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High Frequency Limit of the Helmholtz Equations

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(1)High Frequency Limit of the Helmholtz Equations Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame. To cite this version: Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame. High Frequency Limit of the Helmholtz Equations. [Research Report] RR-3785, INRIA. 1999. �inria-00072875�. HAL Id: inria-00072875 https://hal.inria.fr/inria-00072875 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. High Frequency limit of the Helmholtz Equations Jean-David Benamou , Franc¸ois Castella , Thodoros Katsaounis , Benoit Perthame. N ˚ 3785 Octobre 1999. ISSN 0249-6399. ISRN INRIA/RR--3785--FR+ENG. ` THEME 4. apport de recherche.

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(96) ªe  eK­~eKmŒwse { ·= ‹¬l— 1º7ˆ i1kpu‹}e e < }t?u1¥Te  š?ew°­°eYoT»?½ ™klu · ¦ t  o?u1| 1ºˆ9q a kpwbewyqyo?–9nzkpw a ewq a e}t1u?¥?e  šTeYu‹}e?ˆ  .  . .  . £ a  ¦ a a Œ·§g3t ¬  ¼q„1eq º7a ¬u9kpw"t“£ ­)­~e e}a t?oIoTf&¥Tš?ee™orkpoTqsu tQn  { eq eYo }{ et1|vf&v v3t‘ t“t?t ¥TwyePe q { q ora kpuQqeq kpa u1e%qs¥“evoTš u9  ekporw¥‘n kzt1em‹wbwHo?wyw mŒ–‹ws{ eYe}³3qyŒu9kz¬ t1kpu9uš q a orqHoTq }a }et wy e{ qxkpwu9š qyt†o?u q a{ e©wyeqxš?w kz¥?eoKˆ ¥Io?u9kpˆ w a kpu9šklu1}q t? u1t q {  m‹kl–Œ}em9{ qsklt?kpuu qs qt  e< fQo?orw kpu‹> wX q a 3©e  ·¹e wy¦ et e™ e+q a qst¯e™qseewyqskpfQfQw orqye+·y—Iq ºsa º7e¬ }t1u1q  kl–‹m‹qsklt?ut ¦ q a ewyeq ˆŒu‹oTf&enz|?ˆ < † ‚ † † U >B º · º 8 D · º :B 1 · 2 9 hYj 1 ·42Zº V o < ·-2tMCBŒº ’ 2 ’ B ¼  h j A ‚ G   K L ? < ggg ? ggg H H v   <  j ÎÐÏ5Ñ;ÎÂÒ y. . . . . 2. . .

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(98) k  wynz|?ˆ‹–?|  eYv  t { m‹}kpu9š&q a ef&eq a t { < t ¦ v  t‘t ¦ t ¦ ®XeYf&fQoQg3¬z—?¬ ¼‹ˆ1­~e™fQoI|P­  klqseKq a orqˆ ¦ t  o?u?|  3‹ˆ ^ † ‡tˆ • e‹vd· 9 n n ‚ Š“º 9 — — • —  4· 2 9 rŠqM º,— Š B E IG : o < ·-2[M BŒº 9 o ·42tM BŒº 5 n n ; < f  f ’ l l ‰ x. ?< H. (. G . ! # 0 2. . 2. . %. %. %. %. %. ! &#. %. %. %. . %. !. #. f ! 2# ”7. ‘. ¿e‹vZ· 9 ;[< 2 yº 9 —kn A ;=< 2 e‹vd· 9 ;[< yº ·§g3¬ Œ—Iº awklu#!yg3¬ Œˆ‘­°efQoI|¯q a e  e ¦ t  eKewyqskpfQorqse?ˆ‹q a oTu‹»‘wqst ·¹gŒ¬ ‹—“ºˆ † U >QB Qº D 4· 2 >QB º B 5 · 4 2 H 1 · d 2 º 9 ¼ ghj < ·-2[M BŒº 9 ·42tM BŒº V ’ 2 ’ B < K ? < A ¼ ? < † 2 x w ‹ m v n v 2 n n B n ‚ ‡ ‰ 2  ? < H K L KTL B ‚  x A j  U n¿e9v;· 9 2 yº 9 k— n e‹vZ· 9 ; < 2 sº V ’ B›7 ·§g3¬ 1¼?º ; < A ; < ªaew ­°eKe ­~{ e k {a oI­ ¥TeKa kzm‹nle™wye e{ wyqsoTkpfQwxwxorm‹qyfQklu9v9š qsklt?u²·¹ Œˆ‹¬­~ ‘eº@qy tTeš?o e{ q klanz|Pe  t1–‹­bwykle q a ¥Te+·¹ Œq ¬za —YoT 1qbº¬ q a eo?wywxm‹fQv9qsklt?u ”klfQv9nlkzeYwbq a orqbq a e ¦ m‹u‹}qykzt1u ‰ \ < wxm‹v n 2=n A n B n ‚ ‡ ‰ 2  B 2 ‚ kpq wa orkpu1q qseš  o?–9nle"kpu q a e B¥“o  klo?–9nze?¬ `ba e  e ¦ t  j eTˆZ–1|©q a e { t?f&kpu‹orqye { }t?u1¥Te  š?e u‹}e™q a et  ef©ˆZq a e&ewyqskpfQorqse·§g3¬ 1¼?ºkpfQv9nlkzeYw x. 2. %. %. %. %. %. %. %. %. . n. ! #. (. ! #. ! #. (. %. %. %. . %. . . !. (. ! #. #. 0 2. . ! #. !. ! #. #. ! #. 4.  . !. . H&. ! #. #. <. X. ! #. 032. ! &#. I 37 < ‰. #;.   / 16-=/ . T /";9     & . b` a kpwwyqsevkpwbewxwseYu1qskpornlnz|Po a e ¦  t  fm9anloT¦ qskl t?u\t ¦ q a ef&eq a t { mŒwse { kpu©q a e{ vŒo  orš  oTv a wH!xgŒ¬ o?u { !yg3¬ gQoT–3{ t“¥Te?¬ ˜%wykpu9šQ®XefQfQoQgŒ¬l—T¬l—+qstTš?eq e ­bklq t f"m‹nlo©·§g3¬ ¼?”?º7ˆ1­~e™eo?wsklnl|P–Œt1m‹u ˆ/m‹wskpu9š¯oTwxwxm‹fQv9qsklt?u²·¹ Œ¬ç?ºÜo?u ®XeYf&fQo&gŒ¬l—T¬ç¼9ˆ † n R n´-· 2 BŒº >1·42 G º   ‰  ‡ n  M n 2 B 2 n ´ n · d 2 º 2     A B ; < < B A‡ ‰ A J  < < x K 2 H A  † x w ‹ m v n 2=n n B n  ‡ ‰ 2   >‘p· n 2=n A n B B n º ; < M x KML A KTL m£‹v3e©t?uu9t“m‹­ wyklu‹{ š&klwyq qsa kpu9eš1¾~m9kloTw mŒa } a q a|1½sei‘}} aoTwy­~eo w@ «n 2=klnu9eYj ¨‘m‹n orB nln´kzˆÜq¿|oTu klu{ 2Xn 2v¬ n n B n ¬ `ba e©qse  f$wyqseYfQfkpu9š ¦§ t1f q a e}o?wse n 2vn n B n;š?kz¥?ew&o x ^ }t1u1q  kl–‹m‹qsklt?u­ a kl} a kpwewyqskpfQorqse { –?|?ˆ ^ † B y w ‹ m v 2  ‡ ‰   • >2   — ’ B  KTL ; < KTL !. . #. (. . !. !. # . !. (. ! # 0. &# . # . !. {  pkm‹wbv3et?o?u\wsklqynzoT|P»1–3kpu9t?š mŒu { e 5#{ –1 | A Œ3 ˆ1o?u. ! &#. !. 0. €>   >   ‡ ‚ # !  ‡ ‰  0 # x ‚ \ ‚ A  L ‡  ‚ 6a  H †  ‡ ‰ ! >QB&# ! (. 4. j. ! # . . ! &#. #. 0. †. !. K L. #. B. ! &#. . ’. B. x. (. M. ¬ npwst&q a e+}t?u1q  kl–Œm9qsklt?uQt ¦ q a e+qse  f wyqseYf&f&kpu9š ¦§ t1f q a eK}oTwyen 2=n x. (. ÑXÑ ÕD¡ñ¢£¤. #. #. 0. !. ! #. . (. !. ! #. K L. B&#. . B. ! &#. ; <. ’ BX < I ‰ 3M. n B n.

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(102) ¬z—?­~¬ ¼‹e ¬  eo { kznl| { e { m‹}e²·§gŒ¬ç¼T¼1ºK–1| qyoT»1kpu9š†q a e\nlkpfklq>UXO3Œ¬  /

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