FEM for the poisson problem R april14&16,2003
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1 Model problem 1.1 Formulations 1.1.1 Strong formulation LIDE Find a such that for Q a polygonal domain Generalizat ion We look here at a particularly simple but nevertheless illustrative problem First, we require our domain to be polygonal. More general domains demand at least for accurate treatment parametric(non-affine) mappings that result in elements with curved edges. Although the implementation of such elements is not particularly difficult, it requires more machinery that we can develop in this short series of lectures Second, we consider the homogeneous Dirichlet problem. In one space mension, the difference between homogeneous and inhomogeneous Dirichlet con- ditions was quite small as regards formulation, analysis, and implemention. In yo space dimensions, unless the inhomogeneous Dirichlet data ecewise olynomial, in our case here piecewise linear, we will not longer be able to ex- actly represent ud within our finite element approximation space. In practice one typically replaces u with its interpolant, but this clearly leads to new com plications as regards the theory(a new "variational crime") Treatment of Neumann and robin conditions is a relatively straight-forward extension of what we present here. It is particularly in higher space dimensions that the convenience of natural boundary conditions is most beneficial, since calculation of normals and gradients at the boundary can be complicated(and n many cases, ambiguous ). As regards inhomogeneous Neumann conditio the same issues that arise in higher space dimensions for the inhomogeneous Dirichlet problem also arise for the Neumann problem Third, we restrict ourselves to two space dimensions. In fact, three dimen sions is quite similar, though obviously the implementation(in particular mesh generation)is more complex, and computational cost potentially much higher 1.1.2 Minimization/weak formulation SLIDE 2 Find or find u∈ X such that
❃ ❄❅❇❆❉❈❋❊❍●☛■❏❅▲❑❉❊✤❈❋▼ ◆❏❖P◆ ◗❙❘❯❚❀❱☎❲❋❳P❨✌❩❭❬P❘❫❪❋❴ ❵❜❛❝❵❜❛❝❵ ❞✚❡❣❢✐❤❜❥✌❦♠❧✌❤♥❢✐♦q♣✌rts✼❡✲✉t❤❀❥ ✈❜✇✸①③②✖④✒⑤ ⑥✾⑦⑨⑧❀⑩❷❶❹❸❻❺♥❼❾❽✣❿❻❽❀➀✐❿ ➁❋➂▲➃ ❶ ➄ ➅ ⑦✪⑧❹➆ ❶ ➄ ➇ ➈➉⑧✣➊ ➋➈✞➌❋➆➍➀➏➎❭➐➒➑➔➓❣→✸➐✐➣❜↔➒➑✰↕➉➐✐➙➛↔✐➜③➣✰➝ ➞➏➟ ➠q➡➤➢✕➥➧➦ ➨❷➥➒➩❫➥➒➫➯➭❭➲❝➳❝➵✸➭➤➢P➳➸➡➺➩ ➻q➼➏➽✪➈➺➈✞➾➚❽♥➼❣➌➪➼➏➀✐❿➶➀❇➹♥➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽✪➴❷❸❻⑦⑨➷➬➹❭➽✪➼✠➮❭❺✼❿➱⑧❭➼❣✃➉➼❣➌❻❿❻❽❭➼✲➽⑨➼❐❸❻❸❒⑦⑨➽✪➽⑨❺❀❸➸❿➪➌➪➀➒❿❻⑦✪✃✞➼✠➹❭➌➪➈✞➮♥➽⑨➼✲➷✣➝ ⑥✾⑦⑨➌❾❸P❿✲❮✚❰✦➼Ï➌➪➼✲Ð✸❺❭⑦✪➌➪➼Ï➈➉❺❭➌➱⑩❭➈✞➷➛➀➒⑦✪⑧➧❿➪➈❷➮❜➼➬➹❀➈➉➽⑨➴➺Ñ➉➈✞⑧♥➀✞➽t➝➏Ò❹➈✞➌➪➼ÏÑ➉➼❣⑧❭➼✲➌➪➀✞➽✾⑩✼➈➉➷➛➀➒⑦✪⑧♥❸✶⑩✼➼❣➷➛➀➒⑧❀⑩ Ó ➀➒❿❒➽⑨➼❐➀✞❸P❿ ➋➈✞➌➶➀✞❼❣❼❣❺❭➌❾➀✐❿❻➼➱❿➪➌❻➼❐➀✐❿❻➷➬➼✲⑧➉❿ Ó ⑦➘❸P➈➉➹♥➀➒➌❾➀➒➷➬➼✤❿➪➌❻⑦➘❼▲ÔÕ⑧❭➈➉⑧✼Ö×➀➒Ø➛⑧❭➼❐ÙÚ➷➛➀✞➹❭➹❭⑦✪⑧❭Ñ➉❸Ú❿➪❽♥➀✐❿ ➌➪➼✲❸❻❺❭➽Û❿❇⑦✪⑧Ü➼❣➽✪➼❣➷➬➼✲⑧➉❿❾❸➛❰❙⑦⑨❿❻❽✂❼✤❺♥➌❻✃➉➼✲⑩✒➼✲⑩✼Ñ➉➼✲❸✲➝➍Ý❒➽⑨❿❻❽♥➈✞❺❭Ñ➉❽✒❿➪❽❭➼❹⑦✪➷➬➹❭➽⑨➼✲➷➬➼❣⑧✸❿➪➀➒❿❻⑦✪➈✞⑧Þ➈➋ ❸❻❺♥❼❾❽ ➼❣➽✪➼❣➷➬➼✲⑧➉❿❾❸➏⑦➘❸Ï⑧❭➈➒❿▲➹♥➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽✪➴q⑩✼⑦⑨Ø❇❼✤❺❭➽⑨❿✲❮❫⑦Û❿▲➌❻➼❐Ð➉❺♥⑦⑨➌➪➼✲❸➏➷▲➈➉➌❻➼➚➷➬➀➉❼❾❽❭⑦✪⑧❭➼❣➌➪➴q❿❻❽♥➀➒❿Ï❰✦➼❷❼❣➀✞⑧ ⑩✼➼✲✃✞➼❣➽✪➈✞➹✣⑦✪⑧❷❿➪❽❭⑦➘❸❒❸P❽♥➈✞➌❻❿Ú❸❻➼❣➌➪⑦⑨➼❐❸Ú➈➋ ➽⑨➼❐❼✕❿➪❺❭➌❻➼❐❸❣➝ ß➼❐❼✤➈➉⑧♥⑩➤❮✾❰à➼➚❼❣➈✞⑧♥❸❻⑦➘⑩✼➼❣➌✶❿➪❽❭➼❇❽❭➈➉➷▲➈➉Ñ✞➼✲⑧❭➼❣➈➉❺♥❸✜á❋⑦✪➌➪⑦✪❼❾❽❭➽✪➼✤❿Ï➹❭➌❻➈➉➮❭➽✪➼❣➷✎➝➚âã⑧♠➈➉⑧❭➼❇❸P➹❀➀✞❼✤➼❇⑩❭⑦ÛÖ ➷➬➼❣⑧♥❸❻⑦✪➈✞⑧✚❮❐❿❻❽♥➼✦⑩✼⑦⑨ä❁➼❣➌➪➼❣⑧♥❼❣➼❫➮❜➼✤❿➸❰✦➼❣➼❣⑧➏❽❭➈✞➷➬➈✞Ñ➉➼❣⑧❭➼✲➈✞❺♥❸✖➀➒⑧♥⑩✜⑦✪⑧❭❽❭➈➉➷➬➈✞Ñ✞➼✲⑧❭➼❣➈➉❺♥❸✖á➶⑦✪➌❻⑦➘❼❾❽❭➽✪➼✤❿✾❼❣➈✞⑧✼Ö ⑩✼⑦⑨❿❻⑦✪➈✞⑧♥❸❋❰✦➀➉❸❒Ð✸❺❭⑦⑨❿❻➼➏❸❻➷➛➀➒➽✪➽✌➀✞❸❒➌❻➼✲Ñ➉➀➒➌❾⑩❭❸ ➋➈➉➌❻➷Ï❺❭➽✪➀➒❿❻⑦✪➈✞⑧✚❮❀➀➒⑧♥➀✞➽⑨➴✼❸❻⑦✪❸✲❮♥➀➒⑧❀⑩✎⑦⑨➷➬➹❭➽✪➼❣➷➬➼❣⑧✸❿➪⑦⑨➈➉⑧✚➝Úâã⑧ ❿➸❰✦➈▲❸❻➹♥➀✞❼❣➼➶⑩✼⑦✪➷➬➼❣⑧♥❸❻⑦✪➈✞⑧♥❸✲❮✼❺❭⑧❭➽✪➼✲❸➪❸à❿❻❽❭➼✶⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸àá➶⑦⑨➌➪⑦➘❼❾❽❭➽⑨➼❣❿Ú⑩♥➀✐❿➪➀Ï❶❁åÜ⑦✪❸Ú➹♥⑦⑨➼❐❼✤➼❣❰❙⑦➘❸❻➼ ➹❜➈✞➽✪➴✸⑧♥➈✞➷➬⑦✪➀✞➽t❮❀⑦⑨⑧➧➈✞❺❭➌❋❼✲➀✞❸❻➼✜❽❭➼❣➌➪➼✠➹❭⑦✪➼✲❼❣➼❣❰❙⑦➘❸P➼➏➽⑨⑦✪⑧❭➼✲➀✞➌✲❮❀❰à➼➏❰❙⑦⑨➽✪➽✖⑧❭➈✞❿❋➽✪➈✞⑧❭Ñ➉➼❣➌❒➮❜➼Ï➀✞➮❭➽⑨➼✠❿❻➈❇➼❣æ✸Ö ➀✞❼✤❿❻➽✪➴❷➌➪➼❣➹♥➌❻➼❐❸P➼✲⑧➉❿❒❶❁åç❰❙⑦Û❿➪❽❭⑦✪⑧❹➈✞❺❭➌❒è♥⑧❭⑦⑨❿❻➼➏➼✲➽⑨➼✲➷➬➼❣⑧✸❿➶➀➒➹♥➹❭➌❻➈➯æ✼⑦✪➷➛➀✐❿❻⑦✪➈✞⑧❉❸❻➹♥➀✞❼❣➼✞➝Úâã⑧❉➹❭➌❾➀✞❼✤❿❻⑦➘❼✤➼✞❮ ➈✞⑧♥➼✎❿➸➴✸➹♥⑦✪❼✲➀➒➽✪➽⑨➴✒➌❻➼✲➹❭➽✪➀➉❼✤➼❐❸➛❶❜åé❰❙⑦⑨❿❻❽Ü⑦⑨❿➪❸➚⑦⑨⑧✸❿❻➼✲➌❻➹❜➈✞➽➘➀➒⑧✸❿❐❮✦➮♥❺✼❿❇❿❻❽♥⑦✪❸❷❼✤➽✪➼✲➀➒➌➪➽✪➴❍➽✪➼✲➀✞⑩♥❸➬❿❻➈♠⑧❭➼✲❰ ❼✤➈➉➷➬➹❭➽⑨⑦➘❼❣➀➒❿❻⑦✪➈✞⑧♥❸❙➀➉❸Ú➌❻➼✲Ñ➉➀✞➌➪⑩❭❸❫❿❻❽♥➼✶❿❻❽❭➼✲➈✞➌➪➴➧Ô❝➀▲⑧❭➼✲❰ëê❻✃➯➀✞➌❻⑦➘➀✐❿➪⑦⑨➈➉⑧♥➀➒➽➤❼❣➌❻⑦✪➷➬➼✲ì➉Ù✤➝ í✌➌❻➼❐➀✐❿➪➷▲➼✲⑧✸❿❫➈➋ ➞❒➼✲❺❭➷➛➀➒⑧❭⑧❇➀✞⑧♥⑩➛î❙➈➉➮❭⑦⑨⑧❇❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸❏⑦➘❸❫➀✜➌➪➼❣➽➘➀✐❿➪⑦⑨✃➉➼❣➽✪➴Ï❸P❿❻➌❾➀➒⑦✪Ñ✞❽✸❿PÖ ➋➈✞➌➪❰✦➀✞➌➪⑩ ➼✤æ➺❿➪➼❣⑧♥❸❻⑦⑨➈➉⑧❇➈➋ ❰❙❽♥➀✐❿✦❰à➼➱➹❭➌➪➼✲❸❻➼❣⑧✸❿à❽❭➼✲➌❻➼➉➝✌â×❿Ú⑦➘❸à➹❀➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌➪➽⑨➴▲⑦✪⑧➚❽❭⑦✪Ñ✞❽❭➼✲➌✦❸❻➹♥➀➉❼✤➼➶⑩❭⑦⑨➷➬➼❣⑧❀❸P⑦✪➈✞⑧♥❸ ❿❻❽❀➀✐❿▲❿❻❽❭➼✣❼✤➈✞⑧➺✃➉➼❣⑧❭⑦✪➼❣⑧♥❼❣➼❇➈➋ ⑧❀➀✐❿❻❺♥➌➪➀✞➽à➮❜➈✞❺❭⑧❀⑩❭➀➒➌➪➴q❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸➏⑦➘❸Ï➷▲➈✸❸➸❿▲➮❀➼✲⑧❭➼✤è❜❼✤⑦➘➀➒➽✴❮❫❸P⑦✪⑧♥❼❣➼ ❼❣➀✞➽✪❼❣❺❭➽➘➀✐❿❻⑦✪➈✞⑧✣➈➋ ⑧❭➈➉➌❻➷➛➀➒➽➘❸❙➀➒⑧❀⑩❷Ñ➉➌➪➀➉⑩✼⑦✪➼❣⑧✸❿➪❸❙➀➒❿❙❿❻❽❭➼✜➮❜➈✞❺❭⑧❀⑩❭➀➒➌➪➴➚❼❣➀✞⑧✎➮❀➼✠❼❣➈✞➷➬➹❭➽✪⑦✪❼✲➀✐❿❻➼❐⑩ïÔ❝➀➒⑧❀⑩ ⑦✪⑧ð➷➛➀➒⑧➺➴q❼✲➀✞❸❻➼✲❸✲❮✾➀✞➷➏➮❭⑦✪Ñ✞❺♥➈✞❺♥❸❾Ù✕➝❹Ý➶❸➏➌➪➼❣Ñ✸➀➒➌❾⑩❭❸✜⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸✠➞❒➼✲❺❭➷➛➀➒⑧❭⑧❍❼❣➈✞⑧♥⑩❭⑦Û❿➪⑦⑨➈➉⑧♥❸❣❮ ❿❻❽♥➼✣❸➪➀➒➷➬➼➚⑦➘❸➪❸P❺❭➼❐❸✠❿➪❽♥➀✐❿❇➀✞➌❻⑦➘❸P➼➚⑦⑨⑧✒❽❭⑦⑨Ñ➉❽❭➼❣➌➬❸❻➹♥➀✞❼❣➼❷⑩✼⑦✪➷➬➼❣⑧♥❸❻⑦⑨➈➉⑧♥❸ ➋➈✞➌Ï❿❻❽❭➼✣⑦✪⑧❭❽❭➈✞➷➬➈➉Ñ✞➼❣⑧♥➼❣➈✞❺❀❸ á➶⑦⑨➌➪⑦✪❼❾❽♥➽⑨➼❣❿❒➹❭➌❻➈➉➮❭➽✪➼❣➷ñ➀✞➽✪❸❻➈▲➀✞➌❻⑦➘❸❻➼ ➋➈➉➌Ú❿❻❽❭➼✠➞❋➼❣❺❭➷➛➀✞⑧❭⑧✎➹❭➌❻➈➉➮❭➽✪➼❣➷✎➝ íÚ❽❭⑦✪➌❾⑩➤❮❁❰à➼Ï➌❻➼❐❸➸❿➪➌❻⑦➘❼✕❿➱➈✞❺❭➌❾❸P➼✲➽⑨✃➉➼✲❸❒❿❻➈❇❿➸❰✦➈❷❸❻➹♥➀➉❼✤➼Ï⑩❭⑦⑨➷➬➼❣⑧❀❸P⑦✪➈✞⑧♥❸✲➝➱âã⑧ ➋➀✞❼✤❿✲❮❁❿❻❽♥➌❻➼✲➼Ï⑩✼⑦✪➷➬➼❣⑧✼Ö ❸❻⑦⑨➈➉⑧♥❸✦⑦✪❸ÚÐ✸❺❭⑦⑨❿❻➼✜❸P⑦✪➷➬⑦⑨➽➘➀➒➌❐❮✸❿❻❽❭➈➉❺❭Ñ✞❽❷➈✞➮➺✃➺⑦⑨➈➉❺♥❸❻➽⑨➴▲❿❻❽♥➼➱⑦✪➷➬➹❭➽✪➼❣➷➬➼❣⑧✸❿➪➀➒❿❻⑦✪➈✞⑧ïÔÕ⑦✪⑧❷➹❀➀➒➌❻❿❻⑦➘❼✤❺❭➽➘➀➒➌✦➷➬➼✲❸❻❽ Ñ✞➼✲⑧❭➼❣➌❾➀✐❿➪⑦⑨➈➉⑧❀Ù❫⑦✪❸❙➷➬➈➉➌❻➼✜❼✤➈➉➷➬➹❭➽⑨➼❣æ➤❮❭➀➒⑧♥⑩✣❼✤➈➉➷▲➹♥❺✼❿➪➀➒❿❻⑦✪➈✞⑧♥➀✞➽✚❼✤➈➉❸P❿❙➹❀➈✞❿❻➼✲⑧➉❿➪⑦✪➀✞➽⑨➽✪➴➚➷Ï❺♥❼❾❽✣❽❭⑦⑨Ñ➉❽❭➼❣➌❐➝ ❵❜❛❝❵❜❛➘ò ó☛✉❝❥✾✉❝♦ð✉tô✞s✼❡❐✉❝❤❜❥✰õ❭ö☛÷➺s✼ø♠❧✖❤❀❢✐♦ù♣✾r❝s✼❡❐✉❝❤❜❥✌ú ✈❜✇✸①③②✖④qû ⑥✾⑦⑨⑧❀⑩❷❶✎➄Ü➀➒➌➪Ñ➱➷▲⑦✪⑧ ü✾ý✞þëÿ➃✁ Ô✄✂✆☎✝✂➶Ù ➁✟✞ Ô✄✂➱Ù ✠ ✡☞☛ ✌ ✍✏✎ü✒✑ ✓ ➈✞➌Úè❀⑧♥⑩✣❶✕✔✗✖✑❸❻❺♥❼❾❽➚❿➪❽♥➀✐❿ ➟
(u,)=(u),v∈X SLIdE 3 where ∈H(9)vr=0}≡Hb(2 Vw. Vo de Recall that f need not be in L(Q2): we can consider any e EH-(Q2),for cample a "line source"e(u)= 1 on Te, 0 elsewhere, where Te is some line Q.(Note, he that the delta distributi H-1(2)frg∈R2.) 1.2 Regularity SLIDE 4 If L( Q)and Q is convex Allez2(a)≤C∫ portant for convergence rate tullia =fal vul2+u2 dA, and that e(o ul Note also that in IR the H- norm includes the square of the cross derivatives regula In one space dimension it was sufficient that f be suitably smooth to ensure that u would be in H-( Q). In two space dimensions that is no longer true, due to the potential conflict between what the boundary data tells the derivatives to do and what the equation tells the derivatives to do. however, in the case in which the domain Q is convex, then fEL2(9)is sufficient to ensure that u is in H-(Q)
✘✚✙✄✛✢✜✤✣✏✥✧✦✩★✪✙✫✣✬✥☞✜✮✭✯✣✱✰✳✲✵✴ ✶✯✷✁✸✺✹✼✻✾✽ ✿❁❀❃❂❅❄✝❂ ✲❆✦❈❇❉✣✱✰✗❊✟❋●✙■❍❏✥✧❑☞✣✚❑ ▲▼✦❖◆❃P❘◗❙❊✟❋❚ ✙■❍❏✥❯✜ ✘✯✙✫❱✆✜✤✣✏✥❲✦ ❳✬❨✆❩❬❱❖❭❉❩❬✣❫❪✁❴ ❵❃❛❝❜❞✜ ★✪✙✫✣✬✥❡✦ ❢❣❳❨✆❤ ✣❫❪✁❴✆✐ ❥✯❦✪❧♥♠♥♦✏♣❉♦✾q r❂tst✉✇✈✺✈❯①✄❀❃✉✇① ❤❞② ❂t❂④③ ②✯⑤ ①▼⑥④❂✾⑦②⑨⑧❝⑩ ✙■❍❏✥❉❶❷✿❝❂❸st✉② s⑤✇②❃❹ ⑦✫③❺❂☞❄❻✉②♥❼ ★❷✰❞❊❸❽ ❋ ✙❾❍❏✥t❿❘➀⑤ ❄ ❂④➁✁✉●➂➄➃✚✈➅❂➆✉➈➇■✈➉⑦② ❂ ❹❅⑤✇➊❄④s④❂✬➋➌★✪✙✄✣✏✥✳✦➎➍ ⑤✇②➐➏➒➑ ❿❫◆❖❂❅✈❹ ❂❅✿❁❀❃❂❅❄✝❂❣❿➓✿❁❀❃❂❅❄✝❂ ➏➒➑ ⑦❹✳❹❅⑤➂➔❂→✈➉⑦②❂ ⑤➀➣➇↔②⑦✺①↕❂✕❂④➁❺①↕❂②①☞➋✳⑦② ❍➛➙➝➜✺➞⑤ ①✮❂❣❿❯❀ ⑤✿❝❂❅➟➠❂❅❄t❿➔①✺❀♥✉✇①✆①✄❀❃❂→③❺❂☞✈➡①↕✉➐③●⑦❹ ①■❄❣⑦✫⑥ ➊①■⑦ ⑤●② ⑦❹✳②✚⑤ ①✆⑦② ❊❻❽ ❋ ✙■❍❏✥❝➀⑤ ❄❫❍❙✰✗➢➤⑩ ➙ ➥ ➦➨➧■➩ ➫⑨➭➒➯❝➲❫➳➸➵➻➺♥➼✮➽✪➾ ✶✯✷✁✸✺✹✼✻➆➚ ➢✮♠➶➪✪♣❉♠❃♣❅➹t➘●➴❾➷➒➬❣✛➨➬☞➮❏➱④✃❨❒❐❝❮❷❰ ➬t★❺➬ ➮➄Ï✁➱t✃❨♥❐tÐ ➢↕Ñ ❤ ✰ ⑧❝⑩ ✙■❍❏✥➛✉② ③➶❍❷ÒÔÓ➄Õ☞❦❺♠✁Ö❺♣☞×✒➷ ➬☞✛➻➬❅➮ÙØt✃ ❨♥❐Ú❮✩❰ ➬ ❤ ➬☞Û❃Øt✃ ❨♥❐ ✴ Ü❯Ý ÒßÞ➓à✯❦✪➹✝á④➘✪♠✁áÚÑ✄❦❺➹➌s⑤✇②➟✇❂❅❄■â❺❂②st❂➌❄✝✉✇①✮❂ Ð r❂tst✉✇✈✺✈✼①✺❀♥✉✇①❏➬❣✛➨➬ ⑩ ➮➱ ✃ ❨❒❐ ✦äã❨ ❑ ❩❬✛➨❑ ⑩Ùå ✛⑩ ❪❺❴✆❿Ù✉②③✱①✄❀❃✉✇① ➬t★❺➬ ➮Ï✁➱ ✃❨♥❐ ✦ Ó✝❧❃à æèç ➮❏➱❣✃ ❨❒❐ ★✪✙✫✣✬✥ ➬☞✣✚➬❅➮➱ ✃❨♥❐ q ➞⑤ ①↕❂é✉●✈❹❅⑤ ①✺❀♥✉✇①❏⑦② ➢➤⑩ ①✄❀❃❂ê❊⑩ ②✯⑤ ❄❣➂❆⑦②s❅✈➊③✪❂ ❹ ①✄❀❃❂ ❹❅ë❅➊✉✇❄✝❂ ⑤➀➌①✄❀❃❂✱s☞❄⑤✇❹t❹ ③✪❂❅❄❣⑦✺➟➠✉●①■⑦✺➟✇❂ ❹ ✉❹ ✿❝❂❅✈✺✈➅➙ ì✾í✒î❣ï✗ð ñ✕ï✇ò➨ó♥ô❾õ✏ö➠÷✤î✫ø⑨÷■ù ➢➤⑩ ➢✮♠✳❦✪♠❃♣❯Ó✤à♥➘❺Õ☞♣❯♦✏Ò➅Þ➔♣❉♠♥Ó✤Òß❦✪♠✱Ò➡áûúÚ➘❺ÓÙÓ✝❧✏ü▼Õ☞Òß♣❅♠✁áÚá✝ý♥➘●á ❤ ❥✯♣êÓ✤❧❃Ò➅á④➘✪❥❃➴ßþ➔Ó✝Þ➔❦✬❦●á✝ý✱á✝❦❬♣❅♠♥Ó✝❧❃➹④♣ á✝ý❒➘✇á➄✛✕úÚ❦✪❧❃➴Ô♦▼❥✯♣➛Ò➅♠✕❊⑩ ✙■❍❏✥ Ð ➢✮♠✳á➸úÚ❦➔Ó✤à♥➘❺Õ☞♣➛♦✏ÒßÞ➔♣❅♠♥Ó✝Ò➅❦❺♠♥ÓÚá✝ý❒➘✇á❏ÒßÓ➄♠❃❦➔➴➅❦❺♠❃➪✪♣❉➹Ùá④➹✝❧♥♣✪➷❃♦✏❧❃♣ á✝❦✳á✝ý♥♣➓à✯❦●á④♣❅♠✁á✝ÒÔ➘●➴➨Õ❅❦✪♠✏ÿ♥ÒÔÕ❣áê❥✯♣☞á➸úÚ♣❅♣❅♠✟ú❏ý♥➘✇áêá④ý❃♣❬❥✯❦✪❧♥♠♥♦❃➘●➹④þ→♦❃➘●á④➘éá④♣❅➴ß➴ßÓ❯á④ý❃♣➔♦✏♣❅➹④Ò➅Ö✇➘●á✝ÒßÖ✪♣❉Ó á✝❦➔♦✏❦➓➘●♠♥♦éú❏ý❒➘✇áÚá✝ý❃♣❯♣✁✁❧♥➘●á✝Òß❦✪♠éá④♣❅➴ß➴ßÓÙá✝ý♥♣❯♦✏♣❉➹✝ÒßÖ✇➘✇á④Ò➅Ö❺♣❉Ó❝á✝❦❬♦❃❦ Ð✄✂❦✇úÚ♣❅Ö❺♣❅➹è➷✪Òß♠▼á✝ý♥♣❯Õ❉➘✪Ó✝♣❘Òß♠ ú❏ý❃ÒÔÕtý➶á✝ý❃♣❬♦✏❦✪Þé➘●Òß♠→❍äÒßÓ❫Õ☞❦✪♠✬Ö❺♣☞×✒➷✏á✝ý♥♣❅♠ ❤ ✰ ⑧Ù⑩ ✙■❍❏✥✆⑦❹ Ó✤❧✏ü▼Õ❅Ò➅♣❉♠❺á❘á✝❦▼♣❉♠♥Ó✤❧♥➹✝♣êá④ý♥➘✇á❘✛➆ÒßÓ Òß♠➶❊⑩ ✙■❍❏✥ Ð Ý
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✳❥❅❀❯❱★✦☞✁✠✏☞✧☛✎✢✓✔✗❍✦✞×✆✒✞✕✢✤✔✗✠✬❧✗❚✓✾✗✍✏☞✧✜✦✍✒✢✓■❀✞✕✔✗☞✎✔❖✆✏✖✛✘✉✞✕✔✷✆✏★✗☞❀❍✦☞✌❧✗✔✦✞✕✆✏✞✛✢✓✔✮✢✩❴❞✆✏★✦☞ ➝
space that we hang upon this triangulation, though they are so "obvious"that we sometimes forget their presence. (Note in nonconforming approaches, some of the In addition of the above zeroth-order constraints, there are first-order con straints that must, or at least should, be satisfied in order to ensure that the approximation properties of the associated finite element space are good. The first is regularity; there are many ways to state this condition(and it goes by any names); in short, it requires that the minimum angle of any triangle be ounded away from zero as h tends to zero. This has many implications as regards the geometry and topology of the mesh(e.g, it bounds the number of angles that can share a common vertex, ensures that the length of a short side of a triangle does not tend to zero relative to the length of a long side, .. Note that h, the diameter of the triangulation Th, is the maximum of the h h", in turn, is the diameter-length of the longest edge -of triangle Th The second condition often imposed on Th has already been discussed in the context of one space dimension: quasi-uniformity requires that the minimum of h"k K, divided by the maximum of h", k=1,., K, remain bounded away from zero as h tends to zero the elements must not be increasingly disparate in size as h→0 Finally, there are second-order requirements. We would like our triangulation to place more elements in regions where the solution will vary more rapidly We would like the fexibility to locally adapt and refine the mesh based on a posteriori error indicators. We would like to specify element aspect ratios and perhaps the alignment of the elements in order to capture certain features the solution or to control the conditioning of the stiffness matrix There are, fortunately, triangulation methods that can often honor most of the requests above several very good third-party software packages are available that provide these capabilities. (The most general packages are quite literally triangulator, or, in R, tetrahedron-based; quadrilateral elements do not admit the same level of generality, at least not in as convenient a fashion, as"simplex elements. )Nevertheless, mesh generation remains a major task, sometimes more time-consuming than a calculation itself. For this reason it is often of interest to avoid the construction of a volume-filling mesh if possible(e. g, as in boundary element methods, discussed next in the course 2 Approximation 2.2.1 Space(Linear Elements) SLIDE lula∈P1(Th),ⅤTn∈Th}
ì✏í✗î✓ï✎ð❲ñ✒ò✗î❡ñ❫ó❑ð✹ò✗î✩ô✗õ✧ö✗í▲÷✤ô❘ñ✒ò✦ø✡ì❱ñ✏ù✒ø✛î✓ô✦õ✓ö✗ú✛î✩ñ✏ø✛÷✓ô❛û❖ñ✏ò✗÷✓ö✦õ✤ò❙ñ✒ò✦ð✎ü◆î✩ù✒ð✹ì✏÷❢ý✚÷✤þ❖ÿ✺ø✛÷✓ö✗ì✁✟ñ✒ò✗î❡ñ ó❑ð♠ì✚÷✄✂✧ð✎ñ✏ø☎✂❀ð✁ì✝✆➷÷✤ù✏õ✤ð✌ñ✴ñ✒ò✦ð✎ø✛ù❫í✦ù✒ð✁ì✏ð✎ô✗ï✎ð✟✞✡✠☞☛❳÷✩ñ✏ð❂ø✕ô✉ô✦÷✤ô✗ï✌÷✤ô✌✆➷÷✓ù✁✂❀ø✕ô✗õ❀î✓í✦í✦ù✒÷✤î✤ïPò✦ð✁ì✁û✦ì✚÷✄✂❀ð ÷✍✆❊ñ✏ò✗ð✁ì✏ð✹ï✎÷✓ô✗ì✚ñ✏ùPî✩ø✛ô❖ñ✒ì❱ï✎î✓ô❄þ▲ð✟ù✒ð✎ú✡î✏✎✿ð✒✑✓✞ ✔ ✕ô✮î✟✑✖✑✿ø✕ñ✏ø✛÷✓ô✷÷✍✆❭ñ✒ò✦ð❏î✩þ✣÷❡ÿ✓ð✘✗✁ð✎ù✒÷✩ñ✒ò✌✙❤÷✤ù✁✑✦ð✎ù❲ï✎÷✓ô✗ì✚ñ✏ùPî✩ø✛ô❖ñ✒ì✁û✗ñ✒ò✦ð✎ù✒ð❀î✩ù✒ð✘✚✗ùPì❃ñ✛✙❤÷✤ù✁✑✦ð✎ù❲ï✎÷✓ô✌✙ ì✚ñ✏ùPî✩ø✛ô✤ñPì✬ñ✏ò✗î✩ñ✜✂❂ö✗ì✚ñ✁û❸÷✓ù♠î❡ñ✟ú✛ð✁î✤ì❃ñ♠ì✏ò✦÷✓ö✦ú✢✑✽û❸þ▲ð❪ì✒î❡ñ✒ø✛ì✣✚✗ð✒✑✷ø✛ô✮÷✤ù✁✑✦ð✎ù✬ñ✏÷❄ð✎ô▲ì✚ö✦ù✒ð✧ñ✏ò✗î✩ñ✟ñ✏ò✦ð î✩í✗í✦ù✏÷✤✎✿ø☎✂❏î❡ñ✏ø✛÷✓ô❄í✦ù✒÷✓í✣ð✎ù✏ñ✏ø✛ð✁ì❳÷✍✆✄ñ✒ò✦ð✧î✤ì✏ì✏÷✿ï✌ø✡î❡ñ✒ð✒✑✥✚✗ô✦ø✕ñ✏ð✧ð✎ú✛ð✦✂❀ð✎ô❖ñ✹ì✚í✗î✤ï✌ð✧î✩ù✒ð♠õ✓÷✺÷✌✑✧✞✩★❱ò✦ð ✚✗ùPì❃ñ✹ø✡ì✫✪✛✬✮✭✍✯✌✰✲✱✍✪✴✳✶✵✸✷✺✹❛ñ✒ò✦ð✎ù✒ð✧î✓ù✏ð✻✂❏î✩ô✺ü✉ó❑î➻ü✿ì❳ñ✏÷✉ì✚ñ✒î❡ñ✒ð❂ñ✒ò✦ø✡ì✟ï✌÷✓ô✺✑✿ø×ñ✒ø✕÷✤ô✼✠✲î✩ô✽✑❆ø×ñ✟õ✓÷✺ð❉ì❲þ✺ü ✂❏î✩ô✺ü✉ô✗î✟✂✧ð❉ì✁✔✾✹✽ø✕ô❢ì✏ò✦÷✓ù✏ñ✁û❩ø✕ñ✹ù✒ð✒✿❖ö✦ø✛ù✒ð✁ì✴ñ✒ò✗î❡ñ✹ñ✒ò✦ð✻✂❀ø✛ô✦ø☎✂❂ö✖✂ î✓ô✦õ✓ú✛ð❂÷✟✆❞î✩ô✺ü❄ñ✒ù✏ø✡î✩ô✦õ✤ú✕ð❁þ▲ð þ✣÷✓ö✦ô✽✑✦ð✒✑❋î➻ó❑î➻ü❀✆➷ù✏÷✄✂❁✗✎ð✁ù✏÷❢î✓ì❃❂➮ñ✏ð✁ô✽✑✦ì✧ñ✏÷❄✗✎ð✁ù✏÷✺✞❅★❱ò✦ø✡ì✧ò✗î✓ì❆✂❏î✩ô✺ü✫ø☎✂✧í✗ú✕ø✡ï✎î✩ñ✏ø✛÷✓ô✗ì❀î✓ì ù✒ð✎õ✤î✓ù✁✑✗ì❑ñ✏ò✦ð❁õ✓ð✎÷✄✂❀ð✌ñ✏ù✒ü❘î✓ô✽✑◆ñ✏÷✤í▲÷✤ú✕÷✤õ✓ü❘÷✍✆✄ñ✏ò✗ð✘✂❀ð✁ì✏ò❅✠➷ð✄✞ õ✺✞✕û▲ø×ñ✬þ▲÷✤ö✦ô✽✑✦ì❳ñ✏ò✦ð❂ô❖ö✽✂❂þ✣ð✎ù❳÷✟✆ ñ✏ù✒ø✡î✩ô✦õ✤ú✕ð❉ì✹ñ✏ò✗î✩ñ❁ï✎î✓ô❈ì✚ò▲î✩ù✒ð❀î❆ï✌÷✟✂✫✂❀÷✓ô❢ÿ✤ð✎ù✏ñ✏ð✾✎✽û❊ð✎ô✗ì✏ö✦ù✒ð✁ì✹ñ✒ò✗î❡ñ❂ñ✏ò✦ð❘ú✕ð✁ô✦õ✩ñ✒ò✫÷✍✆❫î✉ì✚ò✗÷✓ù✏ñ ì✏ø☎✑✿ð❳÷✍✆✽î✹ñ✏ù✒ø✛î✓ô✦õ✓ú✛ð❇✑✿÷✺ð✁ì❭ô✦÷✩ñ❭ñ✒ð✎ô✽✑✧ñ✒÷✘✗✎ð✁ù✏÷✹ù✒ð✎ú✡î❡ñ✒ø✕ÿ✤ð❱ñ✏÷✟ñ✒ò✦ð❫ú✛ð✎ô✦õ✓ñ✏ò❏÷✍✆✽î✟ú✛÷✓ô✦õ♠ì✏ø✢✑✿ð✓û✺✞✦✞✦✞❈✔✴✞ ☛❫÷✓ñ✏ð✧ñ✏ò▲î❡ñ❉❂❛û❩ñ✒ò✦ð❃✑✿ø✡î✍✂❀ð✌ñ✒ð✎ù✹÷✍✆❞ñ✏ò✦ð✧ñ✏ù✒ø✡î✩ô✦õ✤ö✦ú✛î✩ñ✏ø✛÷✓ô❋❊✖●✗û❛ø✡ì✬ñ✏ò✦ð✫✂❏î✏✎✿ø☎✂❂ö✖✂ ÷✟✆❬ñ✒ò✦ð❍❂✺■❏✹ ❂■ û✿ø✕ô◆ñ✒ö✦ù✏ô➐û✦ø✛ì✴ñ✏ò✦ð✘✑✿ø✡î✍✂❀ð✎ñ✏ð✎ù▲❑ ú✛ð✎ô✗õ✩ñ✏ò❄÷✟✆❸ñ✒ò✦ð✟ú✕÷✤ô✦õ✓ð❉ì❃ñ❱ð✒✑✦õ✓ð▼❑ ÷✍✆❊ñ✏ù✒ø✡î✩ô✦õ✤ú✕ð▼◆● ■ ✞ ★❱ò✦ð✬ì✚ð❉ï✌÷✤ô✽✑❏ï✌÷✤ô✽✑✿ø✕ñ✏ø✛÷✓ô❪÷✟✆➎ñ✏ð✎ô❙ø✲✂❀í✣÷✤ì✏ð✒✑❏÷✤ô❖❊✖●♠ò▲î✓ì❑î✩ú✛ù✏ð❉î✟✑✿ü❁þ▲ð✁ð✎ô❖✑✿ø✡ì✒ï✌ö✗ì✒ì✚ðP✑❏ø✛ô❏ñ✏ò✦ð ï✌÷✤ô❖ñ✏ð✾✎✺ñ❬÷✟✆❛÷✓ô✦ð❲ì✚í✗î✤ï✌ð✩✑✦ø✲✂❀ð✎ô▲ì✚ø✛÷✓ô✓◗✝❘✦✯✽✱✤❙✴✳✶❚❯✯✌❱✽✳❲✾❳✍✪✴❨✻✳✶✵❩✷❀ù✏ðP✿✤ö✗ø✕ù✒ð✁ì①ñ✒ò✗î❡ñ❞ñ✏ò✦ð❬✂❀ø✕ô✦ø☎✂❂ö✽✂ ÷✟✆ ❂■ û✄❭✫❪❴❫✄❵✦❛✦❛✒❛✾❵✛❜➮û✍✑✿ø✛ÿ❖ø✢✑✿ðP✑❂þ✺ü✟ñ✒ò✦ð❝✂❀î✍✎✿ø✲✂❁ö✖✂✸÷✍✆❞❂■ û✄❭❃❪❡❫✟❵✒❛✦❛✦❛✦❵✛❜❈û❡ù✒ð✦✂❏î✓ø✕ô❁þ▲÷✤ö✦ô✽✑✿ðP✑ î➻ó❱î➻ü❋✆➷ù✒÷✟✂❁✗✁ð✎ù✒÷✷î✓ì❃❂➮ñ✏ð✁ô✽✑✦ì✧ñ✒÷❀✗✁ð✎ù✒÷❢❑ ñ✏ò✗ð❘ð✁ú✕ð✒✂✧ð✁ô❖ñ✒ì❆✂❂ö✗ì✚ñ❏ô✦÷✩ñ❀þ✣ð◆ø✛ô✗ï✎ù✏ð❉î✓ì✏ø✕ô✦õ✤ú✕ü ✑✿ø✡ì✚í▲î✩ùPî❡ñ✏ð✬ø✛ô❄ì✏ø☎✗✎ð♠î✓ì❇❂❍❣✐❤✖✞ ❥ø✛ô✗î✩ú✛ú✕ü✤û❉ñ✏ò✦ð✁ù✏ð❬î✓ù✏ð❭ì✏ð✁ï✎÷✓ô✽✑✌✙❤÷✤ù✁✑✿ð✁ù✽ù✏ðP✿❖ö✦ø✕ù✒ð✦✂❀ð✁ô✤ñPì✦✞❧❦✷ð❭ó❑÷✓ö✗ú☎✑✹ú✛ø✲♠✤ð✄÷✓ö✦ù➐ñ✏ù✒ø✛î✓ô✦õ✓ö✗ú✛î✩ñ✏ø✛÷✓ô ñ✏÷✮í✦ú✡î✓ï✎ð✥✂❀÷✓ù✒ð◆ð✎ú✛ð✦✂❀ð✎ô❖ñ✒ì✧ø✛ô➴ù✒ð✎õ✤ø✕÷✤ô✗ì❁ó✴ò✦ð✎ù✒ð❪ñ✒ò✦ð✉ì✚÷✤ú✕ö✿ñ✒ø✕÷✤ô➮ó✴ø✛ú✕ú✴ÿ❡î✓ù✏ü♥✂❀÷✓ù✒ð❘ùPî✩í✗ø☎✑✿ú✛ü✟✞ ❦✮ð❪ó❞÷✤ö✦ú✢✑✷ú✛ø☎♠✓ð❏ñ✒ò✦ð❃♦✗ð✦✎✺ø✛þ✦ø✛ú✕ø✕ñ❃ü✷ñ✏÷✉ú✛÷✺ï✁î✩ú✛ú✕ü✮î✟✑✦î✓í✿ñ❂î✩ô✺✑❢ù✏ð✦✚✗ô✦ð❀ñ✏ò✗ð❍✂❀ð✁ì✏ò✫þ✗î✓ì✏ð✒✑✮÷✓ô♣✱ q❳✤❙✴✵✮✬✦✪✴✳☞❳✏✪✴✳✴ð✁ù✏ù✒÷✓ù✴ø✛ô✽✑✿ø✡ï✎î✩ñ✏÷✤ù✒ì✒✞❝❦✷ð❁ó❞÷✤ö✦ú☎✑❄ú✛ø☎♠✓ð✟ñ✒÷❙ì✚í✣ð✁ï✎ør✆➷ü❄ð✁ú✕ð✒✂✧ð✁ô❖ñ❲î✓ì✏í▲ð❉ï❥ñ❳ùPî❡ñ✒ø✕÷❖ì❳î✓ô✽✑ í✣ð✎ù✒ò✗î✩í✗ì✟ñ✏ò✦ð◆î✩ú✛ø✕õ✤ô✖✂❀ð✎ô❖ñ♠÷✍✆❱ñ✏ò✗ð❪ð✎ú✛ð✦✂❀ð✎ô❖ñPì♠ø✛ô✫÷✓ù❈✑✿ð✎ù✟ñ✒÷✙ï✎î✓í✿ñ✏ö✦ù✒ð❙ï✎ð✎ù✏ñ✒î✩ø✛ô❄✆➷ð✁î✩ñ✏ö✦ù✒ð✁ì✟÷✟✆ ñ✏ò✗ð♠ì✚÷✤ú✕ö✿ñ✒ø✕÷✤ô❛û✿÷✓ù❱ñ✒÷❏ï✌÷✓ô❖ñ✒ù✏÷✤ú❩ñ✏ò✦ð♠ï✎÷✓ô✽✑✿ø✕ñ✏ø✛÷✓ô✗ø✕ô✦õ❀÷✟✆➐ñ✏ò✗ð♠ì❃ñ✒ørs❩ô✦ð✁ì✒ì✝✂❏î❡ñ✒ù✏ø✲✎✧✞ ★❱ò✦ð✁ù✏ð❁î✩ù✒ð✓û✖✆➷÷✤ù✚ñ✒ö✦ô✗î✩ñ✏ð✎ú✛ü✓û▲ñ✏ù✒ø✡î✩ô✦õ✤ö✦ú✛î✩ñ✏ø✛÷✓ôt✂❀ð✌ñ✒ò✦÷✌✑✦ì❫ñ✒ò✗î❡ñ✹ï✎î✓ô✉÷✍✆➎ñ✏ð✁ô✙ò✦÷✤ô✦÷✓ù❇✂❀÷❖ì❃ñ❳÷✟✆ ñ✏ò✗ð❬ù✒ð✒✿❖ö✦ð✁ì✚ñ✒ì➐î✩þ✣÷❡ÿ✓ð✟✹❡ì✏ð✎ÿ✤ð✎ùPî✩ú❡ÿ✓ð✁ù✏ü✬õ✓÷✺÷✌✑❲ñ✏ò✦ø✛ù❈✑❏✉❖í✗î✓ù✚ñ❃ü✬ì✚÷✟✆➎ñ❃ó❑î✓ù✏ð✄í▲î✓ï❈♠❡î✩õ✤ð✁ì✽î✩ù✒ð❬î➻ÿ❡î✩ø✛ú✡î✩þ✦ú✛ð ñ✏ò▲î❡ñ✟í✦ù✒÷❡ÿ❖ø✢✑✿ð❂ñ✏ò✦ð❉ì✚ð❏ï✎î✓í✗î✩þ✗ø✕ú✛ø×ñ✒ø✕ð❉ì✦✞✥✠☞★❱ò✦ð✫✂❀÷✤ì✚ñ✟õ✓ð✎ô✗ð✎ùPî✩ú❊í✗î✓ï❈♠❡î✩õ✤ð✁ì✬î✩ù✒ð✻✿❖ö✦ø✕ñ✏ð❀ú✛ø×ñ✒ð✎ùPî✩ú✛ú✕ü ñ✏ù✒ø✡î✩ô✦õ✤ö✦ú✛î✩ñ✏÷✤ù✒ì✁û➻÷✓ù❉û❉ø✛ô ✕✈❬✇ û❡ñ✒ð✌ñ✏ùPî✩ò✗ð✒✑✿ù✒÷✓ô✌✙✝þ✗î✤ì✚ðP✑✧✹✏✿✤ö▲î✟✑✿ù✒ø✕ú✡î❡ñ✒ð✎ùPî✩ú✤ð✎ú✛ð✦✂❀ð✎ô❖ñ✒ì①✑✦÷❲ô✦÷✓ñ①î✄✑✌✂❀ø×ñ ñ✏ò✗ð❳ì✏î✟✂❀ð❑ú✛ð✎ÿ✤ð✎ú✦÷✟✆✣õ✤ð✎ô✦ð✁ù✒î✓ú✕ø✕ñ❃ü✓û✤î❡ñ❭ú✛ð✁î✓ì✚ñ❭ô✦÷✓ñ❬ø✕ô❏î✤ì❭ï✌÷✤ô❖ÿ✤ð✎ô✦ø✛ð✎ô❖ñ❭î▼✆✐î✓ì✏ò✦ø✛÷✓ô❛û❖î✓ì✹ý✒ì✚ø☎✂❀í✦ú✕ð✦✎✖ ð✎ú✛ð✦✂❀ð✁ô✤ñPì✦✞②✔✜☛❳ð✎ÿ✓ð✁ù✚ñ✒ò✦ð✎ú✛ð✁ì✒ì✁û✒✂❀ð✁ì✏ò♠õ✤ð✎ô✦ð✁ù✒î✩ñ✏ø✛÷✓ô✹ù✏ð✒✂❏î✩ø✛ô✗ì❛î✝✂❏î✏③❃÷✤ù❛ñPî✓ì✛♠✣û➻ì✏÷✟✂❀ð✌ñ✒ø✲✂❀ð✁ì❧✂❀÷✓ù✒ð ñ✏ø☎✂❀ð✾✙❇ï✌÷✤ô✗ì✚ö✽✂✧ø✛ô✦õ✬ñ✏ò▲î✩ô❀î✬ï✁î✩ú✡ï✌ö✦ú✡î❡ñ✒ø✕÷✤ô❁ø✕ñ✒ì✏ð✎ú✲✆④✞ ❥÷✓ù①ñ✒ò✦ø✡ì✄ù✏ð❉î✓ì✏÷✓ô❁ø×ñ✄ø✡ì✄÷✍✆➎ñ✒ð✎ô✧÷✍✆❩ø✕ô❖ñ✏ð✁ù✏ð❉ì❃ñ❊ñ✏÷ î➻ÿ✓÷✤ø☎✑❏ñ✏ò✦ð♠ï✎÷✓ô✗ì✚ñ✏ù✒ö✗ï❥ñ✒ø✕÷✤ô❙÷✟✆❊î❁ÿ✓÷✓ú✛ö✖✂❀ð✾✙❩✚✗ú✛ú✕ø✛ô✦õ❆✂❀ð✁ì✏ò❘ø✲✆❊í✣÷✤ì✒ì✏ø✕þ✦ú✛ð❃✠➷ð✄✞ õ✺✞✕û✦î✤ì❞ø✛ô◆þ▲÷✤ö✦ô✽✑✦î✓ù✏ü ð✎ú✛ð✦✂❀ð✁ô✤ñ❇✂❀ð✎ñ✏ò✦÷✌✑✦ì✁û✽✑✿ø✡ì✏ï✎ö✗ì✒ì✚ðP✑❙ô✦ð✦✎✺ñ❫ø✕ô❘ñ✏ò✦ð♠ï✎÷✓ö✦ùPì✏ðP✔✴✞ ⑤❝⑥✸⑤ ⑦⑨⑧✩⑧❬⑩✖❶❸❷✝❹✛❺❼❻❸❽✖❹✣❶❿❾ ➀✓➁✢➀✓➁☞➂ ➃✓➄➆➅✖➇✟➈❅➉✣➊❿➋✸➌❸➈➍➅✌➎➐➏❬➑✸➈❏➒❅➈❏➌➔➓P→✤➣ ↔❞↕❏➙✶➛❧➜❄➝ ➞●✘❪❴➟♣➠❍➡ ➞ ➢ ➤✴➥ ➦ ➧✟➨ ➩❬➫➐➭P➯ ➧✝➲✜➳➆➵✒➸☎➺❧➻ ➼ ➠ ➼ ➽✏➾ ➡ ✕➚➶➪ ✠➹◆✓●✄✔✴❵▲➘❉◆✓●✫➡t❊✖●✌➴ ➷
t cr t cy y ∈RR We restrict our attention, as in one space dimension to linear polynomial ap prorimations. The ertension to higher order elements is not difficult. In prac tice, quadratic elements are often the best in terms of accuracy per unit of work We also consider only triangular elements, though in many cases quadrilateral (bilinear or biquadratic)elements have advantages 2.2.2 Basis(Nodal) ∈Xn,;(a)=6,1≤i,≤ nonzero We see how our requirements on the topology of the triangulation are reflected in the space and basis. For example, because we require that elements that intersect n an edge must intersect over an entire edge, we are ens ured that, given two f Note by our node numbering the aj,j=l,., n, are all in the interior; it follows from sp;∈ Xn C X that pi(a)=0rj=n+1,…,n SLIdE 8 Nodal interpretation: U E Xh U=∑vg;(x) u(a)=∑"()=∑6→同=以(a)
➬➮➶➱✍✃➹❐✓❒❏❮✴❰✝Ï✧Ð Ñ✏Ò❉Ó♣Ô✴Õ▲Ö×Ô✾Ø ÙPÚ✾ÛPÜ Ý✁Þ ß ÖàÔ✾á Ù✤Ú✴ÛPÜ Ý✁â ã➔ä Ô ä Ô✾Ø ä Ô✾á✻å✥➬æ ç❢è❆é✛è✴ê✴ë❩é✴ì☞í✾ë❇î✍ï✌é❆ð✏ë❩ë❯è✾ñ✺ë✸ì☞î✍ñ✖ò❝ð✏ê❉ì✶ñóî✏ñ❞è✻ê❩ô✽ð✟í❈è❃õ✏ì✶ö❃è✦ñ✖ê✴ì☞î✏ñ✽ò❝ë❯î✥÷øì✶ñ❞è❈ð✍é✝ô✽î✏÷øù✏ñ➔î✏ö❆ì☞ð✏÷❿ð✁ô➔ú ô❞é✁î✦û✄ì✶ö✫ð✍ë✸ì☞î✍ñ✖ê✦ü❢ý➔þ✖è❃è✁û✟ë❯è✦ñ✖ê✴ì☞î✏ñ❄ë✮î❃þ✌ìrÿPþ✽è✾é✫î✏é✛õ✄è✾é✫è✾÷✲è✦ö❃è✾ñ✺ë☞ê❉ì✢ê✘ñ➔î✏ë❇õ✏ì⑨í✦ï✌÷øë✮ü✂✁✾ñ❍ô➔é✛ð✟í✦ú ë❩ì☞í✁è✾ò☎✄✦ï✽ð✟õ✍é✛ð✏ë❩ì☞í✜è✦÷✲è✾ö❃è✦ñ✽ë✸ê❬ð✏é✛è✡î✝✆❈ë❯è✦ñ✥ë➹þ✖è✂✞❈è✴ê✴ë➆ì✶ñ➐ë✮è✦é✴ö❉ê✩î✟✆❬ð✟í❈í✦ï✌é✛ð✟í✦ù✝ô✽è✦é❇ï✌ñ✽ì✶ë①î✝✆✡✠❿î✏é☞☛✄ü ç❢è❃ð✏÷②ê✦îtí❈î✏ñ✖ê✴ì☞õ✄è✾é❃î✏ñ✺÷øù❖ë❩é✴ì☞ð✏ñ➍ÿ✍ï✌÷✲ð✍é✫è✦÷✲è✦ö✫è✦ñ✽ë✸ê❈ò➶ë➹þ✖î✍ï✟ÿPþ❋ì✶ñ♥ö✫ð✍ñ✽ùtí✁ð✏ê✦è✴ê✌✄✦ï✽ð✟õ✍é✴ì✶÷✲ð✏ë❯è✦é✛ð✏÷ ✍✞✾ì✶÷rì✶ñ❞è❈ð✏é✻î✍é✎✞✾ì✏✄✦ï✽ð✟õ✍é✛ð✏ë❩ì☞í✝✑➐è✾÷☎è✾ö❃è✾ñ✺ë☞ê✩þ✖ð✓✒✤è✫ð✄õ✔✒✏ð✏ñ✽ë❯ð✦ÿ❏è✾ê✾ü ✕✗✖✘✕✗✖✘✕ ✙✌✚✜✛✣✢✏✛✥✤✝✦✥✧✩★✪✚✜✫✭✬ ✮✰✯✲✱✴✳✩✵✷✶ ✸❍❒❉Ó✺✹☞✻✽✼✿✾❁❀❃❂➱ ä❅❄❆❄❆❄✦ä ❂☎❇❉❈✄❰ ❂☎❊①å❋✸❍❒ ä ❂☎❊❧✃❍●✪■P❮ Ó❑❏❅❊▲■ ä◆▼✂❖◗P❈ä❙❘❚❖❱❯❳❲ ❨ïPô✄ô✖î✍é✴ë❬❩✿❭☎❂☎❊✛❰ ç❢è❝ê✦è❈è➶þ✖î✓✠✼î✏ï✌é❝é✛è❪✄✦ï✌ì✶é✛è✦ö✫è✦ñ✽ë✸ê❝î✍ñ❍ë✶þ✽è❇ë❯î✁ô✽î✏÷☎î✛ÿ✟ù❉î✝✆❝ë➹þ✖è✩ë✸é✴ì☞ð✍ñ❏ÿ✟ï✌÷✲ð✏ë❩ì☞î✏ñtð✏é✛è❇é✛è❍❫❝è❈í✦ë✮è❈õ✡ì✶ñ ë➹þ✖è❇ê✸ô✽ð✟í❈è▼ð✏ñ➔õ✎✞❈ð✤ê✴ì✢ê✦ü❵❴❸î✏é✩è✁û➍ð✏ö✩ô❞÷✲è✾ò☎✞❈è❈í❈ð✏ï➍ê✦è❛✠❿è✩é✛è❪✄✦ï✌ì✶é✛è❇ë➹þ✖ð✍ë➆è✦÷✲è✦ö✫è✦ñ✽ë✸ê❝ë✶þ✽ð✏ë ì✶ñ✽ë❯è✾é❈ê✦è❈í✾ë î✏ñ ð✏ñ❅è❈õ✦ÿ❏è❍ö❆ï➍ê✴ë✝ì✶ñ✽ë❯è✾é❈ê✦è❈í✦ë❬î✔✒✏è✾é❖ð✍ñ✼è✦ñ✽ë❩ì✶é✛è❍è❈õ✦ÿ❏è✴ò❜✠❿è❖ð✏é✁è❖è✦ñ✖ê✴ï✌é✛è❈õ➐ë✶þ✽ð✏ë✸ò➶ÿ✍ì✴✒✏è✦ñ ë❍✠❿î ñ❞îPõ✟ð✍÷✩✒✏ð✏÷øï✽è✾ê❈ò➶ð❝✆❈ï✌ñ➔í✾ë❩ì☞î✏ñ❀ì✢ê➶ô✽è✦é❞✆✾î✏é✛í❈è✫í❈î✏ñ✽ë❩ì✶ñ✽ï✽î✍ï➍ê✜î✓✒✤è✦é✘ë➹þ✖è❆ð✤ê❈ê✦î✒í✦ì☞ð✏ë❯è❈õ✥è❈õ✒ÿ✄èPü ❡❉î✍ë✮è❢✞✾ù❅î✍ï✌é✥ñ➔î✒õ✄è❢ñ✽ï✌ö❚✞❈è✾é✴ì✶ñ➍ÿ♥ë➹þ✖è❣●■ ò ❘ Ó ▼✟ä❅❄❆❄❅❄✾ä✟❯ ò❆ð✍é✛è❋ð✍÷✶÷✝ì✶ñ ë➹þ✖è❢ì✶ñ✺ë✮è✦é✴ì☞î✏é❪❤❃ì✶ë ✆✾î✏÷✶÷☎î✔✠➆ê❬✆❈é✛î✍ö✐❂❊ å❋✸❒❦❥ ✸ ë✶þ✽ð✏ë☎❂❊ ✃✏●■ ❮ Ó♠❧♥✆✾î✍é ❘ Ó ❯ Ö ▼✟ä❅❄❆❄❆❄✦ä✿♦❯ ü ✮✰✯✲✱✴✳✩✵q♣ ❡❉îPõ✟ð✏÷❧ì✶ñ✽ë❯è✾é❩ô➔é✛è✾ë❯ð✏ë❩ì☞î✏ñ➆❰ Ï❍år✸❍❒ts Ï❆Ó ❇ ✉ ❊✇✈ ➱ Ï❊ ❂❊ ✃✏● ❮②① Ï➔✃✏●■ ❮ Ó ❇ ✉ ❊③✈❸➱ Ï❊ ❂❊ ✃✏●■ ❮➶Ó ❇ ✉ ❊③✈❸➱ Ï❊ ❏❊④■②⑤ Ï■ Ó Ï❞✃❍●■ ❮ ❄ ⑥
Prs iyc, fs I 2.3. Ra"ll ige-Ritz ofi. all fki( L 1.rEgia tr4 ur a Al Nh 212mea 24+fcry,y∈qua,sr 2.9 1(1 fal Naol 2el hiri aryi inR wi 1 JJJh =h I 武 1≤wj≤ Ahni id)1≤w≤h The procedures, either Rayleigh-Ritz or Galerkin, are 2le812/. to those we de veloped in detail in the one-dimensional case 2. 9.2 Pafticulafi illuot fativl naol tt Uniform Mesh (1,1) Ki 2n i th h 1 len we refer to a uniform mesh, we shall mean the particular triangulation e.(Note the diameter of the elements, and hence mesh, is in fae h)
⑦⑨⑧❍⑩ ❶❉❷❢❸✜❹❻❺❽❼✪❾✰❿t➀✟❹➂➁✡➃ ➄✗➅③➆✩➅✏➇ ➈➊➉✿➋✗➌❍➍➏➎✏➐✰➑✗➒❪➈➓➎❞➔✣→➓➣↕↔❋➙❋➉✜➌❍➍❽↔❃➛✗➎❍➜ ➝✰➞✲➟✴➠✩➡q➢ ➤❝➥❃➦➏➧③➨❆➩③➫✿➭t➯➲➤❛➩✇➳☞➵✿➸ ➺ ➻♥➼ ➥✓➽➾➫✷➚❦➩✇➪ ➶❻➹✿➘❳➴②➷➬✲➮❉➱❞✃♥❐☞✃❜❒❳❮Ï❰✿➱❞✃②❒ Ð ÑÓÒ Ô ÕtÖ➶Ø× Ù➥✓➧③➨❆➽➾Ú➏➩✇➪✩➸ ➺ ➻✌Û❋Ü❣➻❦Ý ➥✔➳➾➩ Ý✟Þ➨ Ý ➮❉➱➺ ➻ ❐☞ß➏❒ ➼ ❰✿➱❞ßt❒à❐âá✎ß Û❋Ü❣➻❋ã ⑦⑨⑧❞ä å❑➀☞æ✲❾❉❸✽❼Ø❿t❼♠ç✥è⑨é❝ê❻❿✜➀✝❹➂➁❜æ ➄✗➅✇ë✪➅✏➇ ➙❋➍➏➜❻➍✲↔❃➉✽➌⑨ì❁➉✜í✣➍ ➝✰➞✲➟✴➠✩➡❱î✓ï ð✗➨❆➳ ➺ ➻ ➱✏ñò❒ ➼ ôó õ☞ö ➷ ➺ ➻ õò÷✩õ ➱✏ñò❒àø❛ß ➼ ÷☎ù ➱✏ñò❒à❐❛ú ➼üû ❐ ã❆ã❅ã ❐✟ý ➸ þ ➻ ➺ ➻ ➼❑ÿ➻ ➺ ➻ Û✁➤ ó þ ➻ ù④õ ➼ ➮❉➱÷❻ù ❐ ÷✩õ ❒✄✂ û✆☎ ú❪❐✞✝ ☎ ý✟✂ ÿ➻ ù ➼ ❰✿➱÷☎ù ❒✄✂ û✠☎ ú ☎ ý✟✡ ☛✌☞✎✍✑✏✌✒✔✓✖✕✗✍✙✘✛✚✜✒✔✍✣✢✙✤✥✍✄✦★✧✩☞✎✍✪✒✆✫✭✬✯✮✯✰✱✍✄✦✳✲✴☞✶✵✷✫✑✦★✧✩✸✹✓✯✒✁✺✥✬✯✰✻✍✪✒✔✼✯✦★✽✾✤✭✬✯✒✔✍✂➩❀✿t➨❅➪❽➳➾➩✱❁❅➥✓➧❂✧❃✓❄✧✩☞✎✓✯✢✪✍❆❅❇✍✹✘❈✍✪✵ ❉ ✍✄✰✻✓✙✏✎✍✙✘❄✦★✽❊✘❋✍✄✧❃✬✯✦★✰●✦★✽❍✧✩☞✎✍■✓✯✽❏✍✪✵❑✘✯✦★▲✹✍✪✽✎✢✣✦✷✓✯✽✌✬✯✰▼✕✙✬✯✢✄✍✖◆ ➄✗➅✇ë✪➅✘➄ ❖❜➉t↔❃➔❅➎◗P❈❘❻➌✏➉t↔✁❙Ó➌✏➌◗❘✪í❅➔❆↔✔➉t➔✣➎✩❚↕➍qì✎➉✽í❅➍ ➝✰➞✲➟✴➠✩➡❱î✜î ❯●✽❱✦❲✄✓✯✒✣▲❨❳❍✍✣✢✔☞✪➸ ❩ ➼ ❬ý ➬ ➷ ❭ý ➼ ➱❞ý➷ ❪ û ❒ ➬ ý ➼ ➱❞ý➷ ❮ û ❒ ➬ ❫ ➼ û✴❴ý ➷ ❵❛☞✎✍✪✽❜❅❇✍❛✒✗✍✷❲✄✍✪✒✹✧❝✓❍✬❞✚✜✽✾✦❲✄✓✛✒✣▲❡▲✹✍✣✢✔☞✶✤✑❅❇✍❆✢✔☞✾✬✯✰★✰❇▲✹✍✙✬✯✽❜✧✩☞✎✍✠✏✎✬✛✒✣✧◗✦✷✕✪✚✜✰✻✬✛✒❛✧◗✒✣✦✷✬✛✽❢✲❈✚✜✰✻✬✯✧✞✦✷✓✯✽ ✬❈❣✙✓ ❉ ✍✖◆✐❤★❥✐✓✯✧❃✍❦✧★☞✾✍❧✘✯✦✷✬✛▲❆✍✪✧❃✍✄✒❧✓♠❲✑✧✩☞✎✍❧✍✪✰✻✍✪▲❆✍✪✽✾✧◗✢✙✤✟✬✛✽❏✘❧☞✎✍✪✽❏✕✙✍✠▲✹✍✄✢❑☞✶✤✟✦❀✢✑✦★✽✆❲✄✬❈✕✪✧♦♥❬❫ ✤✟✽❏✓✯✧ ❫ ◆ ♣ q
0v;0 a r dr dy a 1<i,j< Derivatives of h h 0y:/0x (piecewise constant) (piecewise constant) SLIDE 14 Evaluation of o(api/a)(apj /ax)dA /8x ax s dA ;/8 2/h2 aNw/ax 0p;/x Evaluation of j(api/ay)(api/ay)dA 2/h apE/ay a ps/ay dA= h2 asw/a ay;/8 a w/ay 2000 pi/oy 4/h2 LIDE 16
r❏s❢t★✉✇✈②①❈③ ④✟⑤✄⑥❏⑦✗⑧✣⑨✙⑨✣⑩✷❶✯❷■❸✄❶✯⑦❆❹❺✾❻ ❼❏❽✷❾❧❿❑➀✜➁❇➂➄➃✜➅❧➆✐❾➈➇✖➆✐➀➊➉❹ ➂➋➃✎➅✠❾➍➌✛➀✛➌✑➎❜❾➍➏✴➀✛➏✑➉❹ ➐ ❹❺✌➑➓➒ ➂➔❼✌❽◗→➑ ❿✔→➒ ➁➣➂ ➃➅❞↔→➑ ↔❱↕ ↔→➒ ↔❏↕ ➎➙↔→➑ ↔❱➛ ↔→➒ ↔❏➛ ➉❹ ➜✆➝➟➞ ❿✞➠ ➝➟➡ ➢ r❏s❢t★✉✇✈②①❈➤ ➥■⑧✄⑦✣⑩★➦✯➧✯➨✞⑩★➦✴⑧✄⑨✐❶♠❸ →➑ ❻ ➩↕ ➑ ↔→➑❝➫ ➭ ↔❱↕ ⑥✌⑩✷⑧✙➯✗⑧✪➲✟⑩❀⑨✪⑧❆➯✗❶✛❷✎⑨✣➨❃➧✯❷✾➨♠➳ ↔→➑❃➫ ↔❏➛ ❽⑥✌⑩✷⑧✙➯✗⑧✪➲✟⑩❀⑨✪⑧✹➯✙❶✯❷✎⑨✣➨❃➧✯❷❱➨❃➳ r❏s❢t★✉✇✈②①✯➵ ④➸➦✯➧✯➺➓➻✾➧✛➨◗⑩✷❶✛❷➼❶❑❸❍➽➅ ❽ ↔→➑ ➫ ↔❏↕➁♦❽ ↔→➒ ➫ ↔❏↕➁➾➉❹ ➩ ↕ ➑ ➃➅❞↔→➑ ↔❏↕ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ↔→♦➘ ➫ ↔❏↕ ↔→♦➘❂➴ ➫ ↔❏↕ ↔→➴ ➫ ↔❏↕ ↔→♦➷➴ ➫ ↔❏↕ ↔→➷ ➫ ↔❱↕ ↔→➷✜➬ ➫ ↔❏↕ ↔→➣➬ ➫ ↔❏↕ ↔→➘➬ ➫ ↔❏↕ ↔→➑ ➫ ↔❱↕ ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ➉❹ ➂ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ❐ ❐ ❒✭❮ ➫✛❰❱Ï ❐ ❐ ❐ ❒✭❮ ➫✛❰Ï ❐ Ð ➫❈❰Ï ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ❰Ï ❮ r❏s❢t★✉✇✈②①❈Ñ ④➸➦✯➧✯➺➓➻✾➧✛➨◗⑩✷❶✛❷➼❶❑❸ ➽➅ ❽ ↔→➑ ➫ ↔❏➛ ➁▼❽↔→➒ ➫ ↔❏➛ ➁❏➉❹ ➩ ↕ ➑ ➃➅ ↔→➑ ↔❱➛ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ↔→➘ ➫ ↔❏➛ ↔→♦➘❂➴ ➫ ↔❱➛ ↔→♦➴ ➫ ↔❏➛ ↔→➷➴ ➫ ↔❏➛ ↔→✟➷ ➫ ↔❏➛ ↔→✟➷✜➬ ➫ ↔❏➛ ↔→➬ ➫ ↔❏➛ ↔→▼➘➬ ➫ ↔❱➛ ↔→➑ ➫ ↔❏➛ ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ➉❹ ➂ ➚➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➶ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➹ ❒✭❮ ➫✛❰❱Ï ❐ ❐ ❐ ❒✭❮ ➫✛❰Ï ❐ ❐ ❐ Ð ➫✛❰Ï ➮➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪➱ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ➪ ✃ ❰Ï ❮ r❏s❢t★✉✇✈②①❈Ò Ó➻✜Ô■Ô✹➧✯⑦✣Õ ➩↕ ➑ Ö
Nonzero entries of dentical to finite difference Actually, not quite identical; the FEM matrit is h times the finite difference matric. Obviously, one of the the advantages of the FEm to develop ap- triangulations Th of arbitrary dome eercised in this example. Our purpose rather is to get some sense of how the y;“ interact” to form A 3 Theoretical Analysis 3.1 General results Nor SLIDE 17 Recall llI=a(u, D)=(2(9)=/ Vu12dA Then ah is the projection of u on Xh The proof is identical to that for the one-dimensional case 1.2 H Norm SLIDE 18 H1(g2) Vul+02dA Then l Ju -wnI B: continuity constant(=1)
×✥Ø❋Ù✎Ú✪ÛÝÜ✔Ø■Û✪Ù❢Þ✗Ü✔ß✱ÛÝà❂Ø❈á Ü✗Ø✯â➔ã❇Ø❈á▼äå✾æ ß❀ç✜ÛÝÙ❋Þ✗ß✱èÝé✛ê✌Þ✗Ø✐ë❱Ù✎ß✳Þ✗Û➊ç✜ß✻ì❏ÛÝÜ✔ÛÝÙ✾è✄Û✖à✪í î✑ï✄ð✞ñ✾ò✯ó★ó➓ô✴õ✑ö❏÷✯ð✠ø✄ñ✜ù★ð❃ú❄ù✷û❈ú✪ö✾ð✞ù✷ï✗ò✛óýü✆ð★þ✾ú✐ÿ✁✄✂✆☎✹ò✯ð✞✝✣ù✠✟ ù☛✡✌☞✎✍❞ð✞ù✏☎❆ú✑✡✹ð★þ✾ú✓✒➣ö❱ù★ð❝ú û✯ù✔✭ú✕✝✔ú✪ö❏ï✙ú ☎✹ò✯ð✞✝✣ù✠✟✗✖✙✘✓✚✕✛Ýù✷÷✛ñ✜✡✣ó➓ô✴õ✹÷✛ö❏ú ÷✣✢✁ð★þ✾ú❞ð✩þ✎ú ò❈û✤✛✯ò✯ö❱ð❝ò✦✥❋ú✑✡❍÷✣✢✁ð✩þ✎ú ÿ✁✄✂★✧ ð❝÷➟û❈ú✕✛✴ú✪ó✻÷✪✩➙ò✪✩✬✫ ✩✭✝✗÷✕✟❋ù✏☎❆ò✛ð◗ù✷÷✛ö✮✡✯✚✙ò✰✡✪ú✙û✁÷✯ö✱✥❢ú✪ö❏ú✕✝✔ò✯ó●ð✲✝✣ù✷ò✛ö✗✥❈ñ✜ó✻ò✯ð✞ù✷÷✯ö✳✡✵✴å ÷✶✢➊ò✤✝✕✚✪ù★ð✲✝✗ò✤✝✣ô✁û❋÷✤☎✹ò✯ù★ö✳✡✯✧ ù☛✡❧ö❏÷✯ð ú✪✟❢ú✕✝✔ï✪ù☛✡✪ú✗û✁ù★ö ð✩þ✶ù☛✡✹ú✷✟✶ò✸☎✓✩✌ó✻ú✦✖✹✘❂ñ✺✝✻✩✌ñ✺✝✞✩✎÷✤✡✄ú✼✝✔ò✯ð✩þ✎ú✕✝■ù☛✡✐ð❃÷✽✥❋ú✪ð✾✡✪÷✤☎✹ú✿✡✪ú✄ö✳✡✄ú❄÷✣✢✆þ✎÷✤❀ ð✩þ✎ú ❁✁❂✹❃ù★ö✾ð❃ú✕✝✔ò❈ï✪ð❅❄✆ð❝÷✓✢✄÷✤✝❅☎ äå ✖ ❆ ❇❉❈❋❊✓●✯❍■❊✾❏✁❑✕▲✻▼❖◆◗P❙❘❋▼❖◆❅❚❱❯✁❑✑❯ ❲✾❳✶❨ ❩❭❬❫❪✓❬❵❴✳❛✁❜❞❝❡❬❫❢✜❣❤❜❥✐✮❢ ❦❫❧♥♠✭❧♥♠ ♦q♣sr✉t✤✈✎✇◗①③②✎t✤④ ⑤✭⑥✗⑦✏⑧❫⑨◗⑩✗❶ ❷➍Û✖è✪é✛ê✱ê❹❸❺❸❻❸ ❼✬❸❺❸❺❸ ✍❤❽❿❾✬➀❼✭➁✶❼✺➂❹➃➄❸ ❼➅❸ ✍➆✻➇✑➈❻➉✳➊ ➃➌➋➉ ❸ ➍✯❼➅❸ ✍➏➎ä❉➐ ➑✾➒✎ÛÝÙ ❸❺❸❺❸ ➓✺❸❺❸❻❸➔➃ ß✱Ù✜á →✬➣✸↔➔↕■➣ ❸❻❸❺❸ ➙❱➛➝➜å ❸❻❸❺❸ ➀ ➓➞➃❭➙❱➛➝➙ å ➂q➟ ➙ å ß✱à❂Þ✪➒✎Û ✩✭✝✗÷❥➠✪ú✙ï✄ð✞ù✷÷✯ö Ø✛á➏➙❞Ø❋Ù❋➡å ß✱Ù Þ✷➒✎Û✆ÛÝÙ✎Û✪Ü✪➢➔➤❛Ù✾Ø❈Ü✪➥✁í ➦þ✎ú✓✩✭✝✗÷Ý÷✶✢✆ù☛✡ ß✱ç✜ÛÝÙ❢Þ✔ß❀è✪é✛ê ð❝÷❛ð★þ✾ò✯ð✬✢✄÷✤✝✆ð✩þ✎ú■÷✯ö✌ú✑✫❑û✛ù✏☎❆ú✪ö✮✡✣ù✷÷✯ö✌ò✯ó✟ï✙ò✰✡✪ú✦✖ ❦❫❧♥♠✭❧☛➧ ➨➫➩✌①③②✎t✤④ ⑤✭⑥✗⑦✏⑧❫⑨◗⑩➔➭ ❷➍Û✖è✪é✛ê✱ê❹➯❅❼➅➯ ✍➆➇ ➈❻➉✳➊ ➃ ➋ ➉ ❸ ➍✯❼➅❸ ✍✾➲ ❼ ✍■➎ä➳➐ ➑✾➒✎ÛÝÙ ➯✑➓✺➯ ➆➇ ➈❻➉✎➊➸➵➻➺❵➼ ➲➾➽➚❤➪ ß✻Ù✎á →➣ ↔➔↕➣ ➯✑➙✽➛➝➜å ➯ ➆➇ ➈❺➉✳➊ ➟ ➚ æ è✄Ø✶Û✪Ü✙è✄ß❺➶✶ß✳Þ✣➤❛è✪Ø❈Ù✾à❑Þ✗é❈Ù❢Þ ➀✶➹➴➘ ➂❅➟ ➽ æ è✄Ø❋Ù❋Þ✗ß✻Ù✜➷✎ß✻Þ✣➤❄è✪Ø❈Ù✾à❑Þ✗é❈Ù❋Þ ➀ ➃ ➼ ➂✣í ➦þ✎ú✻✩✭✝✔÷✖÷✣✢❧ù☛✡ ß✱ç✜ÛÝÙ❢Þ✔ß❀è✪é✛ê ð❝÷❛ð★þ✾ò✯ð✭✢✄÷✤✝❧ð✩þ✎ú■÷✯ö✌ú✑✫❑û✛ù✏☎❆ú✪ö✮✡✣ù✷÷✯ö✌ò✯ó♦ï✗ò✤✡✪ú✦✖ ➬
1(T):叫rh) =Lyy,∈R erst co ua,∈ Lydml ay-mmy.mhg ll‖|≤ Chalk2(9) lelr(g)≤ Chaha() le2()≤ Chuah(g) Convergence rate and reg ThT farm af thT pnaafs ar. Try similan iy fact i-Tyticalg ta thT farms af thTamanlspay-iyg pnaafs iy ay T'spadT-im Tysiay. HawT. TngthTbauy-fan lu ThullH1(9) is yaw marl-iffizult ta - Tri t dyatT thT mTsh regularity plays ay apart ayt ralll grlquiray g samT'glyTial taals that wl will yat taktthTtimtta T Tap hTiT. NT. TithTITssgiy thT Iy-gthT'samTh-Iply- TyaTas iy ayT'spaT imlysiay is rlm. Trl WT aay af must rlplarf thT w yarm with thT brakly ry yarang thus p T7mittiyg ITss rlgulan,. HawT. Tngas iu-iaatT- TanliTngthTrlal issuTiy highTh spazT-imTysiays is gTamltry-iy-uTT siygularitiTs. Iy thTaba TwTThplait thT result thatgif E L,dmd ay- m is amy. ThgthTy y TTssarily u E edmd da similan ssumptiay is requir- fan thT-ual prabllm assa ziat- with th 1Stimatil Of munsTgT. Ty fan m yay my. Th u might bTiy dml d- Tply-iyg ay, i gbut it will yat bTthTsasTiy gly Tnal fan ayy, E Ly dmz Just as iy aun simplTThamplTiy ayf'spazl-imTysiay dy whish wrplazta TIta-istrabutiay laa- at a paiyt which is yat a ya-Tothus abtaiyiyg my 1nglyaT' iy thTTy Th aran a rathTnthay hu gif u is yat iycydmd g aay. ThgTyitdsay iy thT yarm will- Tgna- Tta ho fan samTo 1. This THfTat zay bTmitigatT- by a- apti. Il rdfiy Tim Tyt giy which wTplazT pragnlssi Tly smallTh nITnlyts y Tan thTsiygularityg mmplysatiyg fan thTsmallTn Thpay lyt with a smallTn aluTaf h; thTrT ant alsa marl saphistiratT- praz- urls that way bT appliT-. Iy samT prabllms thTsiygularitils-amiyatl hawT Ingiy alitaiy ath1n prabllmsg thT siygulanitils arlf'lss impart ayt- thT'my 1nglyal natT'may bf'slaw but thT aystayts arsa small that thT-Tgra-T pTrfarmayalis yat absTr T- Th tPt fan Try small h day- hTyzT. Tiy small Transl WT yatr that if thT bauy-ary I is samT smaath fuy ztiay dthat isg m yat palygayall g thly high Th-an- Th isaparam Tric farmulatiays will rla. Th thT optimal an- Tn af my. Trg IyzT. 3 Output Fu ctiy=IR eo dul+cogs=eocnl+co uan,∈ dml ay-mmy.mhg
➮✾➱✲✃ ❐❋❒✁❮✺❰✮Ï✶Ð➅Ñ✵Ò✶❒✁❮ÔÓ❡Õ❫Ö✜Ñ✵Ò✣❰✺Ö ×❫Ø☛Ù❵Ø♥Ú Û➫Ü✌Ý✳Þ➏ß❭à✄á✯â③ã✎ä✤åçæ è✭é✗ê✏ë❫ì◗í➔î ï✮ð✉ñ✵òôó àá✉õ✲ö✻÷✾ø➔ù✳ú✱ö➌ûðù✜ü➔ý✑þ➅ÿ ✁✂ ✄☎✂✁✝✆✟✞✡✠☞☛✍✌✎☛✑✏✓✒✍✔✂✕✗✖✙✘ ☛✑✄☎☛✑✏✛✚✜✔✁✕✢✖✓✆✣✞✡✠☞☛✑✌✤☛✥✏✒ ✔✂✕✢✖✦✘ ø✸ù✎ú ☛✑✄☎☛✥✧ ✒ ✔✂✕✗✖★✆✣✞✣✠á ☛✑✌✤☛✥✏✒ ✔✂✕✢✖✦✩ ✪✙✫ ✬✮✭✰✯✍✱✳✲ ✴✵✭✝✶✎✷✸✱✺✹✼✻✽✱✺✶✿✾❀✱❁✹❃❂✰✯✍✱❄❂☎✶✿❅✡✹❆✱❃✻✎❇✢❈❉❂☎✹❋❊●✯■❍❑❏▼▲❖◆P✯●❊◗✭✝✶✤❂☎❈❙❘ ❚❱❯ý❁❲ð➔ñ✜❳ ð❲P❨❯ý✳❩ñ✷ð✜ð❲■❬✯øñý ü➔ýñ✜❭ ❬●❪❳❪✂❫☛øñ ÿ✎❪❺ù❴❲♥ø➔û✑❨❵❪☛ú✺ý✦ù❀❨✜❪❺û✦ø✺❫✞ÿ✤❨ð ❨❯ý❁❲ð➔ñ✜❳❬ ð❲ ❨❯ý➞ûð➔ñ✪ñý❛❬●❩ ðù✎ú☎❪❻ù✢❜❝❩ñ✪ð✗ð❲■❬✓❪❻ù ðù✮ý❞❬❡❩✳ø➔û✕ýqú☎❪❳ý✕ù✢❬❡❪ðù❣❢✎❤ð❃✐ý✕ü✉ýñ ÿ❆❨❯ý❞❥ð❀❦ù✳ú❧❲ð➔ñ ☛✍✌♥♠ ♦❣♣ ✌✤☛✥✏❱✚▼✔✂✕✢✖ ❪✁❬✯ùð❃✐q❳❞ð➔ñý ú✸❪sr✼û❦❫s❨❵❨ð ú✺ýñ ❪❺ü➔ý➝õ♥ùð❨✪ý☞❨❯ý ❳ý❛❬❯◗ñý✼❜❦❫☛øñ ❪✂❨❭ ❩✸❫☛ø❭❬✯ø➔ù ❪❳❩ ð➔ñ❨✪ø➔ùt❨ ñ✷ð❫❻ý✰÷❅ÿ ñý✼✉❦❪ ñ ❪❻ù✸❜✈❬ð❆❳ý✇❜➔ý✦ù✮ýñø✺❫①❨ ð✜ð❫②❬✛❨❯ø❃❨ ✐ý ✐❪✂❫✁❫❫ùð❨✛❨✪ø❆③➔ý❞❨❯ý✇❨❡❪❳ý✇❨ð ú✺ý✦ü➔ý✥❫ð❩ ❯ýñý❀❢ ✪ý✕ü➔ýñ❨❯ý✥❫❺ý✼❬✜❬✦ÿ✸❪❻ù✳❨❯ý❖ý✕ù✎ú➅ÿ✸❨❯ý✵❬✪ø❳ý ✠ ú✺ý✼❩✎ý✦ù✳ú✺ý✕ù✎û✑ý✌ø❀❬✛❪❻ù ðù✮ý❝❬❡❩✳ø➔û✕ý ú☎❪❳ý✕ù✢❬❡❪ðù❄❪✁❬ ñý✦ûðü➔ýñý✦ú❣❢ ④ý û✕ø✸ù ð❲✯ûð❆❦✮ñ ❬✷ý ñý✥❩✸❫☛ø➔û✕ý⑤❨❯ý Û✹á ùð✉ñ❡❳⑥✐❪✂❨❯ ❨❯ý✮❥ñ✷ð③➔ý✕ù Ûôá ùð➔ñ✜❳ÿ✙❨❯✝❦❬ ❩✭ýñ✜❳❪✂❨●❨✜❪❻ù✢❜⑦❫❺ý✼❬✜❬ ñý✼❜❦❫☛øñ✾ò ❢✤❤ð❃✐ý✦ü➔ýñ ÿ✜ø❆❬★❪❻ù✎ú☎❪❺û✦ø❃❨✪ý✦ú✼ýøñ ❫✂❪❺ýñ ÿ❀❨❯ý ñý✦ø✺❫⑧❪②❬❡❬❦ý❖❪❻ù ❯❪✂❜❯ýñ ❬❡❩✳ø➔û✕ý✓ú☎❪❳ý✦ù✢❬●❪ðù✗❬✎❪✁❬✿❜➔ýð❀❳ý✑❨ ñ✜❭❀⑨❪❺ù✳ú❦û✕ý✦ú❵❬●❪❺ù✸❜❦❫❺øñ❪✂❨❡❪❺ý✼❬✼❢❶⑩❥ù❵❨❯ý✵ø✺❥ ðü➔ý ✐ý✻ý✑þ☎❩✸❫ð❪✂❨✎❨❯ý ñý✼❬❦❫s❨❷❨❯ø✺❨✦ÿ✝❪s❲ ò✹ó à❹á õ✞ö✻÷ ø➔ù✳ú✼ö✡❪✁❬➸ûðù✜ü➔ý✑þ➅ÿ❃❨❯ý✕ù❱ù✮ý✦û✕ý✼❬✜❬✷øñ❪✁❫❭ ✌ ó Ûôá õ✞ö✻÷✵õ✲ø❝❬●❪❳❪✂❫☛øñ ø❆❬✜❬❦✸❳❩☎❨❡❪ðù⑤❪②❬ ñý❛✉❦❪ñý✦ú✳❲ð✉ñ ❨❯ý✿ú❦ø✺❫❸❩ñ✪ð❥✸❫❺ý❳ ø❆❬✜❬ð û✥❪❺ø✺❨✷ýú ✐❪s❨❯ ❨❯ý àá ý✼❬●❨❡❪❳ø❃❨✷ý✰÷✍❢ ❹❲sûð❆❦✮ñ ❬✷ý➔ÿ✗ý✕ü➔ý✦ù❧❲ð✉ñ ö ùðù✎ûðù✜ü✉ý✑þ ✌ ❳❪✁❜❯❨❱❥✎ý❞❪❺ù Û✹á õ✲ö✻÷qõ✲ú✺ý✥❩✭ý✕ù✳ú✸❪❻ù✸❜ ðù ò ÷✑ÿ☎❥❦❨✛❪s❨ ✐❪✁❫✂❫❵ùð❨✛❥✎ý❞❨❯ý❖û✕ø❀❬✶ý❞❪❺ù❄❜➔ý✕ù✳ýñø✺❫⑧❲ð➔ñ ø➔ù❭✱òôó à❹á õ✲ö✻÷✍❢ ❺❀❦❬●❨qø❆❬✙❪❻ù ð❆❦✮ñ ❬●❪❳❩✸❫❻ý✯ý✑þ✮ø❳❩✸❫❺ý❝❪❺ù ðù✳ý✵❬●❩✎ø➔û✑ý✯ú☎❪❳ý✦ù✢❬●❪ðù◗õ❙❪❺ù ✐✛❯❪☛û❯⑤✐ý✵❩✸❫❺ø✉û✑ý✿ø ú✺ý✼❫s❨ ø✓ú☎❪②❬◗❨ ñ❪✁❥❦❨✜❪ðù✇❫ðø➔ú ø❃❨sø✛❩ð❪❻ùt❨ ✐✛❯❪❺û❯ ❪②❬❵ùð❨➏ø✻ùð ú✺ý✉ÿ❛❨❯✝❦❬ ð❥☎❨✪ø❆❪❻ù✢❪❻ù✸❜❤ûðù✜ü✉ýñ❜➔ý✕ù✎û✑ý ❪❺ù✵❨❯ý✻ý✕ù✮ýñ❜❭ ùð➔ñ✜❳ ø❆❬❷❻✠ ñø❃❨❯ýñ ❨❯ø✸ù ✠÷✑ÿ❆❪✂❲ ✌ ❪②❬✁ùð❨✎❪❺ù Ûôá õ✲ö✻÷✑ÿ✉ûðù✜ü➔ýñ❜➔ý✦ù✳û✑ýqõ■❬✪ø❭÷ ❪❺ù✈❨❯ý Û➝Ü ùð➔ñ✜❳❑✐❪✁❫✁❫✭ú✺ý✼❜ñø➔ú✺ý❱❨ ð ✠✗❼ ❲ð➔ñ ❬ð❀❳ý❖❽❿❾➁➀❆❢ ❚❱❯❪②❬❹ý✥➂✭ý û✍❨✾û✕ø✸ù☞❥✎ý ❳❪✂❨❡❪✁❜✉ø✺❨✷ýú ❥❭ õ✲ø➔ú✮ø❆❩☎❨❡❪❺ü➔ý✰÷ ñý✑➃✳ù✳ý❳ý✕ùt❨✦ÿt❪❺ù ✐✛❯❪☛û❯❧✐ý✙❩✸❫❺ø✉û✑ýP❩ñ✷ð❜ñý✼❬✜❬●❪❺ü➔ý✼❫❭ ❬❳ø✺❫✁❫❻ýñ ý✼❫❻ý❳ý✦ùt❨✜❬ ù✳ý✦øñ ❨❯ý❵❬❡❪❻ù✢❜❦❫☛øñ ❪s❨❭ÿ✎ûð❆❳❩✎ý✦ù✢❬✪ø❃❨❡❪❺ù✸❜☞❲ð➔ñ ❨❯ý♥❬❳ø✺❫✁❫❺ýñ ý✕þ☎❩ðù✮ý✕ùt❨ ✐❪✂❨❯ ø☞❬❳ø✺❫✁❫❻ýñ ü✤ø✺❫❦ý ð❲ ✠❶✘ ❨❯ýñý✽øñý❞ø❆❫✁❬ð✳❳✿ð✉ñý✈❬ð❩❯❪✁❬●❨❡❪☛û✕ø✺❨✷ý✦ú➄❩ñ✪ð û✑ý✦ú❦✮ñý❛❬✦❨❯ø✺❨✌û✕ø✸ù✮❥✭ý✼ø✺❩✸❩✸❫✁❪❺ý✦ú✰❢♥⑩❥ù➅❬ð❀❳ý ❩ñ✪ð❥✸❫❺ý❳❬⑦❨❯ý❄❬❡❪❺ù✸❜❦❫☛øñ ❪✂❨❡❪❺ý✼❬✿úð❆❳❪❺ù✳ø❃❨✪ý ✘ ❯✮ð❃✐ý✕ü➔ýñ ÿ✤❪❺ù➴û✕ýñ❨✪ø❆❪❻ù ð❨❯ýñ ❩ñ✪ð❥✸❫❺ý❳❬✦ÿ✎❨❯ý ❬❡❪❻ù✸❜❦❫☛øñ ❪s❨✜❪❻ý❛❬ øñý❵❫❺ý✼❬✜❬✇❪❳❩ð✉ñ❨ ø✸ùt❨❞➆➇❨❯ý✼ûðù✜ü✉ýñ❜➔ý✕ù✎û✑ý ñø✺❨✷ý ❳ø❭ ❥✭ý☞❬●❫ð❃✐ÿ❣❥❦❨❝❨❯ý ûðù✢❬●❨✪ø✸ùt❨▼❬ øñý✛❬ð ❬❳ø✺❫✁❫☎❨❯ø✺❨✿❨❯ý✵ú✺ý✥❜ñø✉ú✺ý✦ú❵❩✭ýñ❲ð➔ñ✜❳ø✸ù✳û✕ý✓❪②❬ ùð❨ ð❥✗❬✶ýñü✉ý✦ú✯ý✑þ✮û✑ý✼❩☎❨✿❲ð➔ñ ü➔ýñ❡❭ ❬❳ø✺❫✁❫ ✠ õ♥ø➔ù✳ú ❯ý✕ù✎û✑ý ü➔ýñ❡❭ ❬❳ø✺❫✁❫➅ýñ✪ñ✷ð✉ñ ❬ ÷✍❢ ④ý ùð❨✪ý❿❨❯ø❃❨➈❪✂❲✵❨❯ý➄❥ ð❆❦ù✎ú✮øñ✜❭➊➉ ❪②❬➈❬ð❆❳ý➄❬❳❞ð✗ð❨❯ ❲❦ù✳û✍❨✜❪ðù✙õ❙❨❯ø✺❨➈❪✁❬✦ÿ➞ö➋❪✁❬ ùð❨❵❩ ð❫❭❜ðù✎ø✺❫✏÷❅ÿ✤❨❯ý✕ù ❯❪✂❜❯ýñ❡➌✞ð✉ñú✮ýñ ❪②❬ð❩✳øñø❳ý✥❨ ñ ❪❺û✈❲ð➔ñ✜❳✵❦❫☛ø❃❨✜❪ðù✢❬ ✐❪✁❫✂❫ ñý✦ûðü➔ýñ ❨❯ý ð❩✸❨❡❪❳ø❆❫ ð➔ñú✺ýñ✻ð❲sûðù✜ü➔ýñ❜✉ý✕ù✳û✕ý❆❢ ×❫Ø☛Ù❵Ø☛Ù ➍❧➎❶➏✼➐❸➎✽➏✈➑❶➎sÞ✽➒❆➏✼➓✲ã✎ÞsÝ✸➔♥æ è✭é✗ê✏ë❫ì✮→☎➣ ↔ý û✕ø✺❫✁❫❶↕✦➙✟➛✥➜✓õ✌÷❣➝➟➞✑➜✓ÿ⑧↕ ♣ ➙✡➛✥➜✓õ✌♣ ÷❶➝➟➞✑➜✛❢ ï✮ð✉ñ✵òôó à❹á õ✲ö✻÷✾ø➔ù✳ú✱ö➌ûðù✜ü➔ý✑þ➅ÿ ➠
