It is certainly still possible that u is in H2(Q2)even when Q2 onvex but it is not typically the case. In particular, with a non-convex domain we introduce "re-entrant"corners and associated singularities. The worst case is a crack, where the re-entrant corner has an included angle of 2T; but even in that the solution remains in H( Q)(and in fact is more regular than H(Q) but not as regular as H( 2)). The effect on the finite element conver gence rate will be discussed briefly subsequently. 2 Finite element discretization 1 Triangulation Th∈Th Th: elements, =1,n Our numbering of nodes is purely for convenience of exposition; it will greatly implify our definitions of spaces and bases, and the description of the imposi tion of boundary conditions. In actual practice, in particular for more general boundary conditions, our numbering will not be the bes rds effi ffor direct solvers); but we know we can always renumber at the end through our e(k, a)array In general, finite elements are based on largely unstructured meshes, this has the advantage of flexibility, but also precludes the application of some structured. mesh notions(e.g, FFT or tensorization concepts) Note 3 Triangulation in IR In two(and particularly three) space dimensions, triangulation can be a difficult task. First. the of (the closure of )our tria (open) elements must not intersect (overlap) with each ther: third the intersection of the closure of one element with the closure of nother element must be either an entire edge of both elements or a vertex of both (or null). These conditions figure prominently in the definition of the☎✝✆✟✞✡✠✟☛✌☞✎✍✏✆✒✑✓✞✕✔✗✖✕✘✙✠✚✆✏✞✛✖✕✖✄✜✣✢✤✠✒✠✏✞✕✥✦✖✛☞✧✆✏★✗✑✩✆✟✪✫✞✛✠✬✞✛✔✮✭✙✯✤✰✲✱✴✳✬☞✁✵✓☞✎✔✷✶✴★✦☞✎✔✫✱✸✞✡✠✹✔✦✢✓✆✟☛✎✢✓✔✺✵✓☞✎✻✽✼ ✥✦✾✿✆❀✞✕✆❁✞✡✠❁✔✗✢✩✆❂✆❃✘✺✜✦✞✡☛✎✑✓✖✕✖✛✘✷✆✏★✦☞❄☛✎✑✤✠✚☞✤❅❆☎❇✔❈✜✗✑✓✍✚✆✒✞✛☛✎✾✦✖✡✑✩✍❉✼❊✶✴✞✕✆✏★❋✑✙✔✦✢✤✔✿●❇☛✌✢✓✔✺✵✤☞✌✻✮❍✦✢✓■❏✑✩✞✛✔❈✶❑☞ ✞✛✔✤✆✒✍✏✢✿❍✿✾▲☛✌☞◆▼✚✍✒☞✌●✝☞✎✔❖✆✏✍P✑✩✔❖✆✒◗❂☛✌✢✓✍✒✔✦☞✁✍✒✠❑✑✩✔✗❍❘✑✓✠✒✠✚✢✿☛✎✞✛✑✩✆✏☞✁❍❙✠✚✞✛✔✦❚✓✾✗✖✛✑✓✍✏✞✕✆✏✞✛☞✁✠✁❅✄❯❱★✦☞❲✶❑✢✓✍P✠✚✆❱☛✁✑✓✠✏☞❳✞✡✠✴✑ ☛✌✍P✑✓☛P❨❩✼✤✶✴★✦☞✎✍✒☞✴✆✏★✗☞❳✍✒☞✌●✝☞✎✔❖✆✏✍P✑✩✔❖✆❬☛✌✢✤✍✏✔✗☞✎✍❭★✗✑✤✠❬✑✓✔❏✞✕✔✗☛✎✖✕✾▲❍✿☞✁❍❪✑✓✔✦❚✓✖✛☞❫✢✓❴❛❵✩❜✄❝✺✥✦✾✿✆❞☞✁✵✓☞✁✔❏✞✕✔❏✆✒★✗✑❡✆ ☛✎✑✤✠✚☞❀✆✒★✦☞❙✠✏✢✓✖✛✾✿✆✏✞✛✢✓✔❢✍✒☞✎■❏✑✩✞✛✔✗✠✟✞✛✔✫✭✷❣❡✰❤✱✴✳❪✰✲✑✩✔✗❍✮✞✛✔❢❴✐✑✓☛❥✆❁✞✛✠✟■❀✢✤✍✏☞❏✍✒☞✎❚✤✾✦✖✛✑✓✍✹✆✏★✗✑✓✔❈✭✷❣❡✰❤✱✴✳❥✼ ✥✦✾✿✆✴✔✦✢✓✆❱✑✤✠❬✍✒☞✎❚✤✾✦✖✡✑✩✍❞✑✤✠❞✭❆✯✓✰✲✱✴✳✏✳❥❅❭❯❱★✦☞✬☞✌❦❩☞✁☛✌✆✴✢✓✔❪✆✒★✦☞❲❧✗✔✦✞✕✆✏☞✬☞✎✖✛☞✎■❀☞✎✔❖✆❫☛✌✢✤✔❖✵✤☞✎✍✒❚✓☞✁✔✗☛✌☞✴✍P✑❡✆✒☞ ✶✴✞✛✖✕✖❛✥✣☞♠❍✿✞✡✠✏☛✎✾✗✠✒✠✚☞❉❍❙✥✦✍✒✞✛☞✌♥✗✘❙✠✏✾✦✥✗✠✏☞✁♦❖✾✦☞✁✔❖✆✏✖✛✘✓❅ ♣ qsr✁t✉rP✈①✇③②⑤④✎✇❳⑥⑦✇❲t✧✈⑨⑧⑩r✎❶❸❷✴❹✄✇❱✈✄r✌❺✴❻✬✈✄r✌❼❁t ❽❱❾✚❿ ➀✧➁✗➂❃➃✄➄❳➅❬➆❳➇✚➃❸➈✦➂✚➉❬➄ ➊✣➋❖➌➎➍➐➏✮➑ ✱➓➒ ➔ →❡➣✩↔✤↕✁➣ ➙❳➛ ➜❲➝ ➞❱➟✿➠❁➡▲➟✿➢❀➤P➥✎➠❥➦➎➡❖➧✉➨✚➩❀➡✣➨❉➫✓➥✌➭❁➦✡➭❲➯✣➟✿➠✏➥✎➲➵➳✟➩✌➨❡➠❪➸P➨❡➡▲➺➻➥✎➡✗➦✐➥✎➡✣➸P➥❙➨✚➩❏➥✏➼✌➯✗➨➻➭❥➦➎➽✲➦✐➨✩➡✦➾✬➦➎➽❫➚✄➦➎➲➎➲❸➧✓➠✏➥P➪❡➽❤➲➵➳ ➭❥➦➎➢❫➯❩➲➵➦➩P➳✙➨❡➟✿➠❁➫✤➥✐➶❬➡✗➦➎➽❤➦✐➨❡➡✗➭❀➨❃➩♠➭❤➯✦➪✤➸✒➥✌➭✧➪✩➡✣➫❄➤P➪➻➭✎➥❥➭P➹❫➪❡➡❩➫◆➽➎➘✗➥✧➫✤➥❥➭✎➸✎➠❥➦➯✣➽❤➦✐➨❡➡➴➨✚➩♠➽➷➘✦➥✧➦➎➢❳➯✦➨❡➭❥➦➎➬ ➽❤➦✐➨❡➡➮➨❃➩❀➤P➨❡➟✿➡❩➫✓➪❡➠❥➳✙➸P➨❡➡❩➫❡➦➎➽❤➦✐➨❡➡✗➭✌➱✧✃❥➡➮➪✤➸✌➽❤➟✗➪❡➲▲➯❩➠✏➪✓➸✎➽✲➦✐➸P➥✌➹❫➦➎➡❘➯✦➪✩➠❥➽✲➦✐➸✎➟✿➲✕➪❡➠✴➩✌➨✩➠✧➢❏➨❡➠✒➥♠➧✤➥✎➡✣➥✎➠✏➪✩➲ ➤P➨❡➟✿➡❩➫✓➪❡➠❥➳❈➸P➨❡➡❩➫❡➦➎➽❤➦✐➨❡➡✗➭✒➹❀➨❡➟✿➠◆➡✗➟✿➢❏➤✒➥✎➠❥➦➎➡❖➧✫➚✄➦➎➲➎➲✴➡✣➨❡➽✧➤✒➥✉➽➎➘✗➥✙➤P➥❥➭❥➽❁➪❡➭❘➠✏➥✝➧❖➪❡➠✏➫❡➭✉➥✲❐❋➸✎➦✐➥✎➡✣➸✎➳ ❒➩✌➨❡➠❄➫❡➦➎➠✏➥P➸✎➽✹➭✌➨✩➲➵➺❡➥✌➠P➭✝❮➻➾❀➤✎➟✿➽✹➚❬➥❪❰❡➡❩➨❡➚Ï➚❬➥✉➸P➪✩➡Ð➪✩➲➵➚❬➪❡➳➻➭◆➠✏➥✎➡✗➟✿➢❏➤✒➥✎➠❙➪❡➽✟➽➷➘✦➥✉➥✎➡✣➫✷➽➷➘✺➠✏➨✩➟✓➧❉➘ ➨❡➟✿➠✴Ñ ✰✲Ò❩Ó✒Ô❊✳ ➪❡➠❥➠✏➪✩➳✓➱ ✃❥➡❢➧✤➥✎➡✣➥✎➠✏➪✩➲Õ➹①➶❬➡✗➦➎➽❇➥❘➥✎➲✕➥✎➢❀➥✎➡✗➽✲➭❙➪❡➠✏➥❄➤P➪➻➭✎➥P➫✮➨❡➡❋➲✕➪❡➠✲➧❖➥✎➲➵➳ ✾✦✔✗✠✚✆✏✍✒✾✗☛✌✆✏✾✦✍✒☞✁❍ ➢❏➥✌➭✚➘✗➥❥➭P➾♠➽➷➘✺➦✡➭❁➘✦➪❡➭ ➽➷➘✦➥❳➪✤➫❡➺❡➪❡➡▲➽✝➪✁➧✤➥✬➨❃➩❩Ö✴➥✏➼✤➦✐➤✎➦➎➲➵➦➎➽❤➳➻➹❊➤✌➟✿➽❊➪❡➲Õ➭✎➨❑➯❩➠✏➥P➸✌➲×➟✗➫✓➥✌➭❑➽➷➘✦➥❲➪✒➯✓➯❩➲➵➦✐➸P➪❡➽❤➦✐➨❡➡✉➨❃➩❞➭✎➨❡➢❏➥❱➭❥➽❤➠❥➟✗➸✌➽❤➟✿➠✏➥P➫❡➬ ➢❏➥❥➭✏➘❄➡❩➨❡➽❤➦✐➨❡➡✗➭ ❒➥❉➱Õ➧✤➱✕➹✄Ø❊Ø✹Ù➓➨✩➠✟➽✝➥✎➡✦➭✎➨❡➠❥➦✕Ú❉➪✩➽✲➦✐➨✩➡❢➸P➨❡➡❩➸P➥✝➯❩➽✐➭✝❮✓➱ Û✮Ü✽Ý❥Þ✉ß à❸á➻â✲ã✿ä❸å✄æ✗ç❤ã✽Ý✚â❃Ü✺äÐâ✲ä➴☎è❫✯ ☎❇✔Ð✆❃✶❑✢é✰✲✑✩✔✗❍Ð✜▲✑✩✍✏✆✏✞✡☛✌✾✦✖✡✑✩✍✒✖✕✘➮✆✏★✦✍✒☞✎☞➻✳❪✠✏✜✗✑✓☛✎☞❆❍✿✞✛■❀☞✎✔✗✠✏✞✕✢✤✔✗✠✁✼❱✆✏✍✒✞✛✑✓✔✦❚✓✾✦✖✡✑❡✆✒✞✕✢✤✔Ð☛✎✑✩✔é✥▲☞✮✑ ❍✿✞✕ê❪☛✌✾✦✖✕✆✹✆✒✑✓✠✏❨❩❅✟ë❸✞✛✍P✠❃✆❉✼▲✆✒★✦☞✧✾✦✔✦✞✛✢✓✔❆✢✓❴❲✰➎✆✒★✦☞❀☛✌✖✛✢✤✠✏✾✦✍✏☞❁✢✩❴✒✳❳✢✓✾✗✍❲✆✒✍✏✞✡✑✩✔✗❚✓✖✛☞✁✠❫■❁✾✗✠✚✆✹☛✎✢❡✵✓☞✎✍❳✆✏★✦☞ ❍✿✢✤■❀✑✓✞✕✔➐❝❲✠✏☞✁☛✌✢✤✔✗❍✽✼❞✆✏★✦☞➴✰➷✢✓✜✣☞✎✔✣✳❀☞✎✖✛☞✎■❀☞✎✔❖✆P✠❪■❂✾▲✠❃✆❙✔✗✢✩✆❪✞✛✔✤✆✒☞✎✍P✠✚☞❉☛❥✆❆✰✐✢❡✵✓☞✎✍✒✖✡✑✩✜▲✳❁✶✴✞✕✆✏★Ð☞✁✑✤☛P★ ✢✩✆✒★✦☞✎✍❉❝❛✆✒★✦✞✛✍✒❍✽✼❛✆✏★✦☞❀✞✛✔✤✆✒☞✎✍P✠✚☞❉☛❥✆✒✞✕✢✤✔❆✢✩❴❞✆✏★✦☞❪☛✎✖✕✢❖✠✚✾✗✍✏☞❁✢✩❴❞✢✤✔✦☞❀☞✎✖✛☞✎■❀☞✎✔❖✆✟✶✴✞✕✆✏★✷✆✏★✦☞❪☛✎✖✕✢❖✠✚✾✦✍✒☞❂✢✓❴ ✑✩✔✗✢✩✆✏★✗☞✎✍❏☞✎✖✛☞✎■❀☞✁✔✤✆❪■❁✾✗✠✚✆❙✥▲☞✉☞✁✞×✆✒★✦☞✎✍❘✑✩✔➓☞✎✔❖✆✏✞✛✍✒☞◆☞❉❍✿❚✓☞✉✢✓❴✬✥✣✢✩✆✒★Ð☞✎✖✛☞✎■❀☞✎✔❖✆P✠❀✢✓✍❙✑✮✵✓☞✁✍✚✆✒☞✌✻ ✢✩❴❑✥▲✢✓✆✏★➓✰➷✢✤✍✬✔✺✾✦✖✛✖➎✳❥❅❀❯❱★✦☞✁✠✏☞✧☛✎✢✓✔✗❍✦✞×✆✒✞✕✢✤✔✗✠✬❧✗❚✓✾✗✍✏☞✧✜✦✍✒✢✓■❀✞✕✔✗☞✎✔❖✆✏✖✛✘✉✞✕✔✷✆✏★✗☞❀❍✦☞✌❧✗✔✦✞✕✆✏✞✛✢✓✔✮✢✩❴❞✆✏★✦☞ ➝