Note 6 Proof of L- result (Optional) The proof is by what is known as the "Aubin-Nitsche"trick, an application duality. We will see it again for linear functional error estimates To begin, we introduce an auxiliary problem: find E X= Ho(Q)such that where e is the error u-uh. We now setv=e,so that eedx=a(e,重) a(e,重-工重) B|lln()|一工画n(g2) (continuity) ≤B|llr(g)=|傅l where the last line follows from our interpolation result of Slide 10. Now we note from Slide6 of the last lecture that, since e∈2(2)( (in fact,e∈H(9),重 satisfies 4p H (32)< Cllellz2()(note the strong form for is simply -ra =e) Using this fact and dividing by llellza() gives ell(gx)≤ C hllellHi(g) om which the rest directly follows.(Note C in different expressions need not be the same: C is a generic constant independent of h and u Note the L result appears relatively unimportant(apart from confirming our intuition). However, that is not the case: the fact that llellz2(9) converges faster than ellI(s) has important ramifications in many different contexts (e. g, a posteriori error estimation ). Our proof here needs only continuity not symmetry, not coercivity-and is thus quite general, though the regularity hypothesis on requires more attention in R 1.7 Linear Functionals 1.7.1 Motivation a linear- functional“ output” s is defined by 8=C(u)+c is a bounded linear functionalø➩ù✝ú❖û✏ü ý⑨þ✲ù❄ù✐ÿ❫ù✶ÿ✁￾✄✂✮þ✓û✆☎✞✝✠✟✙ú☛✡✌☞✎✍❙ú✑✏Øù✞✒✔✓✕✟✗✖ ✘✁✙✛✚✢✜✠✣✥✤✞✤✆✦★✧✪✩✁✫✞✬✮✭✯✙✱✰✳✲✴✧✵✩✁✶✸✷✠✤✳✭✯✷✹✰✺✩✻✲✥✙✛✚✽✼✿✾✴❀✛✫✛✧✪✷✕❁❃❂✴✧❄✲✿✩✿❅✌✙✛✚❇❆❈✲✿✣✥✧✵❅✌✶❊❉✛✰✆✷❋✰✆✜✛✜✛●✪✧✵❅❍✰✳✲✿✧❄✤✺✷ ✤✆✦❏■✕❀✠✰❑●❄✧❄✲▲✬❑▼✔◆❖✚✢✭✯✧❄●✪●P✩✥✚❍✚◗✧❄✲❘✰✆❙✺✰❑✧❄✷❚✦✗✤❑✣✯●✪✧✪✷✛✚❇✰❑✣✁✦✗❀✛✷✠❅❯✲✿✧❄✤✺✷✠✰✆●❱✚❇✣✥✣✿✤❑✣✁✚❲✩▲✲✿✧❄❳❨✰✳✲✿✚❇✩❇▼ ✘❩✤◗✫✱✚❇❙❑✧✪✷❬❉❑✭✻✚✁✧❄✷✸✲✥✣✿✤✕■✕❀✠❅❍✚✁✰❑✷❨✰✆❀✕❭✕✧✪●❄✧✵✰✆✣✿✬✢✜✛✣✿✤❑✫✛●✪✚❍❳❋❪❱❫✠✷✱■❨❴❛❵❋❜❞❝❢❡❤❣✐❱❥❧❦✯♠ ✩✥❀✠❅✌✙♥✲✿✙✠✰✳✲ ♦ ❥q♣✱r ❴ ♠ ❝ts ❣ ✐✈✉ ♣①✇✺② ✭✯✙✛✚❇✣✥✚ ✉ ✧✵✩✁✲✥✙✠✚✢✚❍✣✿✣✥✤✺✣✻③❨④⑤③❱⑥✕▼❏◆❖✚✢✷✛✤✳✭t✩✥✚⑦✲ ♣ ❝ ✉ ❉✠✩✥✤❈✲✿✙✠✰✳✲ ⑧ ✉ ⑧ ✂⑨✠⑩✌❶❄❷✠❸ ❝❹s ❣ ✐✈✉✄✉ ✇✺② ❝ ♦ ❥ ✉ r ❴ ♠ ❝ ♦ ❥ ✉ r ❴❺④❋❻⑥ ❴ ♠ ❥ ✤❑✣✥✲✥✙✛✤✺❙❑✤❑✷✱✰✆●✪✧❼✲▲✬♠ ❽ ❾ ⑧ ✉ ⑧❍❿✁➀ ❶❄❷✠❸❏➁ ❴❺④❋❻⑥ ❴ ➁ ❿✯➀ ❶❄❷✠❸ ❥ ❅⑦✤❑✷✸✲✿✧❄✷✞❀✛✧❄✲▲✬♠ ❽ ❾ ⑧ ✉ ⑧❍❿➀ ❶❄❷✠❸✴➂➃ ⑧ ❴ ⑧❍❿⑩❍❶❄❷✠❸ r ✭✯✙✛✚❇✣✥✚✄✲✥✙✛✚✁●✵✰❑✩✑✲★●✪✧✪✷✛✚➄✦✗✤✺●❄●✪✤✳✭✴✩❩✦✗✣✥✤✺❳➅✤❑❀✛✣★✧✪✷✺✲✿✚❍✣✿✜✱✤✺●✪✰✆✲✥✧✪✤❑✷❈✣✥✚❲✩✑❀✛●❄✲✔✤✆✦➇➆✞●✪✧✵■✕✚◗➈❇➉✛▼★❂✴✤✳✭❺✭✻✚✄✷✠✤✆✲✥✚ ✦✗✣✿✤❑❳➊➆✞●✪✧✵■✕✚✮➋✹✤✆✦➄✲✿✙✛✚❨●✵✰❑✩✑✲◗●✪✚❇❅⑦✲✥❀✛✣✿✚➌✲✥✙✱✰✳✲❇❉★✩✑✧✪✷✠❅❍✚ ✉ ❵➍￾➄✂ ❥➎❦✯♠➌❥ ✧✪✷❖✦q✰❑❅❯✲❲❉ ✉ ❵➏❡❤❣✐ ❥➎❦✯♠✑♠ ❉❏❴ ✩✿✰✳✲✥✧✵✩✑❫✠✚❇✩ ⑧ ❴ ⑧❍❿⑩ ❶❄❷✠❸ ❽➑➐ ⑧ ✉ ⑧ ⑨ ⑩ ❶❄❷✱❸ ❥✷✠✤✆✲✥✚➄✲✥✙✛✚✁✩✑✲✥✣✿✤❑✷✛❙✯✦✗✤✺✣✥❳➒✦✗✤✺✣❏❴❤✧✵✩❩✩✥✧✪❳➓✜✠●❄✬❨④①❴✻➔❲➔→❝ ✉ ♠ ▼ ➣❘✩✥✧✪✷✛❙➓✲✥✙✛✧✵✩✁✦q✰❑❅⑦✲❘✰✆✷✠■✹■✕✧✪↔✞✧✪■✕✧✪✷✛❙❨✫✞✬ ⑧ ✉ ⑧ ⑨ ⑩ ❶❄❷✱❸ ❙✺✧❄↔✺✚❇✩ ⑧ ✉ ⑧ ⑨ ⑩ ❶❄❷✱❸ ❽➏➐ ➂ ⑧ ✉ ⑧❍❿✁➀ ❶❄❷✠❸ r ✦✗✣✿✤❑❳↕✭✯✙✛✧✪❅✌✙❋✲✿✙✛✚→✣✿✚❇✩✑✲①■✕✧✪✣✥✚❲❅❯✲✿●❄✬❚✦✗✤❑●✪●❄✤✳✭✴✩❇▼ ❥❂✴✤❑✲✥✚ ➐ ✧✪✷➙■✕✧❄➛❊✚❍✣✿✚❍✷✸✲❘✚⑦❭✕✜✛✣✿✚❇✩✿✩✥✧❄✤✺✷✠✩✁✷✛✚❍✚❲■❋✷✛✤✆✲ ✫➇✚◗✲✥✙✛✚→✩✿✰✆❳➌✚❑❪ ➐ ✧✪✩❘✰❈❙✺✚❍✷✛✚❇✣✥✧✵❅✎❅⑦✤❑✷✱✩▲✲✌✰✆✷✸✲✯✧❄✷✠■✛✚❍✜➇✚❍✷✠■✕✚❇✷✸✲❘✤✆✦ ➂ ✰✆✷✠■❚③★▼ ♠ ❂❘✤✆✲✥✚☛✲✥✙✛✚✹￾➄✂☛✣✥✚❲✩✑❀✛●❄✲❈✰❑✜✛✜✱✚❲✰✆✣✌✩→✣✿✚❍●✵✰✳✲✿✧❄↔✺✚❍●✪✬➙❀✛✷✛✧✪❳➌✜✱✤✺✣✑✲✌✰✆✷✸✲ ❥ ✰✆✜✠✰❑✣✑✲♥✦✗✣✥✤✺❳➜❅❍✤❑✷✕❫✠✣✿❳➌✧❄✷✠❙ ✤❑❀✠✣❘✧✪✷✺✲✿❀✛✧❄✲✥✧✪✤❑✷♠ ▼①➝✴✤✳✭✻✚❍↔✺✚❍✣❲❉✕✲✥✙✱✰✳✲①✧✵✩✴✷✠✤✆✲❘✲✿✙✛✚❈❅❇✰❑✩✥✚❑❪✄✲✥✙✛✚♥✦q✰❑❅⑦✲❘✲✥✙✱✰✳✲ ⑧ ✉ ⑧ ⑨ ⑩ ❶✪❷✠❸ ❅⑦✤✺✷✞↔❑✚❍✣✿❙❑✚❲✩ ✦q✰❑✩✑✲✥✚❇✣✮✲✿✙✠✰✆✷ ⑧ ✉ ⑧⑦❿✯➀ ❶❄❷✱❸ ✙✠✰❑✩❚✧✪❳➓✜➇✤❑✣✥✲✿✰❑✷✸✲☛✣✿✰❑❳➌✧❼❫✱❅❇✰✳✲✿✧❄✤✺✷✠✩✮✧✪✷❛❳❨✰✆✷✞✬➞■✕✧❄➛➇✚❇✣✥✚❇✷✺✲❋❅⑦✤✺✷✺✲✿✚⑦❭✞✲✿✩ ❥ ✚✺▼ ❙✱▼❄❉✯➟❨➠✛➡✳➢✌➤❃➥❍➦❯➧q➡✳➦❯➧✎✚❍✣✿✣✥✤✺✣✢✚❇✩✑✲✥✧✪❳➌✰✆✲✥✧✪✤❑✷♠ ▼✽➨①❀✠✣→✜✛✣✿✤✸✤❑✦✻✙✠✚❍✣✿✚❨✷✛✚❍✚❲■✛✩✢✤❑✷✠●❄✬❖❅⑦✤❑✷✸✲✿✧❄✷✞❀✛✧❄✲▲✬✽➩ ➫➡✆➤★➢✌➭✳➯➓➯❨➥⑦➤➎➦❯➭✠❉ ➫➡✆➤❏➲✌➡❲➥⑦➦✿➲⑦➧➵➳❲➧➵➤❧➭♥➩➸✰❑✷✠■➌✧✵✩❏✲✥✙✞❀✠✩✄➺✸❀✛✧❼✲✿✚✴❙❑✚❇✷✛✚❍✣✌✰✆●➎❉✳✲✿✙✛✤❑❀✛❙✺✙➌✲✥✙✛✚❘✣✥✚❇❙❑❀✛●✵✰✆✣✿✧❼✲▲✬ ✙✞✬✞✜✱✤❑✲✥✙✛✚❲✩✑✧✵✩✁✤❑✷➻❴➞✣✿✚❇➺✸❀✛✧✪✣✿✚❇✩✻❳➌✤❑✣✿✚◗✰✆✲✑✲✿✚❍✷✸✲✥✧✪✤❑✷❋✧✪✷❚➼➽✴✂❑▼ ➾❏➚➶➪ ➹♥➘✥➴✴➷❬➬✔➮➞➱✴✃①➴❘❐➇❒✛➘▲❮➄➴❘➬❏❰✑Ï Ð➇Ñ✵Ò❬ÑqÐ ÓÕÔ✠Ö❲×✗Ø✛Ù✕Ö❲×qÔ➇Ú Û➇Ü✸Ý➵ÞPß❖à✛á âäã➧➫ ➥✌➟✳➦❯åçæ✌è➫➲❍➤➎➧q➡➫➟ã♥é➡✳è✕➤❼➠➇è✕➤⑦ê◗ë✢✧✵✩✯■✕✚⑦❫✱✷✛✚❇■✹✫✞✬ ë✎❝➑ì❍í ❥③ ♠❬îðïítñ ✭✯✙✛✚❇✣✥✚ ì❍í✯❪t❡❖❣✐ ❥❧❦✯♠✄ò ➼➽ ✧✵✩✯✰➌✫✱✤✺❀✛✷✠■✕✚❲■❚●✪✧✪✷✛✚❇✰❑✣✁✦✗❀✛✷✠❅❯✲✿✧❄✤✺✷✠✰✆● ➈❲ó