❝❨✔✘✚❨✴ð✗♥④✖✙✥✌ð③❧❊✎✓✒✆⑨✑❦❥✎♦❢✳ð❪❧✪➈➜⑤❺❨✑✎♦❢❙❨✆✖✙✥✌ð ó