High-order finite element approximations of the Maxwell equations

(1)

$( * ' ,-/1'++ '23-*!$)0

(2)

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

&!" )..+,- B6I"!61"!2O 8=3U 6,% ", 1<) I6 N 6!) 1<) V.< 6**)% J). 6,% J"EO

"F!1).2 "#J),+.8+!)1.N+2

=G:WXYZKX[K\]^K_X]ZK]

&!+,1)%I25A<!E6,,&!+,1G)!0+ )O1<)W)1<)!*6,%.

(3)

$( * ' ,-/1'++ '23-*!$)0 !""#$%&%!'( )**+),-. )/0102300*41* )*3,))/06.-7038-)9*4%:0.)0; *.)/0,<)/*3-)9*4)/030 )*3=,2.-> <8; ?3*4@13@A@B3-.C8=,; *., *<.)*4)/010 -8-*.*4)/023,1<,)-*. *==-))00; )*+0?<+D- D91040.101 *.E3-1,9FG

th

E0+3<,39GHFH,) FIJFK +9 !"!#!$ %&'"&() +*3.*.GL

th

(*70=+03 FMNH -.B<1,?08);A<.2,39

(4)

!"#$%!$&!$'$'$($ )*+ %,! -,./0

(*$ '!% (##"#'(*'#/,%&-"#)&0

(5)

(6)

(7)

! "#$%& '( &)* $%! !# #,(-.!/!( !0( -#%( !0/! 12 .$3(-,).#- 4//3 ,/5 &(-6(7!

0/. 3%/2(& )5 12 #13%(!)57 !0). "#-89 !0/58 0)1 :#- 3-#,)&)57 1( ")!0 !0(

#33#-!$5)!2!#3$-.$(/;0</!!0(=5),(-.)!2#:>"(5!(?/5&:#-.0#")571(!0/!

&#)571/!0(1/!) .)5/5(57)5((-)57"/2).5#!#5%2$.(:$%'$!)! /5/%.#'(:$59

/1(@$/%%2 !0/58:$%!# 12&/)%2.$3(-,).#-A)8(B#! 0(,:#-0). 3/!)(5 (?!)1(

/5& !0(1/52 :-$)!:$%&). $..)#5. "( 0/&9 C). $5.!)5!)57 .$33#-! 0(%3(&1( 7(!

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

!$-5(&:-#1/5AO .$3(-,).#-)5!#/ :-)(5&9

"#$%&%)8(!#(E3-(..12/ 85#"%(&71(5!!#!0(-(.!#:!0(7-/&$/!)#5 #1F

1)!!((? ;949 P(%%)57.? Q9C9R9 O 0)%&(-.? O9R9 ,/5 P)%. /5& A9 C/11(- :#-

!0()-!)1(/5&(S#-!)5 -(,)(")57!0). !0(.).9

/1 7-/!(:$% !# !0( "0#%( TRUA 7-#$3 :#- / :-)(5&%2 /5& .)1$%/!)57 "#-8

(5,)-#51(5!9 A#.!#:!0(-(.(/- 0)5!0).!0(.).:#-1.3/-!#:/5/1')!)#$.3-#D( !

5/1(&03PVA? -$5 '2!0(TRUA 7-#$39 >0(03PVA 1((!)57.? 0(%& -(7$%/-%2?

"(-( (.3( )/%%2 $.(:$% /5& 1#!),/!)57 :#- "0) 0 "#$%& %)8( !# !0/58 !0( "0#%(

!(/1W 4//3? N55#? X$$&? 6)D/2/? C(58? O/5&(-? Y/-.? R%(E? >)!# /5& O0/,/-.09

<$-)57!0(3/.!:#$-2(/-.? 0/,('((5%$ 82(5#$70!#.0/-(#* (")!0/-(%/!),(%2

%/-7(5$1'(- #: #%%(/7$(.9 >0(2 "(-( 7##& #13/52?/%"/2. -(/&2 !#0(%3 ")!0

"#-8 #-!# 0/,(/ 0/! ")!0)5 /.("#-8"/. D$.!?"(%%? / ')! !## 1$ 09 !0/58

!0(1/%%W O(5/?U0-).?6)D/2/?</,)!?R%2#5/?X(1 #?A/- (%?Y)(?B#'/5&O/5&(-9

(8)

!"#$ %&#'($ !%)"*! %,-./*-/)$ !).. 0 12- , 3!!1 41!!)25-67 (-.8!%-.90 !1 (-2 ),,)$ : .5- -.5 ;-2 . 312 <) ./!%) %)-2! -.5 01*, 13 !%) /21*&$ !1 =>-. 312 12/-. 0 ./ 2)/*,-2 0&12! )>).!0 -.5 !1 ?1< 312 9))& ./ 6) ).> 21.6).!-,,8 1.0 1*07 @21<-<,8!%)610!)A) ! >)5 >)20 1.3216B129B-0&,-8 ./312!%)C. >)20 !8D0 <-09)!<-,, !)-67 = -6 )0&) -,,8 !%-.93*, !1 1*2 !2- .)2 ?-0 312 <) ./ -,B-80 %))23*, -.5). 1*2-/ ./$ )>).!%1*/%1*2!)-6B)2)*.51*<!)5,8!%),1*0 )0! . !%),)-/*)7 (8! 6) .'.0 %)5)B1*,5%->)<)).-.-,!1/)!%)25 A)2).!)E&)2 ). )B !%1*! !%) 32 ).50 = 6-5) !%)2) 1*!0 5) B129F G1.$ ?-<0 $ "). 0$ H-!8-$ ")0$ ;,-2)$ @-*,$I)5)2 -$@)!)2$;1.. )$J 6))$C210$( ,-$K-!$H-!L-$@->),$'>-$(-29*0$

M-0$N %-6 -.5O-.-7 P%) 20*&&12! -.532 ).50% & -2)612) !%-.-&&2) -!)57

Q&) -, !%-.90 !1 "). 0 312 -,, !%) R0% %) <1*/%!$ /*!!)5 -.5 119)5S !1 G1.

312 -,B-80 1A)2 ./ 6) -. )E!2- 52 .9 B%). = B-0 -5-6-.! !1 ,)->)S !1 @-*, 312

&21> 5 ./ 52-6- . - %1&),)00,8 *.52-6-! &,- )S -.5 !1 ;,-2) 312 6-9 ./ !%)

"*! % ,-00)0)>).612)3*.!1-!!).57

=-6%-&&8!1!%-.9K 1312> 0 ! ./6) .'.0 %)5)-,610!)>)28! 6)%)%-5

016) B129 !1 51 . !%)K)!%)2,-.507 = B1*,5 , 9)!1 6).! 1.68/2)-! 32 ).50 .

?*5-&)0!$ )0&) -,,8?1/ $M)2T$#)2/U$V21.$?-.5 $M)2--.5QWT7 M !-,,8$=-,01

6).! 1. 68 ).! 2) 3-6 ,8$ . ,*5 ./ 68 (*6$ ?12 $ @% , &$ M 9 $ @)! $ P ! -.5 #X<127 =-6/2-!)3*,!1 !%)6-,,3126-9 ./0*2)!%-!= 1.! .*)5 !13)),-!%16) .68%16) !87 N1B)>)2$!%) &)201. B%1%-0 1.!2 <*!)5 <83-2!%) 610!!1 !% 0!%)0 0 0 68 B 3)$ B%1%-0%-5!1<)-2!%)<2*.!13!%)*&0-.551B.01368 @%" -.5 5- 87 J.5B%1 0$ 13 1*20)$)00).! -,7 Y0139-$68-&1,1/ )0-.5,1>)7

(9)

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

! "#$%&'( $*&#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +

!, -./0123/44/5(1$*&#6! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7

!,! 8136&9/4/ $%&:1;#/$*6:</9&%/0123/44 ! ! ! ! ! ! ! ! ! ! ! 7

!,!, 8136&9/4/ $%&:1;#/$*6:6*# /0123/44 ! ! ! ! ! ! ! ! ! ! ! ! =

!> ?(:/%* 141@@%&2*:1$*&# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!>! A*#*$/'*B/%/# /:/$.&' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!>!, A*#*$/C&4(:/:/$.&' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,

!>!> A*#*$//4/:/#$:/$.&' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! > !>!D E*6 &#$*#(&(6F14/%G*# :/$.&' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! > !D H($4*#/&9$./$./6*6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! I 0 1-234&2-')")* *-22-3",-') -) ,54 )'*"6"33&'" 5 7 ,! "#$%&'( $*&#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7 ,!, 0123/44/5(1$*&#6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! , ,!> E*6 &#$*#(&(6F14/%G*#'*6 %/$*61$*&#*# 6@1 / ! ! ! ! ! ! ! ! ! ! ! ! ! ,> ,!D J(#;/KL($$1$*:/K6$/@@*#;:/$.&'6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ,M

,!I N#14O6*6&9$./'*6@/%6*&#1#''*66*@1$*&#/%%&% ! ! ! ! ! ! ! ! ! ! ! ! >=

,!I! P1C//5(1$*&#*# &#/1#'$3&'*:/#6*&#6 ! ! ! ! ! ! ! ! ! ! ! >=

,!I!, E*6@/%6*&# 1#''*66*@1$*&#1#14O6*6&9$./;4&<146 ./:/ ! ! ! >,

,!Q ?(:/%* 14%/6(4$6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! >I

,!Q! H#/K'*:/#6*&#14 1C*$O ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! >I

,!Q!, ?(:/%* 14'*6@/%6*&#1#''*66*@1$*&#/%%&%! ! ! ! ! ! ! ! ! ! ! >Q

,!+ R&# 4('*#;%/:1%G6 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! D

8 9) ')2-2,4) : -) ,54

H(curl)

; ')<'&+-)%="2-2 <() ,-')2 >7 >! "#$%&'( $*&#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! D7

(10)

! "#$%&'()$*+,)*-&)+& $/-0--)$+0# 12*#3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! 45

!6 78&#3+*8'&12*#*+19$+& $/-0--)$+0# 12*#3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4

!6!: ;$+1$)$'$#1 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4

!6!5 <2,91$)$'$#1! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 46

!4 =82$+>23 0332*#*+19$$%&'()$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 44

!? @'()$'$#1&12*#>$1&2)3+*819$3$ *#>/*8>$8A&%B$))$C0&12*# ! ! ! ! 44

!?!: D23 *#12#0*03E&)$8F2#B$&F+*8'0)&12*#3 ! ! ! ! ! ! ! ! ! ! ! 4?

!?!5 G$&F+*8'0)&12*#*+19$

H(curl)

/ *#+*8'2#,>23 8$123&12*# ! 4H !?! I)$'$#1&)'&182 $3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4H !?!6 E&033C0&>8&108$3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?J !?!4 79$&33$'-)K ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?: !L M*# )0>2#,8$'&8F3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ? "'()*+, '+-+*.(.-./()*+(./ 01- /2**.(-) 45 67 6!: @#18*>0 12*#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?4

6!5 7$33$))&12*#&#>+0# 12*#3(& $3! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ?H

6! D23 *#12#0*03E&)$8F2#>23 8$123&12*#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! LJ

6! !: D$82N&12*#*+19$-2)2#$&8+*8' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! LJ

6! !5 O0'$82 &)P0%$3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! L:

6!6 I%()2 21(&8&'$1$8 &#>$88*8$312'&1$3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! L6

6!6!: =*0#>3+*819$)2+12#,*($8&1*8 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! L6

6!6!5 EQ8>2#,2#$C0&)212$3&#> *#12#021K$312'&1$3 ! ! ! ! ! ! ! ! ! LR

6!6! S(12'&)N&)0$+*819$($#&)1K(&8&'$1$83 ! ! ! ! ! ! ! ! ! ! ! H?

6!6!6 M*#N$8,$# $*+19$=8$TT21K($DE'$19*> ! ! ! ! ! ! ! ! ! ! HL

6!4 O0'$82 &)$%($82'$#13! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! HH

6!4!: U9&8(#$33*+19$(&8&'$1$8 $312'&1$3 ! ! ! ! ! ! ! ! ! ! ! ! ! HH

6!4!5 "3K'(1*12 *#N$8,$# $ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! HR

6!? M*# )0>2#,8$'&8F3&#>*01)**F ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! RJ

7 45 8/#9:9,. ;<=

4!: @#18*>0 12*#! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !:J

4!5 79$B$&F+*8'0)&12*# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !:J?

4!5!: G$&F+*8'0)&12*#*+19$

H(curl)

/ *#+*8'2#,>23 8$123&12*# !:JL 4!5!5 G$&F+*8'0)&12*#*+DE/VIA ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !:JL

4!5! 79$$#$8,K#*8' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !:JR

4! U1&-2)21K*+19$3$'2/>23 8$1$3K31$' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !::J

4!6 72'$/2#1$,8&12*#'$19*>3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !::5

4!6!: <0#,$/W011&'$19*>3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !::5

4!6!5 M*'(*3212*#'$19*>3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !::

4!6! V*0819/*8>$8,)*-&)<2 9&8>3*#$%18&(*)&12*# ! ! ! ! ! ! ! ! !::4

4!4 O0'$82 &)$%($82'$#13! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !::L

4!4!: M*#N$8,$# $&#> *'(&823*#*+($8+*8'&# $ ! ! ! ! ! ! ! ! !::L

4!4!5 O0'$82 &)>23($832*#&#&)K323 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !:55

(11)

' () +,-.()- /)0 12 (332)0/4.()- 565

7.8+.(91/:;< 566

(12)

(13)

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

! "#$%&'( $*&#

!"#$%&'&( !"#$%&#'()!*+

,#&&-./0.1./234 ! "##$ %&'(&%) *+ ,-.)/ ) 1*%2( 3*4&( 4-&

5'67&22 &89'4/*!) ') 4-& :%&'4&)4 ;'4-&;'4/ '2

&89'4/*!<)=*+'224/;&)> 4 4*??&(' 2/)4@)-*7! /!

A/:9%& B>C @ 4-'4 /! 29(&()9 - D&44&%EF!*7! *;E

?&4/4*%) ') G&74*!H) )& *!( 2'7 *% I/!)4&/!H) +'E

;*9)

E = mc

2

> J94 /) 4-/) -/:- )4'!(/!: K9)4/E

L&(M ,%*D'D2. .&)> N-& 5'67&22 &89'4/*!) (&E

) %/D&&2& 4%*;':!&4/ 7'3&)'!(4-&/%?%*?':'4/3&

?%*?&%4/&)> N-&2'7)*+&2& 4%*;':!&4/); O/! 4-&

7'.4-'4P';&)Q2&%F5'67&22+*%;92'4&(4-&;/!

4-&)& *!(-'2+*+4-&BR4- &!49%.O:'3&%/)&4*%'E

(/*4& -!*2*:.@ +%*; 7-/ - /!+*%;'4/*!4& -!*2*:.

2'4&% &3*23&(> 4 7*92( D&3&%. (/S 924 4*('. 4*

/;':/!&4-&7*%2( 7/4-*94'224-&'??2/'! &)@ ;'E

-/!&)'!(4& -!*2*:/&)4-'4 '%&D9/24*!4-&?%/!E

/?2&) *+ &2& 4%*;':!&4/ 4-&*%. '!( -'3& )9 - '

;'K*% /;?' 4*!*9%2/3&)> N-&)&/! 29(& )'4&22/4&

'!()4&'24-4& -!*2*:/&)@;':!&4/ %&)*!'! &/;':E

/!:<5T =) '!!&%)@3'%/*9)-*9)&-*2('??2/'! &)*%&3&!2/:-4!/!:2* '2/)'4/*! !

O

4*!';&D94' +&7> <U&& A/:9%&B>">=

!4%94-@ '2;*)4'22*+4-&2'7)*+&2& 4%/ /4.'!(;':!&4/);?%&('4&5'67&22H)

7*%F> V/) *!4%/D94/*! ?%*3&( :%*9!(D%&'F/!:+*% 47*@)4%*!:2. %&2'4&(@ %&')*!)>

!!"#$$" %&' &)*+,-. */$ )&$ */0

!

!" #$%! ' ()*%!+' "%$' ' " *' +%,,- +!.' $' " .'/ +!0' +!1 %0,' .(/' .' $)+!'23' $' *$' +" ' ,-%

(14)

!"#$%&'() !"#$$!%$% '()*)+'% ,*)$)-(%!',#' %"%.-%/(*0#.'',#*1! ')',%%$% '.)2

"#-*%'( ',%).3)+ 4#56%$$7 8'#.'(*- +.)" ',% ')0$%+' ).*%.#*/ 0.) %%/(*- $) 16(!%9

',%3!,)6# !#'%$$('%/(!,9 #!'%#$', :)":%.9 # $(-,'*(*-!'.(1%;',%'% ,*)$)-3 (*<)$<%/(!

',%$) #$(!#'()* )+#$(-,'*(*-!'.(1%=9#"( .)6#<%)<%*#*/#*4>?! #**%.7

!"#$%&'()*!+'!,-.(/0"12# !" 4!$56657%811#$5&6!#81*56!,9 &1$711,$81161 :

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

C)71?1"% #)6?!,-$81 =5>7166 1A45$!),#)+$1,/")?1#$) &15 +)"(!*5&61 $5#9%

1?1, 781, 5*?5, 1* (5$81(5$! # !# 4#1*< ., 1>5 $ #)64$!), !# ()"1 )+$1, $85,

,)$4,5$$5!,5&615,*),185#$)"1#)"$$) 5//")>!(5$!,-$81#)64$!),&'(15,#)+

5,4(1"! 56*!# "1$!#5$!),(1$8)*!,#$15*< D,&")5*$1"(#%$8!#$81#!#!#5&)4$$81

,4(1"! 56*!# "1$!#5$!),)+$81=5>71661A45$!),#< B81"15"1?5"!)4#1>!#$!,-#4 8

$1 8,!A41#7!$8?5#$6!$1"5$4"1< E8! 8),1!#&1#$$)4#1/"!(5"!6'*1/1,*#),$81

/8'#! 56 /")&61(),1 ,11*#$) #)6?1&4$ 56#)),)$81"% 61### !1,$!; % +5 $)"##4 8

(15)

!"#$#%& (#"# &) !"*$# &) $)(&(#+,-"./$) 0)$ ""-"1") 1" !&* 23+456789 ,

:;"&%0)$ ""-"1") 1" !&* 25678 !, ,--&<# !"*$# ." "2$="=,;;.&>$1, $)?8

.";."#") , $&)&% !"0"-*# &@"*$# &) $)(&(#= A $#,."-, $B"-:)"< &1;( ,

" !)&-&?: !, %&. ". ,$),;;-$ , $&)#&C".#,)(1@".&%,*B,) ,?"#&B".">$# $)?

)(1".$ ,-1" !&*#=

D!" ."1,$)$)? ;,. &% !$# !,; ". ."B$"<# #&1" &% !" @,#$ &) "; # !,

()*".-$" !"23+48567*$# ." $#, $&)&% !"7,><"--"E(, $&)#= F !$## ,?"9 !"

"1;!,#$#$# )& &) ;.&B$*$)? .$?&.&(# *"0)$ $&)# ,)* *".$B, $&)#G !, $#-"% &

-, ". !,; ".#= H, !".9 !"?&,-!"."$# &?$B",)$) ($ $B"&B".B$"<&% !"

7,><"--"E(, $&)#,)* &% !" )(1".$ ,- *$# ." $#, $&)# !, ,." 1&# :;$ ,--: ,;;-$"* &

!"19<$ ! !"%& (#@"$)?&)2*$#8 &) $)(&(#567=D!"&.$?$),-."#",. !;."#") "*

$)#(@#"E(") !,; ".#$#,-#&&( -$)"*=

!" #$% &'()%** %+,'-./01

I" )&< ." ,-- !" -,<# &% "-" .&1,?)" $#19 0.# ,# !": <"." /)&<) @"%&."

7,><"--J# <&./ ,)* !") $) !"%&.1 !": &&/ ,# , ."#(- &% !$# <&./= K"",-#&

!""> "--") "> @&&/ LMN ℄%&.,)$) .&*( $&) &"-" .&1,?)" $ !"&.:=

!"! #$%& '( )*) ,-'.$/0),1&. 2)('-) 3$4%)**

D!".",.", #" &%%&(. "E(, $&)# !, <"."-, ". &@"*(@@"* !"7,><"--"E(,4

$&)#= F !" $1"9 !&<"B".9 !":<"."&)-: !";.$) $;-"# &%"-" .$ $ : ,)*1,?4

)" $#1=

3',1141 *') !" "#" %&' ()* %!&+),! -./ #+0"1 0)&2- " '03&+3+&%'+.-# %+ %!"

". #+0"1 "#" %&' !-&,"4 P)"&% !"#$1;-"# ,;;-$ , $&)#&% !$#-,<$# &*"1&)4

# ., " !, , 5,.,*,: ,?" @-& /# &( "> ".),- "-" .$ 0"-*#= 7, !"1, $ ,--:9

+,(##J#-,<$)*$C".") $,-%&.1$#<.$ ") ,#

∇ · (εE) = ̺,

2Q=Q8

<!"." !"&;"., &.

∇·

*")& "#*$B".?") "9

E

$# !""-" .$ 0"-*,)*

̺

$# !""-" .$ !,.?"*")#$ := D!"1, ".$,- &"R $")

ε

$# !""-" .$ ;".1$ $B$ :,)*,##(1"* &@";$" "<$#" &)# ,) !.&(?!&( !$# !"#$#=

5') /6 7'80%-.17 5-,."%' 6+.+3+#"0 1+ .+% "*'0%4 F# $ <$-- @" ,;;,.4

") #!&. -:9 !$#$# !" -,< !, #;&$-# !" #:11" .:$) !" 7,><"--$,) !"&.:&%

"-" .&1,?)" $#1= A <&(-* #""1 B".: ), (.,- $)*""* $% 1,?)" $ !,.?"#9 S(#

-$/" "-" .$ !,.?"#9 ">$# "*= T( *"#;$ " ()# $) $)? "C&. # & $*") $%: 1,?)" $

1&)&;&-"#9 !":."1,$)"-(#$B"= 7, !"1, $ ,--:9 !"*$C".") $,-%&.1 &% !$#-,<

$#<.$ "),#

∇ · (µH) = 0,

2Q=N8

(16)

!"#"

H

$% &!"'()*"&$ ,"-. (*.&!" '(&"#$(- /"0 $"*&

µ

$%&!"'()*"&$ 1"#2 '"(3$-$&45 67%&(%

ε

8$&$%(%%7'".&/3"1$" " $%" /*%&(*&&!#/7)!/7&&!"&!"%$%5

5$($1$6789$: "#$%&$% '#%$()& *(+,-&++ .(#)(#$(+( ).& *(+,/ 9!$%1#$*2

$1-" $% $*:/-:". $* &!" /#;$*) /< &#(*%</#'"#%8 $*.7 &/#%8 (*. '(*4 </#'% /<

"-" &#$ (-)"*"#(&/#%5 =*.$>"#"*&$(-</#'8$&$%'(&!"'(&$ (--4</#'7-(&".(%

∇ × E = −µ

∂H

_∂t

,

?@5AB

!"#"

∇×

$%&!" 7#-?/##/&(&$/*B/1"#(&/#(*.

∂

∂t

."*/&"%&!"1(#&$(-&$'"."#$:(2 &$:"5

;2%<('78 9$: 0+( ).& 1..($) %($(.#)(2 # '#%$()& *(+,/ 9!" '/%& /3:$/7%

(11-$ (&$/*/<&!$%1#$* $1-"$%&!" (4"-" &#/'()*"&%/1"#(&"5 9!"

'(&!"'(&$ (-</#'#"(.%

∇ × H = J,

?@5CB

!"#"

J

$% &!" "-" &#$ 7##"*& ."*%$&45 =*)"*"#(-8 &!" 7##"*&."*%$&4 &(;"%&!" </#'

J

= σE + J

s

8 !"#"

σ

$% &!""-" &#$ /*.7 &$:$&4 (*.

J

s

$% &!" ."*%$&4 /< &!""D&"#*(-%/7# " 7##"*&5

!"!" #$%& '( )*) ,-'.$/0),1&. &10 ) 2$3%)**

E(D "--#"(-$%".&!(&&!"-(%&/<&!"(3/:"1!4%$ (--( %$%*/& /'1-"&"5 =*%&"(.8

F'1G#"H%-( $*&!" /##" &</#'%!/7-.#"(.I 0+( ).& 1..($)%($(.#)(2#'#%$()&

*(+,#$, "#$%&$%(+( ).& *(+,#+23 %($(.#)(2#'#%$()& *(+,/ E(&!"'(&$ (--48$&

&(;"%&!"</#'

∇ × H = J + ε

∂E

∂t

.

?@5JB

9!" %$)*$, (* " /< E(D "--H% /##" &$/* -$"%$* &!" <( & &!(& ?@5AB (*. ?@5J B */

#"1#"%"*& &!" #"-(&$/* 3"& ""* &!" "-" &#$ ,"-.

E

(*. &!" '()*"&$ ,"-.

H

$* ( .4*('$ (--4 %4''"&#$ (45 9!$% %4''"&#4 .$#" &-4 -". &/ &!" .$% /:"#4 /<

"-" &#/'()*"&$ (:"%(*.&/&!"#"(-$%(&$/*&!(&-$)!&$&%"-<$%(*"-" &#/'()*"&$

(:"5

K/&" &!(& ?@5@BL?@5AB (*. ?@5JB ./ */& /*%&$&7&" &!" '/%& <#"M7"*&-4 $&".

</#' /< &!"E(D "-- "M7(&$/*%5 9!(& )"*"#(--4$*:/-:"%</7# "M7(&$/*% $&! </7#

7*;*/ *%?%""(-%/N$)7#"@5A 5 =*."".8(&,#%&8?@5@ BL?@5AB(*.?@5JB'(4%""'&/

3"/:"#.",*".5 9!$%$%*/&&!" (%"8!/ ":"#8%$* "(%-/*) (%&!"$*$&$(- /*.$&$/*

%(&$%,"%?@5@ B(*.?@5O B8&!"/&!"# & /8&$'"2."1"*."*&"M7(&$/*%8?@5A B(*.?@5J B8

)7(#(*&"" &!(&(--</7#"M7(&$/*%(#"%(&$%,".(&(*4-(&"#&$'"5

P7& /< &!"'(*4 :(#$/7% 1!4%$ (- 1#/ "%%"% &!(& (#" ."% #$3". 34 ?@5@BL?@5AB

(*. ?@5JB8 &!"</ 7%$* &!$% &!"%$%$% /* "-" &#/'()*"&$ (:"%5 9!$% $*:/-:"%&!"

%$'7-&(*"/7%%/-7&$/*/<?@5AB(*.?@5J B8

ε

∂E

∂t

= ∇ × H − J,

µ

∂H

(17)

!"#$ %&# (!")%)(! %&*% %&#%+( !,!(+! -#."/ /*%)/0123435 *!" 2346 5*% )!)%)*. %)7#4 8// 7)!9*/)! /()"*.+*:#/(. %)(!;234<5 *!*./(=#+$)%%#!*/

iωεE = ∇ × H − J,

iωµH = −∇ × E,

234>5 +&#$#

ω

)/ %&#%#7?($*.0$#@ #! 1;

i

2 _{= −1}

*!"*..-#."/ "#?#!"/(.#.1 (!/?* #4

A/)!9 /%$*)9&%0($+*$" / =/%)% %)(!/; =(%& 234< 5 *!" 234>5 *! =# #B?$#//#" */ *

/# (!"C($"#$")D#$#!%)*.#@ *%)(!;$*%&#$%&*!*-$/%C($"#$/1/%#7;

ε

∂

2 _E

∂t

2 + ∇ ×

¡

µ

−1

∇ × E

¢

= −

∂J

_∂t

,

234E5

∇ ×

¡

µ

−1

∇ × E

¢

− εω

2 E

= −iωJ.

234F5

G&#0( $ ")D#$#!%0($7/ 234< 5H234F5*$#7*%&#7*%) *..1 #@ ):*.#!%*% %&# (!C

%)! ( / .#:#.4 I(+#:#$; %&# ?$(?#$%)#/ (0 %&# ")/ $#%)/#" :#$/)(!/ 7*1 =# $*%&#$

")D#$#!% 0($ %&# ")D#$#!% 0($7 .*%)(!/4 J! %&)/ %&#/)/; +# +).. ")/ // ")/ $#%)/*C

%)(!/(0 %&$##(0 %&#*=(:#0( $ 0($7/K 234< 5)! L&*?%#$6;234F5)! L&*?%#$M *!"

234E 5)! L&*?%#$N4

!" #$%&'( *+ *,,'-.(%*/(-0

O# !(+ 9):# * =$)#0 (:#$:)#+ (0 %&# 7(/% +)"#.1C /#" ! 7#$) *. *??$(B)7*%)(!

%# &!)@ #/ 0($%&# P*B+#..#@ *%)(!/4 Q($%&)/ ? $?(/#; +# (!/)"#$ %&#P*B+#..

#@ *%)(!/)! %&#R B0($7

Q(x)

∂q

∂t

+ ∇ · F (q) = 0,

2343S5

+&#$#

Q(x)

$#?$#/#!%/ %&#7*%#$)*. ?$(?#$%)#/;

q

)/ %&# :# %($ (0 %&# -#." :*. #/ *!"

F

(q) = [F

1 (q), F

2 (q), F

3 (q)]

T

"#!(%#/%&#R B4 T*7#.1;

Q(x) = diag (ε

r

, ε

r

, ε

r

, µ

r

, µ

r

, µ

r

) ,

q

=

· E

H

¸

,

F

i

(q) =

· −e

i

× H

e

i

× E

¸

,

+&#$#

e

i

)/%&# ($$#/?(!")!9L*$%#/)*! !)%:# %($4

G&#9#!#$*. !"#$.1)!9 )"#*)/%( $#?.* #%&# (!%)! ( /-#."/ )! %&#P*B+#..

#@ *%)(!/+)%&*")/ $#%# ( !%#$?*$%4 G&#")/ $#%)/#":#$/)(!(0%&#P*B+#..#@ *C

%)(!/)/%&#!?(//)=.#%(/(.:##:#!)0%&#2#B* %5/(. %)(!(0%&#($)9)!*.#@ *%)(!/

)/&*$";($ #:#!)7?(//)=.#;%( (=%*)!4 I(+#:#$;%&#")/ $#%#/(. %)(!+).. (!.1=#

*!*??$(B)7*%# $#?$#/#!%*%)(!(0%&##B* % /(. %)(!4

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

U#% /=#9)!+)%&%&#/)7?.#/%*!"&)/%($) *..1(."#/%7#%&(";,!(+!*/%&#-!)%#

")D#$#! # 7#%&("4 J! %&)/ *??$(* &; * 9$)" )/ .*)" "(+! )! /?* # *!" /?*%)*.

(18)

!"#$%&'() $* ($#++ , - +,%- . $*&- &/ 0 123 ℄ *#( 5,&6 - (5 %#889 5&5"8#,

$*#':($&$* %.5,&6 -#55,&;%.#$%&'5,&5 ,$% (5,&6%- -<9$* ($#++ ,%'+=

>' &/ $* .&($ #55 #8%'+ #(5 $( &/ $*%( . $*&- %( %$( (%.58% %$9= ?* -%(@

, $%(#$%&' &/ + ' ,#8 5,&<8 .( #'- &5 ,#$&,( %( &/$ ' %'$"%$%6 #'- %$ 8 #-( $&

6 ,9 A % '$( * . (/&,.#'95,&<8 .(= B",$* ,.&, ) $* ;58% %$( .%@-%( , $

/&,.+%6 (C ;%<%8%$9%'$* *&% &/$%. @%'$ +,#$%&'. $*&-(%/' - -)#'-$* (

. $*&-(#, ("55&,$ - <9#' ;$ '(%6 <&-9&/$* &,91D3 ℄=

E$ %(#8(&) *&F 6 ,)$* (%.58% %$9&/$* %'* , '$89&' @-%. '(%&'#8 &' 5$&/

G'%$ -%H , ' ($*#$%($* .#%'-%(#-6#'$#+ &/$* . $*&-= E$ '/&, (#(%.58

-%. '(%&'@<9@-%. '(%&'($," $", %'*%+* ,-%. '(%&'()#'-%'$,&-" (#--%$%&'#8

&.58% #$%&'(#,&"'-<&"'-#,% (#'--%( &'$%'"&"(%'$ ,'#88#9 ,(=?*%(.#: ($*

G'%$ -%H , ' . $*&- %88@("%$ -$&- #8F%$* &.58 ;+ &. $,% () <&$*%' $ ,.(

&/ + ' ,#8 &.5"$#$%&'#8 -&.#%'( #'- %'$ ,'#8 -%( &'$%'"%$% (= I*#'+ ( $& 8& #8

&,- ,#'-+,%-(%J $&, C $8& #8 / #$", (&/$* (&8"$%&'%(#8(&5,&<8 .#$% =

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

?* #<&6 -%( "((%&'*%+*8%+*$( $*#$ $& '(", + &. $,% C ;%<%8%$9) &' ' -( $&

#<#'-&' $* (%.58 &' @-%. '(%&'#8 #55,&;%.#$%&' %' /#6&", &/ (&. $*%'+ .&,

+ ' ,#8= ?* .&($'#$",#8#55,&# *%($&%'$,&-" #' 8 . '$@<#( --%( , $%(#$%&'=

?* &.5"$#$%&'#8-&.#%'%(-%6%- -%'$&#'".< ,&/ 88(&, 8 . '$( )&,+#'%(

-%' #' "'($," $", - .#'' , $& G88 $* 5*9(% #8 -&.#%'= E'%$( (%.58 ($ /&,.) $*

G'%$ 6&8". . $*&-#55,&;%.#$ ($* 5*9(% #8G 8-%' # * 8 . '$<9# &'($#'$=

?* ( * . %(5", 898& #8#'-$*"(%.5&( ('& &'-%$%&'(&'$* +,%-($," $", =

E'5#,$% "8#,)#88 88( #'*#6 -%H , '$. (*(%J (= ?* C";$ ,.(

F

(q)

, -" $& 5", (",/# $ ,.(<9$* "( &/$* -%6 ,+ ' $* &, .= ?*%(($ 5%'$,&-" ($*

' -$& 6#8"#$ $* C"; (#$ $* 8 . '$<&"'-#,% (= E' ;$, . 89(5 %#8 #( ()

(" * #( 8%' #, 5,&<8 .( &' !"%-%($#'$ +,%-() $*%( . $*&- , -" ( $& $* G'%$

-%H , ' . $*&-= K"$$* /&,."8#$%&'%(." *8 ((, ($,% $%6 %'$ ,.(&/$*

+,%-($," $", = ?* , 5, ( '$#$%&'&/(&8"$%&'6#8" (#$$* %'$ ,/# (%(#8& #85,& -",

#'-+ ' ,#8%( (($,#%+*$/&,F#,-89$& "'($," $", -+,%-(%' *%+* ,-%. '(%&'()$*"(

'(",%'+ $* - (%, - + &. $,% C ;%<%8%$9= B",$* ,.&, ) $* &'($," $%&' &/ $*

%'$ ,/# C"; ( #'< -&' %'6#,%&"(F#9($*#$#, 8&( 89, 8#$ -$&$* 5#,$% "8#,

!"#$%&'(1LD )MN ℄=

E/) *&F 6 ,)&' ' -($&%' , #( $* &,- , &/# ",# 9&/ $* . $*&-) #("<@

($#'$%#85,&<8 ..#9 . ,+ %'$* 8#((% #8G'%$ 6&8". #55,&# * !

= E'$* (%.58

&' @-%. '(%&'#8 #( ) $*%( #' < -&' ($,#%+*$/&,F#,-89 <9 ;$ '-%'+ $* (%J &/

$* ($ ' %8= K"$%' *%+* ,-%. '(%&'() ;$ '-%'+ $* ($ ' %8, %'$,&-" ($* '

-/&,#5#,$% "8#,+,%-($," $", #'-$*"( &.5,&.%( ($* + &. $,% C ;%<%8%$9&/$*

!" # "$%"&' () $++'&%)$,-+" #.%/ 0$1%)+$&" %$+#),&2.1)3" ,# ).,#$,1&" &%$ !"1%$4!" 5 $!" 1%$

.%(%)# 3#),&!%" "1)3" ,# ).,#

!

6'7 +$# # ) $+82"%" 9 " % &.&!" $((%.$ !2!" %" &!" :" +1#$%" $((%.5 )3$&" 1;'$ .,# &$,&),

" $ ! " ++ <.%" +" 3" ,&=> ? & )#(.# # );+"&.), +01" !)-!" %@.%1" % 3.3" ,&#)," $ ! " ++<.%" +" 3" ,&=

$,1&!0# .,# &%0 &$ .3($ &@# &" , )+!)-!@.%1" %:,)&"A.+03"# !" 3" >B.2" A" %4),&!$& $# "&!"

(19)

!"#$%&'()$)$#*&+"!*",*$-+")$!."&!./ 0*$)1"!2-&3'$)4"#*#*$

*",*5&-+$--$2-$.$!#1#"&! ".#*1#"# (.(1''6!$$+.#& .21!)('#"2'$ $'$)$!#.1. #*$!()$-" 1'

122-&8")1#"&!&9

q

".,"%$!36 $''1%$-1,$.&!'6/

!"!" #$%$&' '(')'%& )'&*+,

:1#*$-#*1! -$2-$.$!#"!,#*$ !()$-" 1' +". -$#".1#"&!36 $''1%$-1,$.; &!$ &('+

3$ #$)2#$+ #& #1<$ 1 +"=$-$!# 122-&1 * 1!+ "!#-&+( $ )&-$ +$,-$$. &9 9-$$+&)

"!#*$$'$)$!#/ 0*".". 4*1#*122$!."! #*$ !"#$$'$)$!#)$#*&+4*$-$;

.")"'1-#&#*$ !"#$%&'()$)$#*&+; #*$ &)2(#1#"&!1'+&)1"!". +"%"+$+"!#& 1

!()3$-&92&.."3'6(!.#-( #(-$+ $'$)$!#./ >!."+$$1 * $'$)$!#;4$1..()$#*1# #*$'& 1'

.&'(#"&!". $82-$..$+ 1. #*$ '"!$1- &)3"!1#"&!&9 '& 1''6 +$ !$+ 31.". 9(! #"&!./

?"#*#*".'& 1'$'$)$!#531.$+)&+$';$1 *$'$)$!#.*1-$.#*$!&+$.4"#*&#*$-$'$5

)$!#./ >94$ &)3"!$#*$$'$)$!#4".$122-&8")1#"&!.;4$*1%$1,'&31' &!#"!(&(.

-$2-$.$!#1#"&!&9#*$(!<!&4!

q

/

0&-$ &%$-1. *$)$#&.&'%$@A/ABC;4$!$$+#&+$ !$#*$.21 $&9#$.#9(! #"&!.;

1!+-$D("-$#*1# #*$-$."+(1'3$&-#*&,&!1'#&1''#$.#9(! #"&!."!#*"..21 $/ 0*$

+$#1"'.&9#*$. *$)$1-$+$#$-)"!$+36*&4#*"..21 $&9#$.#9(! #"&!.".+$ !$+/

E '1.." 1' *&" $; '$1+"!, #& 1 F1'$-<"! . *$)$; ". #& -$D("-$ #*1# #*$ .21 $.

.21!!$+36#*$31.".9(! #"&!.1!+#$.#9(! #"&!. 3$#*$.1)$/

0*".122-&1 *-$G$ #.;"!$..$! $;#*$ '1.." 1'HIJ1!+ '$1-'61''&4.+"=$-$!#

$'$)$!# ."K$. LMN ; OM; OO ; OP℄/ H(-#*$-)&-$; 4$ -$ 1'' #*1# #*$ )1"! )&#"%1#"&!

3$*"!+ &!."+$-"!, )$#*&+. 3$6&!+ #*$ !"#$ %&'()$ 122-&1 * 41. #*$ "!#$-$.#

"! *",*5&-+$- 122-&8")1#"&!./ R( * $8#$!."&!. 1-$-$'1#"%$'6.")2'$ "! #*$ !"#$5

$'$)$!# .$##"!, 1!+ 1! 3$ 1 *"$%$+ 36 1++"!, 1++"#"&!1' +$,-$$. &9 9-$$+&) #&

#*$$'$)$!#4*"'$)1"!#1"!"!,.*1-$+ !&+$.1'&!,#*$91 $.&9#*$$'$)$!#.LST ℄/

U&4$%$-;#*$-$1-$+-1431 <.&9#*$ '1.." 1' &!#"!(&(. !"#$$'$)$!#9&-)('15

#"&!;9&-*62$-3&'" 2-&3'$)."!,$!$-1'1!+9&-#*$J184$''$D(1#"&!."!21-#" ('1-/

H"-.#;#*$,'&31''6+$ !$+31.".9(! #"&!.1!+#*$-$D("-$)$!##*1##*$-$."+(1'3$

&-#*&,&!1'#&#*$.1)$.$#&9,'&31''6+$ !$+#$.#9(! #"&!.")2'"$.#*1##*$.$)"5

+". -$#$ . *$)$ 3$ &)$. ")2'" "#/ H&- #")$5+$2$!+$!# 2-&3'$).; #*". ". 1

'$1-+".1+%1!#1,$ &)21-$+ 4"#* !"#$+"=$-$! $&- !"#$%&'()$)$#*&+./

R$ &!+; #*$-$ ". 1! "..($ #*1# -$'1#$. #& #*$ .#-( #(-$ &9 #*$ 31.". 1!+ ". $.5

2$ "1''6 ")2&-#1!#9&- #*$J184$''$D(1#"&!/ 0*1# ".; #*$&-","!1' 2*6." 1' $'+.

E

1!+

H

1-$!&#!$ $..1-"'6 &!#"!(&(./ 0*$J184$''$D(1#"&!.-$D("-$ #*$#1!5 ,$!#"1' &)2&!$!#&9$"#*$- $'+#&3$ &!#"!(&(.;3(# #*$-$".!&-$D("-$)$!#

9&-#*$!&-)1' &)2&!$!#/ :$2-$.$!#"!,#*$.$ $'+. 4"#* &!#"!(&(.9(! #"&!.1# #*$

+". -$#$'$%$')16-$.('#"! &!%$-,$! $#&.2(-"&(.;!&!52*6." 1'.&'(#"&!./ >!#*$

!"#$$'$)$!# &!#$8#; #*". 2-&3'$) 1! 3$ .&'%$+36#*$ (.$&9 1 .2$ "1' #62$&9

31.".9(! #"&!. 9&-4*" *&!'6#1!,$!#"1' &!#"!("#6".2-$. -"3$+/

!"!- .$/ +%&$%1+1/ 23('45$% )'&*+,

E!"!#$''",$!# &)3"!1#"&!&9#*$ !"#$$'$)$!#1!+ !"#$%&'()$)$#*&+.122$1-.

(20)

!"#$% '( $()')%*+,"-.$(/01234 56756658586 ℄:

;( !$%+<<-'+ !5=">+$( +$( !"#"?($ $'('@",">"( %+%$( !"?($ "",">"(

+(# ?($ "A',)>" % !">"%: B) $( '-#"- ' "(%)-" !" ,' +,$ C '@ !" % !">"5 ="

+,,'= !" A+-$+D,"% ,' + "# + "+ ! ('#" ' !+A" >), $<," A+,)"% E +% >+(C +%

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

A',)>" +<<-'+ !5 !" M*N/01 'A"- '>"% !" ."C,$>$ + $'( '( + !$"A$(F !$F!N

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

H(curl)

N '(@'->$(F/01$%-+ !"-%>+,,: H!$%5 'F" !"- =$ ! !" +##$ $'(+,K"G$D$,$ C '@ M* >" !'#%5 >+."% !"> +("G ",,"( ()>"-$ +, " !($I)"@'-+ )-+ "#$% -" $%+ $'(% '@ !"1+G=",,"I)+ $'(%:

(21)

!" #$%&'() *+ %,)

!"#"$%&'&'()%#*+,*!"*!"-&- +'-&-*-+,,+/# !%)*"#-0 1% !+,*!"-" !%)*"#-

&-%'&'2")"'2"'*%#*& 3"*!%*!%-4""')/43&-!"2 &'5+# -/4$&**"2*+5)""#6#"7&"8"2

&'*"#'%*&+'%39+/#'%3-&'*!":"32 +,- &"'*&: +$)/*%*&+'0

;'<!%)*"#=52&>"#"'**&$"6-*"))&'($"*!+2-,+#%'+2%3!&(!6+#2"#2&- +'*&'6

/+/-?%3"#@&'2&- #"*&-%*&+'+,*!"A%B8"33"C/%*&+'-%#"2&- /--"20 D +$)%#&-+'

4"*8""' *!" $+-* )+)/3%# !+& "- +, E/'("6F/**% GEFH $"*!+2- &- $%2" ,#+$

*!")+&'*+, 7&"8 +, % /#% I %'2 +$)/*%*&+'%3 8+#@0 JI !++-&'( *!" -*#+'(6

-*%4&3&*I6)#"-"#7&'(E/'("6F/**%GKKL6EFH*&$"6&'*"(#%*&+'$"*!+2+,+#2"# +'6

-&-*"'*8&*!*!")+3I'+$&%3 +#2"#+,*!"-)%*&%32&- #"*&-%*&+'54"**"#% /#% I %'

4" %**%&'"2 +$)%#"2 8&*! :B"26+#2"# - !"$"-0 A+#"+7"#5 *!&- +$"- 8&*!+/*

%-&('&: %'*&' #"%-"&' +$)/*%*&+'%38+#@0 D '/$"#& %3M+/#&"#%'%3I-&- &-)"#6

,+#$"2,+#*!&-E/'("6F/**%2&- +'*&'/+/-?%3"#@&'GEFN?H2&- #"*&-%*&+'*+(%&'

&'-&(!* &'*+*!"2&-)"#-&+'%'2 2&--&)%*&+')#+)"#*&"- +,*!",/33I 2&- #"*"- !"$"0

!"%'%3I-&- &- %##&"2+/* +'4+*!*!"+'"62&$"'-&+'%3%'2 *!"*8+62&$"'-&+'%3

,/33I2&- #"*"- !"$"-%'25&'*!"3%**"# %-"5+'/'&,+#$%-8"33%-+''+'6/'&,+#$

$"-!"-0 ;* %3-+)#+7&2"-)#% *& %3 &',+#$%*&+' +' *!" +'7"#("' " +, *!"2&--&)%6

*&+'%'22&-)"#-&+' "##+#/)*+ )+3I'+$&%3 +#2"#OP ,+#*!"+'"62&$"'-&+'%3,/33I

2&- #"*"- !"$"0

<!%)*"#Q2"- #&4"-*!"&$)3"$"'*%*&+'2"*%&3-+,%!&"#%# !&

H(curl)

6 +',+#$6 &'(*"*#%!"2#%34%-&-&'*!":'&*""3"$"'*2&- #"*&-%*&+'+,*!"A%B8"33"C/%*&+'-0

!&-4%-&-&- "-)" &%33I/-",/3 &' *!"2"7"3+)$"'*+,

p

6%'2

hp

6%2%)*&7"$"*!+2-0 ;* %' &' % '%*/#%3 8%I 4" %))3&"2 *+ *!" A%B8"33 "C/%*&+'- &' 4+*!

H(curl)

6 +',+#$&'( %'2 2&- +'*&'/+/- ?%3"#@&' :'&*" "3"$"'* 2&- #"*&-%*&+'-0 D3*!+/(!

*!" +'-*#/ *&+'+, *!"4%-&-&- #"3%*&7"3I8"336@'+8'5 &*&- "--"'*&%3*!%* 8"2#%8

%**"'*&+' *+ % )+--&43" )#% *& %3 2&R /3*I &' *!" &$)3"$"'*%*&+' +, -/ ! %

4%-&-,+# *!"

H(curl)

6 +',+#$&'( :'&*" "3"$"'* $"*!+20 !&- +' "#'- +'" )%#*& /3%# *I)"+, 4%-&- ,/' *&+'5*!" ,% "64%-"2 4/443",/' *&+'5 8!& ! &- )%#* +, *!"

4%-&-,+#)+3I'+$&%3 +#2"#-

p ≥ 3

0 S"-!+85*!#+/(!% -&$)3""B%$)3"5*!%*,% "64%-"2 4/443",/' *&+'- 2+'+*5&'("'"#%35*#%'-,+#$,#+$*!"#","#"' "*"*#%!"2#+' *+%

)!I-& %3 *"*#%!"2#+' &' % +',+#$&'(8%I5 "7"'%,*"# *!"&--/" +, +#&"'*%*&+'

!%-4""' %#",/33I %22#"--"20 D- % +'-"C/"' "5 -)" &%3 %#" '""2- *+ 4" "B"# &-"2

*+(/%#%'*"" *!"(3+4%3 *%'("'*&%3 +'*&'/&*I#"C/&#"2 4I*!"

H(curl)

6 +',+#$&'( $"*!+20 T' " *!%* &- "'-/#"25 *!" :'&*" "3"$"'* 2&- #"*&-%*&+' %' 4" +4*%&'"2

&' *8+ $%&' -*")-0 M&#-*5 +$)/*" *!" 2&- #"*&-%*&+' ,+# "% ! "3"$"'* G%'2 ,% "

&,N? &- /-"2H5 "0(0 4/&32 *!""3"$"'*G%'2 ,% "H $%*#& "-0 K" +'25 %--"$43" *!"

3+ %3 $%*#& "-&'*+ (3+4%3+'"- 8!& ! ,+#$ *!" %3("4#%& -I-*"$+# *!" -I-*"$+,

+#2&'%#I2&>"#"'*&%3"C/%*&+'-*!%* %#"*+4"-+37"2*+ ("**!"2&- #"*"-+3/*&+'0

;'<!%)*"#U58")#+7&2"+)*&$%3)%#%$"*"#"-*&$%*"-%'2%)#&+#&"##+#

4+/'2-,+# -I$$"*#& 2&- +'*&'/+/- ?%3"#@&' GN?H 2&- #"*&-%*&+'- ,+# *!" -" +'26+#2"#

&'2":'&*" *&$"6!%#$+'& A%B8"33 "C/%*&+'-0 A+#" -)" &: %33I5 8" +'-&2"# *8+

7%#&%*&+'-+, -I$$"*#& N? $"*!+2-V *!" &'*"#&+# )"'%3*I N? G;L6N?H $"*!+2

%'2+'" *!%*$%@"-/-"+,*!"3+ %3 3&,*&'(+)"#%*+#&' *!"W/B ,+#$/3%*&+'0 D- %

(22)

! !"#"$%#&#'%!'()#*#!+#(,$-'%()#&!"$'!+.-"!,%#,/-'% !-,#0,##!01'2

&# .4#% !' (-'( 5 61 )&-,-+#(#,# (.+-(# -,#&-,(. 1"-,"$.+&!,(-'(.' (),##2

%.+#' .!'-"789: .+1"-(.!' ;# -1 #.'&,- (. # +-'$89 !+&1(-(.!' ! 1,.'

()# &,#2- $+&(!(. ,#*.+#5 <)#,#0!,#= .( . >.(-" ()-( !1, '1+#,. -"#?&#,.+#'(

()-( - !+&-'$ ()# ()#!,#(. -" ,# 1"( -,# -" ! .' 895 <)#$ -,# -,,.#% !1( !'

(#(,-)#%,-" +# )# @.() ).*)2!,%#, 7

p = 1, 2, 3, 4

: ).#,-, ).

H(curl)

2 !'0!,+.'* &!"$'!+.-";- . 01' (.!' 5

A)-&(#, B !+&-,# ()# %. !'(.'1!1 C-"#,D.' 4'.(# #"#+#'( +#()!% 79C2

EFG:@.() ()#

H(curl)

2 !'0!,+.'*EFG.'()#%. ,#(. -(.!'!0()# # !'%2!,%#, (.+#2%!+-.'G-?@#""#H1-(.!' @.()-&! .;"$'!'I#,! !'%1 (.>.($(#,+5 J)."#

9C2EFG 1K#, 0,!+ -' .' ,#- #%'1+;#, !0 %#*,## !00,##%!+ !+&-,#% @.()

H(curl)

2 !'0!,+.'*EFG=.()- ()#-%>-'(-*#!0-&1,#"$;"! D2%.-*!'-"+- +-2 (,.?5 <). +#-' ()-(=- "!'*- -'#?&". .((.+#2.'(#*,-(.!' )#+#. 1 #%=.(. '!

"!'*#,'# # -,$(! !">#-".'#-, $ (#+-(#- )(.+# (#&L- "#-,-%>-'(-*#!>#,

H(curl)

2 !'0!,+.'*EFG5M(. D'!@'()-( 9C2EFG*#'#,-""$0->!1, ).*)2!,%#, +#()!% @)#,#-

H(curl)

2 !'0!,+.'* EFG . +!,# 1.(-;"# 0!, "!@2!,%#, !'# 5 <)#'!>#"($@#&,!>.%#.'(). )-&(#,. -%.,# ( !+&-,. !'!0()#&#,0!,+-' #!0

()#(@! +#()!% @)#').#,-, ).

H(curl)

2 !'0!,+.'*;- . 01' (.!' -,# 1 #%1& (!&!"$'!+.-"!,%#,

p = 3

5 <)#+!(.>-(.!';#).'%(). )!. #!0;- . 01' (.!' . .( *,!@.'*.+&!,(-' #.' ()#%#>#"!&+#'(!0

p

2-'%

hp

2-%-&(.>#EFG 5 <)#0- ( ()-(@#-""!@0!,'!'I#,! !'%1 (.>.($,#H1.,# &# .-"-((#'(.!'@.(),#*-,% (!()# (.+#2.'(#*,-(.!'+#()!% -&&".#%(!()# #+.2%. ,#(# $ (#+ 5 N).*)2!,%#,&!"$2 '!+.-";- . @-,,-'( ()#1 #!0).*)2!,%#,(.+#2.'(#*,-(.!' )#+# =;1(#?. (.'* ).*)2!,%#, )#+# +-$ 1K#, 0,!+ - (!! #>#,# (.+#2 (#& (-;.".($,# (,. (.!'- ,# 1"(!0()# !'%1 (.>.($(#,+5 J#.'># (.*-(# #>#,-"-"(#,'-(.># 0,!+()#&!.'( !0 >.#@ !0 - 1,- $= (-;.".($ -'% !+&1(-(.!'-" @!,D5 E.'-""$= @# -,,$ !1(

-'1+#,. -"E!1,.#,-'-"$ . (! (1%$()#%. &#, .!'-'%%. .&-(.!'&,!&#,(.# !0()#

#+.2%. ,#(# 9C2EFG )#+#-'% #>#,-"!0 ()#(.+#2.'(#*,-(.!' +#()!% 5 M( .

.' (,1 (.># .' !1, -&&,!- ) ()-( ()# %. &#, .!' -'% %. .&-(.!' &,!&#,(.# !0 ()#

&-(.-" %. ,#(. -(.!'-'% ()! # !0 ()# (.+#2.'(#*,-(.!' +#()!% -,# .'># (.*-(#%

(23)

!"#$% &'() !" #$"%#& '(")#"*# "+,)#-.%* /)*"0 .% #!" 1.#"* .2 #!" (")0"(* .2

3!&*- * 5.(60 -% 7889: !-* -;)'" !)* /""% 0.$%6.)0"0 2(.; )%0 -* )6*. )1)-6)/6" )#

(24)

(25)

!"#$%"!&' (' !""!#()!&' $%%&% !'

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

+(0$%.!' !"/%$)!"()!&'" &1 )*$ 2(34$00

$5-()!&'"

!" #$%&'() %+'$

!"#$%&'(#)&$%*%'+&'%!$,'-',$'.#%&/'!&*&'#0&/'*-&#0 #2")&*&$#%*3'3' 4

&-#2*5%'&$ ! 678 ℄: $% 2*%; *!'! <%$&'4($='-'% ' &$2'4(#2*$% >?@A@B ! /'2'!

6CC : DE℄ *-' )%(#)F&'(3; &/' 2#!& "#")3*- 2'&/#(! *2#%5 "/;!$ $!&! *%( '%5$4

%''-!&#!#3,'&/'&$2'4(#2*$%G*+.'33'H)*&$#%!%)2'-$ *33;I A/$!"#")3*-$&;$!

2*$%3; ()' &# &/'$- !$2"3$ $&; *%( 'J $'% ; $% ($! -'&$!$%5 !$2"3'4(#2*$% "-#F4

3'2!I K#.','-:&/'$-$%*F$3$&;&# '=' &$,'3;/*%(3' #2"3'+5'#2'&-$'!"-#2"&'(

!#2' ! $'%&$!&! &# !'*- / 0#- *3&'-%*&$,'! 3#%5 *5#I ?$%$&'4'3'2'%& >?LB 2'&/4

#(! *-' *% #F,$#)! *3&'-%*&$,': F)& '*-3; '=#-&! .'-' 2*--'( F; &/' 0* & &/*&

!&*%(*-( #%&$%)#)! M*3'-N$% <%$&'4'3'2'%& ! /'2'!5$,'-$!'&# %#%4"/;!$ *3 !#4

3)&$#%!I G#!&*""*-'%&#0 &/'!'*-' &/'!")-$#)!2#('!$% &/'%)2'-$ *3!#3)&$#%

#0&/'0-'H)'% ;4(#2*$%G*+.'33'H)*&$#%!>!''6OP ℄ *%(-'0'-'% '!&/'-'$%BI A/'

-',#3)&$#%*-;!#3)&$#%&#&/$!"-#F3'2.*!&#-'*3$!'&/*& F;)!$%5*"*-&$ )3*-!'&

#0 ,' &#- F*!$! 0)% &$#%! >,' &#- '3'2'%&! !) / *! QR(R3' #- S/$&%'; '3'2'%&!

67O : OT ℄B: $& $! "#!!$F3' &# 2$2$ 2*%; #0 &/' !"' $*3 "-#"'-&$'! #0 &/' G*+.'33

'H)*&$#%!*& &/'($! -'&' 3','3I U''6D ℄ *%( 6T℄I L,'- !$% ': ,' &#- '3'2'%&! /*,'

F''% * ,$*F3' *3&'-%*&$,' &# ?@A@ *%( !&*%(*-( ?L 2'&/#(! $% #2")&*&$#%*3

'3' &-#(;%*2$ !: '!"' $*33; 0#- 0-'H)'% ;4(#2*$% "-#F3'2! .$&/ #2"3'+ 5'#2'4

&-$'!I A/' "-* &$ *3 #%!$('-*&$#%! #0 F#&/ !&*%(*-( *%( ,' &#- <%$&' '3'2'%&!

$% #2")&*&$#%*3 '3' &-#2*5%'&$ ! *-' #,'-'( $% 6OP ℄I ?#- &/' 2#-' &/'#-'&$ *3

(26)

!"#""$%&'&$"("(" %*&'+,#"%- .+/"0*&0+,+%-&#-#(+*,"+#$ &'0("1$&2

'+-#3+#$&/"*+*"(+%-/"(4(&#,%-'"30+#!+3-# *"+3"$%!"$"'+#$5&*!-,!2&*$"*

'"%!&$36 7&."/"*8#"-%!"* !-,!2&*$"* 9: : '"%!&$3 #&* !-,!2&*$"* /" %&* 9;

'"%!&$3 +*"$"/&-$&50*+ %- +( $*+.<+ =36 7-,!2&*$"* 9: :'"%!&$3 5+-(%& "52

5" %-/"(4!+#$(" &'0("1,"&'"%*-"3.!"*"+3!-,!2&*$"*/" %&*9;'"%!&$3><+3"$

&# !-,!2&*$"* ?@$@(" "("'"#%3 ABC ℄ 5&* "1+'0("E ("+$ %& ,(&<+( '+33 '+%*- "3

.-%! *"(+%-/"(4(+*,"<+#$.-$%!3>+5%"* &0%-'+(*"&*$"*-#,E6 !"%-'"2-#%",*+%-&#

3 !"'"3%&3&(/"3F !343%"'3+*"-#%F*# &'0F%+%-&#+((4*+%!"*"10"#3-/"6 !"3"

$-G F(%-"3!+/"'&%-/+%"$%!"$"/"(&0'"#%&5$-3 &#%-#F&F3H+("*=-#>:HEI#-%"2

"("'"#% '"%!&$3 AJK 8 JJ ℄8 %&,"%!"* .-%! 30" %*+( "("'"#% '"%!&$3 ABJ ℄6 L#<&%!

%!" 5*"MF"# 42$&'+-# 5&*'F(+%-&# ANO 8 NP8 QK 8 QR 8 SR℄ +#$ %!" %-'"2$&'+-# 5&*2

'F(+%-&# ARP 8 JR8 NJ8 BP℄ 3-,#-I +#% 0*&,*"33 !+3 <""# '+$"6 T#" &5 %!" '&3%

0*&'-3-#,'"%!&$35&* &'0(- +%"$,"&'"%*-"3-3%!"!-,!2&*$"*#&$+(:H'"%!&$

&5 7"3%!+/"# +#$ U+*<F*%&# ANJ℄8 .!- ! 0*&/"$ <&%! + F*+%" +#$ "G -"#% 5&*

%!" 30+%-+( $-3 *"%-3+%-&#6 L# %-'" -#%",*+%-&#8 !&."/"*8 %!" (&.23%&*+," VF#,"2

WF%%+>VWE'"%!&$%!"+F%!&*3+00(-"$0&3"3+ &'0+*+%-/"(43%*-#,"#%%-'"23%"0

&#3%*+-#%8 .!- ! '+4 %F*# &F% %& <" %!" <&%%("#" = 5&* (&#,2%-'" -#%",*+%-&#6

9F*%!"*'&*"8 I1"$2&*$"* %-'"2-#%",*+%-&# 3 !"'"3 '+4 30&-( %!" !-,!2&*$"* &#2

/"*,"# " &5%!",(&<+(3 !"'"6 L#%!"'"+#%-'"85&*$-3 &#%-#F&F35&*'F(+%-&#3&5

&#/" %-&#2$&'-#+%"$0*&<("'3AJJ ℄ -%!+3<""#3!&.#-# AOR ℄+#$-#ARP ℄ %!+%%!"

%-'"23%"0*"3%*- %-&#'+4<"(&&3"#"$-5."F3"X%*&#,2X%+<-(-%42Y*"3"*/-#,VF#,"2

WF%%+ >XXY2VWE '"%!&$3 &5 &#" &*$"* !-,!"* %!+# %!" 0&(4#&'-+( &*$"* &5 %!"

30+%-+($-3 *"%-3+%-&#6

L# %!-3 !+0%"*8."3%F$4 %!"<"!+/-&F*&5 %!"!-,!2&*$"* #&$+( 3 !"'" .!"#

3"/"*+(&5%!"<"3%23F-%"$%-'"2-#%",*+%-&#'"%!&$3+*"F3"$6 L#0+*%- F(+*8."!+/"

+ (&3"*(&&=+% %!"$-30"*3-&# +#$$-33-0+%-&#0*&0"*%-"3&5%!"VF#,"2WF%%+$-32

&#%-#F&F3H+("*=-#>VW:HE'"%!&$ &'0*-3-#,%!"#&$+(!-,!2&*$"*:H'"%!&$

+#$ %!" XXY2VW '"%!&$6 !" '+-# '&%-/+%-&# 5&* F3-#, %!-3 0+*%- F(+* %-'"2

-#%",*+%-&#3 !"'"-3-%3*"(+%-/"(4."+=%-'"23%"0*"3%*- %-&#6 !-30*&0"*%4-'0(-"3

%!+%." +#*"%+-#!-,!2&*$"* + F*+ 4.-%!&F%(&3-#,'F !&#%!" &'0F%+%-&#+(

.&*= '"+3F*"$+3%!"#F'<"* &5&0"*+%-&#36

!" (-%"*+%F*"&#%!"$-30"*3-&# +#$ $-33-0+%-&#0*&0"*%-"3&5 %!":H'"%!&$

!+3-# *" "#% 4"+*3 <" &'" +<F#$+#%6 Z %!&*&F,! +#+(43-3 &5%!" $-30"*3-&# +#$

$-33-0+%-&#<"!+/-&F*&5 %!":H '"%!&$ 5&*%!"%*+#30&*% "MF+%-&#>3 +(+* (-#"+*

&#3"*/+%-&# (+.E.+3 ,-/"# -# AR ℄8.!- ! +(3&0*&/-$"$ + 0*&&5 5&*"+*(-"* &#[" 2

%F*"38"30" -+((45*&'ANC ℄6 !"3"'-2$-3 *"%"343%"'5&*%!".+/""MF+%-&#!+3+(3&

<""#"1%"#3-/"(43%F$-"$ AO 8BK 8PR ℄6 L#0+*%- F(+*8%!"+F%!&*3-# AO ℄0*&/-$"$ %.&

$-\"*"#%$-30"*3-&#+#+(43"35&*%!"3"'-2$-3 *"%".+/""MF+%-&#&#%"#3&*0*&$F %

"("'"#%36 T#" 5&* %!" -#%"*-&* 0"#+(%4 :H '"%!&$ >LY2:HE &5 %!"3" &#$2&*$"*

.+/""MF+%-&#+#$+#&%!"*5&*%!","#"*+(:H'"%!&$5&*+I*3%&*$"*343%"'6

!" #&/"(%4&5 %!-3 !+0%"* .-%! *",+*$3 %&%!" $-30"*3-&# +#$$-33-0+%-&#<"2

!+/-&F*&5 :H'"%!&$3(-"3-# -# (F$-#, %!"%-'"-#%",*+%-&# -# %!"+#+(43-36 U"

&#3-$"* %!" $-3 *"%-3+%-&#&5 %!" I*3%2&*$"*343%"' *"(+%"$ %& %!"]+1."(( "MF+2

%-&#383&&F*3 !"'"5+((3-# %!" +%",&*4 &5.!+% %!"+F%!&*3 +(( %!"^,"#"*+(:H

(27)

!" $!%&' ()*+$, -($)!*).,-$%* %'/!0! '%!*"'",0' 1*2 (-)!*- 3*,$ .%*%&$

$"%$)$, *) 3!4$, 5-4$ 2*) %!3$6,$($",$"% ()*/0$3 789 ℄; <" ='+$6()*('>'%!*"

()*/0$3 !%! *2%$"3*)$',+'"%'>$*- %*1"*=%&$ *"+$)>$" $)'%$*2%&$,! ($)6

!*"'",,! !('%!*"$))*) %&'"%&'%*2%&$$))*)!"%&$

L

2

6"*)3; ?&$ $ *"+$)>$" $ )'%$ &'+$/$$"$ %'/0! &$,!"7@ ℄2*)%&$ $3!6,! )$%$%)'" (*)%$A-'%!*"; B*)%&$

>$"$)'0CD &$3$E%*=&! &%&$"*,'0CD3$%&*,,! - $,&$)$/$0*"> E!%&'

/$$" &*=" !" 79 ℄ %&'% - !"> F) %6*),$) (*0G"*3!'0 !" %&$ ('%!'0 ,! )$%! '%!*"

)$ -0% !" ' ,! ($) !*"$))*)*2*),$)

O(h

4 ₎

'",' ,! !('%!*"$))*)*2*),$)

O(h

3 ₎

2*)%&$ $3!6,! )$%$ G %$3; <"%&! &'(%$)=$ &*=E%&)*->&"-3$)! '0$4'3(0$ E

&*=%&$,! ($) !*"'",,! !('%!*"$))*) *"+$)>$!" %&$2-00G,! )$%$&!>&6*),$)

HICD &$3$2*)%&$0!"$')'-%*"*3*- 2*)3 *2%&$J'4=$00$A-'%!*" ;

?&$ )$3'!"!"> (')% *2 %&! &'(%$) ! *-%0!"$, ' 2*00*= ; <" K$ %!*" L;L =$

)$ '00 %&$ G %$3 *2 %!3$6,*3'!" J'4=$00 $A-'%!*" '", )$,- $ !% %* %&$ 0!"$')

'-%*"*3*- 2*)3; ?&$ ('%!'0,! )$%! '%!*"! /)!$5G)$+!$=$,!" K$ %!*"L;9'",

%&$HI &$3$ 2*)%&$%$3(*)'0,! )$%! '%!*"!"K$ %!*"L;8 ; M"$6,!3$" !*"'0'",

%=*6,!3$" !*"'0 B*-)!$) '"'0G ! ! '))!$, *-%!" K$ %!*" L;N E '", %&$' * !'%$,

"-3$)! '0)$ -0% E'0*">=!%& *3$*%&$) "-3$)! '0%$ % E')$()$ $"%$,!" K$ %!*"

L;O ; P$)$ =$ $4'3!"$ %&$ /$&'+!*-) *2 %&$ ,! ($) !*" '", ,! !('%!*" $))*) !"

%$)3 *2%&$3$ & !Q$ ($)='+$0$">%& '",%&$ !Q$ *2%&$ %!3$ %$(; B!"'00GE=$

-3-(*-) *" 0- !*" !"K$ %!*"L;R ;

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

S$/$>!"=!%&,$)!+!">%&$,!3$" !*"0$ %!3$6,*3'!"2*)3*2%&$J'4=$00$A-'6

%!*" !" %&$ %&)$$6,!3$" !*"'0 ,*3'!"

Ω ⊂ R

3

; T*0,2' $ G3/*0 &$)$ )$2$) %*

+$ %*) F$0, E !;$; F$0, !"

R

3 _{→ R}

3

; S!%&%&$ $ "*%'%!*" %&$J'4=$00$A-'%!*"

)$',

∂D

∂t

= ∇ × H − J,

∂B

∂t

= −∇ × E,

UL;@V

∇ · D = ̺,

∇ · B = 0,

UL;LV

=!%& &')>$,! %)!/-%!*"

̺(x, t)

E(* !%!*"+$ %*)

x

= (x, y, z) ∈ Ω

E%&$"'/0'*($)'6 %*)

∇ =

³

∂

∂x

,

∂

∂y

,

∂

∂z

´

'",%!3$

t

; ?&$+$ %*)+'0-$,A-'"%!%!$ ')$%&$$0$ %)! F$0,

E(x, t)

E%&$$0$ %)! 5-4 ,$" !%G

D(x, t)

E%&$3'>"$%! F$0,

H(x, t)

E%&$3'>"$%! 5-4,$" !%G

B(x, t)

'",%&$$0$ %)! -))$"%,$" !%G

J

(x, t)

; B*)3'"G'((0! '%!*" !%! )$' *"'/0$%*' -3$%&'%%&$3'%$)!'0 ')$! *%)*(! E0!"$')'",%!3$6!"+')!'"%;

?&- %&$ G %$3*2$A-'%!*" ! 0* $,=!%&%&$0!"$') *" %!%-%!+$)$0'%!*"

D

= ε

r

E,

B

= µ

r

H,

UL;9V

=&$)$%&$ '0')A-'"%!%!$

ε

r

(x)

'",

µ

r

(x)

')$%&$($)3!%%!+!%G'",($)3$'/!0!%GE )$ ($ %!+$0G; B-)%&$)3*)$EM&3W 0'=

(28)

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

σ(x, t)

0 1# #2( ', ($),#,3%'4),"'#, ! 5#+4 #5 ($) 6 7&)!! )8- ('#," 9:0; <=9:0:<> &) ?+"( ',(+#%- ) ('!%)% . +' 2!)" (# +)@+)"),( ($) %'4),"'#, ! ?)!%"0 1$) "@) ' ! ,#( ('#,"

ε

˜

0

,%

µ

˜

0

"( ,% 5#+ ($) %'4),"'#, ! @)+4'(('.'(/ ,% @)+4) 2'!'(/ #5 . --40 A/-"',B ($),#+4 !'")%"@ ) ,%('4). +' 2!)"

x

=

x

˜

L

,

t =

˜

t

˜

L/˜

c

0 ,

&'($ +)5)+), )!),B($

L

˜

,%%'4),"'#, ! "@))%#5!'B$(', . --4

˜

c

0 = 1/

√

˜

ε

0 µ

˜

0

> ($)@$/"' !?)!%" +)4 %),#,3%'4),"'#, !($+#-B$ ($)+)! ('#,"

E

=

_˜

E

˜

Z

0 H

˜

0 ,

H

=

H

_˜

˜

H

0 ,

J

=

_˜

J

˜

H

0 / ˜

L

.

C)+)

Z

˜

0 =

p

˜

µ

0 /˜

ε

0

,%

H

˜

0

+) ($) 5+))3"@ )',(+',"' '4@)% , ) ,% +)5)+), ) 4 B,)(' ?)!%"(+),B($> +)"@) ('.)!/0 D'($ ($) #,"('(-('.)+)! ('#," 9:0E <> )8- ('#," 9:0: < +) F-"(($) #,"'"(), / #,%'('#,"5#+9:0;<0 1#"))($ (@#',(>&)#,!/,))%(#( G)($)%'.)+B), )#59:0; <> @@!/9:0: < ,%9:0E < ,%+) !'")($ (($)+)"-!( ,()8- ('#,+)@+)"),(",#($',B)!") 2-( $ +B) #,")+. ('#,> &$' $ "$#-!% !& /" $#!%0 H#,")8-),(!/> " !#,B " ($) ','(' ! #,%'('#,"" ('"5/9:0: < ,%($)?)!%").#!.) #+%',B(#9:0; <>($)"#!-('#, ( ,/('4)&'!! !"#" ('"5/9:0:<0 I('"($)+)5#+)),#-B$(# #,"'%)+#,!/

ε

r

∂E

∂t

= ∇ × H − J,

µ

r

∂H

∂t

= −∇ × E,

9:0J< ', &$' $ ($) #,"('(-('.) +)! ('#," 9:0E < +) !"# ', !-%)%0 K" 5#+ ($) 2#-,% +/ #,%'('#,">#,)'4@#+( ,("@) ' ! ")'"($ (#5@)+5) ()!) (+' #,%- (#+"9LMH<0 1$)")+) %

ˆ

n

× E = 0,

n

ˆ

× H = 0,

9:0N<

&'($#-(& +%@#',(',B,#+4 !.) (#+

n

ˆ

0 A)(&)),4 ()+' !',()+5 )">',($) 2"), ) #5"-+5 ) -++),(" ,%"-+5 ) $ +B)>($)5#!!#&',B #,%'('#," +). !'%

ˆ

n

× [[E]] = 0,

n

ˆ

· [[ε

r

E]] = 0,

9:0O<

ˆ

n

_{× [[H]] = 0,}

n

ˆ

_{· [[µ}

r

H]] = 0,

&$)+)

[[u]] = u

+

− u

−

%),#()" ($) F-4@', ($) ?)!% . !-)

u

0 1$) )7@+)""'#,"9:0O < +)@+)"),(($) @$/"3 ' ! @+#@)+(/ ($ ( ($)( ,B),(' ! #4@#,),("#5 2#($ ?)!%" +) #,(',-#-" +#"" %'P)+),(4 ()+' !">&$)+) "($),#+4 ! #4@#,),("4 /2)%'" #,(',-#-"0

(29)

!" #$% '()$(*'*% +,-./0$( 1$% /.)$%,)$'( $( %2, . !"##$%&'(")! )*! +%,-)'%.+ )% )*! /"&0!,, !1-")'%.+ '. +#" ! -+'.3 )*!*'3*4 %$5!$.%5",5'+ %.)'.-%-+6",!$7'.(!)*%5'.)$%5- !5'.89: ℄ ".5<-$)*!$+)-5'!5 '. 89= ℄ ".5 8>? ℄@ A.)*! <%,,%0'.3 0!B$'!CD $!E'!0 )*! ("'. <!")-$!+ %< )*'+ 5'+4 $!)'+")'%.@ ! %.+'5!$ )*! /"&0!,, !1-")'%.+ '. )*!3!.!$", 5%("'.

Ω ⊂ R

3

F,,!50')* .%.4 %.5- )'E!(")!$'",+G

σ = 0

H".5$!0$')!G :@9 H'.)*!C-&<%$(

Q(x)

∂q

∂t

+ ∇ · F (q) = 0,

G:@IH 0*!$!

Q(x)

$!#$!+!.)+ )*!(")!$'", #$%#!$)'!+J

q

'+ )*! E! )%$ %< )*! F!,5 E",-!+ ".5

F

(q) = [F

1 (q), F

2 (q), F

3 (q)]

T

5!.%)!+)*!C-&@ K"(!,DJ

Q(x) = diag (ε

r

, ε

r

, ε

r

, µ

r

, µ

r

, µ

r

) ,

q

=

· E

H

¸

,

F

i

(q) =

· −e

i

× H

e

i

× E

¸

,

0*!$!

e

i

'+)*! %$$!+#%.5'.3L"$)!+'".-.')E! )%$@ !+!!7)*!.-(!$' ",+%,-)'%. '.)*! %(#-)")'%.",5%("'.

Ω

K

)!++!,,")!5'.)%

K

.%.4%E!$,"##'.3!,!(!.)+J'@!@

Ω ≈ Ω

K

=

K

[

k=1

Ω

k

.

M!$!

Ω

K

$!#$!+!.)+")!)$"*!5$",)!++!,,")'%.'.)*$!!5'(!.+'%.+".5")$'".3-,"$ )!++!,,")'%.'.)0% 5'(!.+'%.+@ N!<%$!<%$(-,")'.3)*!5'+ %.)'.-%-+6",!$7'.5'+ $!)'+")'%.J0!'.)$%5- !)*! +)".5"$5G%$$!<!$!. !H!,!(!.)

Ω

st

= T

d

<%$5'O!$!.)+#")'",5'(!.+'%.+

d

@ P*!+! "$!5!F.!5 "+

T

1 _{= {ξ : − 1 ≤ ξ ≤ 1}}

'.%.!5'(!.+'%.J

T

2 = {ξ = (ξ, η) : − 1 ≤ ξ, η, ξ + η ≤ 0}

'.)0%5'(!.+'%.+".5

T

3 _{= {ξ = (ξ, η, ζ) : − 1 ≤ ξ, η, ζ, ξ + η + ζ ≤ −1}}

'. )*$!! 5'(!.+'%.+@ Q" * !,!(!.)

Ω

k

'+ %.+)$- )!5 BD )*! '.E!$)'B,! ("##'.3

X

k

_{(ξ) : Ω}

st

→ Ω

k

J 0*' * '+ -.'1-! <%$ ".D 3'E!. !,!(!.)@ R%$ 5!)"',+ +!! )*! !&)!.+'E!B%%78S: ℄@ !.%05!F.!)*!F.')!!,!(!.)+#" !"+

V

h

=

n

q

k

N

∈

¡

L

2 (Ω)

¢

2d

: q

k

N

(X

k

(ξ)) ∈ P

p

d

(Ω

st

), k = 1, . . . , K

o

,

G:@>H 0*!$!

L

2 _(Ω)

'+)*!+#" !%<+1-"$!'.)!3$"B,! <-. )'%.+%.

Ω

".5

P

d

p

(Ω

st

)

5!.%)!+ )*!+#" !%<

d

45'(!.+'%.", #%,D.%('",+%< ("&'(-(%$5!$

p

%. )*!$!<!$!. !!,!4 (!.)

Ω

st

@ T'. !)*'+#%,D.%('",+#" !'+"++% '")!50')*

N =

(n + d)!

n! d!

(30)

!"#$%!& '(

ξ

i

∈ Ω

st

)*+ # !*& '-!". +'/+0.$'&"&0+ (&! #$1#2-# 2+%!$34 !0&#$(

L

i

(ξ)

%#((& 2'/-!.2/ '/+(+ !"+(5

L

i

(ξ

j

) = δ

ij

,

*&'/

δ

ij

=

(

1

&6

i = j,

0

&6

i 6= j.

7#8& 2'/+1#2-# 2+%!$3 !0&#$(#('-&#$ 6. '&! (# ".(& 2'/+0#%%& 2

X

k

_(ξ)

)

*+#%%-!9&0#'+'/+(!$.'&! #''/+

N

!"#$%!& '(*&'/& +# /+$+0+ '#(

q

k

(x, t) ≈ q

k

N

(x, t) =

N

X

i=1

q

k

i

(t) (L

i

(x))

2d

∈ P

p

d

(Ω

k

),

*/+-+

q

k

N

(x, t)

&('/+: &'+ +$+0+ '#%%-!9&0#'&! )# "

q

k

i

(t)

-+%-+(+ '('/+(!$.4 '&! #' !"#$%!& '

x

j

∈ Ω

k

;

7/+ "&('-&<.'&! !6 '/+ !"+( &( # 8+3&((.+ 6!- '/+%-!%+-'&+( !6 '/+& '+-%!4

$#'&! ) +(%+ &#$$3 6!-=+-3 /&2/4!-"+- #%%-!9&0#'&! (; >' &( <+(' 0+#(.-+"<3 '/+

1+<+(2.+ ! ('# '#((! &#'+"*&'/'/+1#2-# 2+%!$3 !0&#$(2!& 2'/-!.2/#%#-4

'& .$#-(+'!6 !"+(; 7/+1+<+(2.+ ! ('# '(/!*(?.('/!* $!(+#2&=+ %!$3 !0&#$

#%%-!9&0#'&! &('!'/+<+('%!$3 !0&#$#%%-!9&0#'&! ; 7/+0!('%!%.$#- /!& +(

6!- !"+( & (%+ '-#$@

hp

+$+0+ '0+'/!"(#-+ '/+A+8+'+ %!& '(BCD ℄# " '/++$+ 4 '-!('#'& %!& '(BFG )HI ℄; >' (/!.$" <+ !'+"'/#' #$'/!.2/ '/+A+8+'+%!& '(/#=+

'/+ <+(' & '+-%!$#'&! %-!%+-'&+( J$!*+(' 1+<+(2.+ ! ('# 'K & # '-&# 2$+ 6!-!-4

"+-(

p ≥ 9

) !"&('-&<.'&! 6!-#'+'-#/+"-! /#((!6#-<++ %-!=&"+"; L J#$0!('K !%'&0#$ "&('-&<.'&! !6'/+ +$+ '-!('#'& !"+()/!*+=+-) &(2&=+ 6!- # '-&# 2$+&

BFG ℄ # "6!-# '+'-#/+"-! & BHI ℄; M!-+!=+-)'/++$+ '-!('#'& %!& '(#$(!%+-6!-0

($&2/'$3<+''+-6!-!-"+-(

p ≤ 8

& '-&# 2$+(; 7/+"&('-&<.'&! !6'/+(+ !"+(& '/+ ('# "#-" '-&# 2$+ &( (/!* & A&2.-+ N;I 6!- !-"+-(

p = 2, 4, 6, 10

; O+ #$(! !'+ '/#' '/+ !"#$"&('-&<.'&! ( & # '-&# 2$+ # "'+'-#/+"-! *&'/ #

L

2

4 !-0 !%'&4

0#$1+<+(2.+ ! ('# '*+-+"+'+-0& +"& BID ℄# "BIG ℄; P!*+=+-)'/+(+ !"+()&

! '-#('*&'/'/+A+8+'+# "+$+ '-!('#'& %!& '()"! !'/#=+# +"2+"&('-&<.'&!

*/& / # <+ &"+ '&:+" *&'/ Q#.((41!<#''!4R# !<& %!& '(; O+ -+6+- '! BSN ℄

6!-6.-'/+-!=+-=&+*! !"#$J# "0!"#$K(%+ '-#$@

hp

0+'/!"(;

7! 6!-0.$#'+ '/+ "&( ! '& .!.( Q#$+-8& ( /+0+) *+ :-(' & '-!". + '/+ $! #$

& +-%-!". '# "&'(#((! &#'+" !-0!

Ω

k

#(

(u, v)

_Ω

k

=

Z

Ω

k

u

_{· v dx,}

_kuk

2 _Ω

k

= (u, u)

_Ω

k

# "! &'(<!. "#-3

∂Ω

k

#(

(u, v)

∂Ω

k

=

Z

∂Ω

k

u

_{· v ds.}

O+0.$'&%$3JN;CK*&'/'/+$! #$'+(' 6. '&!

φ ∈ P

d

p

(Ω

k

)

) /!(+ '!<+'/+(#0+ & '+-%!$#'& 2 1#2-# 2+ %!$3 !0&#$(

L

i

(x)

6!- '/+ '-&#$ <#(&( 6. '&! () "-!% '/+

(31)

!"#$ $&"'

k

()* &)'#+$('# ,- "($' ./#$ #0#1#)'

Ω

k

'. .,'(&) '2# .)'&)!.! 3#(45.$1!0('&.)

µ

Q

∂q

∂t

, φ

¶

Ω

k

− (F , ∇φ)

Ω

k

= − (ˆ

n

· F , φ)

_∂Ω

k

,

∀Ω

k

⊂ Ω

K

.

6789:

;#'2#) $#"0( # '2# .)'&)!.! /($&(,0#

q

3&'2 &' *& $#'# .!)'#$"($'

q

N

< ()* '2##=( ' >!=

F

3&'2'2#)!1#$& (0>!=

F

b

'.( .!)'5.$'2#1!0'&?/(0!#*'$( # (''2##0#1#)',.!)*($-8 @&)(00-< &)'#+$('&.),-"($' 5.$'2# # .)* '&1#$# !0'