A hybri high-orer locking-free metho for linear elasticity on general meshes Daniele Antonio Di Pietro, Alexanre Ern o cite this version: Daniele Antonio Di Pietro, Alexanre Ern. A hybri high-orer locking-free metho for linear elasticity on general meshes. Computer Methos in Applie Mechanics an Engineering, Elsevier, 2015, 283, pp.1-21. <10.1016/j.cma.2014.09.009>. <hal-00979435v2> HAL I: hal-00979435 https://hal.archives-ouvertes.fr/hal-00979435v2 Submitte on 23 Jul 2014 HAL is a multi-isciplinary open access archive for the eposit an issemination of scientific research ocuments, whether they are publishe or not. he ocuments may come from teaching an research institutions in rance or abroa, or from public or private research centers. L archive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recherche, publiés ou non, émanant es établissements enseignement et e recherche français ou étrangers, es laboratoires publics ou privés.
2 r r r r t r r st t2 r s s P tr 1 r r : rs t2 t r t r r rs té P r s st P s P s s s r r r 2 str t s r tr r2 r r r t r r st t2 t r s r s t r r t s t s2 tr s t t s2st tr 1 t t st r s r r t r 2 s r tr r2 r r k ě 1 t s s s t t t r s s s t st r r s 2 r q r s r s r r s t 2 r 2 r s s 2 s t r str t t s2 tr r t r s s t r s t r s r 2 s 1 s r s s r t r s ss s s t s r t rs r r st 3 t r r r rr r st t s r r r t r 2 r r t L 2 r t s t t t r r t s r r pk`1q pk`2q r s t 2 r s t s t s r s s t r t r s ts r r r 2 t P st s t t st r 2 s s 2 r s r st t2 r s s r tr r2 r r r t s tr t t Ω Ă R P t2,3u t t 2 r 2 r s r t s tr r st t2 r σ f Ω σ 2µ s u `λp uqi Ω, u 0 BΩ, t µ ą 0 λ ě 0 s r é ts s t t s2 tr rt t r t r t r s rt t s2 tr r t t t r s s r s r t r2 t s t s t r s t2 r tr t r rr s t r : r r s r
r r2 t s r X Ă Ω t 2 p, q X } } X r s t 2 t st r r r t r L 2 pxq t t t t t t 1 s tt X Ω s r t t s s r L 2 pxq L 2 pxq ˆ t f P L 2 pωq t r t r s sts u P U 0 : H0 1pΩq s t t r v P U 0 p2µ s u, s vq ` pλ u, vq pf,vq. t s t t t r t r 1 t r t r ss t λ Ñ `8 r q r s t s r t s t t r t 2 r r s t tr r r t r s s r r r 1 t s r t s t s t s t s ss t 2 ts t t r r t r t t s t s s s t t r r rt2 t t r r t2 r s t t s t ts r s q t t r r rt2 s t r rr r st t s t t r r t r s t t λ t r r r t s r rt2 s t r t r s t r s t r t r 3 1 rt t r 2 r t t s s ss t 3 t s t s r t s t r ss t r s st sq r s s s s rs t t r s 2 t st r s r r s s r t 2 rs r rt s t t t t 2s s s r t 3 t t s t t 2 t r r s s ss 2 t r 2 r 2 r ts r t r s t s t 1t t t st r r t rã 2r2 3 r t s t t rã r 1 r st t2 t r P tr r tr s r 3 t t st r r r 3 1 rt s rr s 2 t t s s r t t ss t t s r t s t r rr r r r s t s r 1 2 ts r str t r r t 2 r tr r2 r r t s r s s r r s tt t t r rt r t r rã r 33 r r t t rs tr r rt t t r r r st t2 t r rt 1 t s t s r s r t s r tr r2 r r t r q s r ss r st t2 r s s s s P t2, 3u s t r t s r s r s r 2 r k ě 1 r st rt t s t s r s r s r s t r 2 s t s ts s t r k s t t t r s t st s tr s r t s2 tr r t r r str t r t rs t s t 1 s r s s s t s 2 t t t s r t r r t r s t s s t t r r rt2 s st sq r s st 3 t t t 2 r s t t t s s s2st s ss s s t r str t st 3 t r t rs s s st s t r t t s t t r s s t str 2 r ss t r2 t s t r2 s t s s r t r t r s t 2 st t s t r t s2st s t s2st r t r 2 s r r k ě 1 t s s r st t st r r s 2 r q r s r s r r s s s s
s r t 3 t s s r t r s s s s r r s tt t t st 2 rs t t s r r r s 3 r 2 r s t s r t r r t r s t r t t t 3 t t r s t t t t rs 2 r r r s s t t s s r str t r t rs s t t r r t rt t r s t t s rst t s s r str t t s2 tr r t r t st t s r 2 s t t t r 2 s 2 t t s s2 tr r t r t r s s s t r r r s t r str t r t r st t t s t r t t s st t t r 2 t s t st 3 t r r s s s 2 s str t r r s t st 3 t r s t t s r s r t2 2 s t r r r r 1 t r rt s s t t t t s s 2 s s t r s r r st t2 r r t r t rs r r s s t r str t s s r r r s ss r s t t 1 ts r s s t ts t r s r s t r r t t r 2 t s2 tr s t t s2st tr 1 t str 2 s2 tr str str ss t s rs s t t 2 t s t s s t s s 2 s t st s st rs t s s s r t 2 s s t 1 s 2 t r s s s s r s t s t s t t r str s t rr r st t s r r pk ` 1q pk ` 2q r s t 2 r 2 r s s r t 2 s t rst r s r t t t r str t r t rs t s r r r s ts t t t t ss t st q 2 s r t r s t t t st s t s2st s t r s s r s t t s r s t t r st t s t t t s rt 1 r s t s t 2 r t 3 r s t s s r s r s t 2 st r st r ss r t r 2 r s s t r r s t r s r 3 s s t r t t ss s s q s t s r t s s s s r s ts t tr t s t r t t s t s t s t s2 tr r t r r str t r t rs s t 2 t s t 2 t 2 r rt s t s r t rs t t st 3 s r t r r st t t s r t r s ts s ss t r r t rr r 2s s r r rr r st t s r r pk ` 1q t r 2 r pk ` 2q t L 2 r r s t s t s t s ss t t s ts r s t r 1 s 2 s s r r t r t r s ts t t 2 t r s t t r s P st tt ss s s q s t s s t r 2 r t t ss s s q t r t H Ă R` t t s t s s 3 s 0 s ts q t t
s s q s p h q hph r r h P H h s t t t2 s t 2 r ts r s h t u s t t Ω Ť P h h max Ph h h st s r t t r 2 r r s t s s t Ω s t s s t p 1q s s r t r t r 1 st 1, 2 P h s t t B 1 X B 2 s t r r t r 1 sts P h s t t B X BΩ s r2 s t t r s s t 2 i h t s t r2 s 2 b h t h : i h Y b h t r P h s t 2 h r P h : t P h Ă B u t s t s t s 2 t r2 r P t 2 n t r t t t m s s s r m P t0...u s t 2 m t ss s s q s 2 t t t s s q p h q hph s s s r h P H h ts t s s s h s t t 2 r s h s s s t r s h t r 1 sts r r ą 0 t s r r t2 r t r t h s t t t t s r h P H s 1 S P h t r h S r s r S h S ď r S s r r t2 r h P H P h S P h s t t S Ă h ď h S t t r r t2 t s s r s s t ss s s q r t s tr r s ts t t t s r r t r h P H P h P h s r t h t s s t t 2h ď h ď h. r r t t r 1 sts t r N B ě p `1q s t t t 1 r s t s @h P H, maxcarp q ď N B. P h t s t r t s a À b t q t2 a ď Cb t C ą 0 t h µ λ t ss 2 t s r r t2 r t r r t é s r t rs t st t r t q t s s t st t t r rt s t t t r ss t s r s ts t s s t r s s r s ts ss s s q s r 1 st r rs C tr C tr,c t t h s t t t s r t tr q t s r P h t }v} ď C tr h 1{2 }v} @v P P l p q, @ P, }v} B ď C tr,c`h 1 }v}2 `h } v} 2 1{2 @v P H 1 p q, r P l p q t s t s s 2 t r str t s t r t 2 s t t r ď l s t t r t t r s ts t r t t t r 1 sts r r C app t t h s t t r P h t 2 π l t L2 rt r t r P l p q t q t2 s r P h s P t1,...,pl `1qu v P H s p q v π l v H m p q `h 1 {2 v πl v H m pb q ď C app h s m v H s p q @m P t0,...,ps 1qu.
P ré q t2 s r P h v P H 1 p q s t t ş v 0 }v} ď C P h } v}, r C P π 1 r 1 ts r r r t s s C P st t t r s t s r r t2 r t r r P h s t " ż ż * Up q : v P H 1 p q v 0 ss v 0, r ss t s t s s2 tr rt t r t r t r s r q t2 s r P h v P Up q } v} ď C K,2 } s v}, r C K,2 ą 0 s t h st t t r s t s r r t2 r t r r t t s s r t t r P h v P Up q }v} ď C K h } s v}, t C K C K,2 C P s s v P H 1 p q t r 1 sts Γ Ă B t Γ 1 0 s t t v Γ 0 r str t r t rs t s s t t s t r t t s2 tr r t r r str t r t rs r s r r t t 2 r k ě 0 1 r P h t s s s # + ą U k : Pk p q ˆ P k 1p q. r v P U k s t t t v pv,pv q P q r s r r str t r r r t I k : Up q : H1 p q Ñ U k s s t t r v P Up q Iv k pπv, k pπvq k P q. 2 tr r t t P h rst s t r str t r t r r k : Uk Ñ Pk`1 p q r s v pv, pv q P q P U k t rk v P Pk`1 p q s 2 s t s r tr t r r w P P k`1 p q p s prvq, k s wq p s v, s wq ` ÿ pv v, s wn q.
k = 1 k = 2 r r 2 r s r U k r r s r 2 t st t s t ˆk `1` s r s2st s s 3 r t q r t r str k `1 t t s2st s s 3 2 3 s t2 st s r t t t t t r t s s s t t st t w s r 2 t s t t s q 2 r s r t r 2 t ts r kv s t r s r t ż ż ż rv k v, ss prvq k ÿ ż 1 2 pn bv v bn q, s r s t t pr kik v vq P Up q r v P Uk s s t r t t s r t s2 tr r t r t r E k : Uk Ñ sp k`1 p q s t t Ev k : s prvq. k t r t t r w P P k`1 p q pe k v, s wq : p s v, s wq ` ÿ pv v, s wn q pv, s wq ` ÿ pv, s wn q, r t s q t s 2 t r t 2 rts s t t r t 2 t s t r str t r t r 2 s r 1 t r t r r kik : Up q Ñ P k`1 p q t s t s t 2 s st 2 t r 1 t r rt s t s r t r s t t s r 1 t r rt s r r k Ik t k ě 0 P h r 1 sts r r C ą 0 k t t h s t t r v P H k`2 p q }r k I k v v} `h 1 {2 }rk I k v v} B `h } s pri k v k vq} `h 3 {2 } spri k v k vq} B ď Ch k`2 }v} H k`2 p q. Pr t v P H k`2 p q s t t s r k Ik r t t
r w P P k`1 p q p s r k I k v, s wq pπ k v, s wq ` ÿ pπ k v, s wn q pv, s wq ` ÿ pv, s wn q, s s w P P k 1 p q Ă P k p q p s wq n P P k 1 p q t r t 2 rts t r t s 2 s p s r k I k v s v, s wq 0 rt t2 t s t t @w P P k`1 p q. } s pri k v k vq} inf } s pw vq} À h k`1 p q }v} H k`2 p q, wpp k`1 r t st q t2 s r t r 1 t r rt2 π k`1 t s k `2 m 1 t s r 2 t t r kik v v P Up q rst s r t t t ş ş rk Ik v πk v ş v t ż ss pri k vq k ÿ ż 1 P 2 pn bπv k πvbn k q ÿ ż ż 1 2 pn bv vbn q ss v. s r s t s r s q t2 t t r t r t t }r k I k v v} À h } s pr k I k v vq} À h k`2 }v} H k`2 p q. s t t s tr q t2 2 r s q t2 t t r t r t t h }r k I k v v} 2 B À }rk I k v v} 2 `h 2 } s pr k I k v vq} 2 À h 2pk`1q }v} 2 H k`2 p q. 2 t h 3 {2 } spr kik v vq} B s t 2 tr π k sv s t r s t tr q t2 t r 1 t r rt2 π k t s t s v t s k ` 1 m 0 t s r t tr q t2 t carp q t s r r t2 r rt2 t t t t s r kik v P Pk p qˆ s t t }π k sv s r kik v} ď } s pv r kik vq} r t P h s r t r r t r D k : Uk Ñ Pk p q s s t t r v pv, pv q P q P U k q P Pk p q pdv,qq k : pv, qq ` ÿ pv n,qq p v,qq ` ÿ ppv v q n,qq. 2 t r 2s s s t t r r rt2
Pr s t t r rt2 r s r t r r t r t k ě 0 t P h r t s Up q L 2 p q I k U k D k π k P k p q Pr t P h t v P Up q s t v : I kv r t t r q P Pk p q pπ k p vq,qq p v,qq p q,vq ` ÿ pq,v n q p q,πvq k ` ÿ pq,πv n k q p q,v q ` ÿ pq,v n q pdv,qq k, r s t t q P P k 1 p q Ă P k p q q P P k 1p q s r t r s ss t s s t tr t r r s t s r t r st s ts s ss t s ss t t k ě 1 r r s t P h t r r U k ˆUk s t t r v,w P Uk a pv,wq : 2µpE k v,e k wq `λpd k v,d k wq ` p2µqs pv,wq, t st 3 t r r s pv,wq : ÿ h 1 pπk prv k v q,π k prw k w qq, r r v pv, pv q P q P U k Rk v P Pk`1 p q s s t t R k v : v ` pr k v π k r k vq. t r r t s s t r r t s r r r rr t t t s v r r t r str t r k v t t t t s rr t s t t s r r t s r k ě 1 st 3 t r r s tr s st sq r s t2 t L 2 rt r t P k 1 p q t r t v pr kvq s t2 s t r t s r st t2 t t s t t t s r r r s t s2 tr r t r r str t r t rs t rr r 2s s s rt t r s t s st t t r tr t s r t str s r s U k r v P Uk }v} 2 ε, : } sv } 2 ` v 2 ε,b, v 2 ε,b : ÿ h 1 }v v } 2.
t 3 t ss k ě 1 r s r r η ą 0 t h µ λ s t t r P h v P U k t st t2 r rt2 s η}v} 2 ε, ď }E k v} 2 `s pv,vq ď η 1 }v} 2 ε,. r r r v P H k`2 p q t r 1 t r rt2 s s pi k v,i k vq 1 {2 À h k`1 }v} H k`2 p q. Pr t P h t v P U k r t rst q t2 w v r t t } s v } 2 pe k v, s v q ` ÿ pv v, s v n q ď }E k v} 2 ` 1 2 } sv } 2 `N B C 2 tr v 2 ε,b, r s t 2 r3 q t s 2 t s r t tr q t2 r t st t r t r t s s r s t t 2 r P } s v } 2 À }E k v} 2 ` v 2 ε,b. h 1{2 }v v } ď h 1{2 }v π k R k v} `h 1{2 }π k R k v v } h 1{2 }π k pv R k vq} `h 1{2 }π k pr k v v q} h 1{2 }π k pv R k vq} `C tr h 1 }rk v π k r k v} r s t tr q t2 t rst t t t t v P P k 1 p q v P P k 1 p q t s t t t t π k s r t r t t R k t s r t tr q t2 t t r r 2 t w P H1 p q r t w RM 1 pş wq ` 1 pş sswqpx x q r x t s t r2 t r s r t t π kw RM w RM s k ě 1 r t t }w π k w} }pw w RM q π k pw w RM q} ď }w w RM } À h } s w}, r s t t t t π k s L2 r t r r s q t2 s pw w RM q P Up q 2 t s t w r k v s t s r r t2 r rt2 r t t r k t st t r t r t s st t s C h 1 tr }rk v πr k v} k ď 2 C tr C K }Ev} k, s t t h 1{2 }v v } À h 1{2 }π k pv R k vq} ` }E k v}. q r s r P s t carp q s t v 2 ε,b À s pv,vq ` }E k v} 2. rst q t2 t 2 s r r t s q t2 s s s r r ts s tt r t s r t2
t s r r v P H k`2 p q s r t t r P h P h 1{2 }π k pr k I k v π k vq} ď h 1{2 }R k I k v v} h 1{2 }pr k I k v vq π k pr k I k v vq} À h 1{2 }r k I k v v} `C tr h 1 }rk I k v v} À h k`1 }v} H k`2 p q, r s t t t t π k s r t r t t t s s I kv t rst t t Rk t s t tr q t2 t s r t tr q t2 t t t t π k s r t r t t r t r 1 t r rt2 r kik t s r r t2 r rt2 t rt t t s t r 2 s r t carp q r r s r t r r s r s s st r t t t s 2 s ss 2 s s s t 2 t s t t r s 2 # + # + ą ą U k h : P k p q ˆ P k 1p q. P h P h r v h P U k h s t t t v h ppv q Ph, pv q Ph q r P h t 2 L t r str t r t r t t s t s U k h t t rr s s U k r v h P U k h L v h pv, pv q P q P U k s r t r2 t t s t r 1 t 2 t s tt t r2 s s t ) U k h,0!v : h ppv q Ph, pv q Ph q P U k h v 0 @ P h b, r r s a h s h r ss s s s r v h,w h P U k h a h pv h,w h q : ÿ ÿ a pl v h,l w h q, s h pv h,w h q : s pl v h,l w h q. P h P h t r s s r t 3 2 s t r r l h U k h s t t r v h P U k h l h pv h q : ÿ ż f v. P h s r t r r s u h P U k h,0 s t t r v h P U k h,0 a h pu h,v h q l h pv h q. str s r s s t t r v h P U k h }v h } 2 ε,h : ÿ P h }L v h } 2 ε,.
Pr s t r } } ε,h } } ε,h 2 s r U k h,0 Pr t s s t s t t r v h P U k h,0 }v h} ε,h 0 2 v 0 r P h v 0 r P h st rt 2 s r t t }v h } ε,h 0 s t t @ P h, s v 0 v v 0 @ P h. r r2 t P h t P X b h s v 0 t t U k h,0 r t t v 0 t s v 0 r s q t2 s v 0 v 1 0 r 1 P zt u s v v 1 0 t t r t r t 1t 2 r ts t rr s s t r s t s r t r t t ts s s t r r2 s ss ss k ě 1 r v h P U k h t q t2 s η}v h } 2 ε,h ď ÿ! ) }EL k v h } 2 `s pl v h,l v h q ď η 1 }v h } 2 ε,h, P h t η r s t r s s q r s s Pr q t s r 2 s r P h s ss t s r t 1 r Pr s t r } } r s s r t r q t s r s s t s s rt t r r tr q t2 r t t r r t str r r s t t } } ε,h r r v h P U k h,0 }v h} ε,h Á C}v h } t r r C ą 0 t h v h r str t r t t s s v h s pv h q v r P h rr r 2s s s rr r st t t u P U 0 u h P U k h,0 t t q s t s t r s t 2 t rr r u h pu h r pu h P U k h,0 s s t t pu h ppπ kuq P h, pπ kuq P h q s r t t pu h s t r t 1 t s t u s t r t s s t L pu h I kpu q s r t rr r t r 2 r s t t r v h P U k h }v h } 2 en,h : a hpv h,v h q. t t r s r t t }v h } 2 en,h ě p2µηq}v h} 2 ε,h t s r t t r r t2 t 1 t s t s r s s s H m p h q r s s t t r m q t t r } } 2 H m p h q ř P h } } 2 H m p q r r ss k ě 1 t t r r t2 u P H k`2 p h q u P H k`1 p h q t r 1 sts r r C ą 0 t h µ λ s t t 2µ}u} p2µq 1 {2 }u h pu h } en,h ď Ch k`1 H k`2 p h q `λ} u} H k`1 p h q.
r r st t r 2 Ω 1 t s r t t µ}u} H 2 pωq `λ} u} H 1 pωq ď C µ }f}, r C µ ą 0 t s r r Ω µ t t λ r r 2 r k ě 1 t r r t2 s t µ}u} H k`2 pωq `λ} u} H k`1 pωq ď C µ }f} H k pωq Pr r v h P U k h,0 s r t t }v h } 2 en,h a hpv h,v h q ď $ & }v h } en,h ě p2µηq 1 {2 }v h } ε,h r t t p2µηq 1 {2 }v h } en,h ď % sup w h PU k h,0, a h pv h,w h q. }w h } ε,h - ˆ }v h} ε,h. sup a h pv h,w h q. w h PU k h,0,}w h} ε,h 1 2 t s q t2 t t rr r pu h pu h q s 2 s p2µηq 1 {2 }u h pu h } en,h ď sup E h pw h q, v h PU k h,0,}w h} ε,h 1 t s st 2 rr r E h pw h q : l h pw h q a h ppu h,w h q E h pw h q r r w h P U k h,0 s t t }w h } ε,h 1 t t f σ Ω t r t 2 rts t s r t t l h pw h q ÿ # 2µp s u, s w q `λp u, w q ÿ + pσn,w w q, P h r s t t t2 t r str ss t t t r s t t r t w 0 r P h b t r t t ř P h ř pσn,w q 0 w qu : r kl pu h r kik pu q t t E kl w h r P h r t t pel k pu h,el k w h q p s qu, s w q ` ÿ p s qu n,w w q. r 2 t q : π k p uq t t Dk L w h r P h r t t r r rt2 r I k r t t pd k L pu h,d k L w h q pπ k p uq,d k L w h q p u, w q ` ÿ pπ k p uq, pw w q n q, s w P P k 1 p q Ă P k p q s t r t rr s t r s t 1 r ss E h pw h q r t t E h pw h q ÿ # 2µ p s pu qu q, s w q ` ÿ + p s pu qu qn,w w q P h ÿ λpp u π k p uqqn,w w q p2µqs h ppu h,w h q : 1 ` 2 ` 3.
st t 1 s t 2 r3 q t2 t r 1 t r rt2 r kik t s r r t2 r rt s t r t t 1 À 2µh k`1 }u} H k`2 p h q }w h} ε,h. Pr s r 2 r 2 s t r 1 t r rt2 π k 2 s 2 À λh k`1 } u} H k`1 p h q}w h } ε,h. st t 3 r r t s2 tr2 s t t2 s h t t 3 ď p2µqs h ppu h, pu h q 1 {2 s h pw h,w h q 1 {2 À 2µh k`1 }u} H k`2 p h q }w h} ε,h, r s t st t t rst t r t s t st t t s t r s s q t s t st t t r t s s t r s t r r 2 rr r st t r s u t t r s2 tr r t r str t E k h u h s t t E k h u h E k L u h r P h r r2 st t s2 tr r t r str t r t ss t s r t s 2µ}u} p2µq} s u Eh k u h} ď Ch k`1 H k`2 p h q `λ} u} H k`1 p h q, r C ą 0 s r r t h µ λ Pr r P h t tr q t2 2 s } s u E k L u h } ď } s pu r k I k uq} ` }E k L ppu h u h q}, s s pr k Ik uq E k L pu h rst t r t r t s s s t s s r L 2 rr r st t r t s t t s s t t } } r t s t rr r ss t r r t2 t r r g P L 2 pωq t q s t ς g Ω, ς 2µ s z `λp zqi Ω, z 0 BΩ, s t s s t r r st t µ}z} H 2 pωq `λ} z} H 1 pωq ď C µ }g}. rst s r t s t rr r e h s t t e h : u π k u P Pk p q r P h
r L 2 rr r st t r t s t r t ss t s r t t r r t2 ss t t s 2µ}u} }e h } ď Ch k`2 H k`2 p h q `λ} u} H k`1 p h q, r C ą 0 s r r Ω µ t t λ h Pr t r r t 2 a À b t q t2 a ď Cb t r r C ą 0 t h λ t ss 2 µ } } en,h ě p2µηq 1 {2 } } ε,h } } en,h ě p2µqs h p, q 1 {2 r r t rr r st t t t t e h : u h pu h P U k h,0 Bpu,kq : 2µ}u} H k`2 p h q `λ} u} H k`1 p h q }e h } ε,h `s h pe h,e h q 1 {2 À h k`1 Bpu,kq. s r t 1 r2 r t g : e h rr s s t z ς t r t 2 rts t s s e h e r t t }e h } 2 ÿ # ÿ pe, ςq p s e,ςq ` ÿ + pe e,ςn q, P h P h r s t t t2 t r t ς r ss t r s t t r t t t t t e 0 r P h b t pz h : ppπk zq P h, pπ kzq P h q P U k h,0 s t t L pz h I kz s r t t r P h } s pz r k I k zq} `h 1 {2 } spz r k I k zq} B À h }z} H 2 p q, } z π k p zq} `h 1 {2 } z πk p zq} B À h } z} H 1 p q, s pi k z,i k zq 1 {2 À h }z} H 2 p q. st t s r s st t r s ts r t r 1 t r rt s π k st t s r s a hpe h,pz h q E h ppz h q t E h ppz h q l h ppz h q a h ppu h,pz h q r r t s s # «ÿ }e h } 2 p s e,ςq ` ÿ ff + pe e,ςn q a h pe h,pz h q ` E h ppz h q : 1 ` 2. P h r P h s t t E k L e h t w r k L pz h t t D kl e h t q D kl pz h r t t a h pe h,pz h q ÿ # p s e,s k q ` ÿ P h pe e,s k n q + ` p2µqs h pe h,pz h q, t S k : 2µEk L pz h `λpd k L pz h qi P t s 1 r ss t 1 t s s t 2 r3 q t2 t t r t t t t t s h pe h,pz h q ď s h pe h,e h q 1 {2 s h ppz h,pz h q 1 {2 s h ppz h,pz h q ř P h s pi kz,ik zq t s r r t2 r rt s 2 1 ď }e h } 2 ε,h `s hpe h,e h q ( # 1{2 ÿ! ) + 1{2 ˆ }δ pzq} 2 `h }δ pzq} 2 B ` p2µq2 s piz,i k zq k, P h
t δ pzq : ς S k L pz h p2µq s pz r k Ik zq ` λp z πk p zqqi t t st t e h t r 1 t r rt s z t r r t2 st t r t t 1 À h k`2 Bpu,kq }z} H 2 pωq `λ} z} H 1 pωq À h k`2 Bpu,kq}e h }. s r 2 pσ, s zq pf,zq 0 s l h ppz h q ř P h pf,π k zq r t t E h ppz h q ÿ P h ÿ! ) pσ, s zq p2µqpel k pu h,el k pz h q λpdl k pu h,dl k pz h q P h p2µqs pl pu h,l pz h q ` : 2,1 ` 2,2 ` 2,3. ÿ P h pf,π k z zq 2,1 s r t t E kl pu h s pr kik uq Ek L pz h s pr kik zq t s s D kl pu h π k p uq Dk L pz h π k p zq t Pr s t s t rt t2 r t r t t 2,1 ÿ )!p2µqp s pu ri k uq, k s pz ri k zqq k `λp u π k p uq, z π k p zqq. P h s t r 1 t r rt s r kik πk r s t 2 t t t r s t u s t t t r s t z s t 2,1 À h k`2 Bpu,kq }z} H 2 pωq `λ} z} H 1 pωq. rt r r s s pl pu h,l pz h q s pi k u,ik zq ď s pi k u,ik uq1 {2 s pi k z,ik zq1 {2 r s t t 2,2 À h k`2 Bpu,kq}z} H 2 pωq. 2 s π k s s t s k ě 1 r t t pf,πk z zq pπ k f f,zq pπ k f f,z π1 zq s t r r t2 st t t 2,3 À h k`2 }f} H k pωq }z} H 2 pωq. 2 À h k`2 Bpu,kq}e h }. st t s s t t r t s 2 r L 2 st t r u t t r s t r str t s r k h u h R k h u h s t t r k h u h r k L u h R k h u h R k L u h r P h r r2 L 2 st t s t r str t s r t ss t s r t s 2µ}u} maxp}u rh k u h}, }u Rh k u h}q ď Ch k`2 H k`2 p h q `λ} u} H k`1 p h q, r C ą 0 s r r t λ h
Pr r P h t tr q t2 2 s }u r k L u h } ď }u r k I k u} ` }r k I k u r k L u h } : 1 ` 2, 1 s r 2 st t s st t 2 rst s r t t r v P H 1 p q r t v pv v RM q`v RM t v RM 1 pş vq` 1 pş ssvqpx x q r x t s t r2 t r pv v RM q P Up q r t r s q t2 t t }v} ď C K h } ss v} ` }v RM } s 2 r3 q t s s t ż ż }v} ď C K h } ss v} ` 1{2 ˇ v ˇ ` 1{2 h ˇˇˇˇ ss vˇ ˇ. 2 t s st t t v r kik u rk L u h r s t 2 r3 q t2 t t r t s r r t2 r t t ÿ 2 À h }EL k ppu h u h q} ` }πu k u } `h h 1{2 }πu k u } : 2,1 ` 2,2 ` 2,3. t r s 2,1 2,2 r st t s r s r s t 2 2 t st t 2,3 s t tr q t2 t s r t tr q t2 t r t t 2,3 À h }L pu h L u h } ε, ` }π k u u }, t }u r k h u h} s r r s }u R k h u h} s r t t r s t ss π k }u R k h u h} ď }u r k h u h} ` }π k pr k h u h uq} ` }π k u u } t s r 2 s ď 2}u r k h u h} ` }π k u u }, t t r st 2 t s s t s ss t t s ts r s t r r s ts r 2 t rr r st t s r t r s s t 2 s t r 1 t s t s r t ss s r t st s r 2 t r s P st s s t t t rt t st t t t s sts s t s s r t 2 s s t t r t str t t P h t 2 x t t r s t t s st r s t r t sts t r2 t r s s s s s r P l p q l P tk,k `1u t tt Al : tα pα i q 1ďiď P N }α} l 1 ď lu # ź B l : i 1ξ α i,i ˇ α P A l, ξ,i : x i x,i h @1 ď i ď +, t s s B l s s 2 s t tr s t s r t r s pξ,i q 1ďiď s s r t 2 s t r t sp l p q s t t
2 t rt s r t B l r 2 r P h s s B k r Pk 1 p q s 2 s t r s t t r s s t t r t t r s t t s st r s s s rp k 1 p q s t 2 rt s r t q t s r r s s s t t str t t B k`1 r q r t s t q r t r s t B k s t s 2 s 2 s r t st r r t s str ts r t r r s s t 3 r r t r t s t s r 2 t r q r t t t t s s t x t s t s r r t 2 s 2 s r t st t t B k`1 t 3 r r t r t s s2 tr rt t r t s r s r t r s s r r 2 R 3 r 3 r r t s r t r2 t s s r 2 s r t r P k 1 p q r P h b r r t r t t t s s 1 t t s t ts t tr s s st r q r t r r s r t t s t q r t r r s t t t r s s t s s s ss r t s t ts r tr s r q r s 2 2r s ts r st r t r r s r t s s t t s s s t t t s r s 2 2 r s t r q r t r s r 2 r 2 t r t t s s s str t r r t s s s r ts t tr r r q r r s t r s s s r r r 2 s t r s s s tr t t 2 r q r r t t st r 2 t st t s r s s t t s r s t r t µ 1 λ P t1,1000u s t u pu 1,u 2 q s t t u 1 sinpπx 1 qsinpπx 2 q ` 1 2λ x 1, u 2 cospπx 1 qcospπx 2 q ` 1 2λ x 2, f pf 1,f 2 q s t t f 1 2π 2 sinpπx 1 qsinpπx 2 q, f 2 2π 2 cospπx 1 qcospπx 2 q. s t s s r t t λ Ñ `8 s t t r t t t t t f s t λ t s t st s s t t r 2 t t t st t s r r λ s r t t r s s s t r t t tr r rs s s t r t 2 1 s 2 s r t str ss rr r s t st t r t s t rr r s st t s }u Rh ku h} r r2 r r t s s 2 r r λ 1 λ 1000 r s t 2 r r t t t t r t r t s s t s r r s r r t rs s 2 s t t t t t t s r r t2 r s s r s r t t rr r s t λ t r r t r s t t r s P t str t r t s r t t s r t 2 s r t ss 2 t τ ass t r t str t t tr t s t t r r a h
r r r rs 1 s s r t r 1 t t t s t ss 2 t tr 1 t s t t τ sol rr s t t s t t r s2st t t s r s t r r t s r 2 t r r2 r s2st s rr s t r s t t L 2 rt r t rs π k πk r s s t r st s 2 t r 3 t s2st s 2 s s s r t r t P t r t sts r s q t 2 t t r r 2 t r P r t 3 q t t r r t t t s t r s t t st r t ts t t r P st t r t r t τ ass {τ sol s t carp h q s r t tr r 1 s s r N B s r s t 2 t s st t r st t s r t t r t s t r t τ ass {τ sol r 2 r s s s r s t r 1 t 2 τ ass 9carp h q τ sol 9carp h q 3 {2 s s t t r t st s s t t t s 1 t t t t P t r s t t str ss s t rr rs s t t t t P t τ tot : τ ass ` τ sol s r r s t t s t r r s s t r r s t t r t s s 1 t r t r r t2 t 1 t s t t st r r t t r s t s s t st r s r P t s s t r st r t r t r t rr r s r t st s 1t s r t t st s 2 s t r t r t r r rr t s s r r r s r t t t r2 t s t q s r ss t r t µ 0.375 λ 7.5ˆ10 6 r s r s t r2 r s t t t r t q t t s t t t A r t s x A p48, 52q t r t s t t s t s r s t s 2 t r s u h pu h,1,u h,2 q P P k p hq qu h pqu h,1, qu h,2 q P P k`1 p h q s t t u h u r P h qu h Rh ku h r r2 t r ss r p h px A q s r 2 t q t t2 λd kl u h r t s s t s t t A P B t s ts 1 st r tr r 2 s t r s s tr r rs r t 2 1 s s q s t 2 t s s t r t t r r s t r
10 1 10 4 10 5 10 6 10 7 10 8 1.89 2.97 3.93 4.95 10 4 10 6 10 8 10 10 2.91 3.98 4.97 5.99 tr ss rr r tr r s 2 s t rr r tr r s 2 10 4 10 5 10 6 3.54 4.28 4.85 10 4 10 5 10 6 10 7 10 8 3.71 4.82 5.99 10 7.2 10 1.8 10 9.2 10 1.8 tr ss rr r rs s 2 s t rr r rs s 2 10 1 10 5 10 7 10 9 2.65 3.02 4.22 5.17.5 10 1.5 10 5 10 7 10 9 10 11 3.01 4.07 5.06 6.01.5 10 1.5 tr ss rr r 1 s 2 s t rr r 1 s 2 r rr rs s h r λ 1
10 1 10 4 10 5 10 6 10 7 10 8 1.93 2.98 3.94 4.97 10 4 10 6 10 8 10 10 2.92 3.98 4.98 5.98 tr ss rr r tr r s 2 s t rr r tr r s 2 10 4 3.55 10 4 10 5 3.82 10 5 10 6 10 7 4.3 4.79.2 10 1.8 10 6 10 7 10 8 4.85 5.65.2 10 1.8 tr ss rr r rs s 2 s t rr r rs s 2 10 1 10 5 10 7 10 9 2.69 3.04 4.26 5.15.5 10 1.5 10 5 10 7 10 9 10 11 3.01 4.06 5.05 5.99.5 10 1.5 tr ss rr r 1 s 2 s t rr r 1 s 2 r rr rs s h r λ 10 3
10 1 10 1 10 0 10 0 10 1 10 2 10 3 10 4 10 2 10 3 10 4 r r s 2 1 s 2 r τ ass {τ sol s carp h q 10 1 10 1 10 5 10 7 10 9 0.84 1.21 1.78 2.35 10 5 10 7 10 9 1.05 1.13 1.64 1.98 10 1 10 0 10 1 10 2 r r s 2 10 1 10 0 10 1 10 2 1 s 2 r tr ss rr r s τ tot s 10 4 10 6 10 8 10 10 10 12 1.3 2.25 2.84 1.63 10 1 10 0 10 1 10 2 r r s 2 10 4 10 6 10 8 10 10 10 12 1.19 10 1 10 0 10 1 10 2 1 s 2 1.52 1.97 2.3 r s t rr r s τ tot s
44 16 A 48 r r t r s r t st s s t st s t tr r s 2 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 56 92 7.390 7.414 16.712 16.717 5.123 224 352 7.312 7.319 16.566 16.560 5.860 896 1,376 7.279 7.282 16.498 16.496 6.614 3,584 5,440 7.265 7.265 16.468 16.468 6.957 14,336 21,632 7.255 7.255 16.449 16.449 7.051 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 56 92 7.320 7.325 16.586 16.571 6.219 224 352 7.282 7.283 16.501 16.498 6.911 896 1,376 7.268 7.268 16.474 16.474 7.077 3,584 5,440 7.264 7.264 16.467 16.467 7.086 14,336 21,632 7.260 7.260 16.460 16.460 7.093 2t s t s r t s t st s s s r r q t t s t s t r t st tr r s s ts qu h,1 px A q 7.2596ˆ10 0 qu h,2 px A q 1.6460ˆ10 1 p h px A q 7.0928ˆ. r s ts t s s t t t r s t rs r t r s ts rs r s s t t t r t s t s r2 t r s ts s 1 t t r t t r 1 t s u h qu h t s t s r s t 2 r s s r s t s t s r r t s s 2 r s s r s t r 2 t s t s t r t rs st t r t st 1 s r t s r tt r r t r r t t r s 2 r t s t s t t r2 ts t r s t s t t t r s t r r 2 r t s s r t t tr r t s
s t st s t rs s 2 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 289 612 7.334 7.343 16.590 16.594 6.855 1,156 2,380 7.291 7.296 16.513 16.516 7.011 2,601 5,304 7.280 7.282 16.494 16.494 7.068 4,624 9,384 7.273 7.275 16.482 16.483 7.068 7,225 14,620 7.270 7.270 16.476 16.476 7.079 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 289 612 7.315 7.316 16.547 16.546 7.112 1,156 2,380 7.285 7.286 16.499 16.499 7.088 2,601 5,304 7.276 7.276 16.484 16.484 7.086 4,624 9,384 7.267 7.267 16.471 16.471 7.084 7,225 14,620 7.264 7.264 16.466 16.466 7.086 s t st s t 1 s 2 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 22 62 7.275 7.326 16.563 16.621 4.321 76 220 7.270 7.302 16.510 16.526 5.554 280 824 7.268 7.284 16.485 16.493 6.457 1,072 3,184 7.265 7.270 16.472 16.475 6.895 4,192 12,512 7.261 7.262 16.463 16.464 7.036 carp h q carp h q u h,1 px A q qu h,1 px A q u h,2 px A q qu h,2 px A q p h px A q 22 62 7.291 7.264 16.533 16.497 5.975 76 220 7.275 7.283 16.491 16.493 6.808 280 824 7.267 7.269 16.473 16.475 7.120 1,072 3,184 7.261 7.261 16.462 16.462 7.088 4,192 12,512 7.259 7.259 16.459 16.459 7.084
r rs 1 r rs 1 16.7 7.25 quh,2pxaq 16.6 16.5 quh,1pxaq 7.3 7.35 7.4 10 2 10 3 10 4 carp h q 10 2 10 3 10 4 carp h q r rt t r 3 t r t s t t t A r r s r t st s s r t t sts r r t s s s s s s t 2 t s s r t t t r 22 ts 280 ts 4,192 ts r r r t r t rs st t r t st 1 s s r s t st s s r t r r r s ts t t t s t Rh ku h r r2
r s r rã 2s s s 1 t t r r st t2 2 r s r s 2 r s t P tt s ts r t t P ré q t2 r 1 s rã r 33 r rt ts r r st t2 r s r rã 2r2 3 t t r t r t t s r 2 s s t P 2s rã t s r t 3 t t r r st t2 t r r r r s q t s r s H 1 t r s t tr r r r t t t s r r r st t2 t r r s 3 r 2 r 3 t s t s r 1 t s r t s r s r r t r s r r 3 1 P rt r r t t t s r s t st t r2 t s q t s é t s st t rt s r r t s rs rt t tr 1 2s s t s P tr r t t s ts s t s r t s t é t q s t s r r r r P tr r 2 r tr r2 r r 1 t s r t r s s tr s r s s tt Pr r t P tr r r r tr r2 r r t st s r t 3 t s r s s s r str t r t rs t t s t P s P tr r t Pr s r st r r s r t 3 t s s r s r s s r t t s P tr r 1t s t r 3 1 rt s t r s s t t t q s r ss r st t2 t s t t r t Pr r t rt r s st t s t s t ts st 2 s t t s r t tt P 2 r 1 t t s s s t tt t 1 2 r P s rs s t s r t r 3 1 rt t t t st t2 t r r rt r s r t 3 t s s r s tr s r s r r s 2 r ér r t rs t s r 1 t s s 2 s r r s q t s t r t s t s
r P 2 q t s r r r s 3 3 r t r t st r r rr r st t r r 1 t t t s r st t2 t str ss s2 tr2 r 3 r r t t str t r 2 r s s r t s q t t P 2s r P r t tt r t t r t rs 2 r 3 s t s r t r r st t2 t r t r t s r rr r t 2 t ts r st t2 t t t s r t t t r s r r t r s t t