Finite difference discretization of elliptic equations: Fd Formulas and Multidimensional problems Lecture 4
✂✁☎✄✆✁✞✝✠✟☛✡☞✁✍✌✎✟✑✏✒✟✓✄✕✔✑✟☛✡☞✁☎✖✗✔✑✏✒✟✘✝✗✁☎✙✛✚✜✝✗✁✣✢✤✄ ✢✤✥✧✦✩★☎★☎✁☎✪✫✝✗✁✣✔✬✦✧✭✯✮✎✚✜✝✗✁✣✢✤✄✕✖✜✰✱✩✡ ✫✢✲✏✴✳✵✮✕★✶✚✷✖✸✚✷✄✺✹ ✻✼✮✕★✞✝✗✁✣✹✫✁✣✳✽✟✾✄✕✖✴✁✣✢✤✄✎✚✷★✧✿❀✏✒✢✤❁✕★✣✟✓✳❂✖ ❃❄✟✓✔✘✝✗✮✺✏✒✟❆❅
1 Finite difference formulas 1.1 Problem definition We have seen that one of the necessary ingredients in devising finite difference appro imations is the ability to accurately appro cimate derivatives in terms of differences. Here we will consider two approaches to developing such appro i Given l+r+1 distinct s∑v is of opt imal order of accuracy Lagrange interpolation Tw Undetermined coeficients Accuracy of a Finite Difference Approximation 6yv)=O△r) for all sufficiently smooth functions v, we say that the difference scheme is pth 1.2 Lagrange interpolation SLIDE 2 polynomials ()=x(x-x-)…(=x-1(x-2+(x-x) We note that, by construction, L, (a) takes the value of one at a and zero at all the other r+l points Lagrange interpolant Li(a)
❇ ❈✩❉❋❊●❉✍❍✠■☛❏❑❉✣▲▼■✛◆❖■✛❊✺P◗■❘❈✺❙❚◆✘❯✬❱●❲✣❳✤❨ ❩❖❬❭❩ ❪✎❫❵❴✘❛✷❜❭❝✴❞ ❡❢❝❤❣❥✐✛❦♠❧♥❦❭❴✾✐ ♦q♣sr✉t✴✈✂✇ ①●②❥③❵④⑥⑤⑥②❚⑦✣②✍②⑨⑧❶⑩❷③♥④❸⑩✜❹⑥⑧q②✯❹❻❺❚⑩✉③❵②❼⑧q②✍❽✍②☎⑦✍⑦⑨④⑥❾☎❿➁➀✉⑧➃➂❸❾✞②✞➄❸➀➅②✣⑧➆⑩➅⑦✲➀✉⑧❄➄➇②⑨⑤✶➀➈⑦✍➀✉⑧➃➂❥➉✘⑧➆➀✉⑩➊②✯➄⑥➀➋✛②⑨❾➌②⑨⑧q❽✍② ④✞➍➎➍q❾➌❹❋➏➎➀✉➐➑④❸⑩➒➀➅❹❸⑧♥⑦➑➀➈⑦❼⑩❷③♥②➁④➎➓✣➀✉➔→➀✉⑩➣❿●⑩♠❹✕④➎❽✍❽✣↔↕❾➌④❸⑩➊②⑨➔→❿➙④✞➍➎➍q❾➌❹❋➏➎➀✉➐➑④❸⑩➊②✆➄➇②⑨❾☎➀✉⑤➛④❸⑩➒➀✉⑤⑥②☎⑦➜➀✉⑧❄⑩♠②✣❾☎➐❼⑦✯❹❭❺ ➄⑥➀➋✷②✣❾➌②⑨⑧q❽✍②✣⑦⑨➝✆➞❼②✣❾➌②➁➟✾②➜➟❖➀✉➔✉➔➠❽✞❹❸⑧♥⑦☎➀➅➄➇②⑨❾➜⑩➣➟✾❹➙④✞➍➎➍q❾➌❹✶④➇❽✍③♥②✣⑦✫⑩♠❹➙➄➇②⑨⑤➛②⑨➔➡❹✍➍q➀✉⑧s➂❶⑦☎↔❵❽✍③❄④✞➍➎➍q❾➌❹❋➏➎➀✉➢ ➐➜④⑥⑩➣➀➅❹⑥⑧♥⑦⑨➝ ➤➦➥➨➧➇➩⑨➫●➭❵➯➳➲◗➯❀➵✤➸↕➥➈➺❻➻✞➥➡➫➆➼☎➻✛➽q➾➇➥➨➫s➻✞➺✲➚➅➪✒➶➆➹❭➘❭➪✒➶q➹➷➴✗➬❸➘⑨➮❋➮⑨➮✣➘➌➪❵➱s➘⑨➮⑨➮❋➮✣➘❭➪q✃✶❐✣❒↕❮❵➫❵➸✆➻✞❰♥➩✲Ï✑➩⑨➥➨Ð➇❰s➻✞➺◗Ñ⑥ÒÓ ➺➌Ô❵➼✍❰✫➻✞❰❵Õ⑥➻ ÖÒ✷× Ö➪Ò✂ØØ Ø Ø Ù✶Ú✒Ù✶Û Ü ✃ Ý Ó Ú ➶q➹ ÑÒ Ó × Ó ➥➈➺➠➾❸Þß➾➇➽↕➻✞➥➡à➜Õ❸á❤➾➎â✞➸♥➩⑨â➠➾❸Þ❖Õ➇➼❋➼✣Ô♥â✍Õ➇➼⑨ã➇ä å❚➵ æÏ✑➾➑Õ➇➽♥➽♥â✞➾➎Õ➎➼✍❰♥➩❋➺❋ç è➳é✓ê♥ë➆ì➛ê❵í✗ëqî➙ï➅íñð✶îsì✶ò✓ó➆ô➒ê↕ð❋ï➒ó➆í è➳õ✕í✠ö✗îsð❋îsì⑥÷➳ï➒í✗î➃öùø➇ó✒îsú➳ø➇ï➒î➃íñð❋û ü▼ý❤þ☎ÿ✁ ✂☎✄✆✄✞✝✠✟☛✡☞✄✍✌ ý✏✎✑✡✁✒✔✓✖✕✗✓➌þ☎ÿ✙✘✚✓✜✛◗ÿ✢✟➎ÿ✢✕✣✄➎ÿ✙✂✥✤✦✤✧✟➎ý✢★✩✓✖✪✫✡❤þ✬✓❻ý✞✕ ✭Þ❻❒ ➚ ÖÒ✛× Ö➪Ò Ø Ø Ø Ø Ù❋Ú✴Ù ✮ ✯ ✃ Ý Ó Ú ➶➆➹ ÑÒ Ó × Ó ❐✱✰✳✲✫➚✖✴❼➪✏✵s❐☎➘ Þ❷➾➇â◗Õ➇á➡á❤➺➌Ô✷✶➁➼✣➥➨➩⑨➫s➻➌á➨ã➁➺❭à➑➾➃➾❸➻✞❰✫Þ❷Ô❵➫❵➼☎➻✞➥➡➾➎➫❵➺✑×➆❒↕Ï✑➩➦➺✞Õ➛ã➑➻➌❰➆Õ⑥➻➠➻➌❰♥➩❥➸↕➥✹✸q➩❋â➌➩❋➫❵➼✣➩✤➺✞➼✍❰♥➩⑨à➑➩➦➥➨➺✦✺✼✻➒➻✞❰ ➾➇â✍➸↕➩❋â➠Õ➎➼⑨➼⑨Ô♥â✞Õ❸➻➌➩➎ä ❩❖❬✖✽ ✾❀✿❂❁✘❫✼✿ß✐❃❁✘❝ ❦❭✐✓❧↕❝✒❫✏❄✤❴✓❜❅✿✠❧♥❦❻❴✾✐ ♦q♣sr✉t✴✈❇❆ ❈Õ➇Ð➇â✍Õ❸➫♥Ð➎➩✛➽q➾➇á➨ãs➫❵➾➇à➑➥➨Õ➇á➨➺ ❉ Ó ➚❷➪q❐❊✰ ➚➅➪ ✯ ➪✴➶➆➹➒❐●❋❍❋❍❋✶➚❷➪ ✯ ➪ Ó ➶✒➬⑨❐⑨➚❷➪ ✯ ➪ Ó ➴✗➬⑨❐●❋❍❋❍❋✶➚❷➪ ✯ ➪➆✃➛❐ ➚➅➪Ó ✯ ➪ ➶➆➹ ❐●❋❍❋❍❋✶➚❷➪Ó ✯ ➪ Ó ➶✒➬ ❐⑨➚❷➪Ó ✯ ➪ Ó ➴✗➬ ❐●❋❍❋❍❋✶➚❷➪Ó ✯ ➪✃ ❐ ①●②➑⑧q❹❸⑩➊②➁⑩✉③❵④⑥⑩✖■✜➓⑨❿➙❽✍❹⑥⑧♥⑦☎⑩➣❾☎↔❵❽✣⑩➣➀➅❹⑥⑧✼■✱❉Ó ➚❷➪ñ❐ ⑩♠④❑❏➎②☎⑦✯⑩✉③❵②➜⑤⑥④⑥➔→↔❵②✆❹❻❺➜❹⑥⑧ñ②✫④❸⑩ ➪ Ó ④⑥⑧ñ➄✙▲✶②✣❾✞❹➙④⑥⑩ ④⑥➔✉➔✗⑩✉③❵②❚❹❸⑩✉③❵②✣❾ ➲✜➯ ➭ ➍♥❹❸➀✉⑧❵⑩➒⑦⑨➝ ❈Õ❸Ð➎â✞Õ➇➫♥Ð➇➩✷➥➡➫s➻➌➩❋â➌➽q➾➇á➈Õ❸➫s➻ ×ñ➚❷➪q❐ ▼ ❊✰ ✃ Ý Ó Ú ➶q➹ ❉ Ó ➚➅➪q❐♠×Ó ➵
The Lagrange interpolant is the polynomial of lowest degree that passes through A dml 1. 2.1 Example Second order Lagrange interpola =列儿=出-1+=出=吗+ (21)(=2)v+1 (x;+1-2j-1)(a+1-2j) Assuming a uniform grid m= 1(First derivative) 6}-1050 -2△a2-2k2 For ward 4 2= Backward For example a second order backward difference appro imation can be written v=(3 )/(2△x) 6
◆☞❖✏P❘◗❚❙❑❯✢❱❲❙❨❳❩❯❩P❭❬❪❳✼❫❴P✍❱❛❵✼❜❨❝✹❙✢❳✼❫❊❬❡❞❢❫✜❖✏P✔❵✏❜✢❝❤❣❨❳✐❜✢❥❦❬❧❙✢❝✦❜✬♠❘❝✹❜❨♥✱P✍❞♦❫q♣rPs❯r❱tP✉P❀❫✜❖✏❙✢❫●❵✼❙☛❞✉❞❍P♦❞✈❫✜❖✞❱t❜✢✇r❯①❖ ❙❨❝❪❝✗❫❪❖✼Pq❵✼❜❨❬❪❳✠❫❧❞❀②✜③✏④✆⑤✬⑥①④①⑦✍⑤❛⑧⑩⑨❷❶✑❸✬⑤❍❹❍❹❑❹❅❺✏❻ ❼✐❽❩❾❪❿❚➀❇➁ ➂✑➃✼➃✏➄t➅☛➆✷➇➉➈➋➊❨➌t➍ ➎❩➏ ⑥ ➎③➏ ➐ ➐ ➐ ➐ ➑①➒✩➑❑➓✑➔ ➎❩➏✙→⑥ ➎③➏ ➐ ➐ ➐ ➐ ➑①➒✩➑❑➓ ⑨ ↔➣ ④ ➒❂↕✠➙ ➎❩➏❘➛④ ➎③➏ ➐ ➐ ➐ ➐ ➑❑➒✩➑①➓ ⑥①④ ➜➞➝➍❑➄t➍❍➟✜➅r➄❲➍r➠ ➡④➏ ⑨ ➎❩➏❘➛④ ➎③➏ ➐ ➐ ➐ ➐ ➑❑➒✩➑ ➓ ❹ ➢✐➤❡➥✩➤❧➢ ➦➨➧✩➩✏➫➯➭❂➲❧➳ ❼✐❽❩❾❪❿❚➀✁➵ ➸➍❍➌ ❸●⑨➺❺❀⑨➼➻ ➠ ②✜③④ ↕✩➽ ⑤t③④ ⑤t③④t➾ ➽ ⑦ ➸ ➍❑➚✍➅✆➪✼➶✚➅r➄✉➶✷➍❍➄✔➹✗➊✢➘r➄✉➊✢➪✼➘r➍❃➇➉➪❩➌t➍❑➄t➃✐➅r➴❡➊✢➪❩➌ ⑥☞②✜③☞⑦➷⑨ → ➬ ➑r↕✐➑❑➮❅➱ ➬ ➑r↕✐➑❑➮✖✃✞❐❴➱ ➬ ➑ ➮❴❒❩❐ ↕✐➑ ➮ ➱ ➬ ➑ ➮❴❒❩❐ ↕✐➑ ➮✖✃✞❐ ➱ ⑥④ ↕●➽❰❮ ➬ ➑✢↕☞➑❍➮❴❒❩❐✬➱ ➬ ➑r↕✐➑❑➮✖✃✞❐❴➱ ➬ ➑ ➮ ↕☞➑ ➮❴❒❩❐ ➱ ➬ ➑ ➮ ↕✐➑ ➮✖✃✞❐ ➱ ⑥④ ❮ ➬ ➑r↕✐➑❑➮❴❒❩❐❴➱ ➬ ➑✢↕☞➑❍➮❅➱ ➬ ➑❍➮✖✃✞❐❲↕☞➑❍➮❅❒❩❐❅➱ ➬ ➑❍➮✖✃✞❐♦↕✐➑❑➮❅➱ ⑥①④t➾ ➽ ❼✐❽❩❾❪❿❚➀❇Ï ➂❃Ð❲Ð✬Ñ✼➈⑩➇➉➪✏➘Ò➊⑩Ñ✏➪✏➇✹➟✜➅r➄❲➈Ó➘✆➄t➇❡➶ Ô ⑨❷➻ ②❧Õ➇✹➄✉Ð✬➌❃➶✷➍❍➄❲➇✹Ö❨➊✢➌t➇➉Ör➍ ⑦ ➡ ➽④ ↕✩➽ ➡ ➽④ ➡ ➽④t➾ ➽ × ⑨Ø⑧✈❶Ù➻ ❶ÛÚ Ü❲Ý➑ Ü Ý➑ ❶ ➽ ÜtÝ➑ Õ➅r➄❲Þ➞➊✢➄✉➶ × ⑨Ø⑧ ❶ ➽ Ü❲Ý➑ ß ➽ Ü❲Ý➑ à➍❍➪❩➌t➍❑➄t➍①➶ × ⑨Ø⑧ ❮ ➻ ➽ Ü❲Ý➑ ❶ Ü Ý➑ Ú Ü❲Ý➑ á➊✆➚✉â✞Þq➊r➄❲➶ ã✗❜❨❱✙P❲ä❩❙✢❥✑❵☞❝✹På❙å❞❍P✉æ✉❜❨❳✐♣ç❜❨❱t♣✆P✍❱✚è❲❙✆æ❲é✢♥✱❙❨❱❲♣❇♣✢❬ê❃P❍❱tP✍❳☞æ✉Pë❙✉❵r❵✐❱❲❜❍ä✆❬❪❥✥❙✢❫✖❬❧❜✢❳➺æ❲❙✢❳Ùè✉P✙♥✧❱♦❬❪❫✖❫❴P✍❳ ❙☛❞✔⑥✞ì④ ⑨í②✖î✢⑥①④✔❶çï✆⑥①④ ↕●➽ ❮ ⑥①④ ↕ Ü ⑦❲ð✷②✖ñ✆ò❦③✐⑦❍❻ ❼✐❽❩❾❪❿❚➀❇ó Ô ⑨✳ñ ② ➸➍①➚✍➅r➪✠➶ë➶✷➍❑➄t➇➉Ö❨➊❨➌❲➇✹Ö✆➍ ⑦ ➡Ü ④ ↕✩➽ ➡Ü ④ ➡Ü ④t➾ ➽ ➽ Ý➑①ô ❶ Ü Ý➑❑ô ➽ Ý➑❑ô à➍❑➪✆➌❲➍❍➄❲➍❑➶ ñ
We note that since we are starting with three points, our interpolating polynomial is second order, and therefore the second derivative is constant everywhere. In general, to appro ime derivative of order m we will require at least m +1 Fornberg's algorithm The o, can be computed very efficiently using a recursive algorithm developed by Fornberg(generation of Finite Difference Formulas on Arbitrarily space ics of ce ion,51,184). 1.3 Undetermined coefficients SLIDE 7 Insert Taylor expansions for v abou Uy=20+(x;-x1)+5(x-x1)2 determine coefficients om to maximize accuracy. 1. 3.1 Example sLide 8 m=2, l=r=1, i=0,(uniform spacing A r 62 62(+△x6+分2++ Equating coefficients of k=0 621+6+6 k=1 Δr(6-621) (62+621)
õ☎ö✱÷☞ø❨ù❴öqù❪ú✼û❨ù✐ü♦ý❪÷✐þ✉öqÿ✱ö✔û✁❲ö❊ü✉ù❴û✁♦ù❛ý❪÷✄✂➨ÿ✧ý❪ù✜ú❭ù❪ú☎tö✉ö✝✆✏ø✢ý❪÷✼ù✖ü✟✞❚ø✡✠☎✣ý❪÷✼ù❴ö☛☞✆✼ø✁✌➉û❨ù❛ý❪÷✄✂✍✆✼ø✁✌✏✎❨÷☞ø✁✑⑩ý❧û✁✌ ý❡ü❘ü❍ö❲þ✉ø✢÷✓✒✙ø✡✔✒rö✕✖✞qû✢÷✓✒➋ù✜ú✏ö✕tö✘✗✍ø✁tö☛✞✱ù✜ú✏ö❀ü❍ö❲þ✉ø✢÷✓✒✙✒✆ö☛♦ý✛✚❨û❨ù❛ý✛✚❨ö❭ý❡ü❭þ✉ø❨÷✼ü♦ùsû✢÷✼ùqö✕✚❨ö☛✜✎❨ÿ✗ú✼ö☛tö✣✢✥✤♦÷ ✂❩ö✍÷☞ö☛tû✡✌✦✞✑ù❴ø û✟✆✧✆✓tø✩★rý✛✑➋û❨ù❴ö✚û✪✒rö✕♦ý✛✚☛û✢ù✖ý✛✚❨ö✚ø✫✗Òø✁✟✒rö✕✭✬ ÿ✱ö✥ÿ✧ý✛✌✛✌✮tö✖✯✕✠✷ý✛tö✙û✢ù✰✌✹ö✉û☛ü♦ù✍✬✲✱✴✳ ✆✼ø❨ý❪÷✼ù✖ü✕✢ ✵✭✶ ✷✹✸✻✺✜✼✾✽ ✿❀✸☎❁✣❂❀❃❀✼✁❁✣❄✰❅❆❈❇☎❉❊❄✝✸☎❁●❋❍✺❍■❑❏ ▲◆▼P❖❘◗✁❙❚✲❯✕❱❳❲❩❨ ❖ ❯☛❬❳❭❫❪P❴P❵ ❖✩❛✙❜❳❖✩❝✔❞❈❖✕❡❯✕❢❖❲✄❵✔❣❞ ❴✐❤✔❢❥❲P❦❫❱ ❝✔❖ ❯☛❴❝ ❤❍❢❜❳❖ ❱✡❣❧❦❳❬❝ ❢❥❵▼❭ ❛P❖✕❜❳❖❣❥❬✧❪❖✣❛ ❨ ❞✹♠❬ ❝❲♥❨❖✩❝❦♣♦❍qsr✜t✓r✖✉✜✈☎✇✘①✛②✁t③②●④✥⑤❑①❥t✐①✘✇⑥r❈⑦⑧①❥⑨⑩r✖✉☛r✖t✻❶☛r❫⑤✓②✁✉✔❷❩❸✡❹✛✈✡❺❻②✁t❽❼✰✉●❾✁①✘✇✘✉✖✈✡✉✔①❧❹➀❿♣➁✡➂⑩✈☎❶✜r●➃ q◆✉✟①❊➃✧❺✩➄P➅✪❱✡❵▼P❖❭❫❱✡❵✔❢➆❯✕❤◆❬❳➇✮➈✍❬✧❭➉❪✐❴☎❵✟❱✡❵✔❢❧❬❳❲⑩➄✐➊ ✳ ➄ ✳✩➋✡➌♥➍✜➎ ➏❀➐✘➑ ➒➔➓⑧→✰➣✻↔☎➣⑩↕✐➙➜➛✔➓✰➣✝→➞➝➠➟◆➣✻➡➢➝✻➛✫➣✝➓✍↔P➤ ➥✓➦✄➧✛➨✝➩♣➫ ➭ ❵✖❱❝ ❵➯➇❝❬❳❭ ➲✄❙s➳ ➲❳➵❙ ➸ ➸ ➸ ➸ ➺✣➻⑩➺✩➼❀➽ ➚➾ ❚ ➻❀➪➹➶ ◗❙ ❚ ➳ ❚❫➘ ➴❲✐❤ ❖✕❝❵ ▲ ❱❞ ❣❧❬❝◆❖☛➷❪✐❱❳❲✐❤❍❢❧❬❳❲➹❤➬➇➮❬❝ ➳ ❚ ❱❳❨➹❬✧❴☎❵ ➵❩➱➔➵➹✃ ➳ ❚ ➱➔➳✁❐ ✱ ➳♥❒❐ ♦➵ ❚✥❮ ➵➹✃ ➍✝✱ ✳ ✶ ➳♥❒❐❒ ♦➵ ❚✰❮ ➵✓✃ ➍❍❰✍✱ ➘✕➘✕➘✩Ï ❛☎❖❵ ❖✕❝❭❫❢❥❲❖ ❯✕❬ ❖☛❡❯☛❢❖❲✄❵✟❤ ◗✁❙❚ ❵✔❬❈❭❈❱➷ ❢❧❭➉❢❧Ð❖ ❱✧❯✕❯☛❴❝❱✧❯❞❳➎ Ñ✓Ò❧Ó✝Ò❊Ñ Ô⑧Õ⑩ÖP×♣ØÚÙ❊Û ➥✓➦✄➧✛➨✝➩✹Ü ✬ ➱ ✶ ➄✐Ý ➱➔Þ❘➱ ✳ ➄☎ß ➱áà ➄⑩♦❊❴P❲P❢❥➇➮❬❝❭â❤✔❪✐❱❳❯✕❢❥❲✐❦❫ã➵➍ ➳❐❒ ❒ ➱ ◗❰➪⑩ä ♦➳❐ ❮ ã➵✐➳❐❒ ✱æå➺✣ç ❰ ➳❐❒ ❒ ❮ å➺✩è é ➳❐❒ ❒ ❒ ✱êå➺✕ë ❰❍ì ➳✐í ì✖î ❐ ✱ ➘✩➘✕➘ ➍ ✱ ◗❐❰ ➳❐ ✱ ◗❰ ä ♦ ➳❐ ✱ ã➵➹➳❐❒ ✱ å➺✩ç ❰ ➳❐❒ ❒ ✱ å➺✩è é ➳❐❒ ❒ ❒ ✱ å➺✩ë ❰❍ì ➳ í ì✟î ❐ ✱ ➘✕➘✩➘ ➍ ➥✓➦✄➧✛➨✝➩✹ï ð➬ñ❴✐❱✡❵✔❢❧❲P❦❫❯☛❬ ❖☛❡❯☛❢❖❲✄❵✖❤✥❬✡➇ ➳ í❧ò î ❐ ó❈➱áà ô ◗❰➪✻ä ✱õ◗❐❰ ✱❽◗❰ ä ➱áà ó❈➱ ✳ ô ã➵ ♦ ◗❰ ä ❮ ◗❰➪✻ä ➍ ➱áà ó❈➱ ✶ ô å➺✣ç ❰ ♦ ◗❰ ä ✱❽◗❰➪✻ä ➍ ➱ ✳ ö
In general we will start with k= 0, and increase k until a sufficient number of independent equations is generated so that all the coefficients are uniquely Soly to recover the same second order central difference approximation to a second derivative previously derived. Note 3 Multidimensional finite difference formulas The Lagrange interpolation and undetermined coefficients approach we have seen can be easily extended to multiple dimensions Lagrange interpolants can be constructed in multiple dimensions by combining one dimensional Lagrange polynomials. Given a lattice of (L+r+1)x(d+u+1) points truct the fol polynomials for a ty point j, k 1)…( LK(y) (y-y-a)…(y-孙k-1)(y-yk+1)…(y-y) (yk-y-a)…(yk-yk-1)(yk-yk+1)…(yk-ya) The lagi interpolant is thus obt ained 0(x,y)=∑∑L3(x)L(y) L ()Li(y)
÷✜ø♣ù✄ú☛ø➠ú☛û✔ü✡ý✍þ➬ú✾þ❀ÿ✛ý✛ý✁✄✂ü✡û✄✂❘þ❀ÿ☎✂☎✆✞✝✠✟☛✡✌☞ ü✡ø✎✍ ÿ✛ø✎✏✕û✔ú✖ü✑✕ú✒✝✔✓☎ø✕✂☞ÿ✛ý➯ü✖✄✓✘✗✙✏✕ÿ❊ú☛ø✚✂❘ø✕✓✜✛✒✢✟ú✕û ✣✥✤ ÿ✛ø✎✍✧ú✧✦✐ú✕ø✎✍❳ú✕ø✕✂❫ú✩★✘✓✐ü✪✂✘ÿ ✣ ø✌❩ÿ✫❈ù✧ú✕ø✓ú✕û✔ü✪✂ú✩✍✬ ✣ ✂☎✆✐ü✭✂❻ü✡ý✛ý✮✂☎✆✐ú✯✏✣ú✰✗✙✏✕ÿ❊ú✕ø✕✂✰ ü✡û✔ú✱✓☎ø✐ÿ✲★✳✓✐ú✕ý✵✴ ✍❳ú✳✂⑥ú☛û✄✛➉ÿ✛ø✓ú✩✍✷✶ ✸✺✹✼✻✾✽✼✿✷❀ ❁✭❂❃❅❄ ✟ ❆ ❇❉❈❂❋❊ ❁✭❂● ✟■❍ ❏ ❇❉❈❂❑❊ ❁✭❂❄ ✟ ❆ ❇❉❈❂ ✂✣ û✔ú✩✏✣✭▲ ú☛û✒✂▼✆Pú◆✕ü✭✛❈ú✒✕ú✩✏✣ø✎✍ ✣ û❖✍✧ú☛ûP✏✟ú✕ø✕✂☞û✔ü✡ý◗✍✁ÿ❘sú✕û✔ú✕ø✎✏✖ú❩ü✩✦✷✦➠û✣✳❙ÿ☎✛❈ü✭✂☞ÿ ✣ø❚✂✣ ü✱✕ú✩✏✣ø✎✍ ✍❳ú✕û✜ÿ▲ ü✭✂☞ÿ▲ ú❯✦➠û✔ú ▲ÿ ✣✓✺✜ý✵✴P✍❳ú✕û✜ÿ▲ ú✩✍✷✶ ❱❯❲ ❱❨❳ ❩✞❬❅❭✄❪✯❫ ❴❛❵✚❜❝❭❖❞❢❡❅❞✲❣❚❪✪❤❥✐❦❞✥❬✺❤♠❧✜❜♠♥♠❤❋❞❖❭✄❪✯❡❅❞▼♦✮❪✪♣✼❪✪❤rq✼❪ts❝❬✺♣✑❣❚❵✕❜✉❧✜✐ ✈①✇✿③②❋④✪⑤✷⑥✩④✪⑦✕⑤✷✿P⑧✾⑦⑩⑨❖✿❦⑥❖❶✎✹✷✻✫④✭⑨❷⑧✾✹✼⑦❸④✪⑦✕❹✙❺✌⑦✕❹✌✿✘⑨❖✿❦⑥❖❻❼⑧❽⑦✌✿❦❹❸❾✳✹✺✿✘❿◆❾✘⑧❽✿✳⑦⑩⑨❷➀➁④✪❶✌❶✌⑥❷✹✼④✼❾✇✠➂✿ ✇④✑✽✼✿ ➀❖✿✳✿✳⑦✱❾✳④✷⑦t➃✎✿➄✿❦④✼➀❝⑧❽✻❽➅◆✿✘➆✺⑨❖✿❦⑦✕❹✜✿➇❹➁⑨❷✹❼❻➈❺✌✻✾⑨❖⑧❽❶✌✻❽✿➈❹✜⑧❽❻➉✿❦⑦✕➀❖⑧✾✹✼⑦✕➀✳➊ ②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿❨⑧❽⑦✼⑨❷✿✳⑥❷❶✚✹✼✻❽④✷⑦⑩⑨❷➀✮❾✳④✷⑦➁➃✎✿➋❾✘✹✼⑦✕➀✥⑨❷⑥❖❺✚❾✄⑨❖✿➇❹➁⑧❽⑦P❻➈❺✌✻✾⑨❖⑧❽❶✌✻❽✿➋❹✜⑧✾❻❼✿❦⑦✕➀❝⑧❽✹✷⑦✚➀①➃⑩➅t❾✘✹✷❻❉➃✌⑧❽⑦✌⑧✾⑦✕⑤ ✹✷⑦✕✿①❹✌⑧✾❻❼✿✳⑦✚➀❝⑧❽✹✷⑦✕④✷✻✌②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿◗❶✎✹✷✻❽➅⑩⑦✕✹✷❻❼⑧❽④✷✻❽➀❦➊➍➌➎⑧✾✽✼✿✳⑦❼④➏✻❽④✪⑨❝⑨❖⑧✫❾✘✿①✹✷➐♠➑✰➒✳➓◆➔→➓ ❆➇➣↕↔ ➑✰➙❥➓➁➛①➓ ❆➇➣ ❶✎✹✷⑧❽⑦✼⑨✩➀ ➂✿①❾❦④✪⑦✒❾✘✹✼⑦✕➀❝⑨❖⑥❷❺✕❾✄⑨➍⑨✇✿①➐▼✹✷✻❽✻✾✹➂⑧❽⑦✌⑤➏✹✷⑦✕✿✘➜❢❹✜⑧✾❻❼✿❦⑦✕➀❝⑧❽✹✷⑦✚④✪✻✌②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿✁❶✎✹✷✻❽➅✺⑦✌✹✷❻❼⑧✫④✪✻✫➀❥➐▼✹✷⑥◗④➝⑨✥➅✺❶✜➜ ⑧✫❾✳④✪✻❅❶✎✹✷⑧❽⑦⑩⑨✁➞ ❊ ✝ ➟◗➠➡ ➑❈ ➣ ✟ ➑❈ ❍ ❈ ❃✚➢ ➣↕➤❦➤✳➤ ➑❈ ❍ ❈ ➡ ❃❅❄ ➣ ➑❈ ❍ ❈ ➡❖➥ ❄ ➣↕➤❦➤✳➤ ➑❈ ❍ ❈✚➦ ➣ ➑❈ ➡ ❍ ❈ ❃→❄ ➣↕➤❦➤✳➤ ➑❈ ➡ ❍ ❈ ➡ ❃→❄ ➣ ➑❈ ➡ ❍ ❈ ➡❖➥ ❄ ➣❅➤✳➤✳➤ ➑❈ ➡ ❍ ❈➦ ➣ ➟✁➧➨ ➑▼➩ ➣ ✟ ➑▼➩ ❍ ➩ ❃↕➫ ➣↕➤✳➤❦➤ ➑▼➩ ❍ ➩➨ ❃❅❄ ➣ ➑✲➩ ❍ ➩➨➥ ❄ ➣↕➤✳➤❦➤ ➑✲➩ ❍ ➩✷➭ ➣ ➑▼➩➨ ❍ ➩ ❃↕➫ ➣↕➤❦➤✳➤ ➑✲➩➨ ❍ ➩➨ ❃❅❄ ➣ ➑✲➩➨ ❍ ➩➨ ➥ ❄ ➣❅➤✳➤✳➤ ➑▼➩➨ ❍ ➩➭ ➣ ✈①✇✿➄②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿➝⑧❽⑦⑩⑨❖✿❦⑥❖❶✎✹✷✻✫④✪⑦⑩⑨①⑧❽➀①⑨✇❺✕➀✮✹✷➃✌⑨❷④✪⑧❽⑦✌✿➇❹t④✼➀ ➯➲ ➑❈ ❊ ➩ ➣ ✟ ➦ ➳ ➡❖➵ ❃✎➢ ➭ ➳ ➨➵ ❃↕➫ ➟➠ ➡ ➑❈ ➣ ➟◗➧➨ ➑▼➩ ➣ ➲➡ ➨P➸ ➺✿➈⑦✌✹✷⑨❖✿➋⑨✇④✭⑨➎➃✺➅✱❾✳✹✷⑦✕➀❝⑨❖⑥❷❺✕❾✘⑨❖⑧❽✹✷⑦ ➟➠ ➡ ➑❈ ➣ ➟◗➧➨ ➑▼➩ ➣ ⑨✩④✪➻✷✿➇➀❯④➁✽✑④✷✻✾❺✕✿➋✹✪➐r✹✼⑦✌✿❉④✪⑨➉➑❈ ➡ ❊ ➩➨ ➣ ④✪⑦✚❹ ➼✿❦⑥❖✹❼④✭⑨❯④✷✻✾✻❅✹✷⑨✇✿❦⑥✮❶✚✹✼⑧✾⑦⑩⑨✩➀✮⑧✾⑦t⑨✇✿➄✻❽④✪⑨❝⑨❖⑧✫❾✘✿✼➊ ❳
A general partial derivative can be approximated as bmm=am=∑∑bb 户kmb7n()L(y)0 Therefore, defining the weights L i (a)Lk(y) The method of undetermined coefficient extends to multiple dimensions in a straightfor ward manner if we consider multidimensional Taylor series expan assuming a uniform△zand△g △x)22 U +k△ 28 +G△rk△y) azay I(a) A finite difference approximation of the form can be obtained by inserting the Taylor series expansions for ujk and determining Oik coefficients by the coefficients of the different derivative terms ct differe The finite difference approximations considered here are called divided diffe ences approximations. More sophisticated, and accurate, difference approxima tions are also possible. These approximations are called compact approxima ∑v d a ll knowi at ion to the first deriva erify, using Taylor expansi h that this if fourth order accurate for a sufficiently smooth function, 1. e. p=4
➽➚➾✷➪❦➶✌➪✳➹✩➘✪➴↕➷✕➘✷➹❝➬❷➮❽➘✷➴→➱✜➪✳➹❷➮❽✃✑➘✪➬❖➮❽✃✷➪➏❐✳➘✷➶P❒✎➪➋➘✪➷✕➷✌➹❖❮✑❰✜➮❽Ï✒➘✭➬❖➪➇➱➁➘✼Ð Ñ↕Ò①Ó→Ô✺Õ Ñ✎ÖÒ Ñ✎× ÔÙØ Ñ✎Ò①Ó→Ô♠ÚÕ Ñ✎ÖÒ Ñ✎ÖÔÜÛ ÝÞ ß❖à➍á✚â Þã äà♠á❅å Ñ↕Ò①Ó→Ô Ñ✎ÖÒ Ñ✚× ÔPæß✷çÖ↕è æä ç×✌è↕Õß äPé ê①ë➪❦➹❖➪✳ì▼❮✷➹❷➪✷í✜➱✜➪✘î✚➶✌➮✾➶✕➾❼➬ë➪➄ïð➪✳➮❽➾ë➬❷Ð ñÒ①Ô ß ä Û ÑÒ①Ó→Ô Ñ✎ÖÒ Ñ✎× ÔPæß çÖ↕è æä ç ×✌è✭òò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø ïð➪ ë➘✑✃✷➪ ÑÒðÓ❑Ô ÕÚ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø➍ù ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ê①ë➪PÏ❼➪✘➬ë❮✺➱✙❮✷ì❯ú✌➶✕➱✜➪✳➬❖➪❦➹❖Ï❼➮❽➶✌➪❦➱✙❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✒➪✳❰⑩➬❷➪✳➶✕➱✕Ð➈➬❷❮ÙÏ➈ú✌➴✾➬❖➮❽➷✌➴❽➪✱➱✌➮✾Ï❼➪✳➶✚Ð❝➮❽❮✷➶✕Ð❼➮❽➶✠➘ Ð❝➬❖➹✩➘✪➮❽➾ë➬❝ì▼❮✼➹❖ï①➘✪➹✩➱ÜÏ✒➘✪➶✌➶✕➪✳➹✒➮✾ì➏ïð➪③❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹✒Ï➈ú✌➴✾➬❖➮✫➱✜➮❽Ï❼➪✳➶✕Ð❖➮✾❮✼➶✕➘✪➴ ê➘✑ü✺➴❽❮✷➹✒Ð❝➪❦➹❖➮❽➪❦Ð❼➪✳❰✜➷✕➘✪➶✜ý Ð❖➮✾❮✼➶✕Ð✳þ ê①ëú✕Ð❦í✌➘✷Ð❷Ð❝ú✕Ï➉➮❽➶✌➾✒➘➉ú✌➶✌➮✾ì▼❮✷➹❷Ï ÿ❉Ö③➘✪➶✚➱Pÿ❉×↕í✌ïð➪ ë➘✑✃✷➪ Õ ß ä Û Õ ó✾ô➇õ✄ö ÷✩õ❖ø ✁✄✂ ÿ❉Ö Ñ✎Õ Ñ✎Ö ò ò ò ò ó✾ô õ ö ÷õ ø ✁✆☎ ÿ➈× Ñ✎Õ Ñ✎× ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ❉Ö✎è✞✝ ✟ Ñ✠✝✘Õ Ñ✎Ö✝ ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ➈Ö ☎ÿ❉×✌è Ñ✡✝✳Õ Ñ✚Ö✎Ñ✎× ò ò ò ò ó✾ô õ ö ÷õ ø ✁ ç ☎ÿ❉×✌è✞✝ ✟ Ñ✠✝✳Õ Ñ✎× ✝ ò ò ò ò ó✾ô õ ö ÷õ ø ✁☞☛✌☛✍☛ é ➽ î✕➶✌➮✾➬❖➪➋➱✌➮✏✎↕➪✳➹❷➪✳➶✕❐✳➪➄➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶➁❮✷ì❋➬ë➪➏ì▼❮✷➹❷Ï ÑÒðÓ❑Ô Õ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø Ø ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ❐✳➘✷➶➏❒✚➪r❮✼❒✜➬❷➘✷➮✾➶✕➪❦➱➏❒✺ü❯➮❽➶✕Ð❖➪✳➹❖➬❖➮❽➶✌➾①➬ë➪ ê➘✑ü✺➴❽❮✷➹→Ð❖➪✳➹❷➮✾➪➇Ð❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð↕ì▼❮✷➹→Õß ä ➘✪➶✕➱➏➱✜➪✳➬❖➪✳➹❷Ï❼➮✾➶✕➮✾➶✌➾ ➬ë➪①ñÒ①Ô ß ä ❐✘❮✺➪✘û◆❐✘➮❽➪✳➶⑩➬✩Ð♠❒⑩ü➈➪✌✑⑩ú✕➘✭➬❷➮✾➶✕➾❯➬ë➪❨❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✩Ð❥❮✪ì✕➬ë➪❨➱✜➮✒✎✎➪❦➹❖➪❦➶⑩➬➍➱✌➪✳➹❷➮✾✃✭➘✭➬❷➮✾✃✼➪◗➬❷➪✳➹❷Ï✒Ð✳þ ✓✕✔✗✖✙✘✛✚ ✜✢✔✤✣✕✥✧✦✡★✤✖✪✩✗✫✭✬✮✘✰✯✱✘✰✲✳★✱✘✴✦✱✥✵✥✶✯✱✔✰✷✸✫✹✣✺✦✗✖✻✫✞✔✤✲✵✼ ê①ë➪✒î✚➶✌➮➬❷➪➁➱✜➮✒✎✎➪❦➹❖➪❦➶✕❐✘➪t➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶✕Ð➏❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹❖➪➇➱ ë➪✳➹❷➪◆➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù➱✜➮❽✃✺➮❽➱✜➪➇➱Ü➱✜➮✒✎✎➪❦➹❝ý ➪✳➶✚❐✘➪❦Ð①➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð✳þ✵✽✯❮✷➹❷➪➝Ð❖❮✷➷ë➮❽Ð❝➬❖➮✫❐✳➘✪➬❖➪➇➱❅í✌➘✪➶✕➱t➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪✷í✺➱✜➮✏✎↕➪✳➹❷➪✳➶✚❐✘➪➏➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð➋➘✪➹❷➪◆➘✪➴✫Ð❝❮✯➷✎❮✼Ð❷Ð❖➮✾❒✌➴❽➪✷þ ê①ë➪➇Ð❝➪◆➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð➄➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù❐✘❮✼Ï❼➷✕➘✷❐✘➬➈➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð✮➘✪➶✕➱t➬❷➘✿✾✷➪➝➬ë➪➏ì▼❮✼➹❖Ï ÝÞ❀ ß❖à♠á✎â ❀ ❁Ò Õ ❁ÖÒ ò ò ò ò ô à ô✌❂ Ø ÝÞ ß❖à♠á✎â ñÒ ß Õ ß❄❃ ➘✪➶✚➱t➘➉➷✕➘✷➹❝➬❷➮❽❐✳ú✌➴❽➘✷➹✮ï✁➪❦➴✾➴✗✾✺➶✌❮✭ï✮➶✱❐✘❮✼Ï❼➷✕➘✷❐✘➬✮➘✷➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✭➬❷➮✾❮✼➶◆➬❷❮❉➬ë➪➏î✕➹✩Ð✥➬❨➱✜➪❦➹❖➮❽✃✭➘✭➬❷➮✾✃✼➪➎➮✫Ð ❅ ❆ çÕ✤❇ß Ó❉❈ ✁❋❊Õ●❇ß ✁ Õ✤❇ß✘á ❈ è Û Õ ß Ó❉❈■❍ Õ ß✘á ❈ ✟ÿ❉Ö é ❏➬❑➮✫Ð→➪➇➘✷Ð❖ü❨➬❖❮❯✃✷➪✳➹❷➮✾ì▼ü✷í➇ú✕Ð❝➮❽➶✌➾ ê➘✑ü✺➴✾❮✼➹❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð❅❮✷➹→❮✷➬ë➪❦➹❖ï✮➮✫Ð❖➪✷í❦➬ë➘✪➬❑➬ë➮❽Ð❋➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶ ➮✾ì❋ì▼❮✼ú✌➹❝➬ë ❮✼➹❷➱✌➪✳➹❨➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪❯ì▼❮✼➹❨➘❼Ð❝ú✜û◆❐✳➮✾➪❦➶✼➬❷➴✾üPÐ❝Ï❼❮✺❮✪➬ë ì▼ú✕➶✕❐✄➬❷➮✾❮✼➶→í✌➮✉þ ➪✼þ✗❑ Û ❊ þ ▲
2 Poisson equation in 2D Vu(a, g)=f(a, y) in Q we have seen that one of the criti cal requirements to obtain optimal a-priori error estimates is that the solution be sufficiently regular so that the derivatives appearing in the leading terms of the truncation error eist. In id the regularity of the solution depends exclusively on the regularity of f (see Fourier analysis in Lecture 1). In multidimensions however, the regularity of the boundary plays also a very important role and, for general domains involving corners or non- regular boundary data, the solution may not be very regular. This topic will be dealt with in greater detail in the finite element lectures 2.2 Discretization SLiDE 11 ,j-1 △x=m,△y=m,x1=i△x,v=j△y 2.3 Approximation SLIDE 12
▼ ◆P❖❘◗✍❙❉❙✵❖❘❚❱❯P❲❨❳❬❩❪❭✶◗❫❖❘❚❴◗✍❚❵▼❪❛ ❜■❝✻❞ ❡☞❢✗❣✐❤❦❥✞❧♠❥✻♥✧❤ ♦✡♣●qsr✉t✺✈✰✇ ①❦②④③❫⑤⑦⑥✭⑧❉⑨✻⑩❷❶✧❸❺❹⑦⑥✹⑧✉⑨❻⑩♠❶ ❼❾❽ ❿ ⑤❬❸☞➀ ➁✿❽ ➂ ②③➄➃➆➅③ ➅ ⑧③❘➇ ➅ ③ ➅ ⑩ ③ ⑨ ❹➉➈✛➊✗➋ ➌➉➍➏➎❷➐➒➑➒➍✴➓✍➍➔➍❫→↔➣✭➎❷➐➒➣✢↕✰→✡➍➉↕✻➙➏➣✭➎❷➍➉➛✍➜✙➝s➣➞➝✹➛➔➐➒➟■➜❻➍➔➠✍➡♠➝s➜❻➍✍➢➤➍✍→➥➣➞➓➏➣➦↕✕↕✿➧✍➣➨➐➒➝s→➩↕➭➫✠➣➞➝s➢❄➐➒➟➯➐✰➲❾➫✡➜✙➝✹↕➒➜✙➝ ➍✍➜✙➜❻↕➒➜➳➍✙➓✙➣➵➝s➢➤➐✰➣➦➍❫➓✐➝➸➓➄➣✭➎❷➐➒➣➺➣s➎➥➍❪➓✍↕➒➟➻➡♠➣➵➝✹↕➒→✕➧➭➍✐➓➔➡❫➼↔➛✍➝✹➍❫→➽➣➞➟➻➾④➜❻➍➦➚✿➡♠➟✒➐➒➜❪➓✍↕④➣s➎➥➐➒➣➺➣s➎➥➍✪➪✿➍✍➜✙➝s➑➶➐✰➣➞➝s➑➒➍❫➓ ➐➭➫✱➫❷➍➔➐✰➜✙➝s→●➚❘➝s→❨➣✭➎❷➍➄➟✒➍➔➐✿➪✰➝s→●➚❘➣➨➍❫➜✙➢❘➓✮↕✻➙✮➣✭➎❷➍❦➣➞➜✙➡♠→✠➛➭➐✰➣➞➝✹↕✰→✛➍✍➜✙➜❻↕➒➜❦➍➭➹✱➝➸➓✙➣➦➘➺➴✙→✴➷✌➬➮➣✭➎❷➍➄➜❻➍➦➚✿➡♠➟✒➐➒➜✙➝s➣➵➾ ↕✞➙④➣✭➎❷➍④➓✍↕✰➟➻➡♠➣➞➝✹↕✰→➱➪✿➍➨➫❷➍✍→✡➪➒➓❄➍➭➹✤➛❫➟✏➡✤➓➔➝s➑➒➍✍➟➻➾✃↕➒→❐➣✭➎❷➍➤➜➭➍➵➚✿➡♠➟✒➐✰➜✙➝s➣➞➾➉↕✞➙ ❹❮❒➓✍➍➔➍✢❰❉↕➒➡♠➜✙➝✹➍✍➜➏➐➒→✡➐✰➟➻➾➶➓✙➝➸➓ ➝s→❨Ï✉➍➔➛✍➣➞➡♠➜➭➍➳➷➔Ð✿➘➺➴❫→➉➢❘➡♠➟✏➣➞➝✹➪✰➝s➢➤➍✍→❷➓✙➝✹↕➒→➥➓✮➎➥↕➒Ñ✧➍✍➑➶➍✍➜➔Ò✳➣s➎➥➍✐➜❻➍➦➚✿➡♠➟✒➐➒➜✙➝s➣➵➾➤↕✻➙❦➣✭➎❷➍✢➧➭↕✰➡♠→✡➪✱➐➒➜✙➾➄➫✡➟✒➐✰➾➶➓ ➐➒➟➻➓❫↕Ó➐✛➑➶➍✍➜✙➾❬➝s➢➯➫➥↕➒➜✙➣➨➐➒→➽➣➯➜❻↕➒➟❾➍❨➐➒→✠➪➶Ò✉➙❫↕✰➜✐➚●➍❫→✠➍❫➜➭➐➒➟✧➪✿↕✰➢➤➐✰➝s→❷➓➤➝s→➥➑➒↕➒➟➻➑Ô➝s→●➚✕➛➔↕➒➜✙→✠➍❫➜➔➓❄↕➒➜➤→✡↕✰→➥➲ ➜❻➍➦➚✿➡♠➟✒➐➒➜❘➧➔↕✰➡♠→✡➪✿➐✰➜✙➾✛➪✱➐➒➣➨➐➶Ò■➣s➎➥➍❘➓❫↕✰➟➻➡♠➣➵➝✹↕➒→✕➢➤➐✰➾❨→✡↕✰➣➯➧➭➍❘➑➒➍❫➜✙➾✴➜❻➍➦➚✰➡♠➟❾➐➒➜✍➘➉Õ✠➎✤➝➸➓✢➣➨↕➭➫✠➝✹➛④Ñ✶➝s➟s➟✶➧➔➍ ➪✿➍➔➐✰➟➻➣➺Ñ✶➝s➣✭➎❬➝s→❬➚✰➜➭➍➭➐✰➣➦➍✍➜✢➪✿➍✍➣➨➐➒➝s➟✉➝s→✄➣s➎➥➍✧Ö✧→➥➝s➣➨➍✢➍✍➟✒➍✍➢➤➍✍→➥➣×➟✒➍➔➛❫➣➵➡♠➜❻➍❫➓✍➘ ❜■❝➞❜ ❡❺❥❻Ø●Ù✠Ú➥❢✗❧❷❥✞Û✗Ü✵❧❷❥✞♥✧❤ ♦✡♣●qsr✉t✺✈❷✈ Ý ⑧✛❸ßÞ➥à á✿â❉ã♠ä Ý ⑩❄❸ Þ❷å æ■â❉ã❷ä ⑧➽ç❉❸☞èÝ ⑧ ä ⑩➶é❦❸➱êÝ ⑩ ❜■❝➞ë ì↔í❦í➄Ú❷♥❉î✮❥❻ïðÜ❉❧❷❥✻♥✧❤ ♦✡♣●qsr✉t✺✈✿ñ ò ➁✱ó✮ô❫õ❷ö✰÷➤ø❷ù❾ô➤ú✍ú✌ú ➅ ③❫û ➅ ⑧③ ü ü ü ü ç✭ý é➄þ û✰çâ❉ã ý é➯① ÿ✰û✰ç✭ý é ➇ û✰ç ✁ ã ý é Ý ⑧③ ➅ ③❫û ➅ ⑩ ③ ü ü ü ü ç✹ý é þ ûç✹ý é â❉ã ①❋ÿ➒ûç✹ý é ➇ ûç✹ý é ✁ ã Ý ⑩ ③ ✂➁✿ó Ý ⑧ ä Ý ⑩☎✄❻÷➤ö✿ù✒ù ✆
2.4 equations Wzz-lyy=f suggest li, tui n,=01 2.4.1 Example SLIDE 14 i=0i=1i=2i=3i=4 The 9 unknowns are collected into a column vector unknown i, by, somehow ar bitrarily, stacking the unknowns row by row. Thus, i=(01,1, i2, 1, i3, 1, 1, 2, 12, 2, .. 13, 3] 1 4
✝✟✞✡✠ ☛✌☞✟✍✏✎✒✑✔✓✖✕✘✗✚✙ ✛ ✜✣✢✥✤✧✦✩★✫✪ ✬✮✭✰✯✱✯✲✬✳✭✵✴✶✴✸✷✺✹✼✻✾✽✔✿❀✿✫❁✱✻❃❂❄✻✲❅✱❅❆❅ ✬❈❇✭✵❉❋❊❍●❏■ ❑▲✬◆▼ ❇✭✰❉✡■ ❑P❖ ❇✭✵❉❘◗❙●❏■ ❑ ❚❱❯❳❲ ✬❨❇✭✰❉✡■ ❑✾❊❍●✟✬❩▼ ❇✭✵❉❘■ ❑✘❖ ❇✭✰❉✡■ ❑✶◗❬● ❚❱❭❪❲ ✷ ❇✹❉❘■ ❑ ❇✭❳❫ ■ ❑ ✷ ❇✭✰❴ ■ ❑ ✷✺❵ ❛ ❜❩❝❞❜✩❡ ❇✭❉❘■ ❫✚✷ ❇✭❉❘■❢ ✷❣❵ ❛ ❜✐❤❥❜✐❦ ❧ ♠ ❇✭ ✷ ❇✹ ♥❬♦q♣❍♦❘r s✚t❬✉✔✈◆✇②①❘③ ✛ ✜✣✢✥✤✧✦✩★⑤④ ❦✌✷⑥❡⑦✷❣⑧ ❚❱❯ ✷ ❚❱❭ ✷⑩⑨ ❶✰❷✔❸▲❹❞❺❪❻❪❼⑤❻✰❽⑤❾❿❻➁➀✏➂➄➃✾❸✸➅➆❽⑤➇✥➇q❸➆➅❆➈➉❸➆➊❱➋✥❻➁➈➌❽❱➂➍➅➆❽⑤➇➎❺❪➏➍❻➑➐⑤❸➆➅✶➈➌❽⑤➃➒❺❪❻➓❼➄❻✵❽⑤❾❿❻ ❇✭ ➔②→✶➣↔➔ ➀❆❽⑤➏↕❸❄❷➁❽⑤❾✐➂➄➃❏➙ → ➋✥➈➛➃✾➂➄➃❏➋✥➇➣↔➔ ➀❏➈➉➂❀➅❄❼➄➋✥❻✣➜✚➈✡❷✔❸▲❺❪❻❪❼⑤❻✰❽⑤❾❿❻✔➀❿➃✾❽➄❾ →❆➣ ➃✾❽➄❾✘➝➞❶✰❷➓❺➓➀ ➔ ❇✭ ✷➠➟ ❇✭❙●❏■ ●➢➡ ❇✭✵❲❆■ ●➢➡ ❇✭✰➤❆■ ●↔➡ ❇✭❬●❏■ ❲⑤➡ ❇✭✰❲✶■ ❲✫➡❆➥✱➥❆➥✶➡ ❇✭✵➤❆■ ➤⑤➦ ➝ ✛ ✜✣✢✥✤✧✦✩★✫➧ ♠ ✷ ❛ ⑨ ❲ ➨➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩ ➩ ➫ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ✬➯❛ ✬➯❛ ✬➯❛ ➭ ➲➵➳➳ ➳ ➳ ➳ ➳ ➳ ➳ ➳ ➳ ➳ ➳ ➸ ➺
2.4.2 Numbering iDE 16 (i, j) becomes component (m+i) Th 0 -lh g 0 Ihf(xD)卩了 6 Bene ie i d nntd taat fiadmm p tad fnknfinh camld ahhdg lid tbn ii ly cnfg nh d taam-nfih tmfiatca cahdi fitr Id(p x pffilrmck tuti tas nmar fitta Irrckh nf cd(m x mff ani tad Iani fiti ta tmtath cahd fitr Idtm+
➻❬➼q➽❍➼➾➻ ➚✌➪✒➶➘➹P➴❀➷⑤➬➛➮❍➱ ✃✵❐✣❒✥❮✧❰✩Ï✫Ð ÒÑ Ó ÔÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÖ Ò❬×❄× Ñ Ø Ø Ø Ò✵Ù❀× Ñ Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ò✰Ù➄Ú Ñ Û❆ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ý Þ ßÑ Ó ÔÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÖ Ñß ×❄× Ø Ø Ø ßÑÙ❀× Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø ßÑÙ➄Ú Û❆ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ý à✡á Þãâ➓äæå✵ç✱è✶é❀êëç✱ì✮è✶é❀êëí❳é❀î✔ç❆î✣ï àâ❀ðòñ á ä ➻❬➼q➽❍➼❋ó ô➍õ❘ö❬÷➢ø✐ùûú❪ü❆➷⑤➬❘ý ✃✵❐✣❒✥❮✧❰✩Ï✣þ ÿ Ó ÔÕ Õ Õ Õ Õ Õ ÕÖ ÿ✁ ñ ✂☎✄✝✆ ✞✟✄✝✆ ✠ ✡✝✡☛✡ ✠ ✞✟✄☞✆ ÿ✌ ñ ✂✍✄☞✆ ✞✟✄☞✆ Ø Ø Ø Ø Ø Ø ✠ Ø Ø Ø Ø Ø Ø Ø Ø Ø ✠ Ø Ø Ø Ø Ø Ø ✞✟✄☞✆ ÿ✌ ñ ✂☎✄☞✆ ✞✟✄☞✆ ✠ ✡✝✡☛✡ ✠ ✞✟✄☞✆ ÿ✁ ñ ✂☎✄☞✆ Û❆ÜÜ Ü Ü Ü Ü Ü Ý ô➍õ❘ö❬÷➢ø àð✏✎ ðäPï✒✑✔✓✖✕✗✓✖✘☎✙❀é✫î✚✘✍✛✵ê✜✘➄ï✔✑✒✓✣✢ ÿ✌ Þ ✄✝✆✥✤ à✧✦ ✎ ✦ ä✩★ ÿ ✤ à✪✦ð✏✎ ✦ð✌ä ✃✵❐✣❒✥❮✧❰✩Ï✍✫ ✬ ✛qé❪è✮✭✥✯✚ç☞✰❳î✱✓➵ï✒✓qé❀î➁ì ✲✴✳✶✵✸✷ ✹✻✺✽✼ ✾✿ ✿ ✿ ✿ ✿ ✿ ✿❀ ❁ ❂ ✷❄❃ ❅❆❅❇❅❈❃ ❂ ✷ ❁ ❂ ✷❊❉ ❉ ❉ ❉ ❉ ❉ ❃ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❃ ❉ ❉ ❉ ❉ ❉ ❉ ❂ ✷ ❁ ❂ ✷ ❃ ❅❆❅❆❅❋❃ ❂ ✷ ❁ ●❆❍❍ ❍ ❍ ❍ ❍ ❍ ■❑❏ ▲◆▼ ✵❖✷ ✹✻P✝✼ ✾✿ ✿ ✿ ✿ ✿ ✿ ✿❀ ✷◗❃ ❃❘❅❆❅❇❅❙❃ ❃✸✷ ❃ ❉ ❉ ❉ ❉ ❉ ❉ ❃❚❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❃ ❉ ❉ ❉ ❉ ❉ ❉ ❃ ✷❯❃ ❃❱❅❇❅❆❅❲❃ ❃❘✷ ●❆❍❍ ❍ ❍ ❍ ❍ ❍ ■ ÿ❨❳✘❀ì❩✘❱å❬✘➄î✚✕❪ç✽✕☎ì❃ï✔✑✒❭➁è✶ï✔❭✱✑❄ç ô➍ú✔➮✻❪❴❫↕➬✧❪❍ü☛❵ ✤❛✂✦ ñ❝❜ ❞❢❡❑❣✐❤❦❥❆❡❑❥✪❧✱♠❦❥♦♥♣❧✚❡☞❣ ðrq ✦ ❥✪❧✱❡❑s✗❣✉t✈❤❦♥✶❣✚✇❑①✮♠❦❣③②✮❡❢♠④✇✮✇✝❡☞⑤✜②✝⑥⑦❡✮⑧⑩⑨❶❣✚❥❆❤ ÒÑ ②☞❷⑩①✒❤☎⑥❸s✗⑤❹❣✱✇ ❺♠☎❥❶❧✚❡ ❺ ❥❶❧✚♠❦❣ ❺❤☎♥❻✇✌⑨❶❣❢♥♣❧✗⑨✧①✒❧❼①✒♠❦✇✝❡ ÿ ♥✶⑨❶⑥❶⑥❴②✮❡❾❽✦ ✎ ✦✴❿ ②☞⑥⑦❤✽①✮t❾❥❺⑨✧⑧❦⑨✧♠✝➀➁❤❦❣✴♠❦⑥➂♥✶⑨❶❥✪❧➃②✝⑥⑦❤☛①✮t❦✇♦❤➅➄ ✇✩⑨⑦➆✽❡➇❽ð➈✎ ð ❿④➉ ♠☎❣✐⑧✥❥❶❧✚❡➊②✮♠❦❣✴⑧❦♥✶⑨✧⑧❦❥✪❧❼⑨❶❣❼❥✪❧✉⑨✖✇➊①✮♠④✇✝❡➊♥✶⑨❶⑥❶⑥✻②✮❡ ✂ðòñ③❜➁➋ ➌
2mm vPD Proyer-y Aigg. 4ve 4 El]* eli d M] be eiligy verix ea m=∑a ∑∑ u fi A vPD f fiaRe g4 4231 rir lb o i el tis m升 a}{ 41, egr 4'i r3nr31114g,3N=i e m4ri i 147aea Ai. baHia. s, b m4lril { aido t ae!4ig li L3r flu. e]ve 4,:4gvtl3jd itpi der: mbers ady tea beg ai Nev4ggr3m oevl re e,4,| w23] errr it3b4i]ea by iter]t,e el 4 aierelve d w a
➍➂➎⑦➏♣➎⑦➏ ➐❴➑♦➒❚➑➔➓❦→✚➣↕↔✈➓④➙✮➛ ➜✴➝✱➞✌➟✩➠④➟✩➡❇➞✝➢➥➤➧➦➨➞✒➩✝➫✚➭❦➡◆➯✧➤❦➲✚➟❛➳➸➵➺ ➻ ➼➵ ➽➯❶➾❶➾✚➝✱➭☎➚④➞♦➭➪➫✗➲✚➯✧➩✝➫✚➞✟➟☞➤☎➾❸➫✗➡◆➯✧➤❦➲➪➶✐➹✔➤☎➚☛➯✧➘✈➞✮➘❹➡❶➝✚➞✁➢✜➭❦➡◆➹✩➯✣➴ ➳➥➯✖➟♦➲✴➤❦➲✚➷❇➟✩➯❶➲✉➬☎➫✗➾⑦➭☎➹✝➮↕➱☞➡❐✃✮➭☎➲❮❒✮➞➪➞✮➭④➟✩➯❶➾❸➠✥➚④➞✝➹✩➯❰↕➞✮➘✥➡❶➝✚➭❦➡✴➦☞➤❦➹➊➭❦➲❬➠✜➚④➞✮✃✝➡❇➤☎➹❩Ï Ð Ñ✐Ò➁Ó❶Ô❴Õ×Ö✍Ø ÏÙ♣➳✟Ï ➻③Ú☎Û✻Ü ÝÞß Ü àÝÛ♣Ü á ß Ü âäã å➪æ❬ç♣èÏ Þ á✁é Ï Þ ê Ü✩ë á✽ì çîí ã å➪ï✗ç❴è Ï Þ á✁é Ï Þ ë á ê Ü ì ç☞ð ñ✟ò☛ó✚ô☞ò Ï Ù➳✟Ï õ×ö✚÷✥ø✪ù✈ú✟û☎ó✉ü➇Ï þý ö è➳❨ÿ✁ ➐❴➑♦➒ì ➳➸➵➺ ➻ ➼➵ ✂ ➺➵ ↔☎✄✝✆✟✞✝➙✠✞ û☎ó☛✡➸ÿ☞✍✌✏✎✆✒✑✌↔ ➱✪➦✔✓✖✕✘✗ ➭❦➹✒➞ ➾⑦➭☎➹➬➁➞✩Ð✚✙✟➭☎➫✉➟✒➟✩➯✧➭☎➲ ➞✝➾❸➯❶➢➪➯❶➲✴➭❦➡◆➯✧➤❦➲✘✛➧➤❦➹✍✜✔✢✤✣❹➯✖➟➊➞✒➴☞➶✱➞✝➲✱➟✩➯❶➚❦➞✦✥★✧ è✗✪✩ ì ✫♣➤❦➹❼➭❙➢❾➭☎➡➹✩➯✣➴ ➤➧➦❢➟✮➯✭✬✽➞✯✮✱✰✲✮ ➡❶➝✚➞ ✃✮➤❦➟✮➡✜➤➧➦❾➶✱➞✝➹✪➦☞➤❦➹✩➢❹➯❶➲➁➬✳✙✟➭❦➫✉➟✮➟✩➯✧➭❦➲ ➞✝➾❸➯❶➢➪➯❶➲✴➭❦➡◆➯✧➤❦➲❨➯✖➟ ➶✐➹✒➤✒➶✐➹✒➤❦➡◆➯✧➤❦➲✐➭☎➾♦➡❇➤✴✮✶✵✈➮➥➱✧➦❼➡✪➝✱➞⑩➢✜➭❦➡◆➹✩➯✣➴ ➯✖➟ ❒✒➭☎➲✐➘✈➞✒➘ ➽➯❶➡✪➝❨❒✮➭❦➲✴➘➽➯✧➘❦➡✪➝✸✷Ð✥➡✪➝✱➞ ➢✜➭❦➡◆➹✩➯✣➴ ➟✩➡➹✩➫✚✃✝➡◆➫✗➹✔➞❢✃✮➭❦➲ä❒✮➞❼➞✒➴☞➶✐➾⑦➤☎➯❶➡❇➞✮➘ ➡❇➤⑩➹✒➞✒➘☎➫✚✃✮➞❢➡✪➝✱➞❼✃✮➤❦➟✮➡➪➡❇➤✹✮✶✷ç ➮✺✙✟➭☎➫✉➟✒➟✩➯✧➭☎➲ ➞✝➾❸➯❶➢➪➯❶➲✴➭❦➡◆➯✧➤❦➲ ➽➯❶➾❶➾î❒✮➞➃➘❦➯✖➟✝✃✝➫✉➟✮➟☞➞✮➘❼➯❶➲❙➢❹➫✚✃✒➝❮➬✍➹✔➞✮➭❦➡❆➞☞➹✥➘✈➞☞➡❆➭❦➯❶➾î➯❶➲➸➦✮➫✗➡◆➫✗➹✔➞✜➾➞✒✃✝➡◆➫✗➹✔➞✩➟✝➮✹✫♣➤❦➹➸➤☎➫✗➹✜➢❾➭☎➡➹✩➯✣➴❑➳Ð ✮ ➻ ✗ç ➭❦➲✴➘✻✷ ➻✽✼✗ í ã ➝✚➞☞➲✴✃✒➞♦➭➪➡❆➤❦➡❆➭❦➾✴✃✒➤❦➟✩➡❛➤➧➦✾✧ è✗✪✩ ì ➮î➱✩➡❆➞✝➹✔➭❦➡◆➯❶➚④➞➨➢✜➞✝➡❶➝✚➤☛➘❦➟➨✃✮➭❦➲ ➭❦➾➟✝➤ ❒✮➞❐➶✴➫✗➹✮➟✩➫✚➞✒➘❀✿✥➡✪➝✱➞☞➟☞➞ ➽➯❶➾❶➾✻➘✈➞❇➶✚➞✝➲✐➘❑➤❦➲❼➡✪➝✱➞❹✃✮➤❦➲✴➘❦➯❶➡◆➯✧➤❦➲ ➲✚➫✗➢✜❒✒➞✝➹✮Ð❛➘❦➯✖➟✝✃✝➫✉➟✒➟✝➞✮➘❑❒✒➞✝➾⑦➤➽➮ ❁✔❂✟❃ ❄❆❅❇❅❇❈❉❅❋❊✸●✍❍❏■▲❑✍▼❖◆P▼ ➍➂➎☞◗➂➎✒❘ ❙↕➓✌✏✎❯❚❲❱➙✠✆✧→✎❨❳➓✽➓❦→❬➓ ❩✁➞✮✃✮➭❦➾❶➾☎➦✮➹✒➤❦➢❭❬❴➞✮✃✝➡➫✗➹✔➞❫❪❾➡❶➝✚➭❦➡î➡❶➝✚➞♦➡◆➹✩➫✚✃✮➭❦➡◆➯✧➤❦➲❮➞✝➹✩➹✔➤❦➹➔➯✖➟➊➤✍❒✝➡❆➭❦➯❶➲✐➞✮➘➸❒☞➠✜➯❶➲✚➟☞➞✝➹✩➡◆➯❶➲➁➬✜➡✪➝✱➞➪➞✔➴✉➭✈✃☞➡ ➟✝➤❦➾❸➫✗➡◆➯✧➤❦➲❮➯❶➲✚➡❆➤✜➡✪➝✱➞❹➘❦➯❴✁➞✝➹✔➞✝➲✐✃✮➞ ➟☞✃✮➝✱➞✝➢✜➞☛➮ Ñ✐Ò➁Ó❶Ô❴Õ❛❵❝❜ é ➺ èæ Þ Û♣Ü❡❞ ï á☛ì❻é ✼❦➺ èæ Þ ❞ ï á✽ì í ➺ èæ Þ ê Ü❡❞ ï á☛ì å➪æ✐ç é ➺ èæ Þ ❞ ï á Û♣Ü ì❻é ✼☎➺ èæ Þ ❞ ï á✽ì í ➺ èæ Þ ❞ ï á ê Ü ì å➊ï✗ç ➻ ➼ èæ Þ ❞ ï á ì é å➊æç ã ✼ ❢✩ ➺ ❢✐æ✩ èæ Þ í✲❣❲❤Þ å➪æ ❞ ï á ì❻é å➪ï ç ã ✼ ❢✩ ➺ ❢✐ï ✩ èæ Þ ❞ ï á í✲❣❲✐á å➪ï ì ❥ ❦♠❧ ♥ ♦q♣sr t ✉✱ù✍ú ➺✶✈①✇ ✩ ② Þ ë á✤③ ✧ èå➪æç ❞ å➊ï ç ì ø✪ù✈ú✟û⑤④✁④⑦⑥ ❞ ⑧ ⑨
