Non Overlapping Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

(1)

HAL Id: inria-00073552

https://hal.inria.fr/inria-00073552

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Non Overlapping Domain Decomposition for Singularly Perturbed Elliptic Boundary Value Problems

Marc Garbey, Laurence Viry, Olivier Coulaud

To cite this version:

Marc Garbey, Laurence Viry, Olivier Coulaud. Non Overlapping Domain Decomposition for Singularly

Perturbed Elliptic Boundary Value Problems. [Research Report] RR-3137, INRIA. 1997, pp.45. �inria-

00073552�

(2)

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Non Overlapping Domain Decomposition for Singularly

Perturbed Elliptic Boundary Value Problems

Marc Garbey, Laurence Viry , Olivier Coulaud

N ˚ 3137

Mars 1997

TH ` EME 4

(3)

(4)

/0 1&2,43# 576089&1;:<,5-%.=>?,! @/AB3 -$

CADFE?GIHJDKELMON1POQSRTDFU%EVMOWG*MYXZ[EN]\^QS_`[ZbacZbMOEdefU1`gDFU4hji

kl%m

nponrq9sutfvwoxfywzO{v}|~n{|f{vwov}Vz{v}|~

n8{

m

npon|oYfy}nn

|K'n{xoYz{

l

zK|

{ n

n

l n l

n~fpKsz

pKK!sqfJzKn

S ¡p¢*£c¤O¡K¥§¦

nJzK~zy¨f©n{

l

nª%x~z

|O«¬xz

{n

|~fvSzOy}{n

~zO{v}®Kn

|fn x

n9¯°|

{ l

n|Fy±xF«

{v}|~|O¯8v±~ffxfy±z

n

{Vx

²

zO{v}|~³

| ²

y¨npo´pµ

¦

n

l

|O¶·{

l

zO{¸¯°|

zK~zK

|

v±z{n

l

|Kv}n|O¯

{ l n

|o¹zOvw~

n|oYº|Kv}{v}|~%»|f~fnJ|

²

{VzOvw~fz¯¼zO{|f~K®Kn

Knp~K{Bv}{n

zO{8v¨®FnJ8

l

npon¶v}{

lj½¿¾

À'ÁpÂ}ÃOÄOÃKÅgÆ

¾K½

{ l zO{

n8|Ky}®Kn{

l n ²

|x~

z

ÇywzpFn

pµ kl

nY|~F®Kn

Fnp~fnÈv}xn

y}vw~fnpz

¶v}{

l

nn{{|9{

l

nBvw~fxfywz

n

{Vx

²

zO{v}|~z

zKon{n

É

vw~#{

l

n¸¯°|Ky}y}|O¶vw~fYnp~fnKÊS{

l

nzKoYfy}vÌË«

pzO{8v¨|f~¯¼zO{|

v}ÎÍÏ ÉÐ

µ ¦

nKv}®Kn-

l z

!n{vwoYzO{n2|O¯c{

l

nvw~F{n

¯¼zOnÑº|Kv}{v}|~zK~

|~F®Kn

Knp~F{

z{n¯g|

zK~

l

|fo|KKnp~fn|xf

|foYzOvw~

n|foY|K8v¨{8v¨|f~Çvw~³|~fn

v±o´np~fv}|~zOy-VzOnÈz¶-ny}y

zOvw~³{'¶-|

v±o´np~fv}|~zOyzOnÈ

| ²

y}npo9|~³z

v}VÒcµ

¦

nJnf{np~

|fx

nVxfy}{{|

l

n{n

|O«

Knp~fn|fxf

|oYzv±~

n|o¹|Kv}{v}|~jz

v}vw~fv±~z#vwoYfy}vÓËcn

o|

ny |O¯ÔzF~Õny}n{

|oYzO~fn{8v¨

| ² y¨npoÖµ

¦ n

npº|

{|~vwoYfy}nponp~F{VzO{v}|~

nVxfy}{-¶v}{

l

Ë1~fv}{n

vÌ×n

np~fnØzF

|OvwoYzO{8v¨|f~f

zK~

Ë1~fv}{nny}nponp~F{|

nYÏÚÙ

¾*ÛKÜ

Â}ÁÞÝ Ð µ

ßÖàáâ'ãä1¢OåS K¥

vw~fxfywz

n

{x

²

zO{v}|~fp»

²

|x~

z

Õywz*Kn

p»%{

zK~fv}{v}|~³ywz*Kn

*»zOoYF«

{|F{v}zK~zOy}8v¨*»

|o¹zOvw~

n|oYº|Kv}{v}|~%»¿v}{n

zO{v}®Kno´n{

l |

æ°çè

é'ê'ëpì

è

é í î±êïbðpñ

òó2ôpõÓö÷¼ø¼ùbõÓú*û÷ü¿ýÓþ8ÿ*÷%÷Úøgô þ8ø 8ôõýýÓ÷Úÿ*ø þ8ôpô ÷÷*÷Vþ8øK÷2ýÓþ8ô ÿ*ôpõ}ö!wý" ô# $}ø

%

ü%÷Úôpúbø¼÷&±ôpúg÷¼øgøOû÷8õ' ôOþý(p÷*)÷'ùgù 8ÿpø'÷'ù+&±ô$, 8ø-þVúbõ'-ÿO÷ù**÷* 8øgø¼þ8õô ÷/.0&,1%ü(2*öõø3¼õøgõý, $}ø

4

&526)6&578!9 8øgø¼þ8õô ÷/.0&,1%ü(2: 8ÿpýÓþ8ÿ2ý' øgõÓþ: $}ø

Unit´e de recherche INRIA Lorraine Technopˆole de Nancy-Brabois, Campus scientifique,

615 rue de Jardin Botanique, BP 101, 54600 VILLERS L ` ES NANCY (France)

(5)

!,4 Ô" Ô&

#, 5 $·93 -" $ 89&1´, $ È %&,3JÔ&1!$

$%, $

à à ¥

¸|xfYzK~zOy}f|~fYyÓzOy}K|

v}{

l

on

nª1x~z

|«¬xz

{n

|f~fvÔº|x

n8|x n

n

| ² ym npon

nºn

{Vx

²

zO{8v¨|f~f vw~fxfy}v

m

n

npµ¸|xfo|f~K{

|~fxfnº|x

x~

l

|Kv}´z

zKf{

n n

ywzo

n{

l | n n

n|foY|K8v¨{8v¨|f~

n

|o¹zOvw~fnK»O|~9|

²

{v}np~K{x~9

l

npo¹z xfv|f~K®Kn

Kn

zFfv

n«

onp~F{-VzK~f

nywzO z{v}|~Yn{ xfvºn

on{

n

n|x

nywz9|xf

l

ny}vwov}{nKµz|f~K®Kn

Knp~fnn8{

Vxºn

y}vw~

npzOv

nK»%zKx#np~f|fxYy}n¯¼zO{npx

ÓzFoYfy}vÓËcpz{v}|~

x#

l

npoYzn8{Ôx~ÍÏ É8Ð

»

É

n{VzF~K{y¨n

z

zKo

m

n{

n

nn

{Vx

8²

zO{v}|~9vw~fxfy}v

m

n

nKµ¸|xf

|~~f|~fx~fn-n{vwoYz{v}|~

nv}n

nÑywzº|O«

v}{v}|~#|ff{vwoYzOy}n

nByvw~K{8n

¯Þznnp~K{

ny}n

|oYzOvw~fnB|

nVº|~

zK~F{

m

zywz9|xf

l

nBy}vwov}{nn{y¨n

|o¹zOvw~fnÑnf{

n v}npx

»pn{

n-ywz®fv}{nn

n|f~K®Kn

Knp~fn

zK~fTy}n-pzO

Ìx~fn

n|oYº|Kv}{v}|~

n

|o¹zOvw~fn

l

|o|K

m

np~fnnp~

vwonp~f8v¨|f~ÇBn{-np~

v±o´np~fv}|~9Vx

x~

v}xfnKµÑ¸|xf

n{np~

|f~f

~f|K

nVxfy}{VzO{zKxjpzO

n#y±z

n|oYº|Kv}{v}|~

n

|oYzOvw~fn l

n{

n

|F

m

np~fn xfny|~|

²

n

®Kn

zK~fx~Ço|

m

ny}nYvwoYfy}vÌË

n

ny¨n{

|oYzOf~

n{v}VonKµ~FË1~%»T~f|fxfJzK~zOy}f|~f9y¨n

nxfy¨{zO{

| ²

{8np~xfº|x

x~fnv±o¹fy

nponp~F{VzO{v}|~³xf{v}y}v}VzK~F{x~fn´zF

|OvwoYzO{8v¨|f~Õnp~

Û Æ "!Á*ÀÁ

½$#

Á&%('

½

Æ[Á)%

xfv}np~*!ÁpÂ+!Á,JÁ

½

Å-%('

½

Æ.%ØÏ°]|

xfy}n¯

Ð

/·ä¿¡p â¤10+à ¥

ºn

{Vx

²

zO{8v¨|f~Õ8v±~ffxfy¨v mn

nK»|xf

l

ny}vwov}{nK»2|xf

l n

n{

zF~fv}{v}|~%»1zF~zOy}n

zO8 oYf{8|K{v2 xfnK»

n|oYº|Kv}{v}|~

n

|oYzOvw~fnK»%o

n{

l |

n-v}{

n z{v}®Kn

(6)

,ÃKÆ Á , %Æ[ÅgÆ Ý À1Æ Â}ÃKÀ-ÁpÀÅ À ÃFÅgÆ -À Â}Á ,%

&c5¹,S&1!

kl

n#K|zy |O¯{

l

v}´zFn

v}{|{Vx

Ç{

l

n¹|~K®Fn

Knp~fn

zO{n|O¯{

l

n#ª1x~z

|]¬xz

{8n

|~fv

zOy}K|

v}{

l

oÏ Ð

zKfy}v}n

{|9{

l n l

|o|FKnp~fn|xfzK~

l

n{n

|KKnp~fn|fxf

|oYzv±~

n|fo9«

º|Kv}{v}|~¹|O¯1y}vw~fnpz

8n|~

| n

ny}y}vwf{v}vw~fxfywz

ºn

{Vx

²

z{v}|~Y

| ²

y}npo ¶v}{

l

©n

||

n

nKnp~fn

zOv}npµtf|Kywxf{v}|~f|O¯ vw~fxfywz

n

{Vx

8²

zO{v}|~Õ

| ²

y}npo´Ï

Ð

{' fv}pzOy}y}

v}Vfywz*

zFfv

®Oz

vwzO{v}|~fÕzO

|K~z

|O¶

nFv¨|f~fp»{

l

n|O«¼pzOy}y}n

¾KÜf½cÛ

ÃKÀjÂ¨Ã Á*À%µ Þ{v}{

l

np~

l

zOy}y}np~fKvw~fÈ{|

nv}~z~xo´n

v}pzOy%

l

npo´n{

l

z{ÑFv¨®FnÔz9x~fvÌ¯°|

ozF

|OvwoYzO{8v¨|f~Y|O¯S{

l n

|Fy±xf{8v¨|f~%µYtfv±~ffxfy±z

ºn

{Vx

²

z{v}|~Ç

| ²

y}npo

l

z*®Kn

²

nnp~Ç{x

v¨n

n{np~f8v¨®Fny¨v±~zOoYF«

{|F{v}ÈzK~zOy}fv}

²

xf{{

l n

nVxfy}{|O¯Ñ{

l

nnÈ8{Vx

v}n

l

z*®KnY~f|K{9Kn{

²

nnp~³¯¼xfy}y}0nfy}|Kv}{n

{|

np~

l

zK~fn~xo´n

v}pzOy%|oYxf{VzO{8v¨|f~f!Ï°nnB¯°|

nzKoYfy}n

q»

!»

F"TzK~

{ l

nv

n¯gn

np~fn

Ð µ

ª¿|Ky}y¨|O¶vw~f#{

l

non{

l |

v±~Ö{

l

nzKºn

$#

$%

!

!»f¶-nB8{npoYzO{v}pzOy}y}Õxfn{

l

nB|~fnpf{

|O¯ vw~fxfywz

n

{x

²

zO{v}|~Ö{|

nv}~Ön&Èv}np~K{zK~

zOpx

z{n

|oYzOvw~

n|oY|Fv}{v}|~|FyÓ«

®Kn

¯g|

{ l

n8nI

| ²

y¨npo´pµª v

{p»S{

l

n¹Vx

²

|o¹zOvw~fY|O¯|oYxf{Vz{v}|~³|

nV|~

{|{

l n

'

À'Á

Ü

Â}ÃKÀ ÛK¾

,JÃFÆ

½ % '

zF~

{ l n '

%?Æ

½fÜ

Â}ÃKÀÂ}ÃOÁ*À%

'

|O¯ {

l

nJt%µ

µ

µ¿nf{p»¿{

l

nJº|Kv}{v}|~|¯{

l n

vw~F{n

¯¼zOn

npnp~

|~Ö{

l

noYzOy}yÎz

zKon{n

¸É

»1zF~

v}

n{n

ovw~fn

vw~Vxf

l

zY¶z*#{

l

zO{

{ l n¸{

x~fpzO{v}|~´n

|

v¨-zOoYf{|K{8v¨pzy¨y}Y|¯1{

l

nzKon|

n

vw~npzO

l

Vx

²

|oYzOvw~%µª v±~zy¨y}K»

{ l

nzVn{

zO{v}|Y|O¯T{

l

non

l l

zK~fKn

z{v}pzOy}y}#¯

|foz

nfxfy±z

Vx

²

|foYzOvw~{|Yz8v±~F«

xfywz

ywz*Kn

Vx

²

|oYzOvw~%µ

Ú~Y

n®fv¨|fxfy}´x

²

y}v}

l n

¶-|

Ò

$%

'»p¶-n-vw~K®Kn8{v}zO{n

ztf

l

¶z

©-zOy}{n

~zO{8v±~f

|n x n

vwoYfy}nponp~F{n

vw~v}{Ño´|K{Sny¨npo´np~K{Vz

9¯°|

oµ(Þ{¶zOS

l

|O¶~9{

l

zO{ { l n

zO{n-|O¯1|f~K®Kn

Knp~fn

|O¯S{

l

nv}{n

zO{8v¨®FnØ

|fn x

nv}-Vxn

y¨vw~fnpz

¶ l

np~Y{

l n

|o¹zOvw~

n|o¹|Kv}{v}|~#v}

|ºn

y}

nv}~fn

µtxºn

y}v±~fnpz

Y

zO{n|O¯-|~F®Kn

Knp~fnYonpzK~f

l n

nY{

l

z{{

l

nKy}|

²

zOy|f~K®Kn

Knp~fn

z{n

npºnp~

|~

É

zK~

K|fn{|©n

|¶

l

np~

É)+*-,

Ú~#{

l

v}zKn

¶Ñn

npo´|~f{

z{n8v±o´vÓ«

ywz

|n

{v}n|O¯Ñ{

l

nªµÓ¬zOy}K|

v}{

l

ozK~

npoY

l

zO©v}nY{

l

nJvwoYz{|O¯{

l

n

l

|Kv}nY|O¯-{

l n

²

|x~

z

Ç|f~

v¨{8v¨|f~fpµ

¦

n¹|~fv n

y±zv}pzOy8n|~

| n

y}v±~fnpz

ny}y}vwf{v}Õt%µ

µ

¶v}{

l

©n

||

n

nFnp~fn

zOv}n

.ºzK~

z¸{

zK~fov}v}|~

| ²

y}npor{

l

zO{Îz

v}n vw~ny}n{

|oYzO~fn{8v¨

{ l n|

Kµ

¦

nKv}®Kn

v¨F|

|xf|~K®Fn

Knp~fnzK~

zOpx

zO9n{vwoYzO{nvw~Y{

l

nBoYzOfv±oxor~f|

o

vw~#{

l

n¸Ë1~fv}{n

vÌ×n

np~fnB¯

zKon¶-|

Ò ²

xf{-¶-nzKfy}È|fx

nVxfy}{-{|9np~

l

zK~fnBË1~fv}{nBny¨npo´np~K{

|o¹xf{VzO{v}|~fpµ

¼~Çtfn{pµ»¶-nY{x

]vw~

n{VzOv}y}

l

|o´|KKnp~fn|xfzK~

l

n{n

|FKnp~fn|xf

|foYzOvw~

n|fo9«

º|Kv}{v}|~fvw~|~fnVzn

vwonp~fv}|~%µ

¦

nKnp~fn

zOy}v¨©n{

l

n8n

nVxfy}{{8|Èz{'¶-|

vwonp~fv}|~zOy

VznÇ®vwzAzj|o¹z

v}|~ y}npoYoYz v±~ tfn{pµÓµ

¦

n0zKfy}<|x

oYzOvw~

nVxfy}{v±~ tfn{q

{|np~

l

zK~fn#{

l

nY|oYxf{zO{v}|~³|O¯zK~³ny}nponp~F{Vz

³{'¶-|

vwonp~fv}|~zOy-vw~fxfywz

n

{Vx

²

n

{

zK~fVov}v}|~

| ²

y}npo{

l

zO{z

v}nBvw~ny}n{

|o¹zO~fn{v}{

l

n|

Y¶v}{

l Ù

¾*ÛKÜ

Â}ÁÞÝ?µ

(7)

q Ù ÃKÀ Á SÆ[À Â}Ã

: ,5-%. 8 . 4$r? r -$%!!¸Ç+

"!#$%&'!(*)+,-. /%102%

Ú~³{

l

v}n{v}|~%»%¶-n|~fv n

z#y}vw~fnpz

n|f~

«¼|

n

vw~fxfywz

n

{Vx

8²

zO{v}|~³

| ²

y}npo@|O¯

{ l

n!¯°|Fy¨y}|O¶vw~f#{nKÊ

3546798;:

É

7<>=?798@5A&BDC?8

Ï

*FE

ÐHG

7 Ï

*KÐ 8I,J

G 7

Ï'

Ð

8ILK

E

Ïºµw

Ð

¦l

n n É

v}z¹VoYzOy}yº|Kv}{v}®KnÈz

zKon{n

vw~

*+EÉ

J

¯g|

8|on

É

JNM

* µ

| ²

y}npo|O¯{

l

v}

{'nn

l v²

v}{

²

|x~

z

Õywz*Kn

xfVxzy¨y}ÇzO{

²

|K{

l

np~

|¯Ñ{

l

nvw~K{8n

® zOygµ k¸l

v¨9{

v}®fv±zy |~fn

vwonp~fv}|~zyÔ

| ²

y}npo ¶¸v¨y}y

²

n#xfn

zzK~³v}y}y±xf8{

zO{v}|~¯g|

|x

o´n{

l | µ Ú~Çz

{v}pxfywz

¶-n¶v}y}y

l

|O¶

l

|O¶

|n

y}

nv}~j{

l n

|oYzOvw~

n|oYº|Kv}{v}|~ v±~j|

n

{|³Kn{9¯¼zO{

|~F®Kn

Fnp~fn¯g|

{ l

nÔ¬xz

{n

|~fvÌ«¼ª%x~z

|Bv¨{8n

zO{v}®Kn-|Ky}®Kn

¶v}{

l#½c¾

À'Á*Â¨ÃÄ ÃKÅgÆ

¾K½

z

zKo´n{n

µ

¦ n

n{

v¨{-|x

ny}®Kn{|{

l

npzO8n|O¯z9vw~fKy}n

²

|x~

z

#ywzpFn

vw~Ö{

l

n~fnv}

l²

|

l

|f|

|O¯Kµ'O|

vw~f>{|j{

l

nÇzOoYf{|K{8v¨ÇzK~zy¨fv}p»B¶-nÕ

l

|fxfy

Vfy}v¨{{

l n

|oYzOvw~

C

vw~K{8|

{'¶-|³Vx

²

|o¹zOvw~f

CQPSRRUTVW8

Ï1X E Ð

zK~

C.YZH[\TVW8

Ï

*+E

X Ð ¶ l n nWX

M * µ

CQPSRRUT1V

|O®Kn

{ l n ²

|x~

z y±z*Kn

zO{Õ³zK~

C.YZH[\TV

|O®Kn

Y{

l n

|foYzOvw~ |O¯Ø®OzOy}v

v¨{'<|O¯{

l n

nfxfy±z

zK

| fvwoYzO{v}|~%µ

¼~¹|

n

{|9npzOv}y}YKn{

l z

n{8v±o¹zO{nvw~¹{

l

noYzvwoxo ~f|

o»O¶-nz

nF|Kvw~f{|xfn

{ l n!Ë1~fv}{n

vÌ×n

np~fn¯

zKo´n¶Ñ|

Òcµ

¦

nYÒOnnp{

l

nYon

lÕ

nxfywz

vw~³npzO

l

Vx

²

|oYzOvw~ÇzK~

z

zFf{{

l n

|o¹zOvw~

n|foY|O«

v}{v}|~{|{

l n ²

|fx~

z

³ywzpFn

{vÌ×T~fnpµ %n{Yxf

np~f|K{8n^]

K Ï

nVf{pµ_]*`

Ð { l

nÈon

l

8v¨©n

|~

C.YZH[\TV

Ï nVf{

CQPSRRUTV

Ð µ %n{Yxf

np~f|K{8n

² 4aHb

E

A.8

E

#{

l n

v}

n{v}©n

|n

z{|

{ l z{|

nV|~

9{|

4c6

µ ¦

nY¶¸v¨y}yzOy}|

n{

v}{9|x

ny}®Kn9{|Õ{

l

n¹pzOnY¶

l n

nY¶-n

l

z*®Kn

{ l

n9VzKonz o¹f{|K{v}B|

n

|O¯

v

|Kvw~F{vw~npzO

l

x

²

|oYzv±~%»¿vgµn

]

K>d ]`

: X d

e E

Ïºµ Ð

vw~|

n

{|

²

zy±zF~fn{

l

nzKo´|x~K{¸|O¯Î¶-|

Òvw~npzO

l

Vx

²

|foYzOvw~%µ

(8)

¢

¤ 0wàf¡âà ³£< p¤ 1à ³à

¦

n9vw~K{

|

xfn9{

l

nB¯°|Fy¨y}|O¶vw~f#v¨{8n

zO{v}®Kn

|fn x n

!Tvw~|

n

{||Ky}®KnÏµw

Ð

4 a

7

YZH[\T1V

8@5A&BDC.YZH[\TV

G

7Y&ZH[T1V

Ï

*KÐ

8 IJ

G

7YZH[\T1V

ÏX

Ð

87

PSRR/T1V

Ï1X

Ð

4

a 7

K

PSRR/T1V

8@5A&BDCQPSRR/T1V

G

7

K

PRRUTV

Ï

*KÐ 8ILK

G

7

K

PRRUTV

Ï1X

=

]*`

Ð

:(7

K

PSRR/T1V

Ï1X

Ð

]*`

8

7YZH[\T1V

ÏX

Ð : 7YZH[\TV

Ï1X : ] K Ð

] K ,

ÏºµÓ

Ð

k

|Y{Vz

{{

l

n8

l

nponK»f¶-nvwoY|FnzK~z

{8vÓËcvwzOy

²

|x~

z

|~

v}{v}|~z{Ñ{

l

n|Kvw~F{ Xcµ

¦ n

xfn{

l

nÈVzFonË1~fv}{n

vÌ×n

np~fn

l

nponÈvw~jnpzO

l

Vx

²

|o¹zOvw~j¶v}{

l

v v}

l

y}n{

²

|x~

z

|~

v}{v}|~·zO{ X v±~

C.YZH[\T1V

zK~

¸npxoYzK~~

²

|x~

z

>|~

v}{v}|~·zO{ XAvw~

C PRRUTV

µ ¦ n

¶v}y}y-|~fv n

vw~Ç{

l

n#~fn{9n{v}|~³{

l

n¹|K{

l n

º|Kv

²

y}n#

l

|Kv}nK» vgµn#npxoYzF~~

²

|x~

z

|~

v}{v}|~³zO{.X#vw~

C Y&ZH[T1V

zK~

{ l n v

v}

l

y}n{

²

|x~

z

|~

v}{v}|~³zO{.X¹vw~

CQPSRR/T1V

µ

¦

n³¶v}y¨y

n|foY|K8n]{

l

nzF~zOy}v}|O¯{

l

v}v}{n

zO{v}®KnÇon{

l |

vw~·ÅÀ'ÁVÁ %ÅgÁ2 %?ÊË

8{

¶-n

nË1~fn#{

l n ²

n{v±~F{n

¯¼zOnYy}|fpzO{v}|~

²

n{'¶-nnp~³{

l

nYx

²

|oYzv±~f*»

²

zOn

|f~Çz{

x~fpz*«

{v}|~#n

8

|

zK~zOy}fv}p»n|~

y}¶-n n

v}®KnB¯

|o { l n{Vz

²

v}y}v}{'I

|fn

{'Y|O¯S{

l n

v}

n{v}©n

|ºn

zO{|

{ l n

zO{n9|O¯

zKo¹fv±~f|O¯{

l

nÈz

{vÌËcvwzOy

²

|x~

z

Õ|~

v}{v}|~n

| µ zO8{y}Õ¶-n

|o

²

vw~fnY{

l

n8nY{'¶Ñ|

nVxfy}{{8|Kn{ØzF~Õn8{vwoYzO{n|O¯-|~K®Fn

Knp~fnY|O¯{

l

nYv}{n

zO{v}®Kn|Fy¨®Fn

{|¹{

l

nYÁÄOÃ

#

Å2|Kywxf{v}|~|¯T{

l n

vÌ×n

np~F{vwzOyT

| ²

y¨npo Ïµ Ð µ

kl

n{n

l

~fv2xfn|O¯

npo|~f{

zO{v}|~Öv¨ xfv}{ny}nponp~K{z

²

xf{fywz*B¶v}{

l

{'¶Ñ|¹{nB|O¯

Vo¹zOy}y4z

zKon{n

pÊ%Ë

8{Ñ{

l

nVzOnB{npfÏµ Ð

»On|~

{ l

nBVoYzOy}y%vw~fxfywz

n

{Vx

²

zO{v}|~

z

zKon{n

BÉ

µ x

K|zOy2v¨{8|YË1~

{ l n ²

n{zO{

l

vw~³{

l

nz

zKon{8n

VznIÏ1]

EVÉÐ

¶ l v}

l

|O®fv

nVxºn

y}vw~fnpz

|~F®Kn

Knp~fnzK~

|f{vwoYzOySx~fvÓ¯g|

ozK

|OvwoYz{v}|~%µ

!#"

¢ ^¡%$1¡à'&¥

c¡à¢(¼£c¤Oà&Sä%

¡ ä

¦

n³¶v}

l

{|

n{8n

ovw~fnÇ{

l

n³|f{vwoYzyvw~F{n

¯¼zOn0|Fv}{v}|~<z»¶

l

v}

l

ovw~fvwov}©n³{

l n

oYzvwoxo n

| vw~

²

|K{

l

Vx

²

|oYzOvw~fÈx~

n { l

n#|~f{

zv±~F{{

l

zO{¶-n

l

z*®Kn{

l

nÈVzFon

zO8 oYf{8|K{v}|

n

|O¯on

l

|Fv±~F{·vw~fv

nJnpz

l

Vx

²

|o¹zOvw~%µÚ~{

l

v}z

{p»¶-n~fnKy}n{

{ l nz

{vÌËcvwzOy

²

|x~

z

|~

v}{v}|~n

| v±~

l n

np~K{{|¹{

l

nª1x~z

|O«¬xz

{n

|~fvTzOy}{n

~z{n

Ï Ð

|fn x

nKµ k¸l

v¨Bn

|

¶¸v¨y}y

²

n9{VzKÒOnp~#pz

n|O¯ÎywzO{8n

|~%µ

(9)

Ù ÃKÀ Á SÆ[À Â}Ã

%n{

7YZH[\T1V

Ï nf{

7*PSRRUT1V

Ð ²

n{

l n

n{

v}{v}|~|¯

7

{|

C.YZH[\TV

Ï nVf{

CQPSRRUT1V

Ð µ ¦ n

nË1~fn

{ l

n!¯°|Fy¨y}|O¶vw~f#n

|

Ê

Y&ZH[T1V.8

oYzO

7 YZH[\TV>:7 a

Y&ZH[T1V

PRRUTV 8

o¹zO

b

7PRRUTV.: 7 a

PSRRUTV

,

O np~K{n

n

«¼|

n

Ë1~fv¨{8n

vÌ×n

np~fn

l

nponYzKfy}v}n

{|

:

É

=

8

¶v}{

l

nzO{

v

vÓ«

l y}n{

²

|x~

z

|~

v}{v}|~fBKv}®Kn

YZH[\T1V.d

É ]

`K

X ` oYz

7

,

Ïºµq Ð

kl

nØzF~zOy}v}|¯T{

l

nBv±~~fn

Vx

²

|oYzOvw~zK

|OfvwoYzO{v}|~¹¶v}{

l

ov}n

nzO{

²

|x~

z

|~

v}{v}|~fFv¨®FnÈ{'¶-|³{

x~fpzO{v}|~ n

|

p»¶

l

v¨

l

¶-n

l

|xfy

|~f8v n

vw~ z

v}{v}|~ { l n

v}

n{8v¨zO{v}|~n

|

|O¯ {

l

n¸npxoYzK~~

²

|x~

z

|~

v}{v}|~%µ

¦ n l

zp®Fn

PSRRUT1V.d

É ] `` Ï : X Ð `

oYzO

b

7

= ] `` :

oYzO

b

` 7

` E

ÏºµÓ

Ð

¶ l n n 8 = a

` 6 = a

! 6 Ï = a

` 6 Ð ,

¦

nË

8{~f|K{v}n{

l

zO{-{

l

n{

x~fpzO{v}|~n

8

| nË1~fn

z ²

|O®Kn

npºnp~

{

|~fFy¨#|~Ö{

l n

|fn

{'#|O¯Î{

l

n|Kywxf{v}|~{

l

zO{¶-n¶zK~K{-{|YzK

| fvwoYzO{nvw~ÖnpzO

l

x

²

|oYzv±~%µ

%n{

7J

²

n9{

l

n|xf{n

nzK~fv}|~Ö|O¯

7

zF~

#"

Ï EVÉÐ-²

n9{

l

n|

n{|

vgµn

" Ï

EVÉÐ 8 7 Ï

EVÉ8Ð :(7J

Ï EÉÐ

d

%$

Ï

:'&

ÐHE

vw~ { l n ²

|x~

z

AywzpFn

¶v}{

l & 8 K()

6

,¦

nÕ¶¸v¨y}y

l

|O¶@{

l

zO{Y{

l

nÕ{

x~fpz{v}|~ n

| v}

|o´v±~z{n ²

È{

l n ² n l

z*®fv}|

|O¯ {

l

n|

n{|

zO-vw~0Ï

$%

Ð µ

tfn|~

y}Ç¶-n

npoYz

ÒÖ{

l

zO{{

l

nYn

8

| v±~

²

|F{

l

Vx

²

|oYzOvw~fv}9|xfy}n ²

npzKxfn#{

l n

¸npxoYzK~~

²

|x~

z

³|~

v}{v}|~¯g|

{ l

nÕÆ

½c½

ÁpÀ

|oYzv±~v¨9|~fy}ÇzK~ÇzK

|OfvwoYzO{v}|~|O¯z

n

v}®OzO{v}®Kn#v±~{

l n

¾KÜ

ÅgÁpÀ

|o¹zOvw~%µtf|¶-nJ~fnn

{|#|oYxf{8n v

n{8y¨³{

l

nn

|

²

n{'¶-nnp~

{ l

nn z{Y|Kywxf{v}|~ |O¯Ø{

l

n|~K{8v±~fxf|xf

| ²

y}npo zK~

{ l

n#¯g|

o¹zOy-y¨vwov}{|O¯ÈÏµÌ

Ð ¶ l

np~

$

)+* ,f¦

n l z*®KnKÊ

(10)

à$ ]£

ÁpÅ 798

Ï 7P

Ð P J

K ` ÁÅÁ%

¾ ÂÜ

ÅgÆ

¾K½³¾

ÝJÅfÁÝ

¾

ÂbÂ

¾

Æ

½

Â¨Æ

½

ÁÃKÀ %4%ÅgÁ ,

4a

7 PK 8@ A8

E ,, ,&e :

4 a

7P

`

8@ A8

E ,, ,&e :

7J

K8I,J

G

7 K8

7J

` G

7 `

8ILK

7,K

` :

7J

`

] ` 8

7 K:

7 (K

K

] K E

Ïºµ

%KÐ

Á*À'Á

4 a 7 P 8;:

É

7*P

K>:

7*P+= 7P(K

] = 7 P ,

ÁpÅ ÁÅÁ

#¾

,

¾

%ÆbÅ

ÀÆ

Û 8

YZH[\TV

PSRR/T1V

ÆbÅ

3

YZH[\TV 8

PK 8A Ï

ÐHG A8

*+E ,, ,&e

PSRRUT1V 8

P` 8 X = A Ï K(

ÐHG A 8

*+E ,, ,1e

ÁpÅ Ü % % Ü

¾

%ÁÅÃKÅ

e

(K

d

É

ÆbÅ

,

4Á*Å! + 8ÁÅÁ,JÃÄ Æ-,

Ü ,

½c¾

À ,

¾F½

ÅfÁ

#¾

,

¾

%?ÆbÅgÁ

ÀÆ

Û

"T½¿Û

ÁpÀÅÁ 1À'Á$#pÆ

¾KÜ

% ¾

ÅÁ&%Æ%

¾K½

ÅÁ

Û

Æ.%

#

À'ÁpÅgÆ&%ÃKÅgÆ

¾F½

Ã

½¿Û

Ã1À

¾

ÄOÆ,ÃKÅgÆ

¾K½·¾

Ý]ÅÁ

¾

ÁpÀ'ÃKÅ

¾ À %JÆ

½

ÁVÃ

# % Ü ÛF¾

,ÃKÆ

½

' 7_:

7

! Æ.%Ã%4+,1Å

¾

ÅgÆ

#

ÃFÂ[Â ,Æ

½

Æ-,

Ü ,

Á

½

: X)(

É

y}|K4Ï É ( `

Ð,

*9¢pä%ä ( Ê kl

nnfv}{np~fnzK~

x~fv2xfnp~fn|O¯

7

¯g|Ky}y}| ¶¸ ¯

|o·{

l

noYzOfvwoJxo

vw~fvwfy¨nKµ 4n{

7 ²

n{

l

nnzO{ |Kywxf{v}|~9|O¯%{

l n v

v}

l

y¨n{-

| ²

y}npoÊ : É 7 <

="798@

G 7 Ï

*KÐ 8I,J

G 7 Ï Ð 8

ILK

,

%n{

P 8 7P

:

7P

A.8

*+E, ,,&e E,+

8 E E ¶ l n n 7P

v}{

l

nY{

znY|O¯

7

|~{

l n

|o¹|Kv}{B

v

Y&ZH[T1V

PSRR/T1V

µ

(11)

¦ n l

z*®Kn

: É P K K:

% P

K= P( K K

] `K = P

K>8':

É ] `K

7

Ï

P

ÐHE

: É P K ` :

%

P` = P( K `

] `` = P`

8':

É ] ``

7

Ï & P Ð E

J K8

*9G

` 8

*FE

KL8 J ` E

K ` : J `

] ` 8 K : ( K K

] K = ] K

7 ` Ï Ð = ]*`

7 ` Ï

&

ÐHE

¶ l n n P(K

K P

PK G P(K

` &/P

P` G ( K K

K G J `

&

K ` ,

¦

n¹Vfy}v¨{¹{

l

nYn

| P

vw~F{|Õ{'¶-||oYº|~zK~F{

P zF~

P

|Kywxf{v}|~f|O¯-{

l

n¯g|Ky}y¨|O¶vw~f

y}vw~fnpz

f{npo*Ê

: É P K K>:

P K=

P(K

K

] `K = P

KL8 : É ]

`K

7 Ï

P

ÐHE

: É P K ` : P` = P(K

`

] `` = P` 8 : É ] ``

7 Ï& P ÐHE

J K8

*9G

` 8 *

K 8 J `

K ` : J `

]`

8

K:

(K

K

] K E

: É P K K:

P K=

P( K K

] `K = P

K>8

*+E

: É P K ` : P` = P( K `

] `` = P` 8

*+E

J K8

*9G ` 8

*FE

KL8

J ` E

K ` : J `

]`

8 K : ( K K

] K = ] K

7 ` Ï Ð = ]*`

7 ` Ï

&

Ð,

b¯

PK

v}~f|K{BoYzOfv±o¹zOyTzO{-{

l n ²

|x~

z A8

e { l

np~

oYzO

P J

PK

K8

É ]

`K

o¹zO

U7

(12)

B{

l n

¶v}nK»c¶-n n

xfn!¯

|fo

K ` : J ` 8 ]`

] K Ï

K : (K

K ÐHE

{ l z{

P`

pzK~~f|K{

²

nJo¹zOvwoYzy4z{{

l n ²

|fx~

z A8

* µ ¦ n l

z*®Kn{

l n

n¯g|

n

oYzO

P` `

8

Ï'

: X Ð ` É ] ``

oYzO

/7

,

|f~fn& xfnp~F{y}K»¿¶Ñn

l

z*®KnKÊ

oYz1Ï K E `

Ð,

|O¶Ay}n{Ñxf|~fv n { l

nn

| { l

zO{ |on¯

|o·{

l n

v}

n{8v¨©pz{v}|~Y|¯1{

l

n¸npxoYzK~~

²

|x~

z

|~

v}{v}|~%µ

ª

|o{

l

nJoYzvwoxo

v±~fvwfy}nK»¶-n

l

zp®Fn PK ²

|x~

n ² K 8 J ` ,

¼~³z

v}{v}|~

Ï % «¼y¨npo¹onq

Ð Ê K :

(K

K

] K d J `

] K ,

¦

n9y}||Ò´¯g|

P

`Bvw~{

l

nB¯°|Ky}y}|O¶vw~fY¯g|

oÖÊ

P`

8 K P = `

(+P

E

¶ l n n Ï1]`

EVÉÐ

v}{

l

ny±z

Kn

|f|K{|¯T{

l n xz

z{v}|Ky}~f|ovwzOy

` : Ï = ] ``

É Ð = 8

*-,

¦

n9|

²

{VzOvw~

K

zK~

`Ô¯

|o{

l n ²

|x~

z

#|~

v}{v}|~fpµ-ª

|o { l

v}nfy}v¨v}{!¯°|

oJxfywz9¶-n

pzK~Ö|~fywx

n{

l

zO{

P` v}Øz

n

npzOvw~f¹¯Þx~f{v}|~Ö|O¯

A

zK~

{ l

z{pÊ

J ` d

8 É ] K

7 `

Ï Ð,

¦ n l

z*®KnË1~zy¨y}%Ê

oYzOÏ

K E ` E

ÐHE

A

] ÏºµÓ

Ð

K d É ] `K

Ï'

= É ( `

nÏ

:

É ÐHE

Ïºµ

#KÐ

` d É

(K

] `K E

ÏºµÓ

Ð

d É ] K

Ï'

= É ( K

n Ï

:

É Ð,

Ï µw

*KÐ

(13)

np~fnK»¿vÌ¯

É

v}BVx&Èv}np~K{8y¨VoYzOy}yS{

l

np~

`Bv}ØzF~#v±~f

npzOvw~f#¯Þx~f{8v¨|f~Õ|¯

²

zK~

K

z

¶-ny}yzO

z n

n

npzOvw~f9¯Þx~f{v}|~f¸|O¯

² µ g{-v}npzO{|

l

|O¶ { l zO{{

l

nØzOpx

zOYv}

²

n8{

¶ l np~

( É

y}|K

É ( ` ,

kl

n9n

| v}B{

l

np~

d

É,

Ï µwK

Ð

!

$1à¤Oä4å $%¡à &¥

£ &

jä (B£c¢*¡

¤

£$0Säå2£c¢páAà¢O¢*ä1¢O

kl

nY|~F®Kn

Knp~fn|O¯-{

l

nÈon{

l | ³

npnp~

n8np~K{vwzOy}y}Ç|~{

l

n¹¶Ôz*ÕzK~n

| ¶ l v}

l

v}Bv±~F{

|

xfn

z{{

l

nz

{8vÓËcvwzOySvw~K{8n

¯Þzn

|zOz{n-v±~f8v

n{

l

n9Vx

²

|foYzOvw~%µ

kl

ntf

l

¶z

©v}{n

zO{nÑ

|n x

nzKfy}v}n

{|{

l

n{'¶-|Øº|Kvw~K{

²

|fx~

z

®OzOywxfn

| ²

y}npo

: < 8;@

E Ï

*FÐ 8;IJ

E

Ï'

Ð

8;I K E

¯g|

n zFoYfy}nK»

l

zO®Kn

Õº||

n&#v¨np~f

²

npzKxfn¯g|

X E

*+E

TzK~FKv}®Knp~ÖVx

²

|o¹zOvw~³|O¯

$*

»w xfn

vw~{

l

nJtf

l

¶Ôz

©v}{n

z{v}®KnJ

|fn x

nK»

zK~FYn

{x

²

zO{v}|~|¯1{

l n ²

|x~

z

|~

v}{v}|~#zO{

²

n

npzOny}vw~fnpz

y}#vw~fv

n{

l

n¸Vx

²

|O«

oYzv±~

z»

²

µ g{v}~f|y}|~fKn

{ l

npzO8n¯g|

z!n|~

| n

vw~fxfywz

n

{Vx

²

zO{v}|~#

| ²

y}npo

{ l z{

l

z*®Kn

²

|x~

z

ywz*Kn

¶¸v¨{

l

¯¼zO{n|~fnp~F{vwzOy}y}

npzpKµ¼~

$%

Tv}{Bv}

l

|O¶~#{

l

zO{¶-n

oYz*K»Fv±~´¯¼zO{p»

l

z*®Kn¯Þz{ |~F®Kn

Knp~fnB¶v}{

lY

nywzO{8v¨®Fny¨¹VoYzOy}y|O®Kn

ywzKn®Fnp~vÌ¯T¶-nÔzFfy¨¹{

l n

{

zOv}

l

{'¯°|

¶z

tf

l

¶z

©zOy}{n

~zO{n

|n x

n9¶v}{

l

v v}

l

y}n{

²

|x~

z

|~

v}{v}|~f*µ

¦

n9¶v}y}yÎ

|f|O®Kn{

l

zO{-{

l

n9ªµÓ¬r

|fn x

nJo¹zpzOy}|

l

z*®KnB¯¼zO{¸|~K®Fn

Knp~fnJzF~

¶Ñn¶¸v¨y}y

|o¹z

nv}{¶¸v¨{

l { l

nJtf

l

¶z

©ØzOy}{n

~zO{vw~fI

|n x

nKµ

%n{Îxf |f~fv n { l

nªµÓ¬ v}{n

zO{v}®Kn

|fn x

nzKfy}v}n

{|B{

l

n ¯g|Ky}y}|O¶vw~f

l

|o|KKnp~fn|fxf

| ² y¨npo

4

aK P K 8

*9G

J K 8

*_G

K 8

(K

J ` G

4 a` ` 8 *

K ` : J `

]*`

8 K :

(K

K

] K G

` 8 * G

¶v}{

l z

|oYzOvw~

n|oY|Fv}{v}|~Kv}®Knp~

²

0Ï4npoYoYz

Ð

»vgµn 8 : X d É

y}|K

É

(K

,%kl

n

v}

n{8v¨©n

|n

z{|

VzO{v}'ËcnBzYoYzOfvwoJxo

vw~fv±fy}nÈzF~

¶-npzK~#

l

|O¶{

l

z{pÊ

J

` 8

K

K X P J

(K

K

PK

(14)

,ÃKÆ Á , %Æ[ÅgÆ Ý À1Æ Â}ÃKÀ-ÁpÀÅ À ÃFÅgÆ -À Â}Á ,% F

zK~

J ` X P J

P` ,

¦

n9¶v}y}yTpzOy}y

zFoYfvw~fY¯¼zO{|

z

npzOy ´xf

l { l

zO{pÊ

K

J `

E

$ ,

à$ ]£

ÁpÅ Ã 8ÁÅfÁYÆ

½

ÅgÁpÀÝ?Ã

#

Á

¾

%Æ[ÅgÆ

¾K½

8ÁpÅ

ÁÁ

½

ÅfÁ%

Ü ÛK¾

,ÃKÆ

½ % % Ü #

IÅÃFÅ

: X d É

K

`

y}|K

É

(K

, Ü

¾

%ÁJÅfÃKÅ

e

(K

d É

E

ÆbÅ

,

fÁ

½

ÅÁÃ1,1Â}Æ' # ÃFÅgÆ

¾K½

ÝÃ

# Å ¾ À ¾

ÝÅÁ¹Æ[ÅgÁ*ÀÃFÅgÆ#*Á % #

fÁ ,ÁÆ.%

d

(K

,

*9¢pä%ä (

Ê y}n{

7 8 Ï

7'

P Ð P J

K ` ²

n9{

l

n8|Kywxf{v}|~|O¯

4 a

7 PK 8 @ A8

E , ,,1e : E

4a

7 P` 8 @ A8

E , ,,1e : E

7 J K 8 IJ

G

7 K 8

7 (K

J ` G

7 ` 8 I K E

7'

K ` :

7'

J `

]`

8

7

K :

7

( K K

] K E

%n{ÈÏ

P Ð P J

K `

nË1~fn

zO

P 8

7 P : 7P

¶ l n n 7W8

Ï 7*P

Ð P J

K

`Bv}

{ l

n9|Kywxf{v}|~Ö|O¯ÔÏµ

%KÐ

µ

¦ n l

z*®KnKÊ

4ca

PK 8 * A 8

E, , ,1e : G

4ca

P` 8 * A 8

E, , ,1e : G

J K 8

*9G

K 8 ( K

J ` G ` 8

*+G

K ` :

J `

]*`

8 K :

(K

K

] K ,

%n{

²

n{

l n

zKoYfvw~f¯¼zO{|

¯g|

{ l

n|fxf{n

|oYzOvw~¶¸v¨{

l%

v v¨

l

y}n{

²

|x~

z

|f~

vÓ«

{v}|~

nË1~fn

zO¸v±~

$%

»f¶-n

l

z*®Kn

( K K 8 K

¶¸v¨{

l

Kµ

kl

np~

K ` :

J ` 8 ] `

] K Ï

K :

(K

K Ð 8 ] `

] K Ï :

Ð K 8 ] `

] K Ï'

:

Ð ( K

J ` , Ï µw Ð

%n{

²

n{

l n

t#|O¯Ïµw

Ð

»K¶-n¶v}y}yS

l

|O¶{

l

zO{JÏ

J ` Ð

v}

n

npzOvw~fvw~Õo|

xfy}nKµ

P`

pzK~

²

n9¶

v}{n

zO

P ` 8

K P

Ï1] ` EVÉ8Ð

= ` (+P

Ï1] ` EÉÐHE

(15)

¶ l n n Ï1]`

EÉÐ

v}{

l

n9y±z

Kn

|f|K{-|O¯ { l n xz

zO{v}|Ky}~f|ovwzOy

`Q:

Ï

= ] ``

É Ð = 8 *

ª

|o{

l n ²

|x~

z

|f~

v¨{8v¨|f~fp»c¶-n|

²

{zOvw~

K

zK~

`

3 `

8;:

K `

JÏ1]*`

EÉÐ

K 8

Ï Ï]*`

EÉÐ

: Ð Ï = `

(K

Ï1]`

EÉÐÐ

¦

n9pzK~#¶

v}{n

J ` 8 : `

Ï1]*`

EÉÐ

= `

Ï1]*`

EÉÐ

Ï1]`

EÉÐ :

8

(K

J ` E

¶ l n n Yv}JÅÁÃ1,1Â}Æ' # ÃKÅ

¾

ÀSÝÃ

# Å ¾ À ¾

ÝÅÁÆ[ÅgÁ*ÀÃFÅgÆ#*Á ,Á*Å

¾pÛ

|~fn&xfnp~K{y}

8

]*`

] K Ï'

:

Ð : `

Ï1]`

EÉÐ

= ` ( K

Ï1]`

EVÉÐ

Ï]*`

EVÉÐ :

ª

|o

e d É

»f¶-n n

xfn{

l

nzOoYf{|K{8v¨

² n l

z*®v}|

|O¯ È¶

l

v¨

l

v}

d

(K

,

! ¡

¢Oå ^¡à &¥ä %à ¢ ¿à ¤OàA¡ä¡ %à ^ä 01¡

ä rä (¹¡ 1à

&T¢*ä1(0à$ £å

(¼ä%¢ £ & &T¢*ä

]£¡

ä

1àfä1¢*à

4Á*Å

7

8ÁÅÁ%

¾ ÂÜ

ÅgÆ

¾K½³¾

ÝJÅÁ Æ[ÀÆ

#

Â}ÁpÅ 1À

¾

^Â}Á ,

4 7 8;:

É 7 <

=?798@

G 7 Ï

*KÐ 8I,J

E 7

Ï'

Ð

8I K

ÁpÅ 7

Y&ZH[T1V

Ã

½cÛ 7'

PSRRUT1V

Û Á ' ½ Á Û

!YÅÁYÆbÅgÁpÀ'ÃKÅgÆ#pÁ%

#

Á,Á

4a 7

YZH[\T1V

8@ AB(C.YZH[\T1V

E

7

Y&ZH[T1V

Ï

*KÐ

8 IJ

G

7

YZH[\T1V

Ï1X

Ð

87

PSRR/T1V

Ï1X ÐHE

4a7

K

PSRR/T1V

8@ AB(C PSRR/T1V

E

7

K

PRRUTV

Ï

*KÐ

8I K E

7

K

PRRUTV

Ï1X

=

]*`

Ð

:(7

K

PSRR/T1V

Ï1X

Ð

]*`

8 7 YZH[\T1V ÏX

Ð : 7 YZH[\TV

Ï1X : ] K Ð

] K E

(16)

,ÃKÆ Á , %Æ[ÅgÆ Ý À1Æ Â}ÃKÀ-ÁpÀÅ À ÃFÅgÆ -À Â}Á ,% *

Á*À'Á

4 a 7 8;:

É

7P

K>:

7P =?7P(K

] ` = 7P

,

ÁpÅ

7 8 3 7

Y&ZH[T1V

Í B

YZH[\TV 8

P K 8A Ï

ÐHG A8

*+E ,, ,&e

7 PRRUTV

Í B

PSRRUT1V 8

P` 8 X

=?A

Ï K ( ÐHG

A8

*+E ,, ,1e

ÁpÅ! ÁÅÁ,ÃOÄOÆ, Ü ,

½c¾

À ,

¾K½

ÅÁ

#¾

,

¾

%?ÆbÅgÁ

ÀÆ Û

YZH[\TV

PSRRUT1V

ÁpÅ Ü % % Ü

¾

%ÁÅÃKÅ

e

(K

d

É

ÆbÅ

Ã

½¿Û

( É

y}|K1Ï É ( `

Ð,

fÁ

½

7_:(7

Ï

= É Ð8ÐHE

d ( K , Ï µwp

Ð

*9¢pä%ä ( Ê ¦ n l

z*®KnÈ

l | ¶B~³vw~ %npoYoYzÈ{

l

zO{

P 8 7P

:

7*P

A 8

*+E ,, ,&e E + 8 E E

¶ l n n 7 v}{

l

nYnzO{8|Kywxf{v}|~Ç|¯Ô{

l n v

v}

l

y}n{È

| ²

y¨npoÏºµw

Ð » 7P

v}{

l

nY{

zOnY|O¯

7

|~j{

l

nÈ|o¹|Kv}{Y

v YZH[\TV

PRRUTV

zK~

7 8 Ï

7 P Ð P J

K ` 8|Kywxf{v}|~j|O¯´Ïºµ

%KÐ

Vz{v}'Ëcn´ÏµwK

Ð µ

%n{

7YZH[\TV 8 Ï 7P

K Ð P J

Ï nVf{

7PSRR/T1V 8 Ï 7P

` Ð P J

Ð

zK~

YZH[\TV

87

YZH[\T1V

:

7YZH[\T1V

Ï

nVf{

PSRRUTV 87

PRRUTV

:

7PSRRUTV

,ÓÐ

¦ n l

z*®Kn

4 a Ï

YZH[\TV

Ð 8

*FE

|~

C.YZH[\TV

Y&ZH[T1V

Ï1X

Ð 8 ( K

PSRRUT1V

ÏX

Ð G

Y&ZH[T1V

Ï

*KÐ 8 *

4

a

Ï

PSRR/T1V

Ð 8

*FE

|~

CQPSRRUT1V

PRRUTV

Ï'

Ð 8 * G

PRRUTV

Ï1X

=

]`

Ð :

PSRRUTV

Ï1X

Ð

]*`

8

Y&ZH[T1V

Ï1X

Ð :

YZH[\TV

ÏX : ] K Ð

] K ,

¦

n9|~fywx

n9¯

|fo y}npoYo¹z {

l

zO{*Ê

YZH[\T1V

|~Ö

E

|~#

E

zK~

{ l n

n¯g|

n

7_:(7

Ï =

ÉOÐ,

(17)

q Ù ÃKÀ Á SÆ[À Â}Ã

OB|

vw~f#{|Y{

l

nn{8v±o¹zO{n|O¯Ï'

Ð

»f¶-n|

²

n

®Kn{

l

zO{Õoxf{

²

n

l

||K8np~Vxf

l { l

zO{

d

o¹zO4Ï YZH[\T1V

E

PSRRUTV

Ð!,

Þ{v}vw~K{n

n{vw~fY{|Y~f|K{v}n9{

l

zO{{

l

n|~K®Fn

Knp~fnVºnn

|O¯Î{

l

nªµÓ¬·

|fn x

nv}~f|F{Ôz

K|f|

z{

l

n|~K®Fn

Knp~fnVnn

|O¯Ñ{

l

n{

zOv}

l

{'¯°|

¶z

tf

l

¶z

©zOy}{n

~zO{8n´

|n x n

¶v}{

l

ovw~fv±oxo | ®Fn

ywzK%µrªc|

^e (K

d É

y¨|F4Ï É ( K Ð

»¶-n

l

zp®Fn

d

É

Í ( ` Ï É

(K

Ð Ï

$%

»

kl

n|

npo Ð

»-vw~f{npz

|O¯

d

Í ( K ÏÉ

(K

Ð,

xf{È|f~A{

l

n|K{

l n

v

n³{

l

nªµÓ¬v}{n

z{v}®Kn

|fn x

n9v¨z

½¿¾K½'¾

#pÁ*ÀVÂ}Ã%Æ

½

|oYzv±~

n|foY|K8v¨{8v¨|f~]o´n{

l | µ

%n{xfzKfy}{

l

n

n8np~K{

v

v}

l

y}n{'«¸npxoYzK~~¹

l

npon-{|{

l

nÑ8v±o¹fy¨nBo|

ny

| ²

y}npo

:

É

= 8 Ï =

ÉÐ

|F Ï Ð E Ï

*+E

ÐHE Ï

*KÐ

8 E

Ï'

Ð 8

=

ÍÏ'

Ð!,

Ï µwq

Ð

ª v}x

n#

l | ¶¸{

l n

npºnp~

np~fnY|O¯ {

l

nn

8

|

vw~Õo¹zOvwoxo@~f|

o zOz9¯¼x~f{v}|~|O¯ { l n

vw~F{n

¯¼zOnÕ|K8v¨{8v¨|f~A¶v}{

l

{'¶-|Õ{vwon *

v¨8

¦ n l

z*®Kn

l

n?Òn { l

zO{

{ l

n|f{8v±o¹zOyTvw~F{n

¯¼zOnY|K8v¨{8v¨|f~³|

n|~

B{|#{

l n ²

zOywzK~fnY|O¯{

l

n9n

| vw~

²

|F{

l

Vx

² «

|o¹zOvw~fpµ-ª v}x

n

npo|~f{

zO{n{

l

z{{

l

noYzOy}y}n

v}

É

»c{

l

n¯¼zO{n

v}B{

l

n|f~K®Kn

Knp~fn

0.55 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

0.02 0.04 0.06 0.08

position interface

maximum error

epsilon= .01 epsilon=.001 epsilon= .0001

ª v}x

nYKÊS×n{-|O¯º|Kv}{v}|~Öv±~F{n

¯¼zOn9|~oYzOfvwoJxo n

|

|O¯{

l

nv}{n

zO{8v¨®Fn´

|n x

n{|O¶z

{ l

n|Kywxf{v}|~|O¯ { l n

v}

n{v}©n | ²

y}npoµ

¦ n l

z*®Kn

|o¹fy¨n{8n

|x

~fxon

v}pzOy%vw~K®Fn{v}zO{v}|~¹¶v}{

l { l

nB|

v}Kvw~zOycªµÌ¬µK

l

nponB¶

l

v}

l

vw~fywx

n

z

nywzOzO{v}|~9|~!{

l

n-vw~K{n

¯ÞzOn-|~

v}{v}|~

G

ª v}x

n¸

l

|O¶S{

l

n-n×4n{S|O¯c{

l

v}

nywzO z{v}|~

zK~

¶-n|

²

n

®Kn{

l

zO{-{

l

n|f{vwoYzOy

nywzO z{v}|~#¯¼zO{|

K|fn{||~fnzO

É)+*-,

(18)

3 4 5

10 15 20 25 30 35 40

-log10(epsilon)

number of iterations

N= 10 N=15 N=20

ª v}x

n Ê~fxo

² n

|O¯ v}{n

zO{v}|~fÏb¶v}{

l

~f|

nywzO z{v}|~

Ð

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

20 40 60 80 100 120 140 160

relaxation parameter

number of iteration

- : eps=0.005 -- : eps=0.001 -. : eps=0.0001

: : eps=0.00001

ª v}x

nÊSn×4n{¸|O¯

ny±z zO{8v¨|f~Õz

zFon{n

|~~fxo

² n

|¯Îv}{n

z{v}|~f

Öà$(]£ %â ¢ ¤ 0wàf¡ p¤ 1à ³à

¦

nz

n¹K|Kvw~f³{|Õ

l

|O¶ { l zO{{

l

nÈ

l

|Kv}nÈ|¯Ô{

l n ²

|x~

z

Ç|f~

v¨{8v¨|f~fÈzO{{

l

nIz

{vÌËcv±zy

vw~F{n

¯¼zOn X9v}

v}{v}pzOygµ %n{-xf|~f8v n

~f|O¶>{

l

n-ªµÓ¬<on{

l |

¶v}{

l

npxoYzF~~

²

|x~

z

|~

v}{v}|~jzO{ Xv±~

C.YZH[\T1V

zK~

{ l n v

v}

l

y}n{

²

|fx~

z

Õ|f~

v¨{8v¨|f~0zO{ Xvw~

CQPSRR/T1V

µ

kl

n

(19)

l

npon9Kv}®KnpÊ

4ca 7

Y&ZH[T1V 8@

|f~

C

E

7

K

YZH[\TV

Ï1X

Ð

:7 K

YZH[\TV

ÏX : ] K Ð

] K 8 7

PSRR/T1V

Ï1X

=

]*`

Ð

:(7

PSRR/T1V

Ï1X

Ð

]*`

G 7 YZH[\T1V

Ï

*KÐ 8I,J

E

4ca7

PRRUTV 8 @

|~

C

E

7

PSRR/T1V

Ï'

Ð

8I K G

7

PSRR/T1V

Ï1X

Ð

87YZH[\T1V

ÏX Ð,

Ï µwp

Ð

k

|³{Vz

{9{

l

n#

l

npo´nK»T¶-n#vwoY|K8nÕzK~0z

{vÌËcvwzOy

²

|x~

z

Ç|~

v}{v}|~AzO{Y|Fv±~F{Yzµ kl

n

²

n{

l

|Kv}n¯g|

{ l

nBv±~F{n

¯¼zOn|K8v¨{8v¨|f~#v±~#{8n

o|¯ÎzOpx

zO¹v¨

: X d É

y}|K

É ( K

8v±~fn

{ l nB¯g|

o¹zOyy}v±o´v¨{B|O¯ÑÏµp

Ð

v}-v

np~K{8vxfn{8|{

l

nB¯g|

oYzyy¨vwov}{B|O¯ÑÏµÓ

Ð µ

xf{p»f¶-nØz

nF|Kvw~f

{|¹

l

|O¶{

l

zO{-{

l

v}

|n x

nv}{

l

np~

l

v¨

l

y¨³x~f{Vz

²

y}nKµ

1àfä1¢*à

4Á*Å

Ü

%Ã+% % Ü

,ÁJÅfÃKÅ ] K

]`YÃ

½¿Û

] K

É,

fÁ

½

ÅÁÃ1,1Â}Æ' # ÃFÅ

¾

À ÝÃ

# Å¾ À ¾

ÝÅÁÆbÅgÁpÀ'ÃKÅgÆ#pÁ%À

¾#

Á

ÛKÜ

À'Á %ÃKÅgÆ.%-'ÑÁ)%

( ] K

]*`

E

Ã

½¿Û

ÅÁ 1À

¾ # Á ÛFÜ

À'Á

Æ[Å

½c¾

À'ÁpÂ}ÃOÄOÃKÅgÆ

¾F½

Æ% Æ

Â Ü½

%?ÅgÃ Â}Á.

*9¢pä%ä ( Ê ¦

n¹|~fy}0~fnn

{||~f8v n { l

n¹ªµÓ¬

|fn x

nÈzKfy}v}n

{|{

l

n#

| ²

y}npo;¶v}{

l

|o´|KKnp~fn|xf

²

|x~

z

|f~

v¨{8v¨|f~%Ê

4 a

PK 8 * A8

E, ,,&e : E

4 a

P` 8 * A8

E, ,,&e : E

J K 8

*9G

J ` 8 K G ` 8

*+E

K :

(K

K

] K 8 ( K

K ` :

(K

J `

] ` ,

¦ n l

z*®Kn

K ` 8

J `

¶v}{

l

,ck¸l

np~

K :

(K

K 8 ] K

]*`

Ï

K ` :

J ` Ð 8 ] K

]*`

Ï'

:

Ð

(K

J ` 8 ] K

]`

Ï'

:

Ð

(K

K , Ï µw

%KÐ

%n{ ²

n{

l n

tÇ|¯9Ïµw

%KÐ

»%¶Ñn¹¶v}y¨y¸

l

|O¶ { l zO{Ï

K Ð

v}

v}®Kn

Knp~F{nKµ PK

pzK~

² n

¶ v¨{8n

zO

PK 8 K

P

Ï1]

K

EVÉÐ

= ` ( P

Ï]

K

EVÉÐ

(20)

¶ l n n Ï1]

K

EÉÐ

v}{

l

n9y±z

Kn

|f|K{-|O¯ { l n xz

zO{v}|Ky}~f|ovwzOy

`c:

Ï = ]

`K

É Ð = 8

*-,

ª

|o{

l n ²

|x~

z

v}n|~

v}{v}|~fB¶-n

n{n

ovw~fn K

zK~

` G

¶-n

l

zp®Fn

K 8 : `

Ï]

K

EVÉ8Ð

= `

Ï]

K

EVÉ8Ð

Ï]

K

EVÉ8Ð

:

J ` 8

(K

K ,

]ÅfÁJÃ , %Â¨Æ' # ÃKÅ

¾

ÀSÝÃ

# Å¾

À-|O¯ {

l

nv}{n

zO{v}®Knon{

l |

nv}{

l

np~

8 ] K

]*`

Ï'

:

Ð : `

Ï]

K

EVÉÐ

= `

(K

Ï1]

K

EVÉÐ

Ï1]

K

EVÉÐ

:

(K

K ,

¦

n|~fywx

nnpzOv}y}

)(

] K

]`

,

Þ{v} xfv}{Ôº|Kv

²

y}nB{|vw~K{

|

xfnz

nywzOzO{v}|~|~´{

l

nz

{8vÓËcvwzOy

²

|x~

z

|~

v}{v}|~Èz v±~

.T{|

n{

v}n®Kn{

l

n|f~K®Kn

Knp~fn|O¯ ªµÓ¬ on{

l | » l

|O¶Ñn®Fn

{ l

n®OzOywxfn|O¯{

l

v¨z

zFon{n

v}J~f|F{npzO8{|YË1~

v±~Knp~fn

zOygµ x

n ºn

vwonp~F{¶v}{

l

ÏµqcÊ¯Ñª v}x

nq Ð l

|O¶B{

l

zO{

{ l v}

nywzOzO{v}|~#¯¼zO{|

F|n{8|J©n

|ÈzO

É)+*

µ

+022*, $%)+-. /%0 %

Ú~{

l

v}Bn{v}|~%»¶-n|~fv n

zy}vw~fnpz

8n|~

«¼|

n {

zK~fov}v}|~

| ²

y}npo|O¯ {

l

nB¯°|Fy¨y}|O«

¶vw~f#{' ºnKÊ

4QK798;:

É 7 < =?798 @;A&B(C K8

Ï

*+EBÐHG

4

` 8 <

8 5A&BDC

` 8 Ï E ÐHG

7 Ï

Ð

8 Ï

ÐHG 7 Ï

Ð

8 Ï

Ð G

7 Ï

*KÐ

8

*9G

Ï'

Ð 8

*+E

Ï µwp

Ð

¦l

n n É

v}ÈzVoYzOy}y|Fv}{v}®KnÕz

zKo´n{n

» É

*+EÉ

J

¯°|

8|on

É J M *

µ¼~Az

v}{v}|~

¶-nYzOVxo´nJ{

l

nY|oYzO{8v

²

v}y}v}{'Ç|~

v}{v}|~%ÊzOy}y n

v}®OzO{v}®Kn|¯Ñª<® zK~fv}

l

vw~

* µ k¸l

v¨9®Kn