Méthode volumes finis pour la résolution du système de Maxwell 1D sur des grilles raffinées localement
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Méthode volumes finis pour la résolution du système de Maxwell 1D sur des grilles raffinées localement Nicolas Canouet - Loula Fézoui - Serge Piperno. N° 4301 Octobre 2001. ISSN 0249-6399. ISRN INRIA/RR--4301--FR. THÈME 4. apport de recherche.
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(80) oyÃsDr-u~ix{®sqrtvwz oyx{z x{z-icwtvu{uFhv v ili®Ãpr-i,syix hoqsqsDuFvwoyx{z-x{zv oy }'uFwr e-\wrtx{x{}Ogjv ¨hx{x{wz-oyihfhrtz-à irµi±²ihx{® vz«Ã r-ihi vw sDx{u~rt©z-ilx{rtsyi ©ih sq syi e-¨hÀÅg`x{wu grt¨sDuFhvwwoyoqx{v z uF z-i>sD¸-urH Èihvw±²¨hoyx{v ihzÈz }tÈ ¨ i,¨ À¯uihkzpx{v z i oquF z-i>hv syi#i uFrHu{ ±oqrtgjz«ihz-g~woqoysqx{oyihz-r [. Æ% %!? # '@¹#0% µ ° i º » · e h ¨ ` g { u { x z v h ¨ v ´ ¨ q o z w v x t r q o v h i z ! ' } F u , Ô i w o h i w H o h i v F u s t h i # z © { x q s t r j g i ¶ ª t z y o ¨ h i z w v ¨ }Oihºv,x{® rtrt©{z-uFioqvµoqzv x{¨hzpvwmuFl{vwsyihoyx{¶z¥syiv ztihoqg~©i}OuFx{r ih sqsyi i oywi
(81) }v»ihL }oyix{zl!ihrtv zt i ic· ,oÁÉrtr-vw voyuHx{®bz z rl gj e-x{¨hzg`vwuuFoq}'v uF µuFsyz-i e-x{ihoÁvw v i~ rl¨hvw}'r uF i~uFgjÃprµfh±²vwx{ z i ©\ x{r x{sqr-rt ´gje-}ti¨h ´g`¨ª-uÈihztzoy¼Â3v wx{vuFz-iht3zoqvwoysq hoqv ¨o¨l rtihz-v!¨ih}t¨hw vwoq¨vr h oywi uFoyz-x{uFz/´z'¿3uFsDusy x{ih}t r-u
(82) ihi\g~gj}oyihfhx{z- rti\¨Yi }'syi,uFwuFwvwz-Hoy>i wv rt fhzi\gjiiu i\uF i }t|}gjuSx{Hx{wÓ´vz-®Lx ih°0sqs¹oq± uFgjihz'z ihuFz-}'sqHwuFoy wx{i vztuFz-ziihv ³s¯® }trÆ x{ }t we-oy¨h¨hg`v ¨u . '1. D. !. #. _H_a`bc d e f.
(83) 0.8 Calculee Exacte 0.6 Verre. Vide. 0.4. 0.2. 0. −0.2. ' . . −0.4 −2.5. 0. −2. −1.5. −1. −0.5. 0. 0.5. 1. 1.5. - ( 'T P& V & :! / / + ( 0. 2. V& / /
(84) . J
(85) " $%'&. z x{z- o fh i oyhoµsyi wHwv fhgji il7uSLÓYihsqsbihz«©uFwoDuFtsyi ihv . 2.5. +9 ; ! + . /. KLL 5 , 7 / + 9 5 7 ML / ¼ ¿ LN 7 / 9 7 Y 5, / + O 5 zÈoqzpv fh i syilwHwv fhgji wrte'u{Ãr-il¨hsy¨hgjihzpv!z-x{v ¨6n r«g`uFoqsqsDuFi KLL r , 7 / 9 r 7 ML s 5 / + s 5 ¼ ¿ LN r 7 / 9 r 7 s Y 5, / + O s 5 c |9z~ihwv³rtih}tzj}OrtxvwoquFsqzoyvuFÃpzpr-v³i rtz- Çi3Y ÀÅx{ihwvgsyirt syi ¨h wioq©\¨i iihihz`z
(86) }Ov x{ihg~rt}-sy i0 iv ihwgje'uFig~ }-i x{ x{zoqvv ix{ z-iw³vuF¨zpÃv r'0uFwvwrtoyx{z- e'Çu{x{Ãpz`r-x{iYtvwihoysqihsqzrtv syi r KLLL n , 7 / 9 5 / + s y ©x{oy oqz i s v s
(87) v sy o. ML ¼ h p k ¿ r LLN n Y 5 , 7 / + 9 O . o c v v y s.
(88) s / s y ©x{oywoqz i s xst= io wv ii)wv>[ sD¨hu
(89) ©uFz-sqx{r-wihg`0uFsyisyil0oqrtzpztv oq¨hvuFoquF isyiYicLuFv rH¨h~woyihoqzprtv ihi ÀÅ[ u{n i i³ihsqsqrtsyi®pYi³oqzv ¨huFsyiY± oqzv ihw}t fhv ihzpv3}tep· w¼½ oyÃpir-w}Oihgjihihvwzoq©vihgjx{ihg~zv gjijsyi#¿¶x{sDrtu,©uFuFzsyv ihrt0¨hsygjihxvwwihoyÃztr-z-ii
(90) ihi vg`¼½uF iz-w}O¨hvwioyhÃpvwr-oq©iihgjuFihrHz¥v oqzv ¿ihwÀ¯rtu{ i( n
(91) ® z ( n z-x{y ®v i°0±²¨Ãsr's uS·y vwoyx{z ¼ hkp¿ ih©Loyihzv\uFssyx{s y . . /. . . ./. . . . . #_. 687 968:.
(92) KLL n , 7 / 5 /+ M LL LLN n 7 / Y 5, / + x{r 9 r v s
(93) v s y o 9 oq i 5 ( n ( n _
(94) z J %'$,# J
(95) . 9. s sy . s y ©x{oy oqz i s 9 O s s yc. s y ©x{oywoqz i s y7. ¼ ¿. R. . { x t r j h ¨ { © F u q s r h i ~ y s i # ¸ H r ¹ Ç { x z t } h i z D s W u g h j g i Å À { x z h w v y o { x z i ¸ H r c ¨ ª t z y o Æ i ' } F u ½ ¼ p k # ¿ F u y s h t r y s ¨ Æ i y o h o x{u) ©tiv 9 ihz? r-i i,z-©Lx{oDrt7u~¿©ihrtsqz-syii©{x{uFg#syihtrtoq z' uFoy x{i z }'uFx{wvz©ihicv ti ± uFrti#vw}' iuFuFgji
(96) fhvwoqz i~v ihº À¯u{ ii sDuj g`iuFztihoysqfhsq rti
(97) syi rt®oq©{YuFizp#v iÆ©uF¼Âsyihihz|rt z-~x{v x{uFzzpvv 5 B ) szy O ) s 7 º O ) s OUOs O O,O OOs O O,O O KLL ) ss y 9 ) s 5 7 º O ssy c sy - 5 ML s z s y LN ) s y s 9 ) s y O 5 O º 7 5 ) s y O ) s 7 - º O ) s y c OUO,s O yO,O s O#O s O,O OsO s y s sy x = s ihv s y x{zvsyilihzvw i iu©Loqv ¨ ilihsqsqrtsyi6 n ih6v n y Ç s s y syi}Ox{oqzvlg~oqsqoyihr À¯u iÆkx{s¯z-± uF h v i i ( ¨cn ª-zt oq\( ni y Çu O oy)ihzs v,ihiz wv~}'syuFiÆwvwoyhrtu sq oyoyihihzp}v x{ rtr e'iuFg~g`}5uFoqsqwsDrtuF~isDu7z-x{ihzWsqsqrtwsyvwifwr-n¥hvw®0rt s3¨
(98) ¼½©u|x{oq}t sqr-k w¶oyih}'rtuF ic ithihvwg~oyx{}tz syi£ ¿crtÇoq©{z-uFx{zpr-v ~i® ihz x{ztz-ih x{z-jrtz-i ÀÅx{wgrtsDuFvwoyx{z5 oqg~}tsyiÆ}x{rtrtz-iÆwoqsqsyiÈwvwwr-hvwrt ¨i uFz-jsDu °bu
(99) ÀÅx{z-hvwoyx{z«¸-rHÆzrtgj¨hwoyÃpr-iiwv!uFsyx{ ¸-rH 5 n B n y 7 9 5 ) sszy - ) sy4s 7 c µ ° 3 i i v i Y i D s u y o h h ¨ w v y o F u w v y o { x z w ' } F u w v D o F u y s 3 i h i v ¯ s ± q o z v h ¨ F u w v y o { x ` z h i ~ z v h i ~ g } { x p z v o h i z w v y o p à r i ³ F u ~ r e h ¨ ` g u }t ¨ ihzpv ¨ uFz->sDu
(100) }t ihg~oyfh i}'uFwvwoyi® . . . /. . . . . . . . . . . .
(101). . .
(102). .
(103) .
(104). . .
(105) . * '. . J&% & !
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(107) x= ! '9 ? 5 ! B ! 7 ® oqz-wo¯ÇHsDu
(108) ÀÅx{z-hvwoyx{z i\¸-rH«zrtgj¨hwoyÃpr-iihzpvw i sDu~ihsqsqrtsyi 0 ihv 0 -' iwv ¸-rH 50 B0 -' 7 9 ! - ! - º ! O ! O º ! O ! zWuFOx{rtvwoqv uFz->syiu{ r«©po ix= V ihv Y x{zpvz-x{wg`uFsqoy ¨\uFr e-¨hg`u KL ! 9 ! - NK ¸-rH 50mB0 - 7 O ¸-rH 50 O B0 7 Z\ M LN 9 KN Z ! - ¸-rH
(109) 50 B0 - 7 O ¸-rH
(110) 50 O B0 7 \ c ! # \±²iwv oq i KLLL # 9 KLN O º º y O º y º Z L M L ! 9 ! - KLN º ! º- y ! º y ! - º ! \ Z L LLN ! ! - O ! - ! # O ! # - ! \ c . . . . . . . /. ¼ ¿ UD. [. ¼ ¿ &1. x= 9 !-! , + ihvº y 9 º -' ® °sqµoqg~ioqv ioqËg rt i,sDuFJvÍvwpoyJx{Ë }Oz-bi,¡ gjzp-rt¨h¢ vgjuFsq¨hsqwoyÐ oyýä r-r-iiJ®Í}t¡ z« ¨0z- Ëihx{z¢v i v ¨ihi¢ z uFoqgjz- SihË z-wihoyÍvwx{v zÈi ¨hvwW vwx{r ¤¥oy i\ÍÀÂ-x{zpÐÂihÐ v>v µoqz¢ v ihwsy©iihzt oqe' uF g~i}-x{¨hz syi hoqvwvwwoyoyx{Ãz-r-!iuF!rHih v g`uFz-¨hvwoyÃpr-i!wrt>sDu~ rt ÀÅu{i icHv ¨hwoyihrt i r«gj¨hvuFs¯®/Yi e'uFg~ }-!©¨hw oÁª'ihzv\uFsyx{ o 9<; ihv o c 9<; c 0; ^= U= 0;. HG T=. =. K. #;. N 0; U= . T=!J. . . Metal. 0. −1. 1. 2. 'k . z j g x h ¨ q s y o ~ i h i z Ô i , h i D s F u w v y o { x z l ' } F u. D s u g j h ¨ w v e x i i l q o ` g F u i ¹ ® b © i y s i , z { x v F u w v y o { x z i D s u ª-rt i k-Ç-x{z«}Ox i KL . < V& TT /+ # / & '&
(111) (
(112) . ML. LL LN. # . 9. . 9 9 O 9 O c. . # . #_. 687 968:.
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(114) ; 7 9 º O LM ¸-rH
(115) 5 B 7 9 - - º O h - LLL ¸-rH 5 ; B 79<; LN ¸-rH 5 B 79 - º O h - - 5 B ( Í ½Ð JÍ ¢ ¡Ï ¶p¤ FË 0Ë/¡ JÍ ½Ë . ) ! + 2
(116) ¼ ¿ !/, #
(117) 0 5 - º 7
(118) 5 7 O º
(119) 5 7 c z x{ztz-i uFz-syi vuFtsyiuFr Ãpr-ihsyÃr-i!©{uFsyihrt r7YÄ° ihz«ÀÂx{z-hvwoyx{z iº® YÄ° º9; º 9 4 B º 9 h B º 9 ; B ¶ \º p¢ ,¦µzÈgjx{zpvw ii, ¨wrtsqvuFv!ihz rtvwoqsqoyuFzvsyi hwoqv fh i ilwvuFtoqsqoqv ¨ ilÒ0x{zÈihrtg`uFztzµ® zÈz-x{v i = T=. M E !5 QK. I7. U= U= V; 5 * / . " 3< 0 V& TTO& ".. . . . . +T O& . H8. /. ) . 9. . . zÈ}t xLf i
(120) s¯± uFz'uFsqH i i Ä-x{rtwoyih\ihz ¨cª-ztoy . uFzpv [. ./. ) . "9 ! 5 7$# %'& 50 ~ !/, 7. 9s¯± oqv ¨huFvwoyx{z 5 -' 7 syi,gjx i i Ä'x{rtwoyih±²¨hwoqv ) . 9 ) . B { =ÈsDu
(121) g`uFvwwoyi ± uFg~}tsqoÁª'uFvwoyx{z i v x{ztz-¨il}'uF ./. ¼ÅH£¿. . . 9 - 9 & ? a 5 - º 7
(122) 5 7 O º
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(124) ° b l u { x z q o w v y o { x ` z z ¨ i F u q o ! i h i 0 v w H r ` ¬ F u z v i i w v F u t q o q s q o v ¨ ` r e h ¨ ` g j u  ¼ { x z q o w v y o { x z 3 i 0 Ò { x ` z h i t r ` g F u / z ¿ ±²¨hwoqv 2 ; B B
(125)
(126) xY=Èx{z-syiw>o ©¨huF syx{ihz-rt 9 ih}t ¨ ih(z* v ®tihzp°µvi sy}Oi>x{sq©puFz,syl{ihrtgj i >}tuF x{u{}th v i¨h woy wvwi oyÃpsDu
(127) r-i g` uFi vww oyi iwv ® ./. . . O. 5 7J9 -. . ,. - ,c. . nprt}t}Ox x{z- , 2 -® °bi>©{uFsyihrt !}t x{}t i i x{zpv x{ztz-¨i}'uF. 0
(128) , O , - ! 9 * 5 O , 7 - 5 , O , 7 9 c
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(130) Çtsyi e- ¨hg`ujiwv>oqz-wvuFtsyi® , O ,
(131) 9 O$, c z wrt}t}Ox i Ãpr-i 2 /. 9 , -. z-
(132). !. , . O. . ,. 2 . c. YilÃprtoµiwv!uF-wrt i®/Yihou{ e-fh©ilsDu ¨hgjx{z- vwuFvwoyx{z i sDu
(133) }t x{}Oxwoqvwoyx{zµ® SpÏ ½¡ ½Ë ¢ ¡£Ï ¶p¤ FË 0Ë/¡ JÍ ½Ë º
(134) ? º h & a 9 5 O 6 º 7 & . }t}'rtuFsy uFx{vwoyrtx{«9z© ¨hw oÁ ª') ±ihrt«z-®/i«syi x{zW z ih¨h vi uF t¨ }tsqsDoq uFviz-s¯iȱ²¨ ÃoyuF r'r }uFihvw© oyiwx{hoyz x{v zih rti !± rtx{z ±²z x{ z i ie-¨h g`oy ® huH Ç>fhsqv syx{i«i z;i¨h uFvv z-uFtx{sysqi,oqztv«z-©LsD¨o u i i ihuFsDuFz-}'vwjuFoyx{wsyvwz i oq Ãpu{rt~r o, x{ ihe-zpsqoy¨hvwi|oqg`zsDr uu i}thv
(135) ¨}' ihuFzp v!rtx{z-z iÆx{uFt z'vwuFoyihsq Lz v i e'uFwgjx{ztoyÃpr-i x{z ¨hvuFtsqoqv#sDu ihsDuFvwoyx{z i oyw}Oih woyx{z oy h fhv i®¶Ô uFz-
(136) syi`u{ 9 ) - 4 ) !-, KLLN ) ! + O 5 O 6 º 7 Z LL -92q5 ! 4 7 c Å ¼ ¿ \ ! , x{z r->vw sDx{urt©x{i
(137) z uFoqoqz-vwoywx{o¶z|Ãpr-¨h©ilxpsDÃpu r-¨iloyw}hoÁih· ioyx{ zWr-i®wv ±²x{ i uFz-!syiu{¨hz-¨huFs¶ihv\Ãrµ±²ihsqsyi#iwv ±²x{ i#k z¥rtvwoqsqoy i`}x{rtgjx{zvw ih
(138) syi~ ¨wrtsqvuFv}t ¨¨ ihzpv#sDu«gj¨hvwe-x i i¨Ãpr'uFvwoyx{z-¨Ãprtoq©{uFsyihzpv i®¶°¶u M E *E. = U= V;. I7. QK U= U= V; E H + / "
(139) 3 & -& & ? O
(140) \Q
(141) ! * ( ' /
(142) & +& P
(143) 4 $ 4V
(144) \Q &
(145)
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(151) s¯±²¨Ãpr'uFvwoyx{z§Ãpr-i~©¨hwoÁª'ijrtz-i~ÀÂx{z-hvwoyx{z ¨hrtsqoyfh i` x{sqrtvwoyx{z i¨Ãpr'uS· p¢ ,¦ nprt}t}Ox x{z-!Ãprµ± oqsµicHoywv i 5 B 7 )a
(152) ? )a
(153) ? v ihsyÃpr-i 0 B JB 50 !-, B ! + 7 9 ! + 50 !/, B ! +j7J9 ! c s¯±²x{z w rti -vw vwx{oqvwoyr'Ç'uFzx{vjz«gjix{z©vwuF syiihrtÃp r- i uFz-~ sy7 iÆ© ¨h we-oÁª'¨hg`ihzpu v ¼ ¿
(154) ihvjuF}t frtz ¨h©ihsyx{}t}Oihgjihzv i«duLsyx{ 5 B KLLL ( , + 7J9 ( , +j7 !-, ( h , +j7 KLLN O 6 º O ! + Z LL ! 4 7 (+ 5 B ( 5 B - ( 5 B -32q5 H8. [. &1. ./. ML. ,. ,. h. !/, . . \. LLN ( +j7J9 ( +j7 ( h , +j7 KLLN O 6 º O ! + Z LL ! 4 7 , , ! , (,+u5 B ( , 5 B - ( , h 5 B !/, \ -32q5 Yiho0gjx{zpvw i Ãpr-isyij e-¨hg`uÈiwv ±²x{ i Æihz¥v ihg~}-ihv#ihz iw}'u{ijihvÃpr-isyx{ Ãpr-i ? 9
(155) O 6 º 7 syi~ e-¨hg`u«iwv ±²x{ ijk-®bYihogjx{zvw i uFr-woÃr-isDu oyw}Oih woyx{z ¼ x{ztz-¨i~}'uF& & a syi i¨hwwv oq5 i,© ¨v ix{
(156) rtvoqg~ }'iluFg oq ih(gj¿#i iw[
(157) v©t¨hoywihoÁª'z ih ±²Ãpx{r- i, syi i k- ®e-Y¨hg`ihu`oYzµiw±²ivwwv
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