1 Motivation SLIDE 1 Consider a standard second order finite difference discretization of V-u= on a regular g 1.2. and 3 dimensions 1.1 1D Finite differences △m=h bandwidth b= 1 n points Cost of Gaussian elimination O(bin)=O(n) 1.2 2D Finite differences slide 3 i-1,},i+1,j 1 7× points bandwidth b= n Cost of Gaussian elimination 0(b2n2)=0(n")✲ ✳✴☎✵✷✶✝✸☎✹✺✵✻✶✭✴✽✼ ✾❀✿❂❁❄❃❆❅❈❇ ❉✫❊●❋■❍❑❏▼▲❖◆✭P❘◗✕❍❑❙✥◗●❋❚▲❯◗❱P❲▲✟❍❳◆❩❨✭❊●❋❚▲✗❊●P❲▲❖◆❩P✫❬■❋❯❏❭❙✥◆❪▲❖❏❴❫❀◆❩P❳◆❩❋❚❨☛◆☎▲❖❏▼❍✥❨☛P✥◆☛❙❳❏❛❵❩◗❱❙❳❏❛❊●❋✓❊❱❜ ❝❡❞✦❢☛❣✮❤❥✐✛❦ ❊●❋✮◗✦P❳◆❩❧●♠❯♥▼◗❱P♦❧♣P❳❏▼▲ ❦ ❏❛❋rq ❦❚s❖❦ ◗●❋❚▲✮t✦▲❯❏❴✉✈◆✭❋■❍❑❏❛❊●❋❚❍❩✇ ① ②④③ ❤⑥⑤ ⑦✷⑧❑⑦ ⑦✰⑨✍⑩✕❶❑❷❸❶❺❹❖❻❼⑨❥❶❾❽❿❻✰➀❯❻❆❷❸➁➂❻❆➃ ✾❀✿❂❁❄❃❆❅r➄ ➅➇➆ ❊●❏❛❋♣❙❲❍ ➈ 0 1 2 3 4 5 6 0 1 2 3 4 5 6 nz = 13 ➅➊➉✗➅ ✉✈◗❱❙❳P✥❏❭➋ ➌◗❱❋■▲❖➍❘❏❛▲❖❙❳➎④➏ ❤ q ❉✫❊♣❍❑❙❘❊❱❜✑➐➑◗❱♠❚❍✥❍❑❏▼◗❱❋✗◆✭♥❛❏❴✉✈❏❛❋❚◗➒❙✥❏❴❊♣❋➔➓✈→➣➏❢ ➅❆↔ ❤⑥↕➇➙❾➛✙➜ ⑦✷⑧➞➝ ➝✷⑨✍⑩✕❶❑❷❸❶❺❹❖❻❼⑨❥❶❾❽❿❻✰➀❯❻❆❷❸➁➂❻❆➃ ✾❀✿❂❁❄❃❆❅r➟ ➅➊➉✗➅➠➆❊♣❏❴❋❂❙❲❍ ➈ 0 5 10 15 20 25 0 5 10 15 20 25 nz = 105 ➅❢ ➉✓➅❢ ✉✕◗➒❙❳P✥❏❴➋ ➌◗❱❋❚▲❯➍❘❏❛▲➡❙✥➎④➏ ❤ ➅ ❉✫❊♣❍❑❙✙❊❱❜✷➐➑◗❱♠❚❍✥❍❑❏▼◗❱❋✓◆✭♥❛❏❛✉✦❏❛❋❚◗❱❙❳❏❛❊●❋★➓✈→➣➏❢ ➅❢ ↔ ❤⑥↕➠➙❺➛♦➢➡➜ q