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✏ì￾⑨í✦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