Discretization of the poisson Problem in r Formulation april 2, 2003
✂✁☎✄✝✆✟✞✡✠☞☛✌✁☎✍✏✎✏☛✌✁✒✑✔✓✕✑✗✖✘☛✝✙✚✠✜✛✢✑✔✁☎✄✣✄✌✑✔✓ ✛✤✞✡✑✔✥✧✦✒✠✩★ ✁☎✓✫✪✭✬✯✮✱✰✳✲✴✑✵✞✣★✷✶✧✦✸✎✏☛✝✁✒✑✔✓ ✹✻✺✽✼✸✾❀✿✏❁✝❂✱❁❄❃❅❃❄❆
1 Model problems 1.1 Dirichlet 1.1.1 Strong Form SLIDE ain:9=(0,1) Find a h that in Q (1) 1.1.2 Minimization statement Define X≡H(9) Find he w2 d-f This follows from the previous lecture, noting that dA is now dz, and vw 1.1.3 Weak Formulation sLIdE 3 Find u∈ X such that 6J(u)=0 u∈X f vdx Vu∈X Again, this follows f er lecture with Vu. Vu now given by ur Ux
❇ ❈❉❋❊❍●❏■▲❑◆▼☞❉P❖❍■✒●❏◗❙❘ ❚☞❯❱❚ ❲❨❳❱❩❬❳❱❭❫❪❏❴❛❵✡❜ ❝❄❞❡❝❄❞❡❝ ❢✡❣✐❤❦❥❄❧✝♠♦♥✝❥❬❤❦♣ q❄rts✈✉✣✇②① ③⑤④⑦⑥⑨⑧❶⑩❸❷✡❹ ❺✢❻❽❼❡❾➀❿➂➁✸➃➅➄ ➆✌⑩➇❷❫➈➊➉➌➋❱➍❬➎➐➏✻➑➒➏❫⑧❦➑ ➓ ➉❅➔✸➔→❻ ➣ ⑩❸❷↔❺ ➉✽❼❡❾↕➃➙❻ ➉✌❼❱➁✸➃➙❻ ❾ ❿ ➛④⑦➜➞➝↕⑩➇➟↕➠✐❷➡➣✌➄ ❝❄❞❡❝❄❞➤➢ ➥◆➦❡❧✌➦❡♣✘➦❀➧⑦➨➩❣✸➦❡❥❄❧✢❢✡❣➂➨➩❣✸➫t♣▲➫t❧❅❣ q❄rts✈✉✣✇➯➭ ③⑤➠✒➲❬❷➀➠ ➳➸➵➻➺➽➼➾ ❼❀❺➞➃➅➄ ➆✌⑩➇❷❫➈ ➉✻❻✤⑧❶➜➚➝✘⑥➪⑩➇❷ ➶✌➹⑦➘➷➴ ❼❡➬⑤➃ ➮➏➀➠➂➜➒➠ ➴ ❼➱➬⑤➃✟❻ ➁✃➻❐ ➼ ➾ ➬❏❒➔❏❮⑦❰ ➓ ❐ ➼ ➾ ➣❄➬ ❮⑦❰♦Ï Ð❅ÑÓÒ➤Ô✗Õ✒Ö❦×✈×➇Ö❶Ø✽Ô➅Õ➐Ù➒Ö❶Ú✜Û✈Ñ❬Ü✔Ý❅Ù➒Ü✐Þ➂Ò❡Ö❶ßÓÔ✻×➇Ü➐à✐Û❀ß➩Ù➒Ü✒á✗â❄Ö❦ÛãÒ✈âtä➽Û➱Ñ➀å❶Û ❮tæ Ò➤Ô✴â❄Ö❶Ø ❮↕❰ áçå❦â❅è✚éç➬êÒ➤Ô â❄Ö❶Ø♦➬➞➔❫ë ❝❄❞❡❝❄❞❸ì íî➫t➨➀ï②♥✣❥❫❤❦♣➽ð✌ñ❡➨➀❣➂➦❡❥❄❧ q❄rts✈✉✣✇➯ò ➆✌⑩➇❷❫➈➊➉❍ó✚➳ ➋❱➍❫➎➐➏✻➑➚➏❬⑧❦➑ ô ➴↕õ ❼➱➉ö➃÷❻✤❾✻❿ øçù❋ó✧➳ ú ❐ ➼ ➾ ➉❄➔ûù❶➔ ❮↕❰ ❻ ❐ ➼ ➾ ➣✵ù ❮⑦❰ ❿ ø✵ù✴ó✧➳ Ï üä↕å❶Ò✈â➀á÷Û➱ÑÓÒ➤Ô✱Õ✒Ö❦×✈×❸Ö❦Ø✽Ô✱Õ➐Ù➒Ö❶ÚýÖ❶ß➩Ù✵Ü➚å❶Ù☎×þÒ❡Ü✐Ù✵×➇Ü➐à✒Ûãß➩Ù➒Ü✵Ø☞Ò✈Û✈Ñ✻éÿ➉✁➂éÿù â❄Ö❦Ø✘ä⑦Ò✈Þ Ü✐â✄✂✆☎✔➉➔ ù➔ ë ➁
1.1. 4 Notation a(,U) U Minim u=arg min 5 a(u, w)-e() u∈X:a(u,u)=(u),vv∈X 1.1.5 Generalization For any(u)∈H-(9) fndu∈H(9) such that u=arg min 2a(u, w)-e(a),or a(u,v)=C(),VU∈H(2); for example, e(o)=0zo, D)=v(ao) is admissible As indicated earlier, the delta distribution is not admis sible if& Cir be motivated by considering the green,'s function 1. 1. 6 Regularity SLIDE 6 Ife∈H-(9) lal (2)≤C If∈2(9),(v)=/fndr ll()≤Co‖ flea(S Recall ulPH( te 1 If e(o)=So fudr, with fE L2(Q2), we immediately obtain from I|lH1(2)< IellH-1(s)that lalH(Q)S ClIfz (o), since the H- norm is al ways bounded by the L2 norm(there is "more"in the denominator). But from the strong form uz= f we can see that uH2()≤‖fz2()· It thus follows that Co in the
✝✟✞✠✝✟✞☛✡ ☞✍✌✏✎✒✑✓✎✒✔✠✌✟✕ ✖✟✗✙✘✛✚✢✜✤✣ ✥✧✦✆★✪✩✓✦ ✫✭✬✯✮✱✰✳✲✵✴✷✶ ✸✺✹ ✻ ✮✽✼✾✲✿✼❁❀❃❂ ❄ ✬✯✲✵✴❅✶ ✸✹ ✻❇❆ ✲❈❀❃❂❊❉ ❋✍●❍✩✓●❍■❏●❍❑✒▲✿▼◆●❍❖P✩❘◗ ❙ ✶❚▲✿❯❲❱❊■❳●☛✩ ❨❬❩P❭❫❪❴ ✫✟✬✠✮❵✰✳✮❛✴❝❜ ❄ ✬✯✮✧✴ ❞❡✦✒▲✿❢✭◗ ❙✍❣✐❤ ◗✾✫✭✬❙ ✰✳✲✵✴❥✶ ❄ ✬✠✲❦✴❛✰♠❧❵✲ ❣♥❤ ✝✟✞✠✝✟✞♣♦ q♥r❦✕❬r✙st✑✪✉✠✔✠✈❃✑✵✎✒✔✇✌✏✕ ✖✟✗✙✘✛✚✢✜❡① ②✓❖❃❯❁▲✿✩❦③ ❄ ✬✠✲❦✴ ❣✍④⑥⑤ ✹ ✬✇⑦✽✴✆⑧ ★✪✩✪⑨ ❙⑩❣✐④✹✻ ✬✇⑦✽✴✾❶◆❷✪❸❺❹♥▼◆❹✪▲✿▼ ❙ ✶❚▲✿❯❲❱ ■❏●❍✩ ❨❬❩P❻✽❼❽❿❾☛➀✏➁ ➂✹ ✫✭✬✯✮❵✰◆✮✧✴❝❜ ❄ ✬✯✮❛✴➄➃➅❖P❯ ✫✭✬❙ ✰◆✲❦✴➆✶ ❄ ✬✯✲✵✴➇✰ ❧✱✲ ❣✍④✹✻ ✬➈⑦✽✴❛➃ ➉❖P❯✽✦✆➊✓▲P■❏➋✪➌☛✦❃⑧ ❄ ✬✯✲✵✴❥✶➎➍✇➏➐✼ ❽ ✰✳✲✵➑➆✶➒✲✟✬✠❂✻ ✴♠●❍❶❁▲❃⑨✵■❳●❍❶❲❶◆●☛➓✓➌❍✦P➔ →❛➣↕↔✛➙✟➛❿↔✠➜❺➝✿➞➠➟❺➛✍➟❺➝❿➡➇➢➤↔✠➟➐➡❺➥✽➞✛➦✪➟➧➛P➟➐➢➤➞➨➝✍➛❿↔♣➣➇➞✇➡➇↔✠➩➐➫✵➞➈↔✠➭❿➙✄↔♣➣✱➙✭➭❿➞❈➝P➛✿➯↕↔♣➣❺➣➇↔✠➩➐➢☛➟➧↔➲ ⑦➵➳➺➸➻➂ ➥❁➝t➣❏➜❺➝❿➙ ➩❺➟❵➯➧➭❿➞➈↔✛➼❿➝❿➞➨➟❲➛➽➩✆➾♥➜❲➭✿➙✓➣➇↔✠➛P➟➐➡➇↔✛➙✙➚➽➞✯➦✓➟➶➪✾➡◆➟❺➟✆➙➘➹➣❝➲❺➫✵➙✟➜➐➞✇↔✠➭✿➙✟➴ ✝✟✞✠✝✟✞❍➷ ➬➽r❦➮✏➱❬✉✠✑✵s❿✔✠✎❲✃ ✖✟✗✙✘✛✚✢✜❡❐ ➸➉❬❄ ❣✐④⑤ ✹ ✬✇⑦✽✴✆⑧ ❒ ❙ ❒ ❻❼ ❾☛➀✏➁❰❮ÐÏ ❒ ❄ ❒ ❻❈Ñ ❼ ❾❍➀✪➁ ❉ ➸➉❬❄ ❣✐Ò➂ ✬✇⑦✽✴➇⑧ ❄ ✬✠✲❦✴➆✶Ó✸✹ ✻Ô❆ ✲✧❀❃❂ ❒ ❙ ❒ ❻♠Õ ❾☛➀✏➁♠❮✺Ï✻ ❒ ❆ ❒➐Ö Õ ❾☛➀✏➁ ❉ × ❪ Ø➟❺➜❺➝❿➢✛➢ ❒ ✲ ❒ ➂ ❻♠Õ ❾☛➀✏➁ ✶ÚÙ ✲✭Ù ➂ ❻❰Õ ❾☛➀✏➁➘Û ❒ ✲ ❒ ➂ ❻❼ ❾☛➀✪➁ ✶❚Ü✻ ✹ ✲ ➂✼Ý✼ Û ✲ ➂✼ Û ✲ ➂ ❀❃❂ ➴ Þ❡ß➘à➇á✤â ã⑩á❿äæå✏ç✇è✵étê◆à✯ë ➸➉✭❄ ✬✯✲✵✴❥✶ Ü ✻ ✹ ❆ ✲æ❀P❂ì⑧❃í✽●☛▼◆❹ ❆ ❣♥Ò➂ ✬➈⑦✽✴➇⑧✙í❰✦✽●❍■❳■❏✦Ý⑨✵●♣▲❿▼◆✦✒➌☛③❏❖P➓✓▼❲▲✿●❍✩ ➉❯◆❖❃■ ❒ ❙ ❒ ❻❼ ❾☛➀✪➁♠î Ï ❒ ❄ ❒ ❻Ñ ❼ ❾☛➀✪➁ ▼◆❹✪▲✿▼ ❒ ❙ ❒ ❻❼ ❾☛➀✪➁❰❮ïÏ ❒ ❆ ❒✆Ö Õ ❾☛➀✪➁ ⑧❃❶✳●❍✩✪❸✆✦❥▼◆❹✓✦ ④⑤ ✹ ✩✪❖P❯❲■Ó●♣❶æ▲✿➌❍í♠▲t③✵❶✢➓✏❖❃❷✓✩✪⑨✵✦Ý⑨ ➓❦③ð▼◆❹✪✦ Ò➂ ✩✓❖❃❯◆■ñ✬✯▼◆❹✓✦✒❯◆✦✽●♣❶ðò✳■❳❖❃❯◆✦Ýó❁●❍✩↕▼❲❹✓✦❁⑨✵✦✒✩✓❖P■❳●❍✩✪▲❿▼❲❖P❯➇✴➇➔ìô♠❷✵▼ ➉❯❲❖P■➎▼◆❹✓✦❁❶✳▼◆❯❲❖P✩✓❱ ➉❖❃❯◆■ ❜ ❙✼✒✼ ✶ ❆ í♠✦❳❸➐▲✿✩õ❶✳✦✒✦❵▼◆❹✏▲❿▼✁Ù ❙ Ù ❻♠Õ ❾☛➀✪➁✧❮ ❒ ❆ ❒✆Ö Õ ❾❍➀✪➁ ➔➄➸➠▼❛▼◆❹❦❷✪❶ ➉❖P➌❍➌☛❖❿í❁❶❈▼◆❹✪▲✿▼ Ï✻ ●☛✩✤▼◆❹✓✦ ❴
bove slide is(1+C2)1/2. This can also be shown by explicit construction of u (see Lecture 2 from earlier in the course) The fact that u is regular when f is regular(and in IR2, the domain is suitably regular) has very important implications as regards the convergence rate of the of a priori and a poste st es 1.2.1 Strong form SLIDE 7 2=(0,1) Find a such tha u(0) (1) for given∫, 1.2.2 Minimization statement SLIDE 8 {U∈H1(9)|v(0)=0} arg f wdr-g w(1 This follows from the previous lecture, noting that rn g v da is here ju 9v(1). We can also show this explicitly by integrating by parts to find dJ,(u) ∫0v{-x-∫}dr+v(1){uz(1)-g}=0,V∈X 1.2.3 Weak Statement Find u∈ X such that yu∈X +gv(1),VU∈X
ö✿÷✟ø❿ùPú❛û✳ü❍ý♣þ✵ú➄ý❍û➄ÿ✁✄✂✆☎✞✝✠✟☛✡☛☞✌✝✎✍✄✏✒✑✓ý♣û✒✓➐ö✎✔♥ö✿ü♣û✳ø❏÷✟ú➄û☛✑✓ø✖✕✗✔➽÷✙✘➧ú✛✚✢✜✓ü☛ý✣✓✆ý✥✤✦✓✆ø✧✔✪û✁✤✌★☛✩✪✓✫✤◆ý❍ø✬✔➶ø✎✭✯✮ ÿ✠û◆ú➐ú✱✰✢ú✲✓✳✤☛✩✴★❲ú✱✵✱✭✶★❲ø✬✷ ú✒ö✎★❲ü❍ý☛ú✲★♠ý✸✔✹✤☛✑✓ú✱✓➐ø✬✩✴★❺û◆ú✺✟✫✍ ✏✒✑✓ú✻✭✠ö✬✓✳✤✼✤☛✑✪ö✎✤✯✮↕ý❍û✽★◆ú✲✾✬✩✓ü♣ö✎★✼✕✗✑✪ú✛✔❀✿↕ý❍û✯★❲ú✛✾✬✩✪ü❍ö✬★➆ÿ✠ö✎✔✏þðý✥✔❂❁❃✦✝✬❄✺✤☛✑✓ú♠þ✵ø✬✷➧ö✿ý✸✔ðý♣ûìû☛✩✓ý✥✤❲ö✿÷✪ü✥✘ ★❲ú✛✾✬✩✪ü❍ö✬★❅✟✄✑✪öPû❰ùPú✲★☛✘❳ý✸✷❆✜✟ø✬★☛✤❲ö✬✔✙✤♠ý✸✷❇✜✓ü☛ý✣✓➐ö✎✤◆ý❍ø✬✔✪û✾ö❃û❈★❲ú✛✾✙ö✎★❺þ✓û❉✤✌✑✓ú❂✓✆ø✧✔✙ù❃ú✛★✌✾Pú✲✔❊✓✆ú✗★❺ö✖✤❲ú✧ø✬✭✼✤☛✑✓ú ❋✔✓ý✥✤◆ú➶ú➐ü❍ú✛✷❳ú✛✔✙✤✱✷❳ú✳✤✌✑✓ø✵þ❡ö✬✔✪þ●✤☛✑✓ú❍✓✆ø✬✔✏û✁✤✌★☛✩❊✓✳✤◆ý❍ø✬✔❡ø✎✭✱■❆❏▲❑✫▼❖◆✖❑✫▼ðö✎✔✪þP■❇❏✴◆✖◗❅❘❚❙✛❑✫▼❖◆✖❑✫▼❛ú✛★✌★◆ø✧★ ú✒û❯✤◆ý✸✷➧ö✖✤❲ú✒û✲✍ ❱❉❲❨❳ ❩❭❬❫❪✽❴❛❵❝❜❡❞❛❞❣❢ ❤❥✐✣❦✼✐❖❤ ❧✼♠✛♥✖♦❥♣✯q✆r✯♦❊♥✖s t❥✉✙✈①✇✽②✆③ ④ø✬✷➧ö✿ý✸✔✼⑤ ⑥⑧⑦➎ÿ❖⑨✴⑩✲✺✟❶✍ ❷ý✥✔✏þ❸✮❫û❯✩❊✓❅✑❍✤☛✑✏ö✖✤ ❹ ✮❥❺✺❺ ⑦ ✿ ý✥✔❻⑥❼⑩ ✮æÿ❖⑨✙✟❽⑦ ⑨❾⑩ ✮▲❺✪ÿ✁✠✟❽⑦ ❿➀⑩ ✭✯ø✬★✗✾❃ý☛ù❃ú✛✔➁✿❥⑩❯❿➂✍ ❤❥✐✣❦✼✐✣❦ ➃➅➄❖♣➆➄❖s➇➄❨➈✬➉✢♠✺➄❖♦❥♣⑧❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍➌ ④ú❋✔✓ú ➎➐➏❼➑✧➒➔➓❾→●✡❿ÿ➣⑥✗✟↕↔➙➒✟ÿ❨⑨✧✟✄⑦❫⑨❈➛❾✍ ❷ý✥✔✏þ ✮❍⑦❚ö✎★✌✾➇✷❳ý✥✔ ➜➆➝✬➞❝➟ ÿ❖➠❛✟ ✕✗✑✓ú✲★◆ú ➟ ÿ❖➠❛✟✻⑦ ✵ ➡✡ ➢ ➠❺✦➤✧➥ ✝ ❹ ➡✡ ➢ ✿❀➠ ➤✬➥ ❹ ❿❛➠✱ÿ✁✠✟❡➦ ➧▲➨▼✣◗➔➩✳◆✖➫①➫✸◆✖➭❡◗❍➩❅❑☛◆✎➯➲❘➨ ❙➳❏❥❑☛❙✛➵✺▼❖◆✖➸➙◗✆➫✥❙❅➺✳❘➣➸✢❑☛❙✳➻❍➼▲◆✖❘➣▼①➼✙➽❫❘➨ ■✖❘❆➾✲➚✧➪➶❿✹➒ ➤✙➹ ▼✣◗ ➨ ❙✛❑☛❙➴➘✫➸➙◗❅❘ ❿✒➒✟ÿ❯✺✟✛➷➴➬➳❙❀➺❅■✖➼➀■✎➫➮◗✛◆❇◗➨ ◆✎➭⑧❘➨▼✣◗❂❙☛➱✳❏▲➫✃▼❖➺✳▼①❘➣➫✃❐➴❒✛❐❇▼①➼❊❘❚❙➣➽✬❑☛■✎❘❨▼①➼➙➽➔❒✳❐❛❏❊■✖❑✫❘❨◗❶❘❚◆✒❮✄➼▲❰✞Ï ➟✧Ð ÿ✶✮❭✟✄⑦ ➾ ✡ ➢ ➒●Ñ ❹ ✮▲❺✲❺ ❹ ✿✯Ò ➤✧➥ ✂➇➒✟ÿ❯✺✟✫Ñ✲✮▲❺✓ÿ✁✠✟ ❹ ❿❥Ò❛⑦P⑨✴➻▲Ó❀➒➔➓❍➎⑧➷ ❤❥✐✣❦✼✐✸Ô ÕÖ➊✙➉✴×Ø❧✼♠✲➉✢♠✺➊✙s➋➊✙♣▲♠ t❥✉✙✈①✇✽②➍Ù ❷ý✥✔✏þ❸✮➳➓❾➎ û❯✩✪✓❅✑❍✤✌✑✪ö✖✤ Ï ➟Ð ÿ✶✮❭✟✄⑦❫⑨❍⑩ Ó❀➒➔➓✹➎ Ú ➡✡ ➢ ✮❥❺✒➒✎❺ ➤✧➥ ⑦ ➡✡ ➢ ✿❀➒ ➤✧➥ ✂✆❿❣➒✭ÿ✁✺✟❶⑩ Ó✱➒➴➓✹➎Û➦ Ü
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 are
Ý❥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