t cr t cy y ∈RR We restrict our attention, as in one space dimension to linear polynomial ap prorimations. The ertension to higher order elements is not difficult. In prac tice, quadratic elements are often the best in terms of accuracy per unit of work We also consider only triangular elements, though in many cases quadrilateral (bilinear or biquadratic)elements have advantages 2.2.2 Basis(Nodal) ∈Xn,;(a)=6,1≤i,≤ nonzero We see how our requirements on the topology of the triangulation are reflected in the space and basis. For example, because we require that elements that intersect n an edge must intersect over an entire edge, we are ens ured that, given two f Note by our node numbering the aj,j=l,., n, are all in the interior; it follows from sp;∈ Xn C X that pi(a)=0rj=n+1,…,n SLIdE 8 Nodal interpretation: U E Xh U=∑vg;(x) u(a)=∑"()=∑6→同=以(a)
➬➮➶➱✍✃➹❐✓❒❏❮✴❰✝Ï✧Ð Ñ✏Ò❉Ó♣Ô✴Õ▲Ö×Ô✾Ø ÙPÚ✾ÛPÜ Ý✁Þ ß ÖàÔ✾á Ù✤Ú✴ÛPÜ Ý✁â ã➔ä Ô ä Ô✾Ø ä Ô✾á✻å✥➬æ ç❢è❆é✛è✴ê✴ë❩é✴ì☞í✾ë❇î✍ï✌é❆ð✏ë❩ë❯è✾ñ✺ë✸ì☞î✍ñ✖ò❝ð✏ê❉ì✶ñóî✏ñ❞è✻ê❩ô✽ð✟í❈è❃õ✏ì✶ö❃è✦ñ✖ê✴ì☞î✏ñ✽ò❝ë❯î✥÷øì✶ñ❞è❈ð✍é✝ô✽î✏÷øù✏ñ➔î✏ö❆ì☞ð✏÷❿ð✁ô➔ú ô❞é✁î✦û✄ì✶ö✫ð✍ë✸ì☞î✍ñ✖ê✦ü❢ý➔þ✖è❃è✁û✟ë❯è✦ñ✖ê✴ì☞î✏ñ❄ë✮î❃þ✌ìrÿPþ✽è✾é✫î✏é✛õ✄è✾é✫è✾÷✲è✦ö❃è✾ñ✺ë☞ê❉ì✢ê✘ñ➔î✏ë❇õ✏ì⑨í✦ï✌÷øë✮ü✂✁✾ñ❍ô➔é✛ð✟í✦ú ë❩ì☞í✁è✾ò☎✄✦ï✽ð✟õ✍é✛ð✏ë❩ì☞í✜è✦÷✲è✾ö❃è✦ñ✽ë✸ê❬ð✏é✛è✡î✝✆❈ë❯è✦ñ✥ë➹þ✖è✂✞❈è✴ê✴ë➆ì✶ñ➐ë✮è✦é✴ö❉ê✩î✟✆❬ð✟í❈í✦ï✌é✛ð✟í✦ù✝ô✽è✦é❇ï✌ñ✽ì✶ë①î✝✆✡✠❿î✏é☞☛✄ü ç❢è❃ð✏÷②ê✦îtí❈î✏ñ✖ê✴ì☞õ✄è✾é❃î✏ñ✺÷øù❖ë❩é✴ì☞ð✏ñ➍ÿ✍ï✌÷✲ð✍é✫è✦÷✲è✦ö✫è✦ñ✽ë✸ê❈ò➶ë➹þ✖î✍ï✟ÿPþ❋ì✶ñ♥ö✫ð✍ñ✽ùtí✁ð✏ê✦è✴ê✌✄✦ï✽ð✟õ✍é✴ì✶÷✲ð✏ë❯è✦é✛ð✏÷ ✍✞✾ì✶÷rì✶ñ❞è❈ð✏é✻î✍é✎✞✾ì✏✄✦ï✽ð✟õ✍é✛ð✏ë❩ì☞í✝✑➐è✾÷☎è✾ö❃è✾ñ✺ë☞ê✩þ✖ð✓✒✤è✫ð✄õ✔✒✏ð✏ñ✽ë❯ð✦ÿ❏è✾ê✾ü ✕✗✖✘✕✗✖✘✕ ✙✌✚✜✛✣✢✏✛✥✤✝✦✥✧✩★✪✚✜✫✭✬ ✮✰✯✲✱✴✳✩✵✷✶ ✸❍❒❉Ó✺✹☞✻✽✼✿✾❁❀❃❂➱ ä❅❄❆❄❆❄✦ä ❂☎❇❉❈✄❰ ❂☎❊①å❋✸❍❒ ä ❂☎❊❧✃❍●✪■P❮ Ó❑❏❅❊▲■ ä◆▼✂❖◗P❈ä❙❘❚❖❱❯❳❲ ❨ïPô✄ô✖î✍é✴ë❬❩✿❭☎❂☎❊✛❰ ç❢è❝ê✦è❈è➶þ✖î✓✠✼î✏ï✌é❝é✛è❪✄✦ï✌ì✶é✛è✦ö✫è✦ñ✽ë✸ê❝î✍ñ❍ë✶þ✽è❇ë❯î✁ô✽î✏÷☎î✛ÿ✟ù❉î✝✆❝ë➹þ✖è✩ë✸é✴ì☞ð✍ñ❏ÿ✟ï✌÷✲ð✏ë❩ì☞î✏ñtð✏é✛è❇é✛è❍❫❝è❈í✦ë✮è❈õ✡ì✶ñ ë➹þ✖è❇ê✸ô✽ð✟í❈è▼ð✏ñ➔õ✎✞❈ð✤ê✴ì✢ê✦ü❵❴❸î✏é✩è✁û➍ð✏ö✩ô❞÷✲è✾ò☎✞❈è❈í❈ð✏ï➍ê✦è❛✠❿è✩é✛è❪✄✦ï✌ì✶é✛è❇ë➹þ✖ð✍ë➆è✦÷✲è✦ö✫è✦ñ✽ë✸ê❝ë✶þ✽ð✏ë ì✶ñ✽ë❯è✾é❈ê✦è❈í✾ë î✏ñ ð✏ñ❅è❈õ✦ÿ❏è❍ö❆ï➍ê✴ë✝ì✶ñ✽ë❯è✾é❈ê✦è❈í✦ë❬î✔✒✏è✾é❖ð✍ñ✼è✦ñ✽ë❩ì✶é✛è❍è❈õ✦ÿ❏è✴ò❜✠❿è❖ð✏é✁è❖è✦ñ✖ê✴ï✌é✛è❈õ➐ë✶þ✽ð✏ë✸ò➶ÿ✍ì✴✒✏è✦ñ ë❍✠❿î ñ❞îPõ✟ð✍÷✩✒✏ð✏÷øï✽è✾ê❈ò➶ð❝✆❈ï✌ñ➔í✾ë❩ì☞î✏ñ❀ì✢ê➶ô✽è✦é❞✆✾î✏é✛í❈è✫í❈î✏ñ✽ë❩ì✶ñ✽ï✽î✍ï➍ê✜î✓✒✤è✦é✘ë➹þ✖è❆ð✤ê❈ê✦î✒í✦ì☞ð✏ë❯è❈õ✥è❈õ✒ÿ✄èPü ❡❉î✍ë✮è❢✞✾ù❅î✍ï✌é✥ñ➔î✒õ✄è❢ñ✽ï✌ö❚✞❈è✾é✴ì✶ñ➍ÿ♥ë➹þ✖è❣●■ ò ❘ Ó ▼✟ä❅❄❆❄❅❄✾ä✟❯ ò❆ð✍é✛è❋ð✍÷✶÷✝ì✶ñ ë➹þ✖è❢ì✶ñ✺ë✮è✦é✴ì☞î✏é❪❤❃ì✶ë ✆✾î✏÷✶÷☎î✔✠➆ê❬✆❈é✛î✍ö✐❂❊ å❋✸❒❦❥ ✸ ë✶þ✽ð✏ë☎❂❊ ✃✏●■ ❮ Ó♠❧♥✆✾î✍é ❘ Ó ❯ Ö ▼✟ä❅❄❆❄❆❄✦ä✿♦❯ ü ✮✰✯✲✱✴✳✩✵q♣ ❡❉îPõ✟ð✏÷❧ì✶ñ✽ë❯è✾é❩ô➔é✛è✾ë❯ð✏ë❩ì☞î✏ñ➆❰ Ï❍år✸❍❒ts Ï❆Ó ❇ ✉ ❊✇✈ ➱ Ï❊ ❂❊ ✃✏● ❮②① Ï➔✃✏●■ ❮ Ó ❇ ✉ ❊③✈❸➱ Ï❊ ❂❊ ✃✏●■ ❮➶Ó ❇ ✉ ❊③✈❸➱ Ï❊ ❏❊④■②⑤ Ï■ Ó Ï❞✃❍●■ ❮ ❄ ⑥
Prs iyc, fs I 2.3. Ra"ll ige-Ritz ofi. all fki( L 1.rEgia tr4 ur a Al Nh 212mea 24+fcry,y∈qua,sr 2.9 1(1 fal Naol 2el hiri aryi inR wi 1 JJJh =h I 武 1≤wj≤ Ahni id)1≤w≤h The procedures, either Rayleigh-Ritz or Galerkin, are 2le812/. to those we de veloped in detail in the one-dimensional case 2. 9.2 Pafticulafi illuot fativl naol tt Uniform Mesh (1,1) Ki 2n i th h 1 len we refer to a uniform mesh, we shall mean the particular triangulation e.(Note the diameter of the elements, and hence mesh, is in fae h)
⑦⑨⑧❍⑩ ❶❉❷❢❸✜❹❻❺❽❼✪❾✰❿t➀✟❹➂➁✡➃ ➄✗➅③➆✩➅✏➇ ➈➊➉✿➋✗➌❍➍➏➎✏➐✰➑✗➒❪➈➓➎❞➔✣→➓➣↕↔❋➙❋➉✜➌❍➍❽↔❃➛✗➎❍➜ ➝✰➞✲➟✴➠✩➡q➢ ➤❝➥❃➦➏➧③➨❆➩③➫✿➭t➯➲➤❛➩✇➳☞➵✿➸ ➺ ➻♥➼ ➥✓➽➾➫✷➚❦➩✇➪ ➶❻➹✿➘❳➴②➷➬✲➮❉➱❞✃♥❐☞✃❜❒❳❮Ï❰✿➱❞✃②❒ Ð ÑÓÒ Ô ÕtÖ➶Ø× Ù➥✓➧③➨❆➽➾Ú➏➩✇➪✩➸ ➺ ➻✌Û❋Ü❣➻❦Ý ➥✔➳➾➩ Ý✟Þ➨ Ý ➮❉➱➺ ➻ ❐☞ß➏❒ ➼ ❰✿➱❞ßt❒à❐âá✎ß Û❋Ü❣➻❋ã ⑦⑨⑧❞ä å❑➀☞æ✲❾❉❸✽❼Ø❿t❼♠ç✥è⑨é❝ê❻❿✜➀✝❹➂➁❜æ ➄✗➅✇ë✪➅✏➇ ➙❋➍➏➜❻➍✲↔❃➉✽➌⑨ì❁➉✜í✣➍ ➝✰➞✲➟✴➠✩➡❱î✓ï ð✗➨❆➳ ➺ ➻ ➱✏ñò❒ ➼ ôó õ☞ö ➷ ➺ ➻ õò÷✩õ ➱✏ñò❒àø❛ß ➼ ÷☎ù ➱✏ñò❒à❐❛ú ➼üû ❐ ã❆ã❅ã ❐✟ý ➸ þ ➻ ➺ ➻ ➼❑ÿ➻ ➺ ➻ Û✁➤ ó þ ➻ ù④õ ➼ ➮❉➱÷❻ù ❐ ÷✩õ ❒✄✂ û✆☎ ú❪❐✞✝ ☎ ý✟✂ ÿ➻ ù ➼ ❰✿➱÷☎ù ❒✄✂ û✠☎ ú ☎ ý✟✡ ☛✌☞✎✍✑✏✌✒✔✓✖✕✗✍✙✘✛✚✜✒✔✍✣✢✙✤✥✍✄✦★✧✩☞✎✍✪✒✆✫✭✬✯✮✯✰✱✍✄✦✳✲✴☞✶✵✷✫✑✦★✧✩✸✹✓✯✒✁✺✥✬✯✰✻✍✪✒✔✼✯✦★✽✾✤✭✬✯✒✔✍✂➩❀✿t➨❅➪❽➳➾➩✱❁❅➥✓➧❂✧❃✓❄✧✩☞✎✓✯✢✪✍❆❅❇✍✹✘❈✍✪✵ ❉ ✍✄✰✻✓✙✏✎✍✙✘❄✦★✽❊✘❋✍✄✧❃✬✯✦★✰●✦★✽❍✧✩☞✎✍■✓✯✽❏✍✪✵❑✘✯✦★▲✹✍✪✽✎✢✣✦✷✓✯✽✌✬✯✰▼✕✙✬✯✢✄✍✖◆ ➄✗➅✇ë✪➅✘➄ ❖❜➉t↔❃➔❅➎◗P❈❘❻➌✏➉t↔✁❙Ó➌✏➌◗❘✪í❅➔❆↔✔➉t➔✣➎✩❚↕➍qì✎➉✽í❅➍ ➝✰➞✲➟✴➠✩➡❱î✜î ❯●✽❱✦❲✄✓✯✒✣▲❨❳❍✍✣✢✔☞✪➸ ❩ ➼ ❬ý ➬ ➷ ❭ý ➼ ➱❞ý➷ ❪ û ❒ ➬ ý ➼ ➱❞ý➷ ❮ û ❒ ➬ ❫ ➼ û✴❴ý ➷ ❵❛☞✎✍✪✽❜❅❇✍❛✒✗✍✷❲✄✍✪✒✹✧❝✓❍✬❞✚✜✽✾✦❲✄✓✛✒✣▲❡▲✹✍✣✢✔☞✶✤✑❅❇✍❆✢✔☞✾✬✯✰★✰❇▲✹✍✙✬✯✽❜✧✩☞✎✍✠✏✎✬✛✒✣✧◗✦✷✕✪✚✜✰✻✬✛✒❛✧◗✒✣✦✷✬✛✽❢✲❈✚✜✰✻✬✯✧✞✦✷✓✯✽ ✬❈❣✙✓ ❉ ✍✖◆✐❤★❥✐✓✯✧❃✍❦✧★☞✾✍❧✘✯✦✷✬✛▲❆✍✪✧❃✍✄✒❧✓♠❲✑✧✩☞✎✍❧✍✪✰✻✍✪▲❆✍✪✽✾✧◗✢✙✤✟✬✛✽❏✘❧☞✎✍✪✽❏✕✙✍✠▲✹✍✄✢❑☞✶✤✟✦❀✢✑✦★✽✆❲✄✬❈✕✪✧♦♥❬❫ ✤✟✽❏✓✯✧ ❫ ◆ ♣ q
0v;0 a r dr dy a 1<i,j< Derivatives of h h 0y:/0x (piecewise constant) (piecewise constant) SLIDE 14 Evaluation of o(api/a)(apj /ax)dA /8x ax s dA ;/8 2/h2 aNw/ax 0p;/x Evaluation of j(api/ay)(api/ay)dA 2/h apE/ay a ps/ay dA= h2 asw/a ay;/8 a w/ay 2000 pi/oy 4/h2 LIDE 16
r❏s❢t★✉✇✈②①❈③ ④✟⑤✄⑥❏⑦✗⑧✣⑨✙⑨✣⑩✷❶✯❷■❸✄❶✯⑦❆❹❺✾❻ ❼❏❽✷❾❧❿❑➀✜➁❇➂➄➃✜➅❧➆✐❾➈➇✖➆✐➀➊➉❹ ➂➋➃✎➅✠❾➍➌✛➀✛➌✑➎❜❾➍➏✴➀✛➏✑➉❹ ➐ ❹❺✌➑➓➒ ➂➔❼✌❽◗→➑ ❿✔→➒ ➁➣➂ ➃➅❞↔→➑ ↔❱↕ ↔→➒ ↔❏↕ ➎➙↔→➑ ↔❱➛ ↔→➒ ↔❏➛ ➉❹ ➜✆➝➟➞ ❿✞➠ ➝➟➡ ➢ r❏s❢t★✉✇✈②①❈➤ ➥■⑧✄⑦✣⑩★➦✯➧✯➨✞⑩★➦✴⑧✄⑨✐❶♠❸ →➑ ❻ ➩↕ ➑ ↔→➑❝➫ ➭ ↔❱↕ ⑥✌⑩✷⑧✙➯✗⑧✪➲✟⑩❀⑨✪⑧❆➯✗❶✛❷✎⑨✣➨❃➧✯❷✾➨♠➳ ↔→➑❃➫ ↔❏➛ ❽⑥✌⑩✷⑧✙➯✗⑧✪➲✟⑩❀⑨✪⑧✹➯✙❶✯❷✎⑨✣➨❃➧✯❷❱➨❃➳ r❏s❢t★✉✇✈②①✯➵ ④➸➦✯➧✯➺➓➻✾➧✛➨◗⑩✷❶✛❷➼❶❑❸❍➽➅ ❽ ↔→➑ ➫ ↔❏↕➁♦❽ ↔→➒ ➫ ↔❏↕➁➾➉❹ ➩ ↕ ➑ ➃➅❞↔→➑ ↔❏↕ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ↔→♦➘ ➫ ↔❏↕ ↔→♦➘❂➴ ➫ ↔❏↕ ↔→➴ ➫ ↔❏↕ ↔→♦➷➴ ➫ ↔❏↕ ↔→➷ ➫ ↔❱↕ ↔→➷✜➬ ➫ ↔❏↕ ↔→➣➬ ➫ ↔❏↕ ↔→➘➬ ➫ ↔❏↕ ↔→➑ ➫ ↔❱↕ ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ➉❹ ➂ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ❐ ❐ ❒✭❮ ➫✛❰❱Ï ❐ ❐ ❐ ❒✭❮ ➫✛❰Ï ❐ Ð ➫❈❰Ï ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ❰Ï ❮ r❏s❢t★✉✇✈②①❈Ñ ④➸➦✯➧✯➺➓➻✾➧✛➨◗⑩✷❶✛❷➼❶❑❸ ➽➅ ❽ ↔→➑ ➫ ↔❏➛ ➁▼❽↔→➒ ➫ ↔❏➛ ➁❏➉❹ ➩ ↕ ➑ ➃➅ ↔→➑ ↔❱➛ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ↔→➘ ➫ ↔❏➛ ↔→♦➘❂➴ ➫ ↔❱➛ ↔→♦➴ ➫ ↔❏➛ ↔→➷➴ ➫ ↔❏➛ ↔→✟➷ ➫ ↔❏➛ ↔→✟➷✜➬ ➫ ↔❏➛ ↔→➬ ➫ ↔❏➛ ↔→▼➘➬ ➫ ↔❱➛ ↔→➑ ➫ ↔❏➛ ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ➉❹ ➂ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ❒✭❮ ➫✛❰❱Ï ❐ ❐ ❐ ❒✭❮ ➫✛❰Ï ❐ ❐ ❐ Ð ➫✛❰Ï ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ❰Ï ❮ r❏s❢t★✉✇✈②①❈Ò Ó➻✜Ô■Ô✹➧✯⑦✣Õ ➩↕ ➑ Ö
Nonzero entries of dentical to finite difference Actually, not quite identical; the FEM matrit is h times the finite difference matric. Obviously, one of the the advantages of the FEm to develop ap- triangulations Th of arbitrary dome eercised in this example. Our purpose rather is to get some sense of how the y;“ interact” to form A 3 Theoretical Analysis 3.1 General results Nor SLIDE 17 Recall llI=a(u, D)=(2(9)=/ Vu12dA Then ah is the projection of u on Xh The proof is identical to that for the one-dimensional case 1.2 H Norm SLIDE 18 H1(g2) Vul+02dA Then l Ju -wnI B: continuity constant(=1)
×✥Ø❋Ù✎Ú✪ÛÝÜ✔Ø■Û✪Ù❢Þ✗Ü✔ß✱ÛÝà❂Ø❈á Ü✗Ø✯â➔ã❇Ø❈á▼äå✾æ ß❀ç✜ÛÝÙ❋Þ✗ß✱èÝé✛ê✌Þ✗Ø✐ë❱Ù✎ß✳Þ✗Û➊ç✜ß✻ì❏ÛÝÜ✔ÛÝÙ✾è✄Û✖à✪í î✑ï✄ð✞ñ✾ò✯ó★ó➓ô✴õ✑ö❏÷✯ð✠ø✄ñ✜ù★ð❃ú❄ù✷û❈ú✪ö✾ð✞ù✷ï✗ò✛óýü✆ð★þ✾ú✐ÿ✁✄✂✆☎✹ò✯ð✞✝✣ù✠✟ ù☛✡✌☞✎✍❞ð✞ù✏☎❆ú✑✡✹ð★þ✾ú✓✒➣ö❱ù★ð❝ú û✯ù✔✭ú✕✝✔ú✪ö❏ï✙ú ☎✹ò✯ð✞✝✣ù✠✟✗✖✙✘✓✚✕✛Ýù✷÷✛ñ✜✡✣ó➓ô✴õ✹÷✛ö❏ú ÷✣✢✁ð★þ✾ú❞ð✩þ✎ú ò❈û✤✛✯ò✯ö❱ð❝ò✦✥❋ú✑✡❍÷✣✢✁ð✩þ✎ú ÿ✁✄✂★✧ ð❝÷➟û❈ú✕✛✴ú✪ó✻÷✪✩➙ò✪✩✬✫ ✩✭✝✗÷✕✟❋ù✏☎❆ò✛ð◗ù✷÷✛ö✮✡✯✚✙ò✰✡✪ú✙û✁÷✯ö✱✥❢ú✪ö❏ú✕✝✔ò✯ó●ð✲✝✣ù✷ò✛ö✗✥❈ñ✜ó✻ò✯ð✞ù✷÷✯ö✳✡✵✴å ÷✶✢➊ò✤✝✕✚✪ù★ð✲✝✗ò✤✝✣ô✁û❋÷✤☎✹ò✯ù★ö✳✡✯✧ ù☛✡❧ö❏÷✯ð ú✪✟❢ú✕✝✔ï✪ù☛✡✪ú✗û✁ù★ö ð✩þ✶ù☛✡✹ú✷✟✶ò✸☎✓✩✌ó✻ú✦✖✹✘❂ñ✺✝✻✩✌ñ✺✝✞✩✎÷✤✡✄ú✼✝✔ò✯ð✩þ✎ú✕✝■ù☛✡✐ð❃÷✽✥❋ú✪ð✾✡✪÷✤☎✹ú✿✡✪ú✄ö✳✡✄ú❄÷✣✢✆þ✎÷✤❀ ð✩þ✎ú ❁✁❂✹❃ù★ö✾ð❃ú✕✝✔ò❈ï✪ð❅❄✆ð❝÷✓✢✄÷✤✝❅☎ äå ✖ ❆ ❇❉❈❋❊✓●✯❍■❊✾❏✁❑✕▲✻▼❖◆◗P❙❘❋▼❖◆❅❚❱❯✁❑✑❯ ❲✾❳✶❨ ❩❭❬❫❪✓❬❵❴✳❛✁❜❞❝❡❬❫❢✜❣❤❜❥✐✮❢ ❦❫❧♥♠✭❧♥♠ ♦q♣sr✉t✤✈✎✇◗①③②✎t✤④ ⑤✭⑥✗⑦✏⑧❫⑨◗⑩✗❶ ❷➍Û✖è✪é✛ê✱ê❹❸❺❸❻❸ ❼✬❸❺❸❺❸ ✍❤❽❿❾✬➀❼✭➁✶❼✺➂❹➃➄❸ ❼➅❸ ✍➆✻➇✑➈❻➉✳➊ ➃➌➋➉ ❸ ➍✯❼➅❸ ✍➏➎ä❉➐ ➑✾➒✎ÛÝÙ ❸❺❸❺❸ ➓✺❸❺❸❻❸➔➃ ß✱Ù✜á →✬➣✸↔➔↕■➣ ❸❻❸❺❸ ➙❱➛➝➜å ❸❻❸❺❸ ➀ ➓➞➃❭➙❱➛➝➙ å ➂q➟ ➙ å ß✱à❂Þ✪➒✎Û ✩✭✝✗÷❥➠✪ú✙ï✄ð✞ù✷÷✯ö Ø✛á➏➙❞Ø❋Ù❋➡å ß✱Ù Þ✷➒✎Û✆ÛÝÙ✎Û✪Ü✪➢➔➤❛Ù✾Ø❈Ü✪➥✁í ➦þ✎ú✓✩✭✝✗÷Ý÷✶✢✆ù☛✡ ß✱ç✜ÛÝÙ❢Þ✔ß❀è✪é✛ê ð❝÷❛ð★þ✾ò✯ð✬✢✄÷✤✝✆ð✩þ✎ú■÷✯ö✌ú✑✫❑û✛ù✏☎❆ú✪ö✮✡✣ù✷÷✯ö✌ò✯ó✟ï✙ò✰✡✪ú✦✖ ❦❫❧♥♠✭❧☛➧ ➨➫➩✌①③②✎t✤④ ⑤✭⑥✗⑦✏⑧❫⑨◗⑩➔➭ ❷➍Û✖è✪é✛ê✱ê❹➯❅❼➅➯ ✍➆➇ ➈❻➉✳➊ ➃ ➋ ➉ ❸ ➍✯❼➅❸ ✍✾➲ ❼ ✍■➎ä➳➐ ➑✾➒✎ÛÝÙ ➯✑➓✺➯ ➆➇ ➈❻➉✎➊➸➵➻➺❵➼ ➲➾➽➚❤➪ ß✻Ù✎á →➣ ↔➔↕➣ ➯✑➙✽➛➝➜å ➯ ➆➇ ➈❺➉✳➊ ➟ ➚ æ è✄Ø✶Û✪Ü✙è✄ß❺➶✶ß✳Þ✣➤❛è✪Ø❈Ù✾à❑Þ✗é❈Ù❢Þ ➀✶➹➴➘ ➂❅➟ ➽ æ è✄Ø❋Ù❋Þ✗ß✻Ù✜➷✎ß✻Þ✣➤❄è✪Ø❈Ù✾à❑Þ✗é❈Ù❋Þ ➀ ➃ ➼ ➂✣í ➦þ✎ú✻✩✭✝✔÷✖÷✣✢❧ù☛✡ ß✱ç✜ÛÝÙ❢Þ✔ß❀è✪é✛ê ð❝÷❛ð★þ✾ò✯ð✭✢✄÷✤✝❧ð✩þ✎ú■÷✯ö✌ú✑✫❑û✛ù✏☎❆ú✪ö✮✡✣ù✷÷✯ö✌ò✯ó♦ï✗ò✤✡✪ú✦✖ ➬