1 Outline SLIDE 1 Integral Equation Methods Reminder about Galerkin and collocation Example of convergence issues in 1D First and second kind integral equations Develop some intuition about the difficulties Convergence for second kind equations Consistency and stability issues Nystrom Method 2 Integral Equation Basics 2.1 Basis Function Approach 2.1.1 Basic Idea SLIDE 2 ntegral equation: y(a)=G(a, ro(a')ds i=10i Basis function Example Basis Represent circle with straight lines Assume o is constant along each line Note 1 As mentioned earlier, we are investigating methods for solving integral equa tions based on representing the solution as a weighted sum of basis functions Then, the original problem is replaced with the problem of determining the basis function weights In the next few slides we review the basis function approach th g replace the original boundary curve with a collection of straig ht sections, like the example circle in the slide. For the circle example, the result of using this basis is to replace the with a polygon. Then, the charge density is assumed constant on each edge of the polygon. The result is a piecewise constant representation of the charge density on a polygon, not a representation of the charge density on the circle￾ ✁✄✂✆☎✞✝✠✟✠✡☞☛ ✌ ✍✏✎✒✑✔✓✖✕ ✗✠✘✚✙✜✛✏✢✤✣✦✥★✧✪✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘✶✵✷✛✏✙✹✸✯✲✻✺✞✼ ✽✿✾✹❀❂❁❄❃❆❅✰✾✹❇❉❈❋❊✴●■❍✰❏✬❑▲❈■▼◆✾✹❇P❖◗❁❄❃❘❈■❃❆❅❚❙❯●■▼❄▼◆●✰❱❲❈❋❏P❁❄●■❃ ✩✬❳✻✥★❨❬❩❭✧✳✛❪✲✤❫❵❴■✲✴✘✚❛✤✛✏✣✦✢✤✛◗✘✯❴■✛❚✱❜✼✹✼✜✭✞✛◗✼❂✱✳✘❞❝◗❡ ❢✯❁◆❇❤❣✐❏✿❈■❃❆❅❪❣✐✾✜❱✠●❥❃❆❅❦❖◗❁❄❃❆❅☞❁◆❃✏❏P✾✹❧■❇❤❈❋▼✚✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣ ♥✬✾❲♦■✾✹▼◆●❥♣☞❣P●■❀q✾▲❁❄❃❥❏r❍★❁◆❏P❁❄●■❃s❈❋❊✴●■❍✰❏✪❏rt★✾✉❅✰❁◆✈✇❱❲❍★▼◆❏P❁❄✾✹❣ ①✲✤✘✚❛✴✛✏✣②✢✴✛◗✘✞❴❥✛❚❫③✲❆✣❘✼✹✛◗❴■✲✴✘✯✺✖④✻✱❜✘✞✺✶✛◗✫✮✭✯✥✰✙✹✱❜✲✤✘✯✼ ❙❯●■❃✤❣✐❁⑤❣⑥❏r✾❲❃❆❱❲⑦❦❈❋❃❆❅❪❣✐❏r❈■❊★❁◆▼❄❁◆❏⑥⑦❦❁❄❣r❣P❍★✾✹❣ ⑧❪⑨✻✼✹✙❲✣✦✲✤❨⑩✵✷✛✏✙✹✸✯✲✻✺✞✼ ❶❉❁❄❧■t☞●■❇❤❅✰✾✹❇❉❱✠●■❃◗♦❥✾❲❇r❧■✾❲❃✤❱✠✾ ❷ ❸✚✡❹☎✞☛✿❺❼❻❭❽✉✝❿❾➁➀q✂☞❽▲☎✞✟✠➂➃✡➅➄➆❽✉➇✔✟❲➈❯➇ ➉✪➊✐➋ ➌➎➍➐➏◗➑✐➏❿➒❉➓✬➔❵→✴➣★➑⑥↔❯➔➙↕➜➛✬➛❵➝★↔➞➍❭→✤➟ ➠✻➡ ❝ ➡ ❝ ➢❂✥★✼✜✱✳❴❪✗✠✺✯✛✏✥ ✌ ✍✏✎✒✑✔✓➥➤ ➦➧❃✏❏P✾✹❧■❇❤❈❋▼✚✾✹♠✏❍❆❈❋❏P❁❄●■❃✻➨➞➩❂➫➯➭✚➲➞➳➁➵q➸✇➫✳➭✔➺P➭✤➻✒➲⑥➼➐➫✳➭✤➻➯➲⑥➽◗➾➐➻ ✽✿✾✹♣★❇P✾✜❣✐✾✹❃✏❏✿➼✤➚✮➫➯➭✚➲➞➳➶➪➚➹❄➘✞➴ ➼❆➚➹ ➷❭➹ ➫➯➭✴➲ ➬ ➮✠➱ ✃ ❐❭❒✠❮ ➹ ❮✿❰✹Ï➚■ÐÒÑ ➹⑤Ó➚ ❮ SMA-HPC ©1999 MIT Basis Function Approach Basic Idea Example Basis Represent circle with straight lines Assume σ is constant along each line Integral Equation Basics Ô➥Õ✮ÖØ×ÚÙ Û❣❹❀q✾❲❃✏❏P❁❄●■❃❆✾✹❅➎✾✹❈■❇P▼❄❁❄✾❲❇✜Ü✞ÝÞ✾❘❈■❇P✾❦❁❄❃◗♦■✾✜❣⑥❏r❁◆❧✏❈✦❏P❁❄❃★❧s❀q✾✠❏rt★●✰❅★❣➃ß➯●■❇❹❣P●■▼❄♦◗❁◆❃❆❧❚❁◆❃✏❏r✾❲❧■❇❤❈❋▼➞✾✹♠✏❍❆❈✦à ❏P❁❄●■❃✤❣✬❊✤❈■❣P✾✹❅❚●❥❃Ú❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏r❁◆❃★❧❦❏Pt★✾✇❣P●■▼❄❍✰❏P❁❄●■❃➥❈■❣á❈✆ÝÞ✾❲❁❄❧■t✏❏P✾✜❅Ú❣P❍★❀â●❋ß❯❊❆❈■❣P❁❄❣✬ß➯❍★❃✤❱Ø❏P❁❄●■❃✤❣❲ã ä✪t★✾✹❃✻Ü②❏Pt★✾Þ●■❇r❁❄❧■❁❄❃❆❈❋▼❥♣★❇P●❥❊★▼❄✾❲❀❞❁❄❣✞❇r✾❲♣★▼⑤❈■❱❲✾✹❅❼Ý✿❁◆❏Pt➃❏Pt★✾Þ♣★❇r●■❊★▼❄✾❲❀❞●❋ß✴❅✰✾✠❏r✾❲❇r❀❂❁❄❃★❁❄❃★❧❉❏rt★✾Þ❊❆❈■❣P❁❄❣ ß➯❍★❃❆❱✠❏P❁❄●■❃❪ÝÞ✾❲❁❄❧■t✏❏r❣✹ã✞➦➧❃☞❏rt★✾❼❃★✾✠å◗❏✪ß➯✾❲Ý❞❣P▼◆❁⑤❅✰✾✹❣✪ÝÞ✾▲❇P✾✹♦◗❁◆✾✹Ý➁❏Pt❆✾á❊✤❈■❣P❁❄❣Þß➯❍★❃✤❱Ø❏P❁❄●■❃s❈■♣★♣★❇r●❥❈■❱❤t✔ã ä✪t★✾❚✾❲å★❈❋❀q♣★▼❄✾❚❊❆❈■❣P❁⑤❣q❏Pt❆❈❋❏❘ÝÞ✾❚t❆❈②♦❥✾s❊✤✾✹✾❲❃æ❱❲●■❃❆❣P❁❄❅★✾❲❇r❁◆❃★❧❬❁⑤❣✹Ü✪ß➯●■❇❘❈✖ç②à➧♥â♣★❇r●■❊★▼❄✾❲❀❪Ü✪❏P● ❇r✾❲♣★▼⑤❈■❱❲✾➃❏rt★✾❦●■❇r❁❄❧■❁❄❃❆❈❋▼❭❊✴●■❍★❃❆❅❆❈❋❇r⑦Ú❱✠❍★❇r♦■✾❂Ý✿❁◆❏Pt❿❈❪❱✠●❥▼◆▼❄✾✹❱✠❏P❁❄●■❃➥●❋ß✿❣⑥❏r❇r❈■❁◆❧❥t❥❏❼❣P✾✹❱Ø❏r❁◆●❥❃❆❣✹Ü✻▼◆❁❄❖■✾ ❏Pt❆✾è✾❲å✰❈■❀q♣★▼◆✾➥❱✠❁❄❇r❱❲▼◆✾➥❁❄❃➁❏Pt❆✾➥❣P▼◆❁⑤❅✰✾❥ã✄❢★●■❇❘❏Pt❆✾➥❱❲❁◆❇❤❱✠▼❄✾Ú✾❲å✰❈■❀q♣★▼◆✾❥Ü❉❏rt★✾è❇r✾✹❣P❍★▼◆❏☞●■ß✉❍❆❣P❁◆❃★❧ ❏Pt❆❁❄❣❼❊✤❈■❣P❁❄❣▲❁⑤❣á❏P●s❇r✾❲♣★▼⑤❈■❱❲✾➃❏rt★✾❦❱✠❁❄❇❤❱✠▼❄✾❂Ý✿❁◆❏Pt❬❈❪♣✴●■▼❄⑦◗❧■●■❃✔ãqä✪t❆✾❲❃✻Ü✔❏Pt★✾❦❱❤t❆❈■❇P❧❥✾❂❅★✾❲❃❆❣P❁é❏⑥⑦Ú❁❄❣ ❈■❣r❣P❍★❀q✾✹❅❼❱✠●❥❃❆❣✐❏r❈❋❃✏❏✻●❥❃❼✾✹❈■❱❤t❼✾✜❅✰❧■✾➞●❋ß✰❏Pt❆✾ê♣✤●❥▼◆⑦◗❧❥●■❃✻ã✯ä✪t★✾ê❇r✾✹❣P❍★▼é❏✞❁❄❣✯❈✿♣★❁❄✾✹❱❲✾❲Ý✿❁⑤❣✐✾➞❱❲●■❃❆❣✐❏r❈■❃✏❏ ❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏r❈❋❏P❁❄●■❃s●■ß➞❏Pt❆✾❂❱❤t❆❈■❇P❧❥✾❹❅✰✾✹❃❆❣P❁é❏⑥⑦s●❥❃è❈❘♣✤●❥▼◆⑦◗❧■●❥❃✻Ü✮❃★●■❏▲❈❘❇P✾✹♣★❇r✾✹❣P✾❲❃✏❏r❈❋❏P❁❄●■❃s●❋ß❯❏Pt★✾ ❱❤t❆❈■❇P❧❥✾á❅✰✾✹❃❆❣P❁é❏⑥⑦❦●❥❃✆❏rt★✾✉❱✠❁❄❇❤❱✠▼❄✾■ã ë