2.3.1 Jacobi D-(L+U)u+D-f D-(L +U)u+D-f subtracting I(L+u)er 2.3 Gauss-Seidel er+I=(d-dUe=reser It is clear that the above equations satisfied by the error are not useful for prac- tical purposes since eo =u-u, will not be known before the problem is solved We will see however, that by studying the equation satisfied by the error we can determine useful properties regarding the convergence of our schemes candLes 2.4.1 Jacobi SLIDE 1 (0)=(1)=0; u0=0 2.4.2 Gauss-Seidel LIDE 18 0ç✞è❏é✼è♦ê ë✞ì✙í●î❨ï◗ð ñ➌ò✒ó✺ô✼õ✪ö▲÷ ø✮ù■ú✲ûýü þ✧ÿ✞û✁￾✄✂✆☎✞✝✠✟➌ø✮ù✡☎ þ✬ÿ➞û☞☛ ø ü þ✧ÿ✞û✁￾✄✂✆☎✞✝✠✟➌ø✌☎➐þ✧ÿ✞û☞☛ ✍✏✎✒✑✔✓✏✕✗✖✙✘☞✓✏✚✜✛✒✢ ✣✙ù❇ú✁û❙ü✘þ ÿ➞û ￾✄✂✤☎✥✝✠✟ ✦ ✧✩★ ✪ ✫✭✬ ✣✙ù✰ü✯✮✱✰✡✣✙ù ç✞è❏é✼è➑ç ✲▼ì✒✳✵✴✶✴☞✷☞✸✺✹✠ð✄✻✵✹✽✼ ñ➌ò✒ó✺ô✼õ✪ö✙✾ ✿ ✚✜❀❁✚❃❂❄✖❅✕❆❂❃❇❉❈ ✣✙ù❇ú✁û✣ü❊￾☛þ●❋✆✂❍✟ ÿ➞û ✝ ✦ ✧☞★ ✪ ✫✺■❑❏ ✣✙ù✰ü▲✮✱▼❖◆❍✣❨ù P✩◗❙❘❄❚❱❯❳❲❃❨✗❩✁❬❭◗❫❪❴❩✁◗❵◗❛❪✒❨❱❩❉❜✗❝✁❞✁❨❡❨✗❢☞❣❴❩❅◗❤❘✄❝❅✐✒❚❭❚❳❩✁◗❥❘❄❚❫❦❍❨✗❧♠❜☞♥♦◗❫❪❴❨❡❨❳❬✩❬✏❝✁❬✠❩✁❬❆❨♣✐q❝❅◗❵❣✽❚❳❨✄r✗❣✔❲✁r☞❝✁❬✡sq❬❆❩✙❯❳t ◗❥❘✄❯❆❩❅❲❉sq❣✔❬❥s❴❝✉❚❳❨✩❚✈❚✗❘❫✐✇❯✗❨ ✣✔①tü➉ø②❋▼ø❵①✙③✵④❘❫❲❫❲✭✐✇❝✁◗✡❜✗❨⑥⑤✁✐✇❝④✐⑦❜✗❨✄r☞❝❅❬✏❨♣◗❛❪✒❨⑧s✇❬✏❝✙❜❳❲❃❨❳⑨⑩❘❄❚⑥❚❳❝✁❲❶❞✉❨✗❧✙❷ ❸❹❨ ④❘❫❲❫❲✭❚❳❨✗❨✈❪✒❝④❨☞❞✁❨☞❬③ ◗❫❪❴❩✁◗⑧❜❳♥♦❚✗◗❥❣❴❧✁♥✁❘❫✐✽❺❁◗❛❪✒❨❡❨✗❢❳❣❴❩✁◗❥❘✄❝✁✐❹❚❳❩❅◗❤❘❄❚❛❦⑧❨✗❧♠❜❳♥❁◗❫❪❴❨❡❨❳❬✩❬✏❝✁❬ ④❨❡❯✗❩✁✐ ❧✙❨❳◗❻❨☞❬✩⑨♦❘❫✐q❨❡❣✽❚❳❨✄r✗❣✔❲✽s✇❬✏❝❆s❴❨❳❬✩◗❤❘✄❨☞❚❱❬✏❨❼❺❉❩❅❬✏❧✁❘❫✐✽❺❽◗❛❪✒❨♦❯✗❝✁✐❾❞✉❨❳❬❤❺❑❨☞✐✇❯❆❨♦❝❿r➀❝✁❣✔❬❭❚❳❯❆❪❴❨☞⑨♠❨✩❚❳❷ ➁➃➂❛➄ ➅②➆➈➇➊➉➌➋❭➍❿➎❖➏ ç✞è❃➐✁è♦ê ë✞ì✙í●î❨ï◗ð ñ➌ò✒ó✺ô✼õ✪ö❑➑ ❋➈➒✇➓➔➓❝ü➣→ ➒➊￾✄↔❑✟✩ü↕➒➊￾❿→✉✟➄ü✯↔❴➙ ø① ü✯➛ 0.3 0.2 0.1 0 # Iterations log(||e|| L2 ) n=10 n=20 n=40 0 20 40 60 80 100 120 140 160 180 ç✞è❃➐✁è➑ç ✲▼ì✒✳✵✴✶✴☞✷☞✸✺✹✠ð✄✻✵✹✽✼ ñ➌ò✒ó✺ô✼õ✪ö✙➜ ❋➈➒✇➓➔➓❝ü➣→ ➒➊￾✄↔❑✟✩ü↕➒➊￾❿→✉✟➄ü✯↔❴➙ ø① ü✯➛ ➝