(-工v)lk (-工h2)/lrt LIDE 12 h max w h h, max max lu" The first line follows from Rolle's Theorem(which requires WITh EC(Tn), as is the case here). The second line bounds the K=b integrals by hx the mazimum of the integrand. Note, however that we only require w to be defined in the elements, not at the nodes, so if we place our delta distribution loads at nodes this hypothesis is still satisfied for solutions u of our Poisson problem even Since(ln w'rk is a constant, we are effectively appro imating w by a constant Not surprisingly, this will not work very well if w has jumps(infinite) in Th also, the larger the w, the larger the error, since the more w' will vary way from a constant.(In general, if w has strong singularities, lw-IhWlHI(Q will only converge as some fractional power of h) b Exercise 2 Prove the L2 estimate of Slide 10. Hint: write(w- Lh wlrk ral in terms of (w- Lhw'lrk; then express(w-Lhw'T the h SLIDE 13 Ifw∈x,andw∈H2(9,Th)➴✍➷✗➬★➮✑➱➉✃❳❐ ❒ ❒ ❒❺❮✜❰❸Ï✿Ð✄Ñs❰❯ÒÔÓ②Õ Ö❃×Ø ❮✒Ù✍Ò ❒ ❒ ❒❴Ú ❒ ❒ ❒ ❒ÜÛ❪Ý Þ × ❮✜❰⑧Ï✿Ð✄Ñs❰❯ÒÔÓ ÓÔÕ Ö❃×ØàßÙ ❒ ❒ ❒ ❒ Ú ❒ ❒ ❒ ❒áÛâÝ Þ × ❰✮Ó Ó ßÙ ❒ ❒ ❒ ❒ ã➄ä⑧å✾æ✳ç Ýsè Ö❃×Ø Õ ❰éÓ Ó②Õ ê ë ì❑í✙î ÛÖ❃×Ø ❮✜❰⑧Ï✿ÐÑ ❰❯ÒÔÓÔÕ ïÖ✍×Ø ßÙ ã➟ðä ä✫ñ❍äòå✾æ✳ç ì❂í❾î✰óáôáôáô óê å✾æ✳ç Ýsè Ö❃×Ø Õ ❰✮Ó Ó➌Õ õ ï ö✁÷ ø❃ù✆ú✙û③ü✰ý❑þ❾ÿ✁￾✄✂✍ú✆☎✞✝✳ÿ★ÿ✟✝✡✠⑥ý☛☎✰ü☞✝✍✌✏✎✑✝✳ÿ★ÿ✏ú✓✒ý✈ø❃ù✆ú✔✝sü❛ú✞✌✖✕✗✠❾ù✘￾✚✙✰ù❽ü❛ú✔✛✞✜✢￾★ü♦ú❑ý ❰àÕ ÖØ✤✣✦✥ î ❮✚✧Ñ Ò✔★✪✩ý✫￾❺ý þ✜ù✆ú✑✙✩ ý❂ú ù✆ú❩ü❛ú✭✬✍✮✈ø❃ù✆ú❊ý❩ú✔✙✔✝✡✂✰✯➑ÿ✁￾✄✂✍ú✲✱✔✝✡✜✢✂✳✯✳ý þ✜ù✆ú✵✴ Ú î Ñ ￾✄✂◗þ②ú✷✶❳ü✩ ÿáý✸✱✺✹ ä✼✻ þ★ù◗ú✸✌✩✺✽￾✄✌✾✜✢✌ ✝✭☎✶þ★ù◗ú✿￾✄✂✉þ➌ú✗✶❳ü✩✂✰✯❀✮✦❁✾✝✳þ②ú★ ù❂✝✡✠✠ú✺❃✳ú❂ü★ þ★ù ✩ þ❄✠✠ú❅✝✡✂◗ÿ❆✹ ü❛ú✔✛✞✜✢￾★ü♦ú ❰Ó Ó þ❇✝❈✱✰ú❅✯❳úû✵✂✍ú✔✯❈￾✄✂âþ✜ù✆ú ú❩ÿ✐ú✞✌✾ú✺✂◗þý ★ ✂✳✝✳þ ✩ þ❊þ✜ù✆ú❉✂✳✝❊✯❳ú❂ý ★ ý✺✝❋￾☎●✠✠ú✸❍❃ÿ✩ ✙✰ú■✝✡✜ü❏✯❳ú❩ÿÜþ✩ ✯✡￾❺ý❑þ❣ü❑￾✚✱✞✜þ✷￾✚✝✡✂✫ÿ▲✝✩ ✯✳ý ✩ þ▼✂✳✝❊✯❳ú❂ý ★ þ✜ù✘￾❺ý✸ù✘✹◆❍✓✝✳þ✜ù✆ú❂ý✔￾❺ý✤￾❺ý✈ý❑þ❖￾★ÿ★ÿ❦ý ✩ þ❖￾❺ý✜û✁ú✔✯P☎✞✝sü✈ý✞✝sÿ✁✜þ✷￾✚✝✡✂✆ý❄◗❘✝❙☎■✝✍✜ü❉❚✸✝✡￾❺ý✰ý✺✝✡✂❯❍✍ü☞✝❱✱❂ÿ✐ú✺✌ ú✺❃✳ú✞✂❲￾✄✂ þ✜ù✘￾❺ý❉✙✩ ý❩ú❳✮ ❨￾✄✂✰✙✰ú ❮Ð✄Ñ➒❰éÒ Ó Õ Ö❃×Ø ￾❺ý ✩ ✙✔✝✡✂◗ý✰þ✩✂✉þ★ ✠✠ú ✩ ü❛ú❯ú❖❩✮ú✔✙❂þ✷￾✄❃➇ú❩ÿ✁✹ ✩❍❀❍❃ü☞✝✽ ￾✄✌✩ þ✷￾✄✂❬✶ ❰Ó ✱✞✹ ✩ ✙◆✝✍✂✆ý❑þ✩✂◗þ❇✮ ❁❉✝sþ ý❑✜ü✷❍✍ü❑￾❺ý❑￾✄✂✘✶sÿ✁✹★ þ★ù✢￾❺ý❏✠❭￾★ÿ★ÿ❪✂✳✝sþ✑✠❫✝✳ü☞❴❯❃✳ú❂ü❑✹❵✠✠ú❩ÿ★ÿ❫￾☎ ❰Ó ù ✩ ý❜❛❑✜✢✌✑❍✉ý❵✕❰Ó Ó ￾✄✂✢û✵✂❝￾★þ➌ú✭✬❅￾✄✂ ✧ ì Ñ✪❞ ✩ ÿáý✺✝ ★ þ✜ù✆ú✕ÿ✩ ü❖✶➒ú❩ü❽þ✜ù✆ú ❰Ó Ó ★ þ✜ù✆ú✕ÿ✩ ü❖✶➒ú❩ü❽þ✜ù✆ú➏ú❩ü❑ü☞✝sü★ ý❑￾✄✂✰✙♦ú✕þ✜ù✆ú❅✌✤✝✳ü♦ú ❰Ó ✠❭￾★ÿ★ÿ❜❃✩ ü❑✹ ✩✠ ✩✹▼☎✰ü◆✝✡✌ ✩ ✙◆✝✍✂✆ý❑þ✩✂◗þ❇✮P✕❢❡✞✂■✶✗ú✞✂❃ú❂ü✩ ÿ ★ ￾☎ ❰ ù ✩ ý ý❑þü☞✝✍✂❬✶✸ý❑￾✄✂❬✶❀✜ÿ✩ ü❑￾★þ✷￾✒ú❂ý ★③Õ ❰ Ï✸Ð✄Ñ➒❰àÕ ❣❜❤✞✐▲❥❝❦ ✠❭￾★ÿ★ÿ✆✝✍✂◗ÿ✁✹✿✙✔✝✍✂✓❃✳ú❂ü❖✶✗ú ✩ ý➑ý✺✝✡✌✾ú❫☎✰ü✩ ✙❩þ❖￾✚✝✍✂✩ ÿ❂❍❂✝✍✠✠ú❂ü❏✝✭☎ ä ✮ ✬ ❧✦♠✲♥✪♦✍♣❀qsr✚t❀♦❵✉✦✈❫✇☞①✡②❱③✑④◆⑤❂③❉⑥ï ③❳⑦❙④☞⑧å✾æ④☞③❉①✍⑨✵⑩✘❶▲⑧❸❷✢③ ð❊❹❂❺✸❻ ￾✄✂✉þ✞❼✵❽✇◆⑧❆④◆③ ❮✜❰❸Ï✿Ð✄Ñ➒❰éÒ❩Õ Ö❃×Ø æ⑦ æ ❷✢③✞❾✓❿✓⑧❆④◆③❉⑧✟❿❱④◆③✺➀❱✇æ❶➁⑧▲❿❵④◆③✺✇å⑦✑①✍⑨ ❮✜❰ Ï✿ÐÑ ❰éÒ Ó Õ Ö❃×Ø✆➂ ④◆⑤❂③✺❿✦③ç✘➃✇☞③❊⑦☞⑦ ❮✒❰❸Ï➏ÐÑ❰❯Ò Ó Õ Ö❃×Ø æ⑦ ⑧✟❿➄④◆⑤❂③●➅î✔➆ ⑦☞③å⑧✟❿❂①❀✇å➇➃✇◆①❬①❀⑨ ❺ ➴✍➷✗➬★➮✑➱➉✃❀➈ ➉⑨ ❰ ✣➄➊❲➋ æ❿✓❷ ❰ ✣ ➅ ï ❮❖➌➎➍◆➏✆Ñ➒Ò ➋ Õ ❰⑧Ï✕ÐÑ ❰àÕ ❣❤ ✐▲❥❝❦ ã ä➐➒➑ ❰ ➑ ❣❪➓✺✐▲❥ ó ➔Ø ❦ ➑ ❰⑧Ï✕ÐÑ ❰ ➑✺→ ➓ ✐▲❥❝❦ ã ä ï ➐ ï ➑ ❰ ➑ ❣➓ ✐✟❥ ó ➔Ø ❦✲➍ ➣