:
9:
;
,
,4
,
0
'
%*
$
7 6 8
'
5'
'
-
+
&
3
-
$
3,
%
-0
'
"
-
$
3,
%
-0...
3 downloads
136 Views
759KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
:
9:
;
,
,4
,
0
'
%*
$
7 6 8
'
5'
'
-
+
&
3
-
$
3,
%
-0
'
"
-
$
3,
%
-0
-
2
.
1,
%
*0
,
/,
-.
,
+*
&)
(
H
G
U
GH
D
D
H
J ] B
I
KB
OB
D
LD
O
O
KG
R
I
CG
IB
K X
P
H
H
A B
G
[
[
B
V
Ce
Q
C
D
H
YD
E
L
O j
I
W
I
?H
K
F
?D
LD
A PD
K
E
O
TD
CG
]
U
N
K
T
D
H
[
[
PD
D OG
CB
PD
T
H
KG
NI
K
T
D
OJ
K
X
H
D
H
H
I
EB
TD
C
E
E
OQ
O
KG
R
ID
FV
FE
?
C
IB
?A
V
?
E
H
D
IG
V
E
X
OQ X k
FR
Ce
YI
]
U
B
B
H
A G
F
KG
C
H
YD
E
B
O l
PD
C
G A OB
I
VB
PI D
F
S?
ZB IA
CG
PD
C
O
FT
CG
O
OB
C
RG
I
D
B
H
H
H
B
H
G
A U B
EI B
TD
C
E
E
OQ
O
KG H
ID R
V
?I
O X k
I
FR
OQ
CA
T
OB
CG
C
N
KD O
V
T
L
] U
A
O j
I
K
IH
CB
P
K
IP B
KG
F
D
AH
H
VH
[
X
H \
H
OQ G
C
O
IB
IW
A
B
P i @
SK ^ H
A
I h
I
gB
OB
E
^
U
H
D
AH
H
H
V
C
I
W
I
I
I
d
bc
[
H f
B
J
D
EV D
?H
T
TQ D
C
E
Ke
CB
P
K
PI B
KG
F
UV
KI
?W
I
X
H
D
A
G
H
H
H
U
[
[
W?
a
I `
^ K
U
M NB
R
K
P?
^ TD _
C
E
OQ
RD
C
GH
D
LD
O
E
?I
R
CI
ED
L
K
I
Z
P
]
H
H
D
G
H
[
F
L
I O
? >
?I J
I \
I
?
R
CB
I
E
CB
E
RB
J TB
M
IH O
ZAH
F
ED
O
O
KG
R
I
?
PD
T
%$ &'
" ! #
U
J
D
A G
B
BM
A
C
E
N
D
H
IH O
YD
K
I
NV G
C
P
A
IC B
EG
I
CG
IB
D K X
W
I
I
I
EV D
?H
T
PD
TD
C
BM
H
E U H
PI D
TD
C
E
S
OQ
GH
D
RD
C
LD
O
E
PQ D
D OG
NG
C
LB
KJ
A I GH
F
?
ED
C? B
A
? > @
<=
]
]
]
]
]
]
]
]
]
]
]
]
]
]
B
D
A U OB
CG
A
VT D
V
O
W
I
I
?AH
A
]
]
]
]
]
]
]
]
H TD
C
E
OQ
RD
GH
D
]
C
]
]
] D
O
E
D ?[
H
OQ
P
C
I
T
TI
]
Ke
K
B
A KB
TV D
H
LD
B
C
[
[
]
O
]
]
]
]
]
]
S
P
CG
]
]
O
]
P
D
]
]
]
]
]
GH
H
Ke
CB
P
K
IP B
KG
F
D
AH
H
VB
?
?
H TD
C
E
OQ
RD
C
D
LD
[
X
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
P D
N
I OB
P
YI B
LG K
CG
[
A
A B
P
G
W
IB
I
E
ID
?A
VB
VB
CG
D
PG
?
H
PD
[
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
B
OD
G
]
]
]
]
]
]
]
]
]
]
]
]
H
H
J
AH
A
B
G
U B
F
RB
I
TB
E
I
V
OB
TD
C
ED
LD
O
GH
D
RD
C
LD
O
E
J
KB
I
O
]
]
]
]
]
]
]
]
]
]
]
I RB
J
D
H
TB
O
O
Y
NA
D
D
VG
G
B EB
FR
K
[
[
K
B
W
IB
I
E
G
ID
?A
VB
V
NI
I OB
P
YI B
LG
[
]
]
]
]
D CG P
P
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
U
D
K
?W
B
I a
Ce G
E
TD
SI
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
D
H
B
TD
C
E
D
LD
O
RD
C
GH
D
LD
O
E
Ce G
E
TD
SI
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
W
IA
C
I
B
H
[
]
]
]
D
W
A B
IC B
H
OB
I
P
YI B
K
LG
CG
V
IH
OB
I
P
YI B
K
LG
T
AH
B
B
H
RB
I
J
D
TB
O
Ke
TD
E
N
K
T
[
]
]
]
]
]
]
]
]
]
]
]
]
[
SK
A H
]
]
]
]
]
]
]
]
]
]
]
]
OB
E[
H f
U OB
E[
B
A
[
W
H f
B
IA
I h
I
W
IA
H
C
C
H
C H
IC B
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
W
IA
C
IW
A
B
H
B
[
H
H f
SK
A H
I h
U OB I
E[
Ke
CB
P
K
PI B
KG
F
D
AH
H
X
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
B
PD
K
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
D
Ce G
E
TD
SI
]
]
TI D
I h
A
]
[
TI D
C
H
]
W
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
H TD
C
E
OQ
GH RD
C
]
]
P i @
TD
C
H
KQ
?W
I a
I `
D
LD
O
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
U
U
SK U H I
]
]
]
]
]
]
]
]
]
O
IH
Y
KD
A TD
]
]
C?
F
L
O
A
A
H
GH
D
B
D
A CB
H
D
^ D A V
C
? DH
A
^ H V
]
]
X
U
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
D
C?I
F
L
N
W
I
A
GH
D
G
?AH
A
[
A
?H
C
B
O? B
F
N
O
IH
O
A PB
F
LD
KB
OB
OQ
R
IC B
[
H
H f
H
A
EA
O
R
H
D
D
CI B
E
P
SK
[ \
H
PD
TD
C
E
EA
S
OQ
D
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
A
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
I h
U OB I
E[
W
IB
IA
C
,
]
]
X
j
A U H
[
\
]
]
]
]
H
H
BM
P
J
S
OQ
OD
W
B
C H
,
U
A
O
I K
IS
O
EIA
C
P
SK
I
H
B
H
H
U OB I
[
h
H f E[
W
IB
IA
X
]
]
]
]
]
]
]
]
C IA
]
]
]
]
]
]
]
S
X
]
]
]
]
]
[
I
H
PQ D
B
D
CG
D
PG
P
YI B
K
L
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
H
B
PD
K
?H
C
B
O? B
F
N
O
I
U AH
TD
C
E
OQ
GH RD
C
S
F
[
9
]
]
H
U H
O
A
B
H
H
PB
F
S
O
IEA
C
P
SK
A
I h
U OB I
]
h
]
]
A
G
H
D
O
IH
O
A
PB
F
E
H
G CG
I
N
O
IH
ZAH
Ce G
O
YD
KG
NB
?
R
[
C
E \ IH C
I
H
C
H
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
H
H
H
PD
C
O
E
K
S
OQ
R
OD
?
TD
A
DH
B
N
O
IH
O
A
H
PB
F
O
IH
ZAH
K
]
H
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
X
]
KG
]
]
]
]
]
]
]
]
]
]
AH
A
D
B
P? D
FK
I
EA
NB
O
\
]
]
]
]
]
]
]
]
]
]
]
]
]
[ H f
B
H
SK
A H
I h
U OB I
E[
W
IA
C
]
]
]
]
]
]
]
]
]
]
H
B
PD
K
?H
C
U AH
B
O? B
F
O
IH
O
PB
]
]
]
]
A
D
B
S
T B
]
]
]
]
]
]
] H
KG
]
]
]
]
]
]
B
AH
D
H
]
]
]
]
]
]
D
E
C
I
I
O
IH
Y
KD
TD
V
E
AH
D
]
H
CB
H
P?I
IG
Z
I
PD
]
]
]
]
]
H
W
I
I
I J
D
EV D
?H
T
PD
TD
C
E
PQ D
D
H
G
H
OG
?I
R
CB
I
E
A
H
D
J
D
CG
IB
K X
W
I
I
I
EV D
?H
T
PD
TD
C
E
N
O
IH
YD
K
I
VH
OQ
PD
OG
l
)
;
<
(
'
\
H \
Ke B
W
I
O
I
TI D
]
]
* + * ,
H f
E[
W
IB
IA
C
P h
9
]
]
]
]
]
]
]
]
]
]
H
G
H TD
C
E
OQ
GH
D
RD
C
LD
O
E
?I
H
[
[
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
G
J PD
I
I X A B R
L
K
]
O
H
IH
H
S
A
G
H
U X B
S
?I
R
IB A
H
TI D
C
E
IH
O
R
Ke
EJ
I DH
OA
H
ED
PB
A
H f
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
G
G
B
B
OD
F
RB
I
T
]
D
AH
H
B
H
H
B
AH
A
B
U J
E
I
V
O
N
K
CB
P
K
IP B
KG
F
UV
N
K
T
O?
WH
d
bc
X
l
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
D
S
P
B
CG
A OB
CG
A
VT D
V
]
]
D
D
H
D
O
W
I
I
B
?AH
A
?[
O
Ke
B
B A KB K
V
H TD
C
E
Ce G
E
TD
SI
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
D
AH
H
Ke
H
CB
P
K
IP B
KG
F
V
d
bc
X
! #!" %&$ $
(
B
IB
BM
H
H
KD
^ H
KG
R
I
ED
C
X
>
AH
H
UN
D PG
V
EB
TD
C
E
E
OQ
RD
C
GH
D
LD
O
E
N
E
P
H
H
U
&
B
H
D
D
O
E
E
D
H
IG
A
H
D
B
OB
I
V
I
I
I
C
O
C
?A
CG
E
O
IH
YD
KJ
EI
LD
IA
CG
D
G
[
X
)
G
H
H
B
D
B
D
\
X
B
Z X F
?
I
IB
A
LD
OD
E
TD
S
PQ
Ce B
PJ
O
O
YD
E
E
C
I
QD
E
OQ
C
E
@
?
A
H
D
H
J
B
TD
SL
H
KB
RB
O
PB
O
H
YD
KB
K
QB
TD
C
E
N
I
C
B
U J
^ IP D
U
H
AH
NB
O
O
E
PD
PA
V
P
O
D
H
AH
CD
[
X
A
]
H
D
W
IB
IJ
B
A
D
H
AH
CG
IB
O
Ke
C
EI
CG
D
KB
CG
W
I
?
[
K X
IF
V
CG P
R
I
KD
?
^ H I
I
OB
OJ
I
KV
R
I
G
H
IAH C
?
T
I
D
KG
^
U
O D
D
A D
G
H
B
F
KD
TV D
C
E
C
T
A OB
CeI B
CG
FL
T
NB
O
RB
I
O
?
Ke D
C
H
S
OQ
B
CG
R
P
O
IH
O
A PB
F
E
CG
[
@
A
?
[
^
?
CG
ID
O
YD
^
H
A
J
A
T C
B
A OB
H
B
D
CD
P
PD
OB
C
Q
C? D
K
Q
TD
C
E
OQ
GH
D
RD
C
L
B
C
D
IC G
D
CG
ID
O
CG
R
P
^ B
]
D
H
KG
O
Ke
C
[
A
X
C
B
X
+
X
@
D
H
D
U
O
E D
H
YJ
E
OQ
R
C
IB
?A
VH
O
I
KB
C
ID RG
TV D
C
E
I
I
D
OB
E
O
PI
LD
?A
I
Z
BH
B
EQ
C
I
L
A D
A
D
KB
Ce
I
CG
NG
C
TF
T
LD
O
I BH
FR
KG
ED
C
] B
^ N
]
P
CG
D [
?
[
[
H PG
>
]
^
U
B
B
J
IH
H
A OB
B
J
H
QD A
CD
?H
O
K
I
F
IA
F
C
F
?A
CG
FM
O
Ke B
W
H
D
B
A B
A
K
I
?H
K
P
I
VH
O
CQ D
S
I
TQ D
L
C
P
?
@
?
X
T l O
P
WH
OB
S
A
CG
H
M
H
B
B
V[ G IH
E
I
T
P
I H
G
O
WH
A
IC B
E
O
IH
O
A PB
F
EQ
H
G CG
I
N
O
IH
ZAH
O
I
[
[
^
^
^
H
B
D
AH
D
I
B
PD
KB
SP
O?
T
H
PD
?
CB
I
E
CB
E
A D
D
LD
O
Ke D
PJ
I
VH
?@
A
@
?
>
D
J
G
H
AH
B
M
H
A
D
A B
G
S
?I [ G
G
G
ID
A
CD
Ce
N
KG
RG
I
I
C
N
O
I
PV D
TD
C
E
S
OQ
RD
C
GH
D
LD
O
E
A
T
YD
S
O
NG
FH C
C
X
T
P > H
D
H X
A
A
G
D
D
U
D
CG
] B
V
P
I
D
CG
ID
A C? B
I
X
[
,
H X ?J
E
OQ
KD
VH
O
Q
K
Q
O
IH
A
H U KD
T
FY
E
CQ
Q
C? D
NG
BH
CG
D
B S
[
X
O l
AH
H
C
U H
I
L
RB
TB
S
?I
R
C
IB
KB
]
H O[
O
K
D
H
I[
O
Ke
C
?A
X
N
E
KG
I
R
I
D
]
AH
DH
PQ
S
V
LD
I
O
E
P
^ M
B
Ce G
RG
j
P
L
H
H
D
G
H
[
C
C
FKe H
? \ B
D
LD
?
R
CB
I
E
CB
E
?
OB
IS
?I
CB
I
E
] B
U
H
D
CB
E
O
BH
A
P
S
OQ
YB
P
S
I
OQ
FT
C
S
G
J EQ B
EI
O
TI D
NG
C
U
K
[
B
H
Ke
C
j
AH
H \
B
H
KG
I
R
I
PQ
T
?
I
I >
C
C
FKe
D
eC G
O
YD
I
D
B
A
G
D
U H
F
E
C
I
KB L
?
C
KG
H
^ M
? \ B
N
I
OB
A PD
K
IH
TD
E
I
LD
I
D
H
KG
O
C
I
D
KG
C
G
D
R
I
PQ
O
Ke
A
D
Ke B
E
C
I
RG
I
V
P
CD
C
C
FH
P
CG
[
[
[
U
J
G PD
D
OD
ED
E
^
>
L
O j
I
] B
]
]
]
IH
C
SO
C
S
OQ
Ke
C
I
AH
D
H
H
KG
R
I
PQ
T
H \
B
?B
B
Y
?
C
S
N
W
I
H
B
B
?H
K
S
C
I
O
NG
C
A FH
B
U D
V
D ?[
?
I
I X B J
D
]
O
A
[
K X
K
I
ID
O
Ke B
E
C
BM
V[ ID
I
TB
D K X
N
O
IH
PD
C
PB
N
I
OB
PB
TD
VAH
VH
C
QD
VB
O
^
A
D
ZAH
E
C
I
L
KB
V
A
H
H
B
D
H NG
C
N
K
P
N
N
O
IH
ZAH
N
K
T
E
C
I
L
I A
Ce
I A
D
D CG
? j VB
D
G
H
G
H
A G
H
U H
C
I
P
I
*
O @
ED
O
O
P
B A CG
FT
G
D
ED
G
PD ?
?D
P
C
KD H
Ke D ?
?
O
A EI B
C
PB
D
D
H
D
U KB
M
Ce G
O
YD
KG
O
Ke
C
I KG
R
I
[
PQ [
O
Ke B
E
C
G
H
ID
V
?I
R
D
H IAH C
C
F
D
^
D
AH
H
B
NG
KB
C
R
A
IG
F
D ?[
?
] D
O
F
V
D
D
B
CG
T
I
O
C
?
EV D
?
N
E
P
YD
C
P
O
E ^ H
H
+
=<X
H
H
D
W
D
N
O
IH
ZAH
I
CG
D
IB
R
PB
A
A PV D
C
?H
P
I
Q
A
CI B
E
PD
C
O
E
K
P
CG
H
ED
K R
I D C
V
[
?
O
B
PB
A
RB
I
J
D
TB
C
N
K
T
S
OQ
OB
T
S
OQ
TD
S
PQ
S
I
OQ
TD
S
P
[
M
H
D
G
H
?
P
B
CG
ED
B
OJ
D
LD
PA
NV
K
T
ED
O
RD
CB
CG
T
P
eB
KB
LB
KJ
PI D
TD
C
E
NB
N
K
U D
^
A
A B
G
G
H
G
H
U PB
CG
D
G
NB
?
R
D
H IAH C
C
NI B
?
R
C
IB
?A
VH
T
L
W
I
?H
K
F
?D
LD
C? B
?
^
]
[
H
H
D
D
D
H
P
B
CG
H
YD
O
E
Ce G
O
E
O
Ke
C
[
[
H h
D
BH
H
IC G P
E
FRZ
FK
O
NG
C
N
K
P
N
O
]
D
G
D
H
A
D
D
G
A
B
M
OB
I
TJ
D
G
?G
O
K
IH
P
I
NV
O
IH
YB
C
I
M
OB
I
A PD
K
IH
TD
E
IH
?
CB
I
E
X
H
N
O
IH
ZAH
B
TD
C
E
N
K
T
CD
CG
ID
O
YD
KG
O
Ke
C
KG
I
R
I
PQ
Ce G
E
I
PB
IJ
[
B
B
H
AH
B
U H
D
H
CB
E
O
Ke
C
KG
I
R
I
PQ
D
[
V[
W
I
?
K X
FI
S
V
OQ
FR
O
S
OQ
O
E
PD
G
Ce G
R
]
H
H
D
H
H
NB
OB O
?
CB
I
E
CB
E
O
Ke
C
I KG
R
I
PQ
D
PV D
?
[
h
O X
I
FR
H
B
^
D
A
D
A
H
FM
O
Ke
C
I
RB
OJ
Ce
N
OD
VD
O
TF
C
PD
E
C
I
L
KB
S
C
I
PD
OG
N
I
OB
OJ
J
H X
C[
IB
OD
O
YD
SE
QD A
CD
?H
OB
D IG
I
V
N
O
IH
ZAH
D
LD
O
RD
C
N
O
IH
CG P
B YD
S
O
TQ
T
X
O
I
Z
Ke D
P
G
B
H
D
B
D
VD
?
CD
I
C
EV
I
Y
?
[
[
+
H
B
B
G
H
H
B
O
^ H
F
V
D CG
T
ED
TD
D
PG
P
NG
S
I
N
TD
S
O
IB
?
C
S
?I
RA
E
ED
?
?
^ B S?
C
?
V
X
U
A
B
D
H
V
B
C
E
OQ
CA
T
H
OB
]
A
D
H
G
H
C
B
D
S
OQ
TB
K
?A
I
SV
LD
I
O
E
P
ED
CA
T
OB
CG
F
? \ B
D
VD
KB
CG
I
OB
PD
J
Ke D
V
I
^
^
I
O
PB
A
PD
OB C
I
P
L
B
[
B
H
BM
G
B IG
F
O
CG
FL
IA
T
IC
TB
K X
D
M
G
B
IH
R
K
?
I
TD
C
H
H
D
J
D
[
A
S
I
C
R
F
D
ED
C
?
K
IH
P
I
V
CG
IB
K X
W
I
I
I
EV D
?H
T
TD
C
E
TI D
C
E
OQ
D
B
KD
IG
D
D
U H
E
eC G
O
YD
G
H
[
[
\
U RD
C
GH
D
LD
O
E
C
BM
TB
D K X
E
QD A
CD
?
^ C
O
H
EB
T
FO
?I
R
D
H IAH C
C
OQ
T
N
K
T
C
R D
^ NG
T Ce
F
L?
H
KG
]
]
H
CG
^
^
H
G
H
D
[
I
D Ce G
O
P
C
I
\ ?H \
eC G
O
R
C
IB
?A
O
IH
O
A PB
F
S
?I
I X A B R
L
KB
EQ
G
H
H
G
H
H
DH
U D
O
A
D
CD
?
IC G P
T
S
?I
R I
B
LD
I
K
S
I
?I
R
RB
J
A
B
H
[
[
X
E \ IH C
I
S
OQA
C
O
E
TB
N
IH O
ZAH
N
K
T
CG
I
H
D
IH X
H
X
A
D
PJ
I
V
E
TD
IS
NB
O
Ke
C
KG
I
I R
PQ
T
P
H
D
D
ID
L
O
RD
CB
CG
T
W
A
IC B
E
J
\
G
D
U G
I[
O
TD
S
NI
O
IH
ZAH
D
LD
O
P
C
I
?H \
N
K
T
S
N
CG
B
D K X
TD
V
P
RB
TB
N
O
I
]
D
H
D
L
A
J
B
D
BM
U H
K
D
KG
R
I
C
P
QB
O
E
C
I
KB
M
Ce G
O
YD
KG
O
Ke
C
KG
I
R
I
Q
CG
I
[
>
^
D
A
H
H
U H
ZAH
LI D
I
KD
SO
C
H
CG OQ
PJ
I
Ce B
PD
O
I
I
M
ED
?
E
I
N
KD
PJ
V
CG
IB
X
D K X
U
D
H
O D
YD
KG
O
Ke
C
KG
I
R
I
PQ
N
O
J IAH
I
G R? D
C
I
RB
J
D
TB
C
N
O
IH
ZAH
TQ D
S
[
[
]
D
B
H
H
W
I
I
I J
D
EV D
?H
T
EB
TD
C
EB
O
N
PG
D
A
B
D [
[
+
A
D
B
U TD
VH
OQ
YD
EJ
P
O
E
CI D
D EG
^ D V[
?
E \ IH C
I
O
O
[
]
H
C
1
1
3
67
29
38
67
35
234
NG
C
N
K
A
D
P
Re
C
D
AH
C
e
H
M
EV D
?
N
I OB
P
H
:
X
l
;
OQ
J
H
OB
N
PG
^ G
D
H
A QD
V
P
OQ
YB
P
D A OD
CG
P
N
N
K
PB
CG
A ] B
D T[ H
VH
E
C
I
L
KB
ED
V
AH
eC G
R
N
Ce
L
I
I O
?
CG
A
D
IB
R
C
P
D
H
IB
Z
E
O
R
OD
?G
l
A
A
D
G
B
OQ
O
B
A G
H
UP
N
CG
ID
O
Ke B
P
FO
I
DH
PG
NA
TD
LB
D
D Ke D Z
O
RD
CB
CG
T
f X
[
D
J
AH
B
G
D
U J
Ke D
V
FH CI
C
RB
TB
F
ED
C
^ NB V
K X
H
A OB
CG
V
H
H
B KD
I
^ E
^
D
H
eC G
OA
E
X
X
]
]
AH
H
W
Q
A
IC B
E
Ke
C
K
IH
T
AH
D
VD
C
R
^ E
YI D
KD
TV
H
H
H
[
j
A
CG
ID
E
IH O
E
I
V
Ce G
B
A
J
B U B
FK
O ^ N
D K X
DM
PG
C
E
S
I
IJ O
I
TQ
YB
CG OQ
PJ
I
EI B
W
H CD
O
R
K
IC D
T
L H
H
A
D
J
B
H
AH
A
G
UA
IC B
E
IE
OQ
O
Y
E
H
G CG
I
N
O
IH
ZAH
EQ
C
I
KB L
OQ
Ke B
E
C
I l
V
]
A
D
J
H
H
D Ce G
O
YD
KG
Ce
E
OI
Y
KD
T
RB
I
TB
N
O
IH
ZAH
E
C
I
L
KB
OQ
Ke B
E
[
A B
B
?
?
^ C
V
B
H
CG
S
W
A
CI B
E
P
PD
C
O
IH
O
A PB
F
]
I A
[
U
C
ID
V
^ H
P
H IC G
T
KG
? j B
n→∞
P
^ NB
P
QC
[
(
'
G
B
EG
A
]
CG
?H
G
D
KG
I
I R
PQ
O
Ke B
H CG OQ
P
O(n)
RG
IB
O
\
ND A
CG LD
]
O
IH
O
X
A YD
K
I
V S
K X
I
I
NB
L
I
!
$
*
,
n
O
&
W
IB
IJ
C
IS
LD
I
O G
I
O
IH
ZAH
E
[
H
ED
O
O
H
J
TD
V
?I
R
A
KG
R
I
E
C
ID
V
n2 + 2n
A ED
/
+
*
+
G IAH C
?
T
X
M
H
I \
S I
I
E
S
I
OQ
TD
I
VD
H
A
#
' *
'
I
PD
D
G
D
G
D
K
T
n→∞
n→∞
LD
.
VS
OQ
YD
^
A PJ
I
V
n
O(n3 )
B
H
%
D
J
EJ
P
OI
O(n2 )
A
O
*
' !" # * * # * $%
%
$
TD
I
n
'
'
$
&
! '
"!
A
-*
'
*
n
A ∈ Rn×n , x, b ∈ Rn .
^ N
,
*
Ax = b, O(n)
J
B
H
S
?I G
H
BM
P
UO
E
K
S
PQ
KF
O
O
H
H
H
H
D
KG
0 IR
CG
G
W
IE B
A
CI B
E
O
IH
O
A PB
F
S
?I
[
AH
A
U PB
F
S
OQ
'
.
3
0
0
0
\
H \
Ke B
W
I
O
T
I
IB
B
W
A
H
D
IJ
I
O
K
I
O
YD
G
^
D EJ
P
I
H
R
H
^
^
L
I
I O
?
ID
O
L
KB
EQ
H
G CG
I
C
BM
[
IB CG
R
I X A B
AH
J
B
[
?
OJ
ID
P
O
IH
O
H O
Y
A
ED
CG
D
PG
IE
I
D K X TB
I X A B R
L
KB
EQ
H
G CG
I
N
O
IH
ZAH
N
O
IH
T
YD
S
O
F
RB
TB
E
A
IC D
EG
G
B
$
$ $
%&$
#
$
%&$
#
]
A BH
H
B
H
H
AH
G
B
U CB
EG
A
PB
K
L
Ce
C
IB CG
H TD
C
E
F
ED
O
GH
D
RD
C
] F
J
D
B
H
LD
O
E
O
N
D
B
PG
V
PB
K
L
D K X
W
I
I
I
EV D
?H
T
PD
TD
C
E
VF
VA
L
NG
C
P
I
[
>
H
]
H
B
J
D
A NB
D
C >
W
I
I
I
EV D
?H
T
S
TD
C
E
P
IB
?
C
^ B S
TD
C
E
S
OQ
GH
D
RD
C
LD
O
E
P
]
D
H
H
G
G
D
?
? B
NG
C
FM
J
Ke D
V
I
O
O
H
CG P
F
B
TD
C
E
D
LD
C
B
CG P
P
O
IH
O
A
PB
F
[
[
AH
H
R
S
?I
B
A D
H
H H
KG
N
K
T
C G
FYI
[
H
H
D
AH
H
H
H H
N
K
T
PD
C
O
E
K
S
OQ
R
OD
?
TD
C
E
Re
C
T
G
H
U H
O
A PB
F
S
?I
R
C
I
V
K
I
K
PD
C
O
E
K
S
OQ
R
OD
?
H TD
C
] S
OQ
TD
PJ
I
S
V
OQ
CG
R
P
O
I
H
I X A B
H
KB L
E
H
G CG
I
?
N
O
IH
O
A PB
F
D
LD
O
\ H \
Ke B
W
I
O
T
I
N
O
IH
T
H
PG
]
U D
VH
D
PB
K
L
PA
H
[
[
V >
H
H
FM
T
H
H
B
KG
L
O
I
?
P
KB
A
IH
H
CB
E
O
I DH
A
CG
D
]
^
^
^
D
B
H
K
I
KB
OD
IA
CA
B
>
@
?
@
?
A
P j
I
B
VD
VA
B
TB
A CG
e
C
IE
L
I
O
I
?
E
K
IH
T
]
^
U
J
PQ
A IB
LD
OD
E
S
OQ
C
H
EB
S
FL
IA
T
IAH T
B
PD
?
[
\
@
?
@
?
U
H
J
D
Z
L
I
P
T
I
\
X
D
B
H
HX
H
IB
A
LD
OD
E
I K
Q
CG
IB
D K X
W
I
I
I
EV D
?H
T
EB
TD
C
E
]
]
^
A B
H
A
H
@
I LV
O
S?
PQA
V
I
EI
D
H
J
?
X
TI l O
E
A
IB
V
O
EG
CD
S
FL
IA
T
T
NA
P
H
B
J
AH
A
A
J
D
PD
OB
CQ
Q
C? D
K
I
Q
QC B
C
KeF
C
I
^ W
B
P i
B
TD
C
E
D
LD
O
P
C
I
T
I
C
B
X
]
B
D
D
H
B
A
D
H
H
U TB
I
I
C
OB
I
PJ
O
P
FR
I
KD
L
N
O
ED
@
?
VH f
I
O
?IW
B
H
D
H
H
N
Ce B
CG
NG
C
N
K
P
N
^ H
?
CB
I
E
CB
E
O
Ke
C
I
J
D
KG
R
I
PQ
P
W
I
I
I
EV D
?H
T
PD
TD
C
[
D
A B
H
BM
U H
E
PQ D
D
H
G
H
OG
?I
R
D
H
H
IAH C
C
PZ
YI D
KB
J
^ D
CD
LB
KD
V
PD
OG
[
[
[
X
l
]
A
H
A B
J
D
D
eC B
CG
P
O
H
OD
K
F
?
O
D
KB
CG
PD
TD
C
E
S
C
I
E
N
IH O
YD
K
I
V
IM
C
I
PJ
IB
A
]
^
^
U
A
H
H
V D
H
CB
CG
D
CG P
R
I
KD
?
PD
TD
L
S
G
NB
O
[
?
@
@
?
I f RB
E
H
]
^
UA
IB
V
O
EG
A ^ B S
C
M
Ee D
?B
O
E
H
G CG
IS
OQ
GH RD
C
N
O
IH
ZAH
D
LD
O
B
H
OD
W
IA
C
I
U
D
G
D
D
H
B
\
D
LD
O
P
C
I
?H \
N
K
T
KB
W
O I
CD
S
V
IH
OB
I
OJ
K
ZA
I
VH
O
S
B
TD
L
S
P
^
D
H
M
A H
G
H
?
V
B
D
B
J
KB
I
OB
D
LD
?
R
D
H IAH C
C
ED
C
O
FE
CG
O
I
O
Ke
C
I R
K
?
@
?
]
U
D
D [
H
AH
DH
D
J
D
G
IG
e
KB
I
PB
CG
W
I
I
I
EV D
?H
T
TQ D
C
E
N
E
P
L
KD
CG
IB
h
K X
O
PB
CG
H
G
B
G
H
D
U G
P
N
O
IH
O
A PB
F
D
LD
?
R
C
I
V
I K
K
N
O
IH
ZAH
N
OB
I
PD
CG P
F
P
H
CG
Ke
C
^
AH
B
B
D U J
?
T
N
K
T
D
LD
O
O
H
YD
K
T
V
E
N
B CG
@
?
D K X
TD
VD
V
N
OB
I
A
A PD
I
A
H
D
BH
B
B
C U H
IW
A
P i
B
TD
C
H
EG
NG
C
O
I
J
D
RB
O
W
I
I
I
EV D
?H
T
PD
TD
C
E
N
I
CG
, ]
@
? H
J
D
A PD
D
PB C
I
LD
O
TD
F
L
O
I
?
H
D
FM
TZ
Z
PQ
O
H PB
T
O
B
^ J I
KB
I
OB
D
LD
O
O
KG
R
I
^
A
AH
H
A H
GH
D
[
X
CG
IB
@
?
K X
P
P?I D
CD
L
TD
VD
OJ
e
N
K
CB
C
R
I
CD
FM
FM
C
B
PD
F
T
]
D
H
G
D
B
D
U B
M
A IB
LD
OD
E
Ce B
PJ
O
O
YD
E
L
O
I
?J
N
OB
I
E
O
IP
KB
E
O
Ke
C
I
[
i
,
\
O
I
?B
U
A G
D
B
A
H
D
H
H
D
CD O
O
K
FT
E
IAH
F
CB
C
K
J
D I[
O
R
ND Q
?G
F
P
?
O
TD
^ L
^
H NG C?
T
$
$ $
%&$
#
$
%&$
#
0 D
I J
[
\
H (Ω)
&
v
C
H
c kvka ≤ kvkb ≤ C kvka
H
W M
U
FO?
]
]
−(k(x)u0 )0 = f (0, 1), u(0) = u(1) = 0.
]
OA DH
H
ED
OQ
C
\ U H \
T
I
N
O
IH
ZAH
D
LD
D
H UR
OD
?
H TD
C
] S
OQ
TD
PJ
A
H
G
A
A
G
B
O
H
IH
A D
B
D
E
P
H
CG
A OB
CG
V
O
C
R
^ C
FH
T
H
KG
S
TI
I
P
H CG
O
T
H
D
A
H PB
O
B
A
H
H
H
PD
C
O
E
K
S
OQ
R
OD
?B
PJ
I
V DH
OA
KB N
?
CG
I
P
P
H TD
C
E
>
O D
YD
H10 (Ω)
E
N
(0, 1) \
O
TD
H
KB
RB
O
P
O
KB A E
P H
CI D
P
I
?
FC
FT
X
A
G
H
U PB
F
S
?I
H
I
S
PQ D
DH
H
H
G
H
H
J
] D
O
O
H
CG P
F
O
K
WH
S
IB
Z
O
N
K
T
S
QD A
CD
?
O
IH
RB
OJ
O
C
AH
O? D
?
[
,
^
^
"
'
.
. 0
$
)
0
H
H
IJ
PB
CQ
OB
CG
OD
?
OQ
Ke
C
YI D
KD
VH
QD A
C
H
H
I
H
KB
T
IG
e
T
%
^
^
A D D
CG
V
I K
D
CG
IB
D K X
CD
?
H
Ke D
C
I
NG
U D
?H
O
C
BM
RB
D OJ
D X
]
.
I
FO?
G
B
B
H
I
R
I X A B
H
O
O
KG
D
'
.
0
6
I
W
I
FO?
N
V
A
D
H
A
A
G
D
$B
S
TI
I
P
CG
O
H
H
A PB
O
P
K
I
T
PB
G
J
[
K h
\
]
Ce B
RB
OJ
D
D X
^
# KG I
>
^
K
T
C
GH
D
]
H B
P
H
Ce G
H
D A RD
V
O
YD
E
W
I
I
FO?
T
KG
IC
I
@
?
\
]
S
P
N
E
FT
O
T
K
H
B
X
v ∈ H1 (Ω) Γ v|Γ (x) = v(x) x∈Γ
B CG
R
A
A
A
EG
E
I
VD
LD
H
B
CG
I
P
PQ
?
?D
L
H
KB
EQ
CG
V
CQ G
A
OG
E
CI D
"
0
,0
$
]
O
I
B
J
^
D
K X
T
^
I
P
B
CG
H
H
D
D
D
D OD UN
V
S
IB
P
CG
OB
A
CG
A
VH
Z
PQ
S
OQ
O
T
P
P
A D O X
TD
A D
W
I
FO?
PD
T
H
D KG
P
CG
A OB
CG
V
[
] F
eC G
IE
H X
PQ D
CA
Ke
L
I
NG
M E
N
K
P
N
C
\
X
RB
I
D
D
J
AH
H
OJ
R
B
Ce G
U
0V D
Γ
\
B
D CG
A
H
D
H
J
OB
X
I R
N
B
G
B
7
I
T
L
G
H
]
G
A
P
O
H
IH
T
H
D PJ
I
V DH
OA
KB N
?
C
BH
RB
OJ
D
D
KD
P
E
C
B
X
D A U OB
CG
A
Rd
D
D
V
O
$
J
V
A
EQA
O
E
R
I
V
U G
I
Ce
I FR
KD
^
B
D
H
G
O
IH
R
[
A
H
H
EG
EQ
K
TJ
B
ED
P
H CG
O
A PB
]
B
WH
OB
IA
L
O
A
P
I
P
O
H
IH
T
H
D PJ
I
V DH
OA
N
KB
?
NG
C
BH
TB
J
H10 (Ω)
FM
O
A
NB
O
P
Q
AH
B
\
P
CG
H
RB
? >
X
H
VH
KB
?
CV
FT
I
M
T
P
O
IH
O
A PB
F
S
G
H
A
C, c, C1 , c1 , · · ·
P
D VH
O
O
H
I \ IF
T
I
O
IH
O
PB
C[
IB
O
NG
[
T
[
RB
O
IH
H
PG
N
K
T
C
FYI
KG
?I
R
C
I
V
(∇u) dx
CG
U
G
A
U H
AH
D
D
FH C
C
A
I
I RB
J
TB
D
[
ZAH
?
H
K
Ω
12
O
RD
F
P
H
H
H CD
?
O
O
I
J
D
TB
O
Ke
N
H
PD
C
O
E
H1 (Ω)
Ke D O
H
A PB
D PJ
DH
A
VB
O
I
O
O
H
T
YA
P
FC
OQ
PD
X
IH
Ke B
AH
H
O
IH
O
A PB
F
D
LD
O
Ke B
>
2
!
FR
]
F
D
N
O
D
OG
E
ZAH
[
W
I
O
H \
\
H AH
A
B
TD
E
E
I
V
f
ED
C
Z
KD
E
0
?
U
PD
D IG TF
Q
^
O
IH
O
A PB
T
IF
P? D
]
K
I
N
K
T
≤ C(Ω)
V
D
]
H
IH
T
YA
P
A
H
M
] B
C
e
F
S
G
H
OB
CG
D
k · k1
$
E
*
k(x)
Ω
' '
YI D
O
E
C
R
Ce B
LB
G
U B
K
?
TD
C
H
FC E
S
C
M
EV D
?
?I
V
K
S
OQ
Z
FH
A
A
H
^
AH
H
KD
TV
EV
FT
A
Ce G
F
P
]
H
H
PD
C
I
GH
D
O
E
K
S
OQ
R
H
OD
?B
RD OQ
C
LD
O
E
B
J
H TD
C
eD
H I X A B R
L
KB
AH
v ∈ C (Ω) H10 (Ω)
LI
C
O
I
V
I
^ N
W
U
NG
Ce
IE D
B
D
D OJ ?
O
YD
E
I KB
OB
N
K
T
EQ
E
G
D
ED
V
O
[
W
I
O
(u, v)1 :=
'
D
I
I
Z
V
I
]
N
O
X
H
H
PD
C
O
E
K
S
X
ID CG
E
TD
G
k · k H1
A
D U H
VS
M
PD
I
IH
RB
D
OJ
D
H
B
@
OQ
R
D
H
TD
IS
O
E
H
IS
FM
X
O
IH
O
u dx
I
H A KB
L
C
O
EQ
A FH
[
OQ
CA
T
OB
]
?
H
OD
?
ED
[
F
B
KG
Ω
12
,
k · kb c !
0
I
H
M
D
H
KG
F
NA
P
C
K
PD
FT
CG
Ce B
H
TD
H
Z
O
H
0
(
H
A
H
D
[
J
PD
KA
N
e
H
C
G
D
D
E
CG
2
*
O
IL
C
O
I
P
CG
A OB
^
G Ke D
V
L
O
E
H
H
KG
R
I
· L2 (Ω) H1 (Ω) H10 (Ω) 1 ∂Ω H 2 (Γ) H1 (Ω) Γ
B
H
IH
T
PJ
D
1 ≤ k(x) ≤ K. v(x) ∈ H10 (0, 1)
Ω
O
S
EQ
A FH
W
I
O
AH
k · ka
D
ED
A KB
A CG
0V
WH
IA OB
G
D
I
D ?[
G
O
N
O
I Ke B
T
C(Ω)
A
DH
H
J
L
A NI G
E
I
PD
LD
?
A
D O X k
T
:= Z
*
'
/
I
V
OA
L
C
O
H
H AH
P
B
Ce G
h
H
G
Z
X
X
H \
\
N
KB
G
I
E
OQ
R
OD
?G
R
X
K X
D
R
I
H1
*
[
\
T
A
AH
ID
O
P
Q
?
B
B
E
I RB
D
OJ
l
0
v|Γ
D
M
H
U D
S
I
E
^ I
W
I
FO?
J
B
O
BH
^
[
\
] B
W
(u, v)
'
J
0, . . . , k
#
.
I
B
D
B
O VH
A
Ω
ED
T
T?
NA
VH
OQ
TD
SO
[
C
J
TB
C k (Ω)
+
H
H
KG
E
PQ
FK
D
EQ
A FH
A
W
\
O
W
I
FO?
HX
v 2 (x) dx
1 Ω
G
IL
H
Ω R
O
A
H
PJ
I
V
OQ
OB
C
O
D
I
FO?
D
L2 (Ω)
W
D
ZH
EG
I
P
CG
OB
P
CG
A OB
F
L
∂Ω
I
FO?
P
H
KG
R
A
D
I
A CG
V
ED
0
k
CG
OB
A
CG
A
H10 (Ω) V
D
CG
A
u(x) v(x) dx Ω L2 (Ω)
L2 (Ω) C
H
k
P
H
V
Ω
OQ
O
H (Ω)
R Rd
Ω
(u(x) v(x) + ∇u(x) · ∇v(x)) dx,
Ω
∇u(x) · ∇v(x) dx,
Ω
#
%$
"$
!!
%
$
%
$
! #" "
!
#
a(u, v) = f (v) ∀ v ∈ H.
$
'
&
$
+
'
H
$
!
@
D
Z 0 1
(u ) dx
00 2
12
A
^
C
R
Ce
D
≤C Z 0 1
f dx
2
D
12
O D
^
A
A
>
^
B
G
0?
T
L
$
G
H
^
C D
OA
N
DH
KB
(·, ·) kuk ≤ cku0 k
P
YD
E
S
TI
I
ED
P
CG
O
H
H
Z j ID
U
H
Ke D
G
VD
PD
O
Ke
CB
T
Ce
PB
I
CG
TD
V
P
H
H
PD
I
A KG
]
^
E
A
V f IB
N
H
D
B KG
V
P
H CG
O
A PB
O
IH CG
IB
D
LD ZAH
]
G EB
N
D R[
A PB
V
I
O?
O
IH
T
2
U
D
AH
H
FM
C
O
X
A
D
Ce
I FR
KD
V
O
FT
IH O
C
H
RB
I
J
D
G
H
TB
?
R
C
I
V
K
I
K
N
O
IH
ZAH
D
LD
B
KG
O
D
N
[
K h
ZAH
EA
O
S
QD A
CD
?H
O
I
O?
WH
^ N
]
C ] H
^
6
3
6
29
3
U
X
B
KG ]
U F
KD
]
J
$B
?
AH
^
J
[
U
P
C
K
D
H
PD
FT
Y
O
I
M
H
NG
C
N
K
^
]
^
A D
H
O
H
IH
ZAH
NG
C
B IH
T
L
P
CG
OB
A
CG
H
B
D
A D
V
P
D
CG
OB
A
CG
V
I
P
KB
OD
W
I
FO?
EA
NB
O[
O
K
I
I
\
\
A
T U H
AH
D
˜0 H
D O X H H
D
0
A
A
PB
O
I
ED
P
CG
O
H
WH
D
, N
PD
I
BM
I
W
I
FO?
\
D
H
P
D
CG
E
P
A P?I D
FK
I
EA
[ \
X
U
? B
]
]
D
B
D
$
ZAH
I
O
IH
T
YD
S
O
P
C
IC G
D
G
B
P? D
OB
CG
V
$
CG
ID
O
R
C
I
V
K
K I
^
]
N
O
IH
]
B NB
K X
ai,j (x)ξi ξj ∀ ξi , ξj ∈ Rd ˜0 u∈H ˜ 0. ∀v ∈ H
TD
f dx
2
] V
0
1
EG
O
TD
E
FR
I
KD
P
PB
N
B
H
YD
S
O
I
J
NG
H ^ AH V
A
A
CD
?
H
B
ED
P
K
O
D
[
U
G
A BH
G
H
O
IE
N
B
PD
I
KG
F
IE
PQ
?
IE
FL
IA
T
O
IH
O
A PB
F
E
A
CI D
EG
B
LD
IE
\
]
]
H
KG
EQ G
ED
O
R
QD X
^
^
C
R
Ce
A
D
IH
A PD
VD
ED
D YD
E
C
O
G
^
O
Ke
I
P
?
T
L
H
C
O
W H
H
?
C B
NG
C
N
^
E A
A H
IB K
P
V
O
W
I
I
FO?
H
T?I B
K
L
eFC
G
N
I
\
I \
\
]
^
]
U
P
I
D
H
IH O
ZAH
KG
I
H
D
LD
C
H
H
D
H
KD
O
O
P
CG
O
T
IH
CI
FH
CG
H
A
H
E
O
H
IH
ZAH
Y
?
^
H
H
O B
KB
OD
W
I
FO?
OQ
O
R
OB
IA
LD
[
[
OQ [
O
\
]
D
H
U G F
O
IH
ZAH
C
]
]
]
]
U
−
g
A
H
K
I
C
BH
J TB
CG
IB
R
D
A PB
PV
KB
L
C
O
I
I
H
KG
E
O
IH
ZAH
[
#
D
H
A
H
OA D
H
ED
O
PB
I
PD
KG
F
O
O
H
KD
V
PQ
I
EI
N
W
I
FO?
EI
OQ
O
R
OB
IA
[
^
U
M
H
EQ
A
IH
EJ
NI G
C
N
K
P
N
CQ
O
W
I
\
I \
D
]
d d X X ∂ ∂u ∂u (ai,j (x) )+ bi (x) + c(x)u = f, ∂xi ∂xj ∂xj i,j=1 i=1 ∂u u|Γ1 = 0, |Γ = h, ∂n 2 Γ1 ∪ Γ2 = ∂Ω Γ1 > 0
C
k(x) ∈ C 1 (0, 1) PJ
H
K
T
]
KG
H
E
I
H
X ∂u ∂v ∂u a(u, v) = ai,j (x) + bi (x) v + c(x)uv dx, ∂x ∂x ∂x j i j Ω i,j=1 i=1 Z Z ˜ 0 := {v ∈ H1 (Ω) : v|Γ1 = 0}. f v dx + hv ds, H f (v) =
d X
[
[
^
^
A
|(f, u)| ≤ kf kkuk L2 (0, 1) 1 ≤ k(x) Z 1 12 Z 1 12 Z 2 0 2 u dx + (u ) dx ≤c H
I
H
]
C
N
V
E
AH
B
3
D
B
KG
D
a(u, v) = f (v),
D
D
D
V
O
NG
eC B
D
UO
P
CG
* ]
K
P
N
F
T
TQ
PV
?
?
^ E
VB
P
CG
A OB
O
Ke
PB
i,j=1
Pd
I
V
A
YI D
0
C
AH
KD
TV
D
^
O
IH
T
H
D
PD
J
O
RD
CB
]
^
]
F
P
H
CG
O
PB
A
NB
M
Ω
P
D
V
H
A PJ
I
V
BH
O
T
IH
I
O
IH
D
EI
N
Z
CD
I PD
O
BH
KG
F
A
G D
ED
O
EI
PQ
P
CG
A OB
D
CG
A
D
IA
NV
OB
A
0 < c0 ≤
F
EA
O
B
O
Ke
C
NB
P
CQ
FR
I
A
D
CG
T
NG
IH
OB
I
A
H
Ke B
L
?G
3
H
Z
O
F
I
O?
O
I
B
E
PQ
?
BH
PD
v=u
KD
D
U
KG
D
D
I
C
P
H
G
CG P
F
C
FH
T
C
O
B
7
u(0) = 0
WH
V
T
I # KG
D
UO
AH
H \
C
R
]
M
J
D
H
QB
?
O
IH
ZAH
H
B KG
N
D CG
R
CG
D
V
LD
O
PD
OG
^
H
g
u(x) u ∈ C 2 (0, 1), k ∈
Ce
FR
I
KD
V
a(v, v) ≥ ckvk2 H
?
O
RD
CB
\
^
O
T
P
I
J
O
IH
O
A PB
F
C
H
D
D
D
H
¯ 1. f v dx + a v(1) ∀ v(x) ∈ H
+
&
D
CG
T
I
N
\ I \ IF
T
I
N
NA
C
NA
P
C
A
A
TV D
V
X
H10 (0, 1)
.
H
O
IH
ZAH
D
LD
O
X
IH O
H
KS
I h
I
M
PD
A Ce B
∂u (1) = a. ∂x
!
T
O
RB
I
J
H TB
OQ
R
QD
N
P
C
]
N
NV
I OB
PD
A
A
I
KG
F
I
TB
J
]
^
I
N
W
I
FO?
>
J
^
]
M
D
H
J
5
BH
1
9
NG
C
P
QB
O
NB
NA
P
C
K
PD
FT
B
A D
I T
P
C
E
S
I
OQ
H
H
TD
PJ
I
S
V
OQ
CG
R
P
N
O
IH
O
H
D
H
A BH
BH
F
A D
H
I
FO?
OQ
O
R H
OB
IA
LD
D
H
B
A D
B
D
A
D
H1 C
H
E
H
P? D
I KF
EA
D
B
KG
P
M
Ce G
R
A PB
V
C
J IH
EB
^ E
0
^ H
T?I B
K
L
US
I
V
N
K
T
E
N
PD
I
KG
F
E
OQ
R
OB
IA
L
E
PQ
KF
O
NA
TD
LB
K
D
YD
\
^
]
T
CD
B
O
KB
[
[
OQ [
O
K
I[
OQ
Ke D
PJ
I
V
Ce
IFY
OD
]
M
KG
$
G
X
X
D IH O
Z
H
>
D
G CD O
?
E
T
I
W
LD
?
R
C
I
V
I K
K
N
K
T
I
RB
J TB
NB
P
?
FR
KG
E
H
KD
A U PB
]
∀ v(x) ∈ H10 (0, 1).
a(u, v)
B OB
T
T
LD
H
KG
IP
H
K
PD
FT
D
H
H1 (0, 1)
]
PD
OQ
C
O
^
J
LI
C
D
H
E
A
H10 (0, 1)
*
*
#
O
A PB
B OG
FR
IH
O
A
PB
F
E
P
CG
O
P
N
K h
I
V
u(x)
*
C
IN
OD
#
A EB
H
R? D
H
I
D
CG
H
f v dx
# *
&
&
KG
I
N
L
O
IH
S
PQ
TB
R
0
'
]
IH O
B
C
PB
O
I
M
O
1
+
*
K
D PD
FT
[
Ke
I
H
IH
KG
D
NA
¯1 H
*
D
D OD
C
D ZAH
B
U G
?
A
PB
E
R
I
u(0) = 0,
F
X
H
X
0
FR ^ B ?[
EQ G
ED
O
R
^
W
I
B LD
N
W
1
LD
OA
H
ED
\
OB
0
Z
Z
f (u) u∈H
VH
O
O
QD
P
FO?
KG
I
FO?
>
k(x)u0 v 0 dx =
B KG
X
X
TD
]
[
X
0
D
D
IH
H AH
A
D
ZAH
E
Q
k(x)u0 v 0 dx =
H
.
O
N
E
E
N
PD
E[
H f
x=1
,
O
Ce G
H
H
B
f
H
H10 (0, 1)
!
*
,
IH
O
A PB
O
I
V
EQ
E
K
¯1 H
F
D
IH
ZAH
I
D
eC G
O
k(x)
LD
O
Ke B
T? B
K
L
u
1
*
,
KG
F
1
DH
X
Z
+
W
I
u
'
.
*
*
C 1 (0, 1)
#
+
Z
A
[
[
\
X
v(x)
d
Γ2
12
$
$
"
$
#
%$
"$
!!
%
$
%
$
! #" "
%&$
#%"
%&$
!
]
I RB
TB
J
]
]
] g
H
H
Z 1
0 D
U
k(x)u0h (ψhi )0 dx = Z 0 1
f ψhi dx
D
vh
i = 1, . . . , N − 1
U X D
?
U
FO?
I
J \
B
D
U D I[ K
?
Uh Uh
AH
G
X
H
S
?I G
H
I X A B R
L
H
KB
EQ
CG ^
G
D
D
U H
ZAH
?
I
V
C
CG
ID
OA
H
EJ
B
]
∀ vh (x) ∈ Uh .
CD
VD
O
P
X
G
A
[
^
I
N
W
I
FO?
\
J
M
D
H
NB
NA
P
C
K
PD
TF
]
PD
I
[
PQ
]
OQ
K
TI
PD
?
N
J
[
K h
U
H
H
D
H
E
O
H
IH
ZAH
E
OQ
C
O
E
K
O
R
OD
j
E
UA
L
N
O
B
IH
O
H
B
KD
V
PQ
Y
?
C
FH
AH
CD
V
I
W
I
FO?
\
X
A U PB
F
\
O
W
I
I
FO?
B
\
O
O
N
IH
J
D
H
G
D
A IB
O
Ke
C
I
O
CD
X
B
U
J
AH
CD
S
OQ
PB
A
N
RD
P
PB
A
I
M
D
BH NG
C
P
Uh
CB
I X B J
PQ
O
IH
f vh dx
0
^
eC B
K
H
D TG
0
D
H
J
QB
O
KG
F
S
OQ
R
O
I
U
H
PD
C
O
W
I
[0, 1]
CG
]
O
R
OD
g
uh
T
H
RB
O
I
ZAH
?D
]
]
RB
I
TB
J
U \
D I \
?
D
Ce G
]
U
H
H
H
D
H VB
FR
KG
ED
OA
ED
LD
O
E
O
YD
KG
D
]
>
X
D
A D
E
I
A
A H
VD
CG
V
EQ B
H
YJ
V
[
[
U
H
H
OQ
P
Q A
H
AH
VH
O
OQ [
O
K
ID
O
G RD
F
0?
]
D
D
B
D
A D
^
GH
E BH
B
?
I
CD
eC
PB
H IJ
C
YD
E
H
VB
V
EI B
PD
KG
TD
V
NG
C
UW
I
\
D
X
E
B
I \ D
C
OB
CG
[
OD
?
NB
H
S P
OQ
TD
PJ
I
S
V
QD A
C
P
F
EA
O
O
O?
WH
N
O
Ke
C
H
M
D
I
P
D
CG
A OB
CG
A
VH
D
AH
A
A OB
^
CG
C
A
G
FH
EI
ID
?A
C
B
]
C
R D
^ DH
?
C
O
C
N
H
H
OD
CV
H
OB
U
OD
I
VD
C
^
B
UA F
EI
G
ID
?A
VB
V
CG
ID
OA
H
EJ
A
23
6
82
3
D
CG
P
> H X
CG
V
E
Q
]
:
A
I
]
]
U
J
B
H
H
H
PD
C
O
E
K
S
OQ [
O
K
ID
O
G RD
F
?
N
K
T
W
I
I
FO?
H
OQ G
I
$
] G
B
D
B WH O?
P
K
TI
H
G PB
VS
N
PD
I
KG
A
\
B
H
M
H
A
S
PQ
S?
OQ
A BH
A TD
D
TD O
I
^ B V
?
] D
CG
E
C
E
IH
Y
?
C
I
D
H
WH O?
OQA
D
+
PD
C
]
]
]
UA
I
V
N
O
B
IH
ZAH
D
LD
D
D
UO
W
I
H
\
I \
?
EA
O
D eC G
KB
I
B X
D
H
eC G
O
Ke
C
YI D
KD
V
eC
E
HB
>
@
?
YI D
KG
N
K
T
FR
KG
ED
C
EG
AH
KD
VT
V
^
^
^
DH
H
D
H
I
A
A
D
H
O
Ke
C
D
H
KD O
V
T
E
IB
V
O
KG
I
O
O
H
KD
V
PQ
C
FT
H
PD
I
KG
F
O
OB
h
X
(i − 1)h ih (i + 1)h
B
O
Ke
H
G
O
H
P
CG
F
P
CG
O
H
PB
A
B
I
KG
H
+
B
T? B
K
L
[
H
LD
O
E
O
C
D
CI G
O
IH
O
A
PB
F
IY D
Uh ⊂ H10 (0, 1) H10 (0, 1)
T
P
I
C
?
O
T
D
OB
A
I \
D
?
^ B
]
CG
IB
K X
D
\
Uh vh (0) = vh (1) = 0
N
O
] G
D YD
E
H
PG
A CG
0V
D
?B
^ PD
K
FL
S
I
N
F
EA
D
B
O
PD
I
KG
H
D
D
H
Uh
IH
D
A
IB
X
B
Z
I
O
− (i − 1), x ∈ [(i − 1)h, ih], − hx + i + 1, x ∈ [ih, (i + 1)h], 0, . CD
C
H
?
]
O?
WH
D
CG
H
H
H
D
J
TD
S
P
HX
NB
O
N
O
IH
R
F
S
OQ
Ke
C
K
T
PD
KD
F
A
D
H
$
EQA
D
D [
O[
O
K
I
I
CG
ID
O
P
IW H C
?
Ce B
PD
AH
CD
V
\
X
X H
D
]
D
A J
B
D
D
O
D
RD
CB
CG
T
CG
ID
E
ZAH I
O
RB
OJ
O
TD
N
K
T
^ BH
FR
KG
E
[
X
>
]
J
$
H
RB
TB
P
S
OQ
OB
T
D Ce G
C
B L B k T? B
K
O
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]
-
]
N −1
D
H
eC
Q
C
YD
?
B
+
k(x)u0h vh0 dx =
O
=
{ψhi }, i = 1, . . . , N − 1 O
?
C
IB
?A
V f B
E
N
?
H
eC G
B
D K X
H
E
C
E
I
C
\
D
T? B
K
L
OB
IA
LD
I O
KD
D
BH
H
Uh H10 (0, 1) Uh
A PB
Ce G
F
H
[
]
O
I
B
A CB
QD A
C
PB
D
Y
I # KG
E
^
W
I
I
FO?
H
U
O
G
RD
F
0?
EQ
X
V
T
D
FR
C
O
2h
F
]
ψhi (x)
LD
H
VA
A
O
IH
O
PB
A
X
TB
P
K
FL
]
X
0
O
^
E
]
TD A U D
F
LD
C
CG
ID
H
T?I B
K
L
C
FT
1
O
X
V f IB
] G
]
[
?S
AH
D
CG O
?
PD
C
CG
IB
G
T?I B
K
L
FCe
N
O
D
H
TD
SI
O
E
[0, 1]
CG
D
J
B
A YB
I
T
P
I X B J
PD
C
H
D
CI D
A KB
FL ^ KD
D K X
A
H
K
I
N
O
KG
h
ID
D
I[
W
I
O
O
W
I
\
O
H
H
X
[
FCI
O
P
IH
O
A PB
F
^ NB I
K
F
IE
ψh1
H
^
U
CG
I \
O
R
OB
L
C
O
I
I
CG
IA
H
X
D
IB
LD
C
FT
H
A
A
CG
Q
FC
I
N
TD
S
P
A
H
IH O
Ω Z
O[
@
D AH
B
D
LD
E
^
A
ZAH
CQ
O
H
h
H
?
?
P
N
O
EA
O
N
O
IH
E
IB
V
O
Uh
O
K
FI
K
G
I >
]
P
BH
D
?
IH O
ZAH CG
T? B
K
L
CD
N
O
IH
ID
I
O
IH
T
H
D
B
PD
V
ZAH
]
H ZAH
OQ
TD
Ce G
H
D
IH
O
c > 0.
O
\
^
FR
KG
ED
OA
H
AH
HX
@
?
CG
ID
T? B
C ] H
^ NG
eC
TD
IS
G
B
^ C
H
H
?
V
a(·, ·)
IH
O
KD
V
PQ
Ce B
P
U
T
I
N
O
IH
ZAH
D
,
%
SE
C
P
A
^
P
Ce G
H
K
L
CG
G ID
E
A
D
PJ
A
W
I
k(x) ∈ C 1 (0, 1)
LD
F A U PB
H
LD
O
O
Y
.
-0
0
AH
A
D
RD
V
O
YD
E
C
H
I
C
Ω
NG
A K X
I
I
7
2
I
W
Q
IA OB
L
?H
RD
C
S
I
PB
IJ
D
FT
E
I
kukH 1 ≤ C1 kf kL2 .
C
H
H
B
AH
A
A
D
H
NV
K
T
C
FYI
.
1
0
.,
&
W
Q
OB
H
V
kukH 2 ≤ C2 kf kL2 ,
N
K
P
N
E
I
V
O
Q
VW
O I
I
V
L
E
OG
I
l
KG
PD
H
H
0
3,
%
PQ D
K
FL
S
IA
L
X
Ω Γ1
PD
C
O
E
K
H
N
=<X
[
C
O
E
K
S
OQ
R
N
H
˜ 0, ∀v∈H
S
OQ
R
]
O
IH
O
A
PB
F
S
H
OD
O
c(x)
OD
l
OQ
Ke B
W
?
TD
H
I
kukH 2
?B
TD
H
[
I
O
C
CG
C1
C
E
N
K
T
D
PD
OG
N
AH
H \
∂Ω
B
P? D
X
O
IH
O
bi (x)
OB
CG
D
NV B
B
KG
a(v, v) ≥ ckvk21
A D
G
D
O
D
H
AH
O
H
IH
YD
KD
TV
V DH
O
Ke
C
KD I
V
T
eD
K
I
TB
OD
V
FM
O
TD
PJ
I
V
D A FM
C
P
X
g
C
O
I
O?
D
B
WH
N
O
IH
FR
KD
NV
K
T
C
R D
IH
EB
I
RB
^ E
]
]
]
J TB
N
O
IH
ZAH
%
6 6 ψhi
-
1
x h
M
D
N
K
T
EI B
O?
D
WH
EI
OQA
A
IB
V
EI
OQ
PD
D OG
NG
C
N
K
P
NB
P
CG
O
H
H
H
A PB
O
OQ
OB
h
#
%$
"$
!!
%
$
%
$
! #" "
#
%$
"$
!!
%
$
%
$
CG
I
D
N
BH
X
C[
H10 (Ω) ]
v(x, y) I
u|∂Ω = 0. J O
M
IH
O
PB
A
]
]
]
]
H
F
RB
J TB
FM
OA
FE
H
B
B
CG
ID
OA
H
EJ
A
CG
ID
C? G
Y
W
B
T?I
EI B
A
U VK D I
0V
W
I
C
B
BH
^
H
O
W
I
FO?
S
B
[
OQ [
O
K
ID
O
RD
G
\
UC
H
CD
]
M
R? D
C
P
FK
SO
OQ
A PB
H D
D
[
U
F
?D
P
x=0
G P
C
R
^ NG
C
P
TQ B
? K
\
O
H
I
[
O
FO?
D
OG
I I X B J
PD
KG
F
B
[
D
S
?I
?
O
I
WH
OD
?
ED
f
]
A BH
BH
H
PQ
H
CG P
RB
?
P
PD
I
KG
F
S
PQ
?
FR
KG
[
A
A
D
BH
A U OB
CG
V
E
I X
>
]
]
]
]
]
]
]
]
]
I
H
H
W
Q
A
]
$
T
P
I
C
E
I
H
H
J
^ A
A H
W
A B
IC B
E
N
K
T
E
V f IB
] D
O
P
N
C[
D
H
IB
O
L?
K
M
D
CG
ID
OA F
W
A
E
S
NB
FR
KG
S
CQ G
A
V >
FK
O
OQ
PB
A
CG
ID
H
B
H
IC B
A
A
C? G
Y
W
Q
IC B
E
EJ
]
]
^
H
PD
C
O
W
G
H
GH
D
G
D
H
H
D
PD
C
O
E
K
P
CG
O
I
Ke D Z
C
W
I
FO?
S
OQ G
[
X
\
I X B J
S
O
TI
S
Ke D ?I
?
O
$
H
M
H
D
GH
AH
H
DH
IE
N
K
C
I
O
?
H
H
G
D
G
D
Ke D
C
O
IH
R
V
CG
F
VH
O
C
E
I
O
R
QD
PD
C
O
E
X
^
H
G
OQ E
CG
H
H
H
PJ
OI
T
L
O
IH
O
A PB
F
S
?I
H
P
H
G
D
B
B
+
W
I
FO?
D
OG
[
[ \
I X B J
Ke
C
I
O
?
?
?
[
I X A B R
L
KB
S
H
H
H
K U H
S
OQ
R
OD
?
TD
C
E
]
]
]
]
]
B
D
D
UC
TD
I
PG
H
IH O
Z
O
H
CD
P
O
IH
YD
KJ
A
H
PD
TD
C
E
S
OQ
GH
D
RD
C
LD
O
E
JH
KB
I
OB
]
CG
ID
?
]
H
H
G
D
I
V
A
Ce B
J PD
Ke D
V
I
H
H ]
H
T
P D
CB
I
OA
C
Ke B
C
FYI
KG
DH
D
[
X
D
H
AH
A
H U CG
Y
W
Q
IC B
E
N
O
IH
K
T
V
N
K
,
D O X
FT
C
FT
O
IH
K
T
AH
D
VD
O
O
E
A
CD
?
]
$
]
0P
Ce G
F
ZH
D
D
H
PD
C
O
W
I
\
I \
?A
C
?H
B
]
A B
H
X
D
D U X B
V
FO?
\
I X B J
NV
O
IH
YD
KJ
T
E
U
I P
P
OQ G
O
IH
YD
KJ
A
G
H
Ce B
?
EI
FT
H
IH
O
N
k(x)u0h vh0 dx,
IB
O
$
RB
F
P
T
E
A
CI D
]
]
A
CI B
O
TD
H
H
M
X
B A PB
V
CG
D
7
IC B
E
CQ
O
E EB
] K
D FK
EA
\
35
Ah
I
Ω = (0, 1) × (0, 1),
.
TB
I
G
B
B
H
E
BH
KG FR
O
IH
ED
C
>
D PB K
T
C
FH
P
CG
UA
]
2 −1 0 −1 2 −1 1 Ah = 2 h −1 2 −1 0 −1 2
J
M
A
A
A
D
EG
E
I
V
LD
]
H
R? D
C
Uh
FM
O
TD
H
H
P
V
EI
OQA
F
M
NA
D
2 −1 0 −1 2 −1 1 Ah = 2 h −1 2 −1 0 −1 1 H
T
P
I
C
E
I
]
G KB N
?
T
AH
H
D
7
6
5
:
k(x) = 1
P? D
W
P
H
]
CD
NG
BH
$ F
[
NV G
M
6
7
\
ψhi
OB
CG
I
FO?
C
]
]
C
R
D
[0, 1]
D
\
K
PD
FT
u(x, y)
V
M
H
B
U J
EA
O
H
H
C
N
1
I
I \
D
H OQ [
O
E
PB
0
FM
^
W
H
U X B
D
D IC D K
O l
V
K
C
KG
I R
PQ
NG
C
HB
D
H
F
E
N
W
I
1
FM
O
Ke D
P
I
FO?
\
P
H CG
[
D
]
P
CG
O
]
YI D
B
Ce G
R
P
?A
I K
I
CG
TD
G
J
Z
B
D PJ
A
C[
IB
O
$B
RB
? >
7
3
W
I
\
H
PB
]
IH
O
IH
T
H
D
PJ
A
H
O
E
S
X
NB
J
QB
C
?H
E
H
CG
G
I
hAh z, yi =
KG
E
FR
I
VB
O
∂2u ∂2u − 2 =f ∂x2 ∂y
KD
V
^ E
]
G
G OD
1
[
FO?
ED
]
J
x=1
B
D
NB
YD
*
B
V
W
I
\
P
CG
OB
A
N
AH
H
I
D
YJ
I
A FM
B
A PB
O
(n − 1) × (n − 1)
CG
A
O
E
F
]
FO?
D
CG
O
IH
R
Ce B
RB
V DH
OA
V
FCe
R
G
V
P
NG
I
?
NG
Ah
R
V
A
H
−
FN
IL
C
AH
U H
T
D
5
6
2
O I
IH
YD
A
T
H
D
PJ
A
G KB N
?
I
C
ZB
E
B
H
N −1
O
B
N
O
IH
V
ckzk ≤ kPh zk0 ≤ Ckzk, Uh
I
U
IE
IE
H
B
F
V
I
h yi zi , kyk := hy, yi1/2 .
N
[
P
N
K
C
^
A KJ
P
H
OJ
X
X
E
OG
I
ND
V
fh = {fi }i=1,...,N −1 , Z 1 1 fi = f ψhi dx. h 0
D
O
KD
V
D
I
EI B
C
T
Y
D
D
D
H
D
A
W
Ah zh = f h .
R
I
NG
OQA
N
IJ
G KB
PQ
N
K
T
H
YI D
O
E
O
IH
K
PD
C
V
LD
OA
E
FT
P
P
E
H
I
CG
ID
C
N
K
P
N
zi ψhi .
O
PB
H
PB
O
CG
E
X
OB
CG
H
G KB N
?
H
CG
OB
A
CG
EI
O
IH
A
CI B
uh
IH
D
A
?
T
AH
^
ID
TD
SI
D
H
h
Z
O
CD
Ah V
I
1 h H
H
O
P
ND
C
G
[
O
C
O
c
D
V
H
H
]
O
W
I
\
I \
D
E
O
IH
H
Rh
LD
W
Q
A
CI B
K
KB
P
E
\
K
T
IC D
N
O
K
H C? G
Y
{zi }
OJ
H
B
E
C
YI D
P
I
?
OD
?
EI
EJ
D
D
G
EG
IH
T
1 h
KD
V
PD
C
O
O
E
D
?
ED
P
CG
OB
IA
A
P
ND
zh ∈ R
E
G
H
H
] F
EA
O
QD A
CD
?H
O
G
A
CG
AH
D
B
H
PJ
Ah = {aij }i,j=1,...,N −1 , Z 1 1 aij = k(x)(ψhj )0 (ψhi )0 dx, h 0
e
E
H
U
BH
V
V DH
PD
f
i=1
N −1 X
P
N
O
IH
T
H
D
C
C ^ ED
D
J
ED
A
Ph : R h → U h
N
K
A PJ
I
D
BH
I
uh =
H
IH
IE B
C
OB
CG
OD
?
D
RB
OJ
O
A
V
T IC
^
I
i=1
O
O
IH
K
P
ND
V
Te
V
LD
X
H
I
] F
G
J
N −1 X
K
I
^
D
D OA
N
L2
PD
O
H
AH
D
AH
]
O
KD
V
PQ
I
hy, zi :=
IH
K
T
V
OB
D
H
RD
DM
G
KB
?
H
^
RN −1
CG
F
N
K
T
P
EI
N
P
CG
O
T
L
Ah
E
X
>
H
I
O
IH
K
k · k0
TD
PG
A
H PB
O
Rh
IS
O
D
IH
]
h
T
H
D
CD
RN −1
H
fh O
A PJ
I
V
h
uh zh {ψhi } zh ∈ RN −1 PQ
D
D
H
IH O
Z
H
H
H
CG
E
C
E
]
]
]
]
H
J
A D
I
S
OQ
Ke D
PJ
I
NV
K
T
C D
FH
T
KG
I
R O
B
D
IH
Y
?
C
G CD
H
] D
]
C
^ C
H
C
N
O
KD
V
PQ
^ NG
]
I
KG
NG
C
N
O
KD
V
uh = Ph z, vh = Ph y
,
#
%$
"$
!!
%
$
%
$
! #" "
#
%$
"$
!!
%
$
%
$
#
z, y ∈ Rh
Ah
.
n×n
]
0
−I
−I Tn H
O
ku − uh kH 1 ≤ c h kukH 2 .
C
IY
H2 (Ω) D
U G
U
O
O B
NG
C
G
D
I
W
I
FO?
F
[
\
J
H
^
]
US
0 B
h
IH
I
O
AH
H
RB
I
J TB
]
H
H
H
[ P
D
CG
O D
YD
E
H
QD A
D
? CD
^
H KB
I
C
T
NB
PG
D
CG
E
C
E
]
]
Ce
NA
P
C
K
D
D
H
PD
FT
O
Y
A
KD
T
VC
I
]
G
H
O
U D
H
IH
J
H
H
IB
O
YD
KG
D
Ke D ?
?
X
X
]
B EG
O?
D WH
C
IFY
KG
[
@
?
]
^
H CB
Y
K
A
H
TB
O
I
VA
E
A
A H
U G
ID
?A
VB
V
ED
E
I
V BH
FR
KG
ED
OA
H
ED
O
TD
>
^
H
H
^
P
A D
B
[
I
H
UA
IB
V
O
D Ce G
T? B
K
L
FM
O
Ke
C
KD O
V
A
D
D
S
C
N
K
T
H
Ce B
A PD
FK
I
EA
C
N
O
I
B V
P
CG
OB
A
CG
V
CG
D
T D
M
H
H PG
OQ
OD
W
I
C
E
IH
QD A
CD
?
PD
C
O
E
K
\
H
H
B
G
A H
D
H
U
E B
G
ID
?A
VB
VF
ED
C
D
^ D V[
O
Ke
C
I
P
C
K
PD
FT
NG
C
N
K
P
N
O
N
W
FI
C
I
NB
?
D
]
H
W
I
I
FO?
O
C
D
AH
O? D
?
N
O
IH
T
PD
V
CD
CG
G ID
E
I
PB
IJ
P
^ KB
ED
O
TD
FL
[ \
C
B
U B
+
v
C
H
ZAH
F
T
Y
T
Y V
H
P?I D
˜ 0, ∀ v ∈ H2 (Ω) ∩ H
EQ G
E
H
FM
T
E
Ce G
]
G
l
O
N
W
B
P
CG
D
PG
S
OQ
B OD
W
I
E
U
I
A
G
X
T
^
C
H
D
H
B
C A
J
H
A EB
V
EQ
TD
IS
O
ED
O
OB
PD
[
X
]
A D
D
CD
?H
O
N
K
[
G
H
Ke D
?
Ce
Q
C
YD
E
X
U
A ? \
I
I
VH
YB
T
D
B
?
O
T l
N
IH
O
IB
A
^
]
D
A
O
C
?
B
J
AH
CD
H
IH
J
DH
H
D
IB
K?
E
O
RD
CB
CG
T
C ] H
X
M
D
LD
O
J OB
TB
D
LD
K
N
K
T
B ]
X
O
C
R D
YI D ^ E
0
AH
BM
KD
TV
FR
LD T
KG
F
C
ED
OA
H
ED
^
AH
TD
?
N
E
P
O
\
J
NI
W
I
FO?
B
C
R D
^ DH
?
U
H
F
CG P
F
G
D A NB
CD
?H
v ε > 0 (0, 1)
^
H
uh KG
E
D Ce G J
H
Uh
H
D
OJ
I
E
D
?F
V
B
FI
C
G
BH
(v 00 )2 dx
P
B
B
OB
]
Q
T
A
H
I
0
K
^
A
N
I
C
R
CG
ID
C
C
?I
h
TB
A
ED
B OJ
CG
D
^
Ce B
LB
^
FR
>
O
G
H
B
H
RD
F
?
N
K
T
O
IA
Z
P
N
W
IA
FE
O
I
CG
IB
vh ∈ U h
O
O
X
[
X
OA
N
FK
AH
O
KG
ED
OA
H
ED
LD
D
X
A
67
D
P
D
CG
OB
A
CG
V
1
5
3
]
C
H
H
D
O B
NG
E
OA
P
LD
C
N
K h
H1 (Ω)
I
V
LD
A
O
IH
D
B
inf kv − vh kH 1 ≤ c hkvkH 2
IH
B
J
V
vh ∈Uh
O
O
A IB
KD
TV
L
E
I
Ke D
O
E
U
1
]
] H
Y
C
J KG
AH
H
N
PD
FL
AH
B
D
Z
KB
OB
]
PD
C
O
X
I
KG
C
O
CG
IB
K
U
]
D K X
H
IH O
J
IB
A D
P
D
CG
A OB
CG
V
H
OB
I
P
J
G
BH
QB
?
PQ
C
RB
D
D OJ
C
R
FH ^ E
A
IG
A
A
ZB
e
H
VH
+
B
\
B
]
T?
A
NA
F
VW
B
D
CI B
E
FM
O
R
O
I
T
IH
C
BH
RB
D OJ
D X
]
]
]
]
]
]
]
]
]
4 −1 0 −1 4 −1 Tk = , 0 −1 4
?
K D
A
H
I
WH
A
IC B
E
NG
C
N
K
P
N
[
$
T
P
I
C
E
I
H
H
n×n
]
RB
A K X
I
I
B
V
E
FT
@
F
B
D CG
G
PD
2
^ PD
C
O
V
E
D
E P
?
QD A
CD
O
O
R
H
OD
C
O
H
Uh
I
E
H
H
I
P
H
?H
O
Ce G
B
eC G
K X
dx ≤ h
TB
u J
OQ
E
K
H
H
CG
A OB
^
A
RD
V
Ce G
F
H
vh0 )2
C
H
E
K
D
ED
T
]
]
?B
O
E
K
H2 (Ω)
O
E
O?
WH
D
A
H
A CG
0V
B YB
?
]
S
]
H
PD
C
O
H
H
IH
I
FRZ
K
D
C
H
UN
FK
OB
J
X
(v −
0
S
OQ
B
C
R
EB
T
T
L
U AH
D
M
D
H
D
J
B
OQ
R
F
P
CG
O
H
A PB
H1 (Ω)
H
D
IO
^ NG
C
BH
H
I[
OQA
B
N
V
E
<
A
H
OD
?
H TD
C
E
1
R
H
OD
H
H
l
C
?A
]
X
SO
H
G
B
^
J
FM
M
D
H
E
K
$
h>0 kv − vh kH 1 < εkvkH 1 ε
K
O
H
?B
P
QB
?
OQ
O
IH
J
IB
I
FO?
A
P
P
CG
OB
A
IF
I \
\
NA
P
C
K
PD
FT
S
]
H 1 (0, 1)
ZAH
U
R
OD
I
J
>
H
D
Ce G
D IB
O
N
I DH O
A
D
CG
A
H
0
KG
H
TD
C
IA
Z
A
J
OB
D
CG
V
N
K h
T
I
vh ∈Uh
Z
I
X
]
H
P
OQ [
H
B
D
D
B
]
VB
PD
CA
RB
TB
J
D
inf
^ C
H
E
F
?
˜0 H2 (Ω) ∩ H ?D
L
ID
Z
B
]
^
˜ 0, H
FH
H
I
H
U
A
G
CG
IB
O
C ] H ] I K
S
?
V
C
?
?
D
H X
O
R
∀ v ∈ H10 (Ω).
T
]
E
B
E
O
I
I
FO?
O
[
H OQ [
O
K
ID
O
^
?I
I O
Ke D
Ke
H
^
Uh
KG
H
] S
O
V
PD
KD
D
GH
D
K X
A
U
G
GH
RD
F
?S
P
CG
OB
A
L
I
LD
I
KB
] G
I
| {z } h I
B
$
K
H
IA
Z
H
CG
V
J
P
P
S
OQ
P
AH
[
FL
AH
D
CG
O
H
H OQ [
O
K
I
Uh
TF
C? l
Y
P
S
Uh
O
IB
^
?
O
I
Ke D
FL
AH
\
X
Q
VH
O
W
Ω
J
OB
IB
D
A
A
K X
FCI
O
IB
O
D
OB
H
NI
W
I
FO?
NB
^
PD
P
CG
C
H
J
EB
0
. J
IM
W
I
K
B
H
M
D
D
I
FO?
\
A OB
A
V DH
O
f v dx dy
]
OQ
Ke B
CG
D
\
C
S
O
IA
Z
P
K
C
R
OG
Uh
B
D
S
FO?
P
H
D
H IB
Y
K
TG
C
B
]
P
^
H
B
P
C
I
H10 (Ω) Uh
?
V
GH
E
H
?
A
C ] H
N IE
O
IH
RB
]
B
OQA
C
H
OA
H
1
^
]
P
D
J
?
H
D
OJ
IE
IP G
D
CG
A
O
E
E
0
Z
]
H
E
eD
P
^
^ H
E
A
AH O
I
O
GH
IB
V
FC
O
P
P
0
V
H
1
]
−I T2 E
P
IH
O
0
]
]
D IH O
Z
O
\
O
E
?
O
I
Ke D
AH
K
dx dy = Z
]
]
T1 −I 1 Ah = 2 h 0 W
I
I
FO?
[
X
X
O
NG
FL
C
CD
H
[
D
J
B
ED
A
D
D
NG
S
M
Uh
D
IA
Z
P
D
OG
I X B J
[
H FH
A
H
I
∂u ∂v ∂u ∂v + ∂x ∂x ∂y ∂y
G
NB
H
U
i E
N
H
1
K
P
H
D
K
I
C
N
H
∂Ω
T
D AH
H
i
O
IH
P
FE
OB
K
T
0
V
Ah ^
U
UE
B
% 0
RB
OJ
CG
?I
?
u ∈ H10 (Ω) Z 1Z 1
]
B
H
RB
?
]
W
FM
M
Uh
A
CI B
h Uh
T
n
∀ v ∈ H2 (0, 1).
ku − uh kH 1 ≤ c inf ku − vh kH 1 . vh ∈Uh
#
%$
"$
!!
%
$
%
$
! #" "
#
%$
"$
!!
%
$
%
$
D
D
CG U H WH
A
IC B
[
[ D
E U H
Ce G
O
YD
]
G
OQ
U G
I
P
S
CG
H
PJ
I
H
B
A PB O
FY
J
WH
A
^
H
D
A
H
D
E U H
Ce G
O
YD
KG
NB
O
Ke
C
I KG
I R
PQ D
C
R
Ce
IH
D A PD
V
O
YD
KG
O
] U
H
B
G
G
B
H
\
E
OQ
P
C
I
[
H h
?H \
O
NG
C
PD
I
OB
CG
F
TD
C
E
S
QD A
CD
?
N
K
T
^
M
A
B
U J
E
OQ
YD
KG
H
H
KD
?D
LD
E
O
IH
E
I
VB
TD
S
TD
V
W
IB
IA
CG
K
K
NI
X
D
H TD
C
E
eC G
O
G
D
B
P
UA
I ?I G
H
R
G
B
BH
CD
I
C
EV
I
C
RB
D
D OJ
C
R
IC G
H
VC D
^
T
C
E
J
FH
Ke D
V
I
EQ
?
O
T l
]
^
[
[
A
G
H
D
H
B
TD
C
E
Ce G
O
YD
KG
NB
?
R
E \ IH C
I
W
A
CI B
H
H
B
H
F B
G
G
B
TD
C
E
I
O? D
A LD
V
H
B
TD
C
E
?
?
^ H
?I
C
TQ D S E
C
OQ
C
H
OD
OQ
RD
C
E
N
AH
LB
ST
I
Ke B
YD
K
PB
] B
^
D
G CG
I
KG
FM
O
Ke B
H
C
B
O
H
D
\
O
P
C
I
?H \
O
YD
E
EQ
CG
G
I[
D
OB
T
N
O
IH
ZAH
N
K h
]
]
]
]
]
]
]
]
]
A
U NA
V
Ce
IH O
E
I
VD
WH
A
IC B
E
G
[
B
FE
P
T
?
?
N
A
H
B
A
B
R U H
E \ IH C
I
FM
O
H
S F UA
?I
B
G
H
I X A B R
L
KB
S
O
H
H
H
D
J
D
M
Ce
I A
D
CG
C
N
KD
PJ
V
W
I
I
I
EV D
?H
T
TQ D
C
E
I
TQ D
C
E
OQ
RD
C
GH
D
LD
C
H
H
H
U
]
O
G
D
RD
F
H
D
G
A
B
D
G
H
D
O
H
W
IB
IA
C
ID
TD
M
Ce G
O
YD
KG
?
R
[
[
E \ IH C
I
I
CG
ID
E
TD
IS
M
Ce G
G
]
H
H VB
N
W
I
I
E B
G
ID
U
\
H
B
B
C
J
? D
G
NG
FH
A
IH
C
?A
S
PD
TD
C
E
S
OQ
B
H
OD
W
IA
C
I
D Ce G
O
P
C
I
?H \
TQ D
C
K h
?A
A
]
g
V P
RB
?
P
X
]
]
F
RB
CG
?
A
PB
PQ
^ PD
O
E
K
S
OQ
R
J
TB
E
A
H
OQ
D
H
H E U H
OQ
B
H
OD
W
IA
C
I
EQ
TD
IS
O
S
CG
H
H PJ
OI
KG
R
I
ED
Ke D Z
I
X
X
"
0
C
e
V
A
(
H
M
] B
'
.
.
$
)
-
0
%*
'*
,
,
0
I
VS
OQ
A
G
?
Ce
N
EB
V
FM
O
P
CB
IA
H
D
VB
O
O
IH
R
OB
IA
LD
S
?I
R
C
IB
?A
D
H
AH
D
H
ED
TD
C
E
Ce
ZAH I
O
YD
EJ
P
O
N
E
P
H
H
ND
B CG
O
P
'
.
0
"
%*
.
H
[
$
*
0
7*
&
1
'
%
VJ B
J
B
U ED
U
G
B IG
F
"
-0
'
E
?I
A J
B
D
B
G
U CG
H
H PJ
IO
ED
OD
K
K I
IE
E
TI D O
EQ
H
G
H
CG
I
OQ
RD
C
GH
^ ED
]
+
A
X
]
^
g
H
H
H
D
C?
K?
E
O
C
IN
H
GH
D
H
B
B
EB
V
O
P
CB
H
D
AH
IA
V
E
Z
Oe
E
[
[
[
A
J
A G
B
B
A
GH
D
D
C
D
NV G
FH
AH
CD
VH
C?
F
L
O
O
AH
CD
EG
O
RB
I
TB
N
O
IH
ZAH
N
K
T
[
[
X
X
T
P
I
H
H
B
PD
C
O
E
K
S
OQ
^
D
D
G
J
[
H
M
$
C
IN
EB
VD
OA
C
e
EV D
[
D ?[
E
FH
Ke D
V
IH
E
<
P
I
O
O
IH
KG
I R
PQ
O
I
X
D
H
UR
H
OD
?
ED
TD
C
E
N
O
IH
ZAH
I
RB
I
J
D TB
O
\
[
H \ AH
Ke B
W
I
O
T
I
N
O
IH
ZAH
eC G
O
H
BH
M
D
D
AH E U H
P
IH
E
<
P
?
H
Ke D
C
O
NG
C
R
K
B
B
G
G ?J
F
B
B
TD
C
E
N
K
T
RB
TB
J
X
]
J
B
H
A U B
O
O?
D WH
?
C
N
A TD
P
I
V
G
H
H CG
E
PB
P
CG
O
H
H A PB
O
S
O
TI
H
D
B KG
V
]
]
^
P h
A
D
D
B
S
H
AH
OQA
SE
C
Ce G
O
FT
C
?
O
B
G
`
c
c
T l
W
L
O
e
I K
T
I
PQ
C?
O
I
H
GH
B
RB
J
H
D
B
TB
Y
?
C
N
IH O
ZAH
[
^
I
G
RB
I
J
D
TB
O
RD
C
GH
D
H
AH
OA
SE
C
N
O
IH
Z
[
[
U
A
D
PD
V
N
O
IH
KG
R
I
N
K
T
T
FO
? GH
J
B
D
G
D
B
D
D
U
O
P
E
R H
^ H
Z
PQ
?
T
NA
OV
TD
I
O
Ce G
E
TD
IS
Ce B
?
T
H
H
D
^
]
U
H
D
B
D
X
^
B
D
G
H
H
AH
B
G
G
U AH
N
K
T
T
FO
? GH
FH
C
C
F
B
B
TD
C
E
A B
J
B
[
A
A
B
H
J
NI B
FRZ
K
OI
E
V f IB
P
E
H
H
H W
R
KD
IJ B
KB
I
CQ D
?
R
AH
D
IH C
E
O
A
[
H
D
YD
E
EA
O
P
C
O?
WH
NG
C
N
O
KD
V
PQ
N
IH O
ZAH
N
K
T
EB
O
S
CG OQ
PJ
I
$
EA
h = 25−1 h = 50−1
O
IH
Ke B
E D ZAH
H
BM
ID
V
I
P
R
D VH X G IH
C
H
H
H
TQ D
C
E
OQ
H
OD
?
H TD
C
E
I
?I
C B
RB
I
J
H
TB
OQ
OB
T
KG
H
D
A
A
D
D
O D
Ke D
PJ
I
V
I
V
C
R
^ H
^
LD
C
H
H
KD
[
A
X
X
A
B
J
B
B
IH
O
M
IH
R
K
FP
?
C
I
A
TD
P
I
V
P
T
P
U
J
I
O
A
B
M
IH
R
K
IH
FP
A IB
P
\
N
LD
O
O
Y E
K X H
G
I
I
V
CG
IS
OQ
A
N
O RD
T
YD
S
O
IH
U
D Ce G
O
OD
B YD
GH C
H
W
IA
C
KG
^
I
H OQ [
O
K
IF
E
H
G CG
I
E
FR
I
KD
H
H
A
IC D
EG
G
B
h = 100−1
S
H
AH
OQA
SE
C
IB
?
C
G FM
?
]
CQ
O
E
K
E
O
H
IH
E
A
ku − uh kL2 ≤ c h ku − uh kH 1 .
S
[
H
I
V
OQ [
O
O
IH
I K O
P
f ∈
]
^
OQ
]
Ah z = f h
GH
H
D
KG
H
D
[
^
O
Ke
C
EQ
ku − uh kL2 ≤ c h2 kf kL2 .
C?I
Ke
E
O
C
OI
KG
0 IR
O? D
N
K
T
Uh
B
CB
J
B
D
H
NB
O
Ke
[
EQ
H
G CG
I
N
O
IH
ZAH
IC B
B
H
D
G
H
B
G
G
E
T
L
W
IA
SV
?I
R
N
K
T
B
H
^
C
A E \ IH
F
E
A
LD
V
L2 H1
LB
ZH
?
H
H
C
I KG
R
B
K
T
C
ID
V
.
O
O
E
K
D
H
I
^ EB
A
H
2
VH
?
U
O
R
OD
I
PQ
RB
]
H
E
0
IH
Z
Oe
J
TB
S
H
AH
OQA
SE
C
O I
IH
]
C B
RB
T
k(x) = 1
C
IB
B
?G
H
A ZAH
H
P
TD
N
?A
V
O
[
[
EQ
CG
G
H
IC G
2 −1 −1 2 −1 1 Ah = 2 h 0 −1
H
D
H IC G P
T
I
N
O
IH
ZAH
B
G
G
B
I
E
C
FE
C
R
BH
RB
O(N 2 )
OJ
C
H
C
BH
N
K
T
F
B
TD
CG
O(N )
^ C
]
Z
PQ
L
CG
ID
N
NG
BH
T
B
D
C
3
C
P
J
D
T?
NA
V
O(N 2 )
QB
?
O
IH
V
L2 (Ω)
$
#
%$
"$
!!
%
$
%
$
! #" "
%&$
#%"
"$
&
%&$
"
%
!
$
&$
"
&
G
D
X
H
N
L
P
C
R
A CB
D
D
A
B
S
I
Ke D Z
I
V
?
?
X
B
X
A J
B
B
F
?
ID
^
O
Ke
D
H
CB
Z
H PD
T
K
] B
P
T
P
E
P
?
OD
I
EA
L
S
S
B
D
B
UC
A
BM
B
?G
S
?
OD
I
EA
L
P
D
W
I
FO?
S
OQ
O
H
CG
G
J
[
\
X
^
]
]
H
IE B
PD
KG
IE
FL
IA
E
R
I GH
H
YF ^ H
S
O
I X B J
P
Ce
CG
h
U
D
BH
B
H
B
? D
TD
C
SE
?
OD
I
EA
SL
D
?I G
PQ B
O
C
R
^ C
ID
YI D
H
P
?I
ID
Z
O
IH
O
H
EJ
B
B
H
IJ
O
O
P
H CG
C
P
B
CD
W
H
B
[
X
I
V
A
]
N . 2
IH
D PD
L
C
C
^ C
^
CG
C
NA
]
+
H
D
?I G
PQ
IA
C
IW
k≥
NB
FM
T
KG
B
D
?I G
R
^ H
H
H
A
IC B
W
IA
C
A
B
H
[
]
]
V B
P
E
Q
AH
>
$
E
E
I
H
H D
G
B
O
IH
RB
OJ
S
OQ
O
H
CG P
N
K
T
T
LD
[
+
HX
X
X
^
\
G
B
UA
V
E
A
D
CI D
EG
A
J
I \ IF
T
I
N
O
IH
O
^
]
H
O
C
O
E
K
O
R
D
H
H
D
H
OD
D ?[
O
O
T
V[ U B
U
D
A YD
V
B
D
A
P
K
TI
D
H
G PB
V
S
?I
C
N
K
T
C
R
^
AH
H
H
H
C
R D
eC B
LB
KD
VT
V
E
FT
EQ
K
T
X
DH
C
IH
EB
K
DH
H
WH
O
C
0R
^ E
]
^
W
Q
I A
^
H
D
?I
C
H
B
Ce G
C
%
]
G
B
D
?G
PQ
?
Ke D
C
E
A
CI D
EG
S
QD A
CD
?
J
A U B
ED
C
>
D
G
H
H
H
U CB
E
W
I
I
FO?
OQ
O
CG P
OQ
CD
CG
B
RD
(
N
3,
%
-0
2
.
D K
*0
1,
%
)
T
1 2h
D
M
?
[
Z
Oe
E
RB
B E
A KJ
D
E
F
IW
Q
IC B
E
U
D A PB
F
LD
OA
k ≥ N/2 =
I
B
$
FM
A
]
PQ
TB
B
A
IS
G
]
B
D
OJ
X
O?
WH
TD
?I
D Ce G
A
D
G
?
Ke D
E
OQ
OD
W
B
B
AH
G
D
IG
?
K
I
?
[
EQ
H
CG
G
\
X
C
B
]
G
A G
D
H
BH
EQ
H
CG
G
D I[
E
ZAH
O
IH
T
YD
S
I
V DH
?
^
A
B
G
H
D
G
G
R U H
I
W
IH
V
C
FH
J
Ke D
V
I
eC G
O
YD
KG
FM
?
R
\
E \ IH C
I
FM
O
Ke B
E
C
ID
V
H
H
AH
D
D
H
H
NB
P
R
[
D H X V G IH
^ TD
C
E
OQ
RD
C
GH
D
LD
O
] W
A
IC B
E
S
OQ
O
K
T
V
O
Ke
C
H
U
G
A D
J
A
H
G
YI D
KD
S
V
OQ
R
CI
E
E
IS
OQ
Ke D
PJ
I
V
N
K
T
E
EQ
FH
Ke D
V
^ EB I
TD
C
E
H
D
D
E
OQ
B
H
OD
W
IA
C
EI
FL
IA
T
?
I
C
FT H
Ce G
O
YD
KG
B
TD
C
E
L
X
H
G
K
U
D
H
O
D
OB
T
N
K
T
PD
C
O
H
TB
IA
L
S
OQ
O
Y
NA
VG
^ N
T[ H
H
B
U G
F
B
EB
TD
C
E
?
I
%
]
H
H
G
D
NG
C
I
O
CD
TD
P
PQ
Y
CD
EQ
CG
G
D I[
O
OB
T
N
O
IH
ZAH
N
K
T
E
OQ
P
C
I
+
?H \
]
U
BH
H
H
D
A
Ce B
OJ
I
NV J
Ke
O
I
?
TD
C
E
$
TD
P
Q
NG
C
L
CG
H
D
ID
T
O
ED
T
H
J PB
>
\
X
D
[
]
R
T
O
IH
KG
?
?
BH
B
NG
C
P
O I
D
B
H
D
G
H
WI H
W
IA
SV
?I
R
PQ
Ce G
O
YD
KG
NB
O
Ke B
E
[
RB
I
J
D
TB
O
OB
T
N
K
[
U
Ke
C
ID
0V
WH
B
Z f B
O(h−3 ln ε−1 ) O(h−1 ln ε−1 )
?
V
NG
K
G
H
I
H
ID
?
C
TD ^ H
]
h2 1 λk (Ah )| ≤ 4 2
C
[
HB
K
I
W
ID
K
P?
W
IA
C
V
QC
O
N 2
D Ce G
C
P
IY B
zih ≡ zi
I[
D
J
B
O
H
H
O
W
I S
H IB
IA
W
I
Z
A
D
H
H
G
?D
H
H
H TD
C
0E
^ TD _
H
H
ED
[
I
N
O
IH
−1
W
O
FE
A
NG
K
C
?I
C
k≥
O
[
B
M
I
RB
OJ
H
H
G Ke D
\
C
E
[
E UO
E
N
KG
R
I
PQ
EQ
N
K
T
D
OB
C
E
H
TD O
W
B
ZAH
O(h−2 ln ε−1 )
B
RD
C
D
K
LG
B
C
N
O
H
EJ
D
O
C
X
[
I \
D
?H
P
H
D
H
^
J
TB
OQ
B
H
OD
W
IA
I
E
N
K h
N =h
IA
H
C
NG
C
X
IA
CG
TF
ID
T
YB
?D
C
E
D
K
J
U
Ce G
F
wh2 2 Ah
C
O
OB
TB
H
[
X
?
H
H
?B
I X
I
A CG
F
V
H
NI B
O
O
K
PB
CG
D
G
B
|λk (Sh )| = |1 −
N
?
ID
Z
B
Q
O
KG
D
C
H KG
I
H
B
H
^ W
A
CI B
]
[
w = 1/2 Sh = I −
J
G
K
PB
CG
T
]
W
IA
H
W
I
I
O
VH
^
A
A
D
TD
C
E
E
I
V
LD
O
OB
E T
O
Ke B
NB
H
]
OD
LB
0 IT
N
K
T
O
C
O
KB
I
OD
LB
T
D I[
C
I
G
ID
?A
2 < 4 sin2
IS
BM
AH
X
C
G I GH
WH
NG FY
]
H
KB
P
P
I
?
H
I[
Z
V
ψk
I
]
N
J
B ^ ED V
A
+
D
VH f
$
]
A ED
C
H
EB
CG
?
]
? B
^
A J
B
D
B
U
H
H
D
A
B
D
+
X
X
E \ IH C
I
KG
R
I
^ ED
E
?I
U
H
D
H
B
H
D
O
G
D
RD
C
EQ
H
G CG
I
N
IH O
ZAH
N
K
T
S
EQ
TD
SI
O
W
IA
C
I
CG P
R
^
I
KD
?B M
Ce G
O(h−2 ln ε−1 )
P
YI B
K
B
O
IH
ZAH
D
B
U
B PQ D
?
O
TD
I
O
PB
H
A
k
LG
?
E
?I
\
eh = zih − zh ψk
Ke D
?
N
O
IH
T
YD
D
FO?
X
A
H
H
H
H
H
GH
T
L
Ah
H f
A
U D
G
B
]
W
Q
ψk
]
LB
h ψ k = sin(πk jh), λk = 4h−2 sin2 (πk ), k = 1, . . . , N − 1. 2 O
RB
I
A IC B
E
C
?H
G
CQ
O
E
K
PQ
FK
O
O
P
TD
C
H
K
L
PD
C
Ah
S
B
Ah ψ k = λ k ψ k V
J
H
TB
E
O
IH
V
NB
OJ
Ce
Ah
i
O
D
^
D IG FT
O
YD
E
wh2 (Ah zi − fh ). 2
BH
ZAH
C
R
[ HX
D
]
CG
ID
E
W
IB
IA
H
D
A
[
G TD
SI
zi+1 = zi −
^ NG
E
T
Y
F
?
H
D
A PD
V
H
l
C
I
zi+1 = zi − wD−1 (Ah zi − fh ),
N
O
IH
RB
O
O
P
AH CG
T
G
D = diag(Ah ) i = 0, 1, 2, . . .
]
H
H
OJ
OQ
O
P
CG
z0
M NG
C
N
K
P
N
w
ε
[
X
]
G
V
A
CG
ID
E
TD
IS
C
^
C
R
^ E
D
IH
EB
%
^
R
Ce
J
AH
H
D
D IW H O
O
YD
E
ED
O
Ke B
E
C
ID
V
I
^
]
H
H
J
B
D
Ke
H
CB
?
VD
O
Ke
CB
PD
T
K
λmin ≤ π 2 λmax = O(h−2 ) w 1−O(h2 )
π ≤ h2 λk ≤ 4. 4 λk (Sh )
k
%
$
"
$
%
! #" "
*
&
&
'
]
.
Uh → U2h O
r¯
(¯ r vh , u2h )L2 = (vh , u2h )L2 ED
∀ vh ∈ Uh , u2h ∈ U2h .
H
H
H
X
]
^
U
H
D
H
Q
C
FH
U
YB
?
H
F
ED
B
D
T
L
H
G
T
IA
C
^]
FO?
A
IB
O
D
OB
J ]
H
?H
P
C
O
W
I
\
I \
D
?[
J
B
D
H YD
UC
CD
E
D
H
GH
G P
C?
F
KJ
F
F
ED
A FM
D
CD
?[
WH
A
IC B
E[
O
O
Y
J
B AH
A
NG
C
N
K
U
H
H
H
H
H
D
B
D
N
O
IH
Y
KD
A TD
VA
A CB
V
H
CI G P
T
PD
C
O
E
K
S
[
OQ [
O
K
ID
O
G RD
F
?
U
N
K h
]
p := Ph−1 P2h .
PD
C
O
E
K
S
OQ
R
OD
B
BH
>
i
P
S?
P
CG
A OB
P
N
FR
KG
H
D
?
Rh
CG
H
D
I
NB
OJ
Ce
E
P
V
A J
B
D
B
^ ED
E
?I
+ X
D
A
O H
IH
Y
K X
I
I
V DH
]
P
O
A
B
CG
V
^
KG
I
R
I
PQ
D
CG
]
2
6
PD
O
C
D
3;
3
A
$
I
$
C?
F
H
GH
D [
X
6
2
6
^
ED
LB
ZG
6
67
5:
3
4
2
^
V
D
]
]
H
H
G
EG
PD
EJ
IA
D
ED
J
IM
eD
ED
V
NG
C
N
K
T
AH
\
NB
ZAH
^
^
PQ
O
H
IH
Y
KD
T
A D
DH
D
B
C
O
W
I H
\
I \
S?
P
CG
OB
A
CG
PV
O
O
H
A YB
J
BH NG
E
C
QD
V
V[
OB
I
R
H
EB
S
OQ
H
OB
K
TG
J
I DH
PA
H
VH
p : R2h → Ph : R h → U h PD
D
OB
I
E
O
IP
P
NB
E
A
AH
P
C
R D
YI D
KD
TV
^ E
^
]
]
^
]
^
H PD
O
YD
K
D
G
FM
O
TD
S
I
IB
?
?
eC
^
E
B
T[ H
O
EQ
I KG
#
]
O H
IH
ZAH
EI
B
D
T[ H
O
EQ
C
D
B U
I
]
]
FM
P
H CG
C
P
C?
F
L
A
H
GH
D
H
H
^
[
X
H
H
D
D
V DH
O
C
O
E
K
O
R
H
OD
?B
T
N
? B
ID
Z
S
QD A
U > D P C
X
T
B
LD
?
T
L H
N
O
IH
O
^
U
A
H
PB
F
E
O
IH
ZAH
K
S
OQ
R
H
D
D
G
D
U CD
P
CG
OB
A
CG
A
V DH
O
C
O
H
D
W ^ N
IF
C
I
H
D
B
G
NG
C
N
K
P
N
O
R
I
V
C
I
PD
C
H
H
H
OD
?B
O O
H TD
C
E
N
[
K h
R
]
^
E H
H NG
C
N
K
P
OD
0?
E
?
H
LD
O
T l
T
L
^
K
UA
U
A
H
B
H
ZAH I
O
TF
C
Y
?
C
F
E
CG
C
Uh
A PB
NG
C
BH
TB
J
M
O
p Ph
O
W
I
]
^
A
H
D
CB
0V
A
B
O
I
I
U2h
M
IH
H
? DH
A
&
H
X
L
O
Rh P2h : R2h → U2h
K
D AH
L2 UV
67
C
FT
M
]
F
p¯
T
A
$
'
*
5:
3
4
5:
6
6
3
i
E
IC D
'
*
'
G
NG
BH
p
EG
G
B
+
N
O
IH
$
A QD
C
P
zi+1 = zih − p e2h . h
VD
(
'
,
Y
KD
A TD
D
J
QB
O
O
IH
A2h e2h = d2h .
*
'
'
V
P
A
]
H
ZAH
2h
,
'
*
VD A
IE
CQ
Q
C? D
G
D IH O
Z
RB
C[
IB
zh = zih − eh
U2h
&
&
'
H
D
CB
A
X
e
KB
I
CG
O
?
TB
V
dh = Ah zih − fh . U2h ⊂ Uh
'
$
'
, * ! & ' $ * ! * *
'
Rh
H
H
D
B
A
H
FM
EI
G
ID
?A
VB
V
W
I
I
FO?
?
C
N
OB
I
PD
P
[ \
^
[
H
T
P
I
H
ED
O
P
N
P
Ce B
OJ
O
E
H
YD
E
EQ
N
CD
S
CG
H
PD
KJ
F
ED
P
CG
G F
C
? \ B
G
EB
]
H
A
6
5
82
23
J
B
D
ED
1
E
I
FM
T
KG
A Ce B
I
PQ
X
X
O
H
H
H
H
D
A
P O
O D
H
CG
H
H CG
N
OB
I
A PD
C
I
A ? DH
V
NI
O
IH
Y
KD
TD
V
QD A
A CB
V
^ PD
C
O
E
A
G
E
I
Z
Oe
E
C?
O
U H
R
KD
H
G
H
GH
B
B
OB
PD
I
K
IE
ID
?A
VB
P YB
I
S H
? B
ID
Z
^
PQA
H
VD
X
UV > D
KG
D
H
D
ZA
S
?
Ce
Q
C
YD
E
EB
O
eh
X
T
B
A H
H
S
OQ
R
H
OD
?B
TD
C
E
N
O
IH
O
E
I
VH
J
AH
B
H
CB
C
FKe
P
O
FR
KD
V
E
H
CG
G
IB
LD
j
D
B
OD
C
^ N
W
I
FO?
NB
T? B
K
0L
\
]
^
A D
B
BH
H
D
J
N
OB
I
R
H
EB
S
OQ
YB
P
O
S
CQ G
V
P
T
E
K
T
]
]g
]
H
B
P
B
G
D
H
H
A QD
V
P
OQ
O
K
PB
D
B
BH
H
CG
V
O
C
R
CD
A
A LB
V
I
FM
T
K
D
C?I
F
L
W
I
[
A
GH
D
G
H
A ?AH
?
BH
J
NG
C
P
Q
\
[
[
X
U
G
G
H
G
H
A
G
H
H
H
H
O B
LB
Z
TD
C
E
ED
O
GH RD
C
S
F
P
T
ED
?
R
CI
ED
L
ED
?
R
IB
K
? >
]
i zi+1 = zih − pA−1 h 2h r(Ah zh − fh ).
]
'
V
A Ce B
PQ
l
CD
D
eh
+
! * +
.
? j B
e2h
$
,
Uh
!
'
#
]
O
Ah eh = d h ,
'
%
&
J
?
? B
]
^
]
O
>
X
D
F
LB
Z
D
G
]
4
2
3
7
6
3
Ce B
J TB
? j B
H
H
H
D
OB
G
VAH I
V
eC
Q
C
YD
E
O
X
6
67
5:
X
3
B
D
H
#
IH
A
EA
O
E
A
D
Z
CD O
S
N
O
IH
RB
D
D OJ
S
PQ D
^
B IB
V
O
FC
O
L
CG
ID
H T
O
Ce G
O
RD
?
AH
H
D
B
C[
I
V
C
K
C
AH FH E
C
KG
I
NB
^
]
?
H
H
GH
J
H
LB
IH
P
P
?I
N
P
O
i
^
I A
PV D
C
O
H
N 2
*
'
I # KG
eh eh = zih − zh
!
,
,
* ! + # $ ,
*
* #
eh
(
.
'
eh
'
*
#
p : R2h → Rh zh
#
zi+1 h ψk k <
D
H
AH
C?
F
L
O
?I
N
P
O
N
O
A D
B
J
H
IH
K
T
V
N
K
T
EQ
UW
[
[
X
\
]
U
^
J
A
H
D
O
2
3
7
6
3
A
H
D
CB
V
A Ce B
PQ
? j B
B
RD
?
I
N
O
IH
K
O
I
A
PB
I \
D
TD
NV
D
?
N
O
H
IH
Z
Oe
FE
K
T
F
F N
K
T
]
A WH
V
A FM
D
CD
?H
O
Ce
O
A IH
E
I
T
X
]
] H
C
^ H
?
ID
Z
P
C
O
H
OD
EV D
S?
OQ
CD
CG
LB
i
X
X
H
GH
U
D
TD
IS
r : Rh → R2h d2h = r dh
A2h
h 2h p : R2h → U2h ⊂ Uh → Rh .
P −1
p P2h
r¯ :
%
$
"
$
%
! #" "
$
"
%
$
#
hA2h z, yiR2h = ]
Z 0 1 0 k(x)u02h v2h dx, ]
B
zk+1 = ˜zk − pA−1 zk − fh ). h 2h r(Ah ˜
¯ h zkh + M ¯ h g ν − pA−1 r fh = M ¯ h zkh + g¯ν , zk+1 =M h 2h
B
W
H AH
U
Ke D
]
˜zk = S ν zkh + g ν , C
C
G
O
H
DH
RD
FC
Y
ED
G
H
ED
KG
I R
E
I X
D
D
Ke D ?
?
O
NG
PQ
A
O
H
IH
Y
K X
I
I
H VH
KG UN
R
I
zk+1 h
I
A
I
PB
A
f
X
H
0
AH
V
OQ
B
H
D
OD
W
IA
C
IB
?
?
eC
CD
[
B
H
H
AH
D
O
ED I
V
O
OQ
^ E
^
K
T
V
CQ
O
H
OD
EV D
H
V
A
B
A
D
CB
A
V[ D H
PQ D
?
O
I
BM
TD
C
T
N
OB
I
PD
C
I
? DH
A
D
LD
^
]
] g
D
TD
S
TD
V[
E
NA
V
H
H
H
R U H
OD
I
OB
?
] I
J
PI D
C
O
E
K EB
OQ
R
OD
S
D
\
?B
TD
C
FK
EA
D
C ] H ] V
D A PD
CB
H
H
AH
U
I A
J
D
VB
?
?
E
N
K
T
V
2
3
7
6
3
3
34
l
]
A
B
GH
FM
F
L
O
C?I
[
X
]
U
U
A
H
G
H
R
S
B
G
OQ [
O
K
ID
O
RD
F
?
N
K
T
N
OB
I
A PD
C
I
A ? DH
VD A
CB
V
$
j
l
...
E
H
IJ
^ HD V
PD
D
C
G EG
B
?
H
2
5
A2h
I
V
E
A
D
]
I
?
N
O
A
D
E
GH
X
...
I
D A ?AH
?
CB
B
C
gν
?
Z
O
X
F
P
CG
F
TD
O
EQ
X
^
H
6
S
P
[
1 4
?D
C
FKe
D
?H
PB
H
FM
T
H
KG
D TD
IS
G
A
D
AH
4;
3
] G
I
D
1 4
OB
G
P
I
^
W
IB
IA
O
KD
V
PQ
YD
E
TD
C
?
SV
H
D
CG
F
^ ?
J
AH
B
?
B
TD
H
H
H
D
M
H
A PD
I
A
C
O
;
N
K
T
KG U H
1 2
I
VJ
?
?
NG
C
C
E
PB
C
E
O
O
O
A
E
H
X
I
V
B
D
N
K h
3
H
z, y ∈ R2h , u2h = P2h z, v2h = P2h y.
Ce
X
B TD
C
A
H
S
H
H
?I
A
D
H IH
T
P
W
UO
2h
SI
N
[
C
^
PQA
E
GH
H
KB
PQ
D
G
1 4
Q
H
NA
VA
H
zkh
K h
fh zk+1 h O
I
BM
^
A
[
P
B
G LD
?
U NA
A
^ ?
1 4
C
]
D
E
IH
Y
OQ
T
LD
L
TD
S
TD
V
I
L
1 2
H
S
H
UV > D
K X
I
A
GH RD
C
^
]
\
j
2h
YD
^
P
C
K
B
] B
I
F
V
I
S
1 4
E
G
P
YI B
PD
[
B
B
e ^ A ?
H
+
]
A VG
H OQ [
O
h
F
K
EQ
H
CG
I
V
O FT
V
C
D
VH
B
p
QD A
^
I
OB
P
S
F
P
T
D
?
?
NB
D
K
ID
1 4
C
HB
D
K
LG
B
H
˜zk
G
W
O
ED
O
OB
C
R
UA
r
I
CG
IB
A
IC B
P
T
P
NG
TB
J
D
H
NG
C
D
I \
S?
B
A
U
G
?
I
U > D P C
[
IH
H
L?
D
VAH
D
A CB
H
D
P
CG
C
? DH
O
RD
F
h
P
C
R
]
R
D
0E
IY B
K
LG
C
C U H
K
W
P
CQ
FR
I
0V
A OB
CG
A
[
?
N
K
T
l
? 2h
^
C?I
GH
D
A PB
Sh
3 4
[
X
2
Sh = I − wh 2 Ah ˜zk F
Y
ν K X
I
A
CB UR
YI D
AH
H X
D
H
V
G
H
N
O
IH
Y
KD
A TD
VD A
A CB
H
r
T
A
H
I
V
x=0
CV D
T
L
H
[
I
E
SH
B
CG
X
T
D
?I
R
I
OD
OB
FM
V
$
x
G
L
G
D
A2h ?
O
H
CB
A
H
D
J
B ^ ED
A
D
L
A
BH
]
]
6
N
B
] B
TD
S
H
A
V
^
V
P
H
?
C
T
A YD
D
D
B
B CG
R
O
1 2
W
O
NG
O
O
E
B
CI B
E
S
E
B
+ ∀ vh ∈ Uh , u2h ∈ U2h .
J
I
TD
C
H
B RG
R
A
]
?I
E
Z
V
P
CG
r : Rh → R2h
UA
C
FH
R? D
P
] D
O
O
J
ED
O
I
D
]
J
r = p∗ ,
D
B
C
D A
r
V
A
V
P
P
H
NG
O
OB
H EIA
I
H
]
I
VAH
H
X
H
...
D
AH
A CB
P
CG
H
J
TB
C
P
I
I
C
BH
TB
J
H
]
J
B
D
ED
A
O
PB
]
LD
O
O
T
C
FH
C
C
H
P
C
P
A
B
H
D IH O
Z
r∗
G
# KG I
N
K
CD
D
P
ND G
IAH
A
D
O
CD
E
IE
ID
]
]
H
G
I
V
A H
E
?
]
] H
C
D
G
DH
] G
I
1 2
CD
?A
H
D
VD
C
V
A
D
C
?[
H
^
?[
GH
6
CG
H
FO?
A
IB
I
C
IB
?A
OQ
O
Y
NA
D
VG
A
D
BH
−1 hrPh−1 vh , P2h u2h iR2h = hPh−1 vh , Ph−1 u2h iRh
P
F
^
C
A ? DH
H
E
C?I
x
I
A
G
B
VB
A2h G
uh (x)
VD
D
V
ED
B
A
D
CB
V
A
D
1 2
EG
A
O
^
O
? \ B
V
O
H
g
I
C
]
N
K
H
I
1 2 h·, ·iR2h
O
P
J
B
OB
E
V
N
]
D
D
IH O
Z
O
CD
$ FM
T PD
C
...
I
U2h CG
M
\
H
HB
A P
O
H
AH
D
IH
K
IC B
T
V
N
D
O
OB
h
r∗ = Ph−1 P2h = h·, r∗ ·iRh .
T
+
H
W
I
FO?
B
T
G
B
EG
A
>
K
T
p
? ^ B
C?
[
L
D
]
N
O
FR
H
2h
AH
GH
D
X
D
B
O I
A
OQ
O
Y
E
T
A
?
H IH
K
D AH
C
R
Ce
K
PB
KD
1 2
H
]
F
H
D
T
V
P
^
IH
Y
H
6
O
G
D
O
SK
VA
C
A ? DH
VH
[
IH
PD
IA
V
1 2
IH
D
H
O VH
P
CI G
h·, ·iRh
Z
O
P
CG
L
B
B
A
I h
I
N
PD
I
KG
F
H
[
KD
T
L
2h
V
D
O
@
NG
OQ B
I
A
r = 12 pT ,
OB
]
D
]
BH
T
D
V
hr·, ·iR2h
A
B
A
CB
C
TB
T
2h
G
D
RB
TB
% A
H
D
X
TB
V
]
CD
A
3 ;
D
A VB
PQ
H
CG
PV
: ;
3
13
(
K
1 2
NG
Ce B
PB
]
C ] H
^ H
[
X
N
K h
O
6
H
J
TB
C
H
A2h X
GH
D
J
2h
TF
C?
A
J
h r
h
F
B
F
L
p∗
6
A
H
p
V
E
h x
1 2
^ ? x
A2h := r Ah p.
A2h .
%
$
"
$
%
! #" "
$
"
%
$
#
¯ (k) ) ∈ {0, s2 c2ν + c2 s2ν }, k = 1, . . . , N − 1, λ(M k k k k h 2 ¯ ( 2 ) ) = 2−ν . λ(M h G
A
O
H
GH
[
U
PQ
D
LD
C
H
F
L
O
?I X
A D
B
[
X N
K
T
W
IA
C
S
I
B
H
PQ D
J
B
[
X
FCe
G
^ TD
C
E[ H
OQ
GH RD
C
S
A
A H
U AH
CG
P
Q
E
I
VH
FL
IA
J
\
K
I
I
J
I \ IF
T
IB
RB
TB
AH
RB
I
J
D
TB
O
\
[
H \
Ke B
W
I
O
T
I
W
IB
IJ
]
Ah zh = f h ,
IH
C?
D
D KG
[
X
C
IAH
G
[
H
H
AH
D
H
U KB
S
OQ [
O
I K
EQ
H
G
H
CG
I
O
IH
ZAH
NG
C
N
K
P
N
F
E
O
Y
UV
V
M
3,
%
-0
2
.
1,
%
*0
,
/,
.
(
B
H
T
P
B
I[
I
OB
P
IY B
K
LG
]
]
B
H
B
B
H
PD
TD
C
E
S
OQ
RD
C
GH
D
LD
O
E
N
K
T
OA
C
?A
S
^
]
HB
D
G
CD
E
I
PB
H IJ
O
NG
C
P
D
D
B
IW H O
I
?
W
IA
[ X
^
I
H
D
G
B
KG
R
I
CD
Ce G
E
I
PB
I
%
H
D
H
D
C U H
I
W
Q
A
IC B
E
EQA
O
CD
R
I
K
B
CI D
P
TD
C
E
LD
O
GH
B
RD
C
S
F
P
T
W
IA
C
[ A J
B
D
B
H
A
A
B
D
B
H
W
Q
A
CI B
E
EA
O
E
I
VD
LD
O
Ke
TD
E
D
LD
O
OB
T
N
K
T
^ ED
E
?I
+ X
]
I
C
E
I
M
H
I
V
A
A
U D
G
J
I GH
X
I X B J
P
NG
C
TD
I
P
I
A
B
A D
DH
D
O
Ke
C
I
D
H
G
D
CD O
P
CG
A OB
CG
V
O
OB C
I
P
O
CI
O
[
^
]
]
^
]
D
G
D
H
H
X
I
QD A
C
?H
P
OQ
O
CG P
C
R
EG
X
@
?
]
IH
eF A
H
B
J
KB
I
OB
M
eD
ED
VG
EI
N
O
KG
R
I
PQ
IE
EQ
NA
V
>
^]
¯ h k2 ≤ k M ¯ h k∞ kM ¯ h k1 ≤ max (s2 c2ν + c2 s2ν )·2 max{s2 c2ν ; c2 s2ν } . kM k k k k k k k k P
D
H
E
FR
BH
KD
V
O
IH
RB
OJ
S
OQ
O
CG
G
H
D
D
WH
O?
O
R
LD
I
KB
U
G
D
A BH
H
D
A
H
YD
E
U
J
A
A
H
$B
T
Ke
H
ED
P
CG
O
H
H PB
O
NG
eC B
PD
V
^
I
A PD
B
H TD
C
E
D
A
D
B
H
D
D
V
IA
E
I
V
LD
O
Ke
TD
E
LD
E
P
CI B
CB
G
B
LD
O
GH RD
C
S
F
P
T
eC G
E
TD
IS
] I
] G
A
IB
O
S
A EG
D
U
G
D
D
A
B
H
[
[
Ke D
G
VD
PD
D O X k
FT
W
IA
C
IW
Q
IC B
E
EQA
O
O
A Ke B
C
?H
V
I
O?
D WH
N
K h
I
I
V
A
]
D
D
EA
O
OA
H
D
D
D
CB
A
[
V[ D H
A
CD
?H
O
]
P
H
O?
B
D WH
NG
C
N
K
P
N
E
H
PD
I
KG
F
E
OQ
RD
CB
CG
D
h
^
H
O
Ke
D
H
CB
PD
T
K
D
H
P
B
CG
O
H
H
H
A PB
O
O
IH
O
D
KD
V
PQ
E
TD
IS
O
X
H
US
N
K
T
C
R
^ C
D
G
D
FH
T
H
KG
PD
TD
C
E
S
OQ
OD
B
H
W
IA
C
A
D
H
H
D
B
H
D
I
I
I
C
J
TD
C
E
LD
O
GH RD
C
S
F
P
T
CG
ID
E
T
I
[
, X
^
]
ν
CG
B
D ID
Z
h
Ce
Q
C
H
YD
D
?
T
I
O
IH
O
PB
A
Ke
WH
UA
C
S
I
D
D
H
J
B
FM
A
A
eC
A
IH
PD
1 1− ν +1
R
B
eh
] D
O
PQ B
[
E
C
H
J
CB
C
FKe
F
S
G
H
[
CG
IB
H
H K X ?I
T
L
F
T
V
O
Ke
CB
√
O
]
K
L
H
]
H
PD
C
O
E
H
AH
[
?I
H
Z f B
U H
YD
R
BM
h
F
P
T
O
AH
CD
F
AH
J
K
S
OQ
R
P
D
R
I X A B
B
I
P
YI B
^
P
IF
ED
^
H PD
T
K
Ce
IH
A PD
D
OQ
OB
N
K h
H
AH
C
FM
CG P
TZ
V
?I
W
IA
C
S
I
I
P
B
H
BM
YI B
K
LG
[
^
V
H
AH
N
O
IH
K
T
D
VJ
CI
H FH
T
KG
? j B
A
D
UC
F
L
W
I
I
A
GH
D
G
?AH
?
?F
TD
S
AH
H
[
X ]
T
L H
C
T
C
R
^ E
D
IH
EB
$B
H
D
TD
C
E
LD
O
GH
B
H
B
RD
C
S
F
P
T
W
IA
C
IW
A
CI B
0E
[
%
]
N
K
D
F
ED
C
k = 1, . . . , N2 − 1
RD
S
G
B
T
TQ
V
H
O
H
H
B
H
D
A KJ
K
LG
J
AH
2 1 e ν
O
?I
A
H
H
EG
K
Q
K
B
OD
?
O
FR
D
O
?
K X
CD OQ
CG
D
?
K X
B
[
D
RD
?G
ED
C
H
PQ
H
O
Ke B
OD
LB
ID T
O
RD
K X
]
X
V
A
J
B
1 1 |λ(M¯h )| ∼ e ν
CG
H
PJ
Ke D
G
X
]
G
G
B OD I
F
H
ED
TD
L
>
PQ
K
+
H
R
U
?
D
J
O
?
C
G
eB
KB
?
B
H
D YJ
VH
L
O
I
O
IH
O
U
]
GH
^ B
?
O
IN
D
l
B
]
I
YD
VD
O
1 = ν +1
? X
˜eh = p e2h ID
H
H
A
H
TJ
E
F
T
TQ
?
P
KD
V
NG
C
X
C?I
D
J
UI D
K >
I
K X ?I
C
I
E
RB
I
OJ
D
X
¯ hk ∼ kM
IO
N
ZAH
O
KG H K
T B
B
D D
V
C
RB
TB
J
R
^ ED
C
PB
K
ND
>
H
B
H
H
?H
P
OQ
O
[
X
]
]
$
T
M
PD
O
N
IH O
YD
O
x∈[0,0.5]
Z
G
D
CG
D
G
D
AH
C
E
U
C
FH
J
KB
P
CG
D
G
C
A
X
P
I
C
]
A
D
¯ hk ≤ 1 kM 2
RB
A
eB
K
I
CG
D
A LD
A PB
F
NG
P
YI B
K
I
AH
EJ
OB
SI
H
F
[
O(ν −1 )
J
B
^
]
EQ
NG
C
D
C
BM
YD
LG
C
FH
P
H CG
C
H
[
I
T? B
K
L B k
?
O
IB
O
C?I
GH
D
G
H
C
?H
P
S
OQ
H
FE
1 |λ(M¯h )| ≤ 2
TB
C
V
U
?
A
P
CD
D
h
G
N
O
A I DH
A
E
I
D
D
Ce G
OA
H
O
P
CG
s2k ∈ [0, 0.5]
?
]
A2h e2h = d2h H
IH
Y
K X
I
I
V
N
eD
K
CD
U H
R CD
K
T
O
FE
e
H
VH
C
E
A
L
W
I
I
CG
J
F
L
W
I
?I
K X
H
B
EJ
A
k = 1, . . . , N2 −1
H
H
KG
N
^
C
ψk N
C
N 2
I
H
H
]
] H
B
D
G
FL
D
A ?AH
?
C
FH
I
OQ
T
P
C CG
P
D
ID H
I
?AH
OQ
OB
h
¯ ( 2 ) = 2−ν , M h
PQ
OQ
O
P
C
D
O l
BM NG
C
YD
P
CD
D
A
D
N
,
ψk
CG
?
AH
^ A
s2ν k
CG
T
G
D
C
]
H
\
ZA
S
B
¯ (k) M h
C
H
]
C
O
W
I
A
O
W
\
]
N 2
?H
I
I \
D
S
D
G
s2k = sin2 πk h2 , c2k = cos2 πk h2
P
H
H
N
O
O
FE
]
H
0 s2ν k
VH f
OQ
O
?[
k ≤
I
O?
P
X
IH
I \
D
?[
?
[
T
L
c2ν k 0 ¯ (k) , M h
p
H
G
D
WH
D CG
QD A
KJ YD
?
N
K
T
CD
B
H
B
A
D
^
C
C
I
W
I
H
KQ B
E
f
?
K X
D
]
N
KQ B
G
?H
V O
I E
s2k
G
C
B
B
B
] H
C
^ A
c2k c2k
OQ
H
]
E
N
W
IA
LD
A PB
V
N
K
T
s2k s2k
RD
I
FM
A
G
H
M
C
I
K
K I
W
C
EI
I
¯ (k) = M h
Ce
FR
I
]
D
ID
UA
X
B
2×2 ¯h M
r
N
C?
I
D KD
V
O
k≥ ψ k
¯h M ¯ hk < 1 kM
KG
GH
]
YD
CG
Q
¯ h )| < 1 |λ(M ν
¯h M
R
BM
A
¯ h = (I − pA−1 rAh ) Shν . M 2h
I
$
S
Q
ψ N −k
C
T
N
O
I
¯h M x∈[0,0.5]
¯ h )| ≤ max (x(1 − x)ν + (1 − x)xν ) ≤ |λ(M ≤ max (x(1 − x)ν ) + max ((1 − x)xν ) = x∈[0,0.5]
ν+1 1 + . 2
ν → ∞.
k
ν → ∞.
%
$
"
$
%
! #" "
$
"
%
$
#
I A ] I
W
A
IA
V B
C
S
B
H
[
H
OB UP
IF A
G
O D
Ke B
[
U H
ZAH
D
GH
D
U RD
C
LD
O
E
S
NB
FR
KG
S
LD
I
O
E
H
D
ID
V
NG
C
N
K
P
N
D
M
eC G
O
[
U
D
YD
KG
O
Ke B
E
C
[
E
I
V
?
Ce
A
]
^ A F
^
H
AH
L
P
D
GH
FH C
C
X
UO
E
K H
H
A J
B
D
B
EQ
H
G
H
CG
I
O
IH
ZAH
^ ED
E
?I
+ X
^
U
G
H
B
K
E
O
D
H
A PD
F
ED
T
YB
?B
O
OB
I
P
YI B
LG
D
Ke D ?
?
O
D
TD O
O H
[
]
U
CD NB
H
OQ
C D
H
H
H
GH
D
H
KD
C?
K?
E
D
G
B EB
O
S
CG
H
H
B
PJ
OI
CG P
R
I
KD
?
^ DH K
[
[
A
^
E
QD A
CD
?H
O
I
C?I
GH
B
LB
Z
CD
E
I
NG
PB
H IJ
O
^
Y
C
C
R
K
I
H
D
D
YB
OQ B
H
[
A B
D
H
AH
K B
H
TJ
L
F
T
TQ
VW H
$
I
RB
?
P
OD
J
D
EI B
O
E
TB
O
Ke
TD
E
OQ
O B
H
PD
OB
CG
F
?I
K
H
^
ID
Z
N
O
IH
TD I
C
E
D
LD
O
GH RD
C
S
F
P
T
N
K
T
X
D
A
J
B
D
J
H
H
P
B
D
?G
ED
CG
D
PG
E
OQ
Ke
CB
R
EB
C
[
D
H
B
D
H
H TD
C
E
OQ
GH RD
C
D
LD
O
E
C
R
^ E
YB
?
V BH
TB
Y
?
OB
I
P
YI B
EQ
]
K
LG
B
KG
R
I
D K X
RB
TB
FL
?
LD
?A
I
Z
N
K
T
UA
p
C
B
H
ED
O
O
[
C
R
[
[
D
H
G
D
RD
C
O
Ke
C
I
YI D
KD
EB H
T
D
G
U
[
[
+
H
GH
D
H
D
T
V
Ke
C
AH
H
H
YI D
O
E
B
^ H
C?
K?
E
G
B EB
O
S
CG OQ
P
[
[
UA
C
I
D
? DH
A
G
CD
C
I
PB
IJ
^
]
D
O
TD
Ce G
O
H
D
G
H
U
H
IO J
H
D
CG P
R
I
H
B
KD
0?
T
L
W
IA
V
S
?I
R
C
[
W
I
I
FO?
E \ IH
G
B
J PD
PQ
B
D
[
X
I
FM
P
CG
C
H
D
H
GH
D
[
LD
B
H
A
D
CD P
C?
F
L
S A
OQA
C
O
E
U
G
EB
C
FH
P
H CG
C
\
D
C
N
K
H
PB
CG
K U H
O(Nl )
FM
I
A
KG
I R
I
TV D
S
TD
V
] B
H CI G P
T
G
H
G
O(Nl | ln ε|)
PJ
OQ
CG
TD
C
H
D
S
?I
R
O
CD
H
G
B
TD
J TB
V
A EI B
D
D
]
H
$
P
CG
OB
A
?
H
YD
KG
NB
O
Ke
C
KG
I
R
I
SPQ
NB
FR
D
^
O
B
KD
V
PQ
Ce
$ KG
H
D
TD
C
E
LD
O
GH
D
RD
C
I
H
B
OB
J TB
F
eC G
A U B A
S
]
]
M
H
LD
O
E
T
EI
J
FH
] B
N
h
D
LD
I
O
ED
PD
C
R
^ E
U
D
H
H
YB
?
V
EQ
KB
A
H
H
D
CG
S
V
OQ
C
O
E
K
D
KB
I
EA
]
G
H
H
F G
G
B
B
TD
C
E
M
eD
^
A
D
D
B
H
H
A
A
B
H
H
$[
W
IA
C
I
E
I
U AH
Ce
Q
C
YD
E
O
IH
O
A PB
F
E
CG
G
[
I[
A
CD
?B
O
NB
F
L
NB
G
EB
A
ED
V
E
A
IB
V
O
D
] B
^
O
RD
X
C
X
B
]
^
D
LD
O
OB
J
H TB
Z
Oe
H
G
G
B
GH
B
H
J
ED
CG P
R
KD
I
?
E
KQ B
E
C?
FC
O
L
CG
H
H
B
D
ID
T
C
FT
O
?
UF
E
A
VD
O
P
FI
X
3
67
D
G
T
D
A
H
D
B
H
D
O
?I
N
P
H
J
OB
EA
D
OB
?
V
I K
I
W
IA
C
I
CG P
R
KD
I
?
U AH
QF
WH
A
VD
O
OB
T
I
D IAH O
C
D
PD
PV
C
IC G
1
29
38
67
35
234
[
[
:
;
C
H
OB
CG
H
kAh znew −fh k
A
E
H
PJ
E
B
V
E
D
?B
O
[
A
H
D
CB
V
P
CD
G
H
B
PD
H
B
H
[
>
X
A
[
X
O
H TD
C
E
J
M
O
DH
OB
TB
C
N
O
D
KD
V
PQ
L
GH
W
IA
C
I
ED
KG
I R
E
D
D
X
I X
?
[
B
H
[
T
A D
VH
O
IH
O
D
KD
V
PQ
T l
A
QF
FM
F
L
H
H
H
H
GH
B
H
KD
F
C?
O
E
O
G Ke D
H P
O
E
? N
LD D
A
H
IH O
E
I
V
TD
S
AH
H
VB
WH
GH
H TD
C
E
GH
D
LD C?
F
A D
RD O
C
D
LD
O
E
W
IA
C
ID
O
TD
] B
]
^
GH
B
RD O
C
S
L F
H NG
C
N
K
P
N
A
]
?
N
O
IH
Y
K X
I
I
V
e
Z
K
I A
B
C
Ce
TD
IS
O
O
R
I
I
V
A
H
EQ
H
G CG
I
N
IH O
ZAH
D
LD
O
O
Y
K X
I
I
NV
K h
H
B
E
O
H
IH
T
YD
S
O
NG
Ce
R
I
OB
IA
LD
E
YD
E
Q
] D
O
RD
E DH
^
A
D
?H
IH O
Y
K X
I
I
V
?
Ke D
0C
H
H
H
S
OQ
Ke
C
I
H
ZAH
C
OD
O
Ke
CB
PD
T
KG
B
C
%
X
U BH
FR
KD
V
C
R D
IH
[
H
GH
D
EB
C?
F
A
LB
O
RB
I
J
H TB
O
IH
D
B
ZAH
C
^ E
]
0
H TD
C
E
i = 0, . . . , γ − 1 γ
A
γ=2 ?AH
C
ID
V
?[
I
N
A TD
P
I
VS
OQ
TD
[
LD
A
E \ IH C
I
S
L
O
H
W
B
H
J
OQ B
N
Nl
O
B
D
I
OQ
O
OD H O
P
W CG
B
H IA
C
NI
K
T
M
]
[
U
C
E
hl
K
I FR
I
γ=1 K
>
^
B
^
D
GH RD
C
PD
PD
J
IA
C
S
I
PG
D
U
O
R
OD
C
H
O P
T
znew = M GM (l, zold , fh )
IC G
B
^
[
O
T
I
0H
TD
C
E
OQA
Y
O
G
D
PQ
S
BM
H
PD
C
O
\
P
CG
M GM (l, zold , fh ) Nl
KD
V
I
A eC B
X
A PJ
I
S
V
OQ
B CG
R
P
O
IH
O
PB
F
Ce G
K X
A
B
LD
O
E
N
W
I
I
P
W
I
I \
D
?B
P
CG
OB
A
A OB
>
O
ε
K
] I
H TD
C
E
N
W
B
CG
ID
E?
TF
U
P
IF
?AH
NG
H
G
QD A
D
?
N
?
zν
K
M
G
IA
I
PQ D
O
J
B
[
H
D
C
B
ε
I
H
B
A
A
O
A
A
[
q ≤ 0.1
FO?
I
C
I
FE
DH
OQ
F
I
I
VD
O
Ke B
C
OQ
P
A
YI B
]
]
]
J
A CB
$
N
IH O
Y
A CG
K
ei+1 = M GM (k − 1, ei , d)
] I
]
K
?W
O
γ=1 γ = 2 TD
D
]
A
O I
I
D
RB
H
IF
K
LG
D
KG
R
EI B
H
D
VD
C
R
KD
A TD
V
D
CG
A
new
zold = 0 kAh zold − fh k ≥ ε
I a
C
R
B
C
A
B
WH T
V
EB
H
IA
B
q <1
U
J KB
B
D
B
]
H
C
I
?AH
D
KG
I
DH
K
PB
D
^ E
YI D
KD
VH
Z
¯z = zν − p eγ
0
^
?
V
?
C
IB
N
K
O
T l
R
I
^
H
NI
OB
S
I
z
?W
?A
k>0
I
OQ
0O
T
D AH
^
DH
O
A Ke B
IA
V
V
OQ
KG
Ak , k = 0, . . . , l − 1
O
IH
Y
K X
V
]
O
A Ke B
F
F
CB
O
%
I
U0 ⊂ U 1 ⊂ · · · ⊂ U k ⊂ · · · ⊂ U l = U h .
I `
%
]
A
H IB IA S
PI
CB
O H
A
PD O
F
E
TB
I
TB
AH
I
P
γ
^ K
0
H
G
e
EQ
H
G
CG
I
N
O
IH
zold
T
ZAH
^ H I
O
A PD
F
TV
p
I
I
A
V DH
]
[
ED
T
YB
R
I
Al = A h
DH
H
PD
O
O
O
FR
KD
D
B
TD
Ah zh = f h znew = M GM (l, zold , fh ) znew
0
V
A
H
ED
?B
p
H
F
C
D
γ
]
T
WH
E
PQ
→ → → → → R0 ← R1 ← . . . ← Rk ← . . . ← Rl = R h . r r r r r p
¯z = M GM (k, z, b) { k=0
Y
H
D
T
O
e2h
^
?
A
L
ν
eh
U
d = r(Ak zν − b)
O
IB
G
OB
O
IH
V
p
Y
e2h ˜e2h
¯z = A−1 0 b
D
zh U0
z = z zi+1 = zi −Wk−1 (Ak zi −b) i = 0, . . . , ν −1 ν
I
V
˜eh
A DH
H
A
O
ZAH
?
N
IH O
Y
D
LD
O
GH
B
RD
C
S
F
P
T
C V
B
B
H
GH
C?
D
CG
R
O
[
K X
I
I
V
N
W
I
D A ?AH
?B
K
M
]
G
D
TD
IS
A
X
I
IH H
[
f
V
OB
K
V
H
ED
O
Ke
C
KG
I
R
I
PQ
P
I
E?
FT
C
H
B KD
I
e0 = 0
}
znew = M GM (l, zold, fh ) zold =
kzh − znew k ≤ q kzh − zold k
%
%
$
"
$
%
! #" "
$
"
$
%
Cn := k=1,...,l
inf
T
D
$
QF A
WH
A
E
H
OB
I
BH
D
U J
PQ
NG
C
J TB
znew = M GM V (l, zold , fh )
z = z z1 = z0 − Wk−1 (Ak z0 − b)
0
e1 = M GM V (k − 1, e0 , d)
¯z = A−1 0 b
P
U
H
D
O
I
E
H
OB
I
P
YI B
K
AH LG
T
V
E
IT D
G
B
TD
C
E
LD
O
GH RD
C
D
]
A B
]
P
YI B
K
LG
I
H
[
D CG
V
IB
?
C
UA
D
^ H
OB
I
P
YI B
K
LG
T
AH
VB
?
?
H NG
C
N
O
KD
V
PQ
OB
A
H
G
X
H
H TD
C
E
V f IB
O
R
IC
E
E
I
H
H
KD
NG
C
PD
I
OB
CG
F
WH T
A
LD
E
D
V BH
FR K O
KG
ED
?W
I `
A
C
>
]
BM
ED
O
RD
C
GH
D
LD
O
E
P
D
B
CG
1
6
2
4
75
35
NG
C
P
^
U
D
D
GH
D
B
J
QB
O
OD
I
T
A
G
LD
C
C?I
F
L
W
I
I
?AH
A
?H
KG
V
I
eC
N
O
KD
V
QP D
O
YD
E
[
X
]
H
D
AH
H
A
GH
D
G
H
BM
H
W
IB
IA
C
IH
I
P
YI B
K
L
C?I
F
L
W
I
A ?AH
?
T
V
NG
C
N
O
[
[
X
^
J
6
2
B
B
U KD
V
PQ B
4;
56
BM
OD
?
?
?
C
1
C
P
QB
O
D
CG
U
G
M
R
N
B
OB
I
P
YI B
LG K
W
B
H
D
B
AH
IA
SV
C
FO?
S
V
IF
T
TQ
PV
FM
O
O
AH
CD
EG
B
]
FR BM
KG
^
U
U
D
G
H
A
D
F
ED
C
0P
K
?W
I a
F
ED
PA
CV
FH
P
H CG
C
CD P
K
W?
I `
^ I
PQ B
K
L
Cn = 2
[
IY B
K
LG
CG
D
^
H
A
A
B
H
I
\
F
T
TQ
V
I
H
AH
J
J
[
I \ FI
T
N
O
IH
O
A
PB
F
D
LD
OA
H
ED
TD O
E
I
NV
K h
]
^
I
KG
H
C D
κ>1
G
VD
O
A
V
E
O
E
^
] I
KG
H
C D
κ=1
¯z = M GM V (k, z, b) { k=0
P
IF
?AH
ED
IT D
H
H
^
] I
KG
H
C D
κ<1
P
H
O
IH
Y
KD
A TD
OQ
RD
C
^
Ce G
F
D
B
A
$
NB
FR
KG
CI
OQ
YD
EJ
P
T
LD
C
κ = γCn−1
e0 = 0
d = r(Ak z1 − b)
D
LD
\
W -
U
G
O
OD
W
z new
C
A
B
D
O
IH
V
-
D
LD
I
B
H
-
I
PB
IJ
IA
-
B
O
TD
KG
R
I
zold-zν
U
D
l
CG
ID
C
I
W -
OJ
X
CG
ID
I KG
H
O
RD
D
Ce G
W -
O
DH
H
[
DH ZAH
O
RD
C
U
PD
A
E?
C
^ D
]
K
?W
I `
U
N
N
W
I
-
F
[
O
YD
W -
FT
D
O
Ke B
U
$
OB
I
P
A
? DH
V
O
A PI D
F
GH
z new
Nk /Nk−1 . A
O
OB
]
]
] G
I
YI B
K
LG
U
U
ED
G
D
K
?W
I a
O
C
A
KG
NB
U
U
^
RD
GH
TB
E
C
O
$
C
D
J
V
H
]
] G
I
U
]
k V
ID N
O
Ke
C
KG
I
W -
P
NG
C IH
Y
H
R
I
PQ
K
I
I
R
ED
E
I FR
,
N
K
CG
ID
P
RB
I
D
DH
-
B
J
J
TB
A
Ce G
E?
FT
+
-
O
] S
[
N
O
IH
ZAH
B
N
B
H
D
E
C
D
T
N
K
T
W
IA
V
S
G
A
U
U
] B
OQ
CG
H
H
B
PJ
OI
KG
R
?I
H
I
L
KB
S
H
D
TD
C
E
LD
O
I
CD
D
R
C
IH
E
W -
OQ
GH
D
RD
C
H O[
O
K
I
W -
CG
H
H
D
PJ
IO
KG
0 RI
B
LD
O
E
K
?W
I
*
]
Ce G
F
Nk
E
R
I
7
zν z old H
D
]
A
H f
H
H
O
H
IH
T
YA
P
FC
H
H
FM
T
KG
Ce
A
IH
A PD
VD
O
FT
C
] B
TD
C
E
LD
O
GH
D
RD
C
L
]
H
D
C
R
E
I
J
AH
H
RB
OJ
N
T[ H K
K
I
I
X
l
%
U
D
DH
O
E
K
W?
D
B
ID
LD
O
TD
eC G
E?
FT
A
Nmg ^
D
H
B
B
B
H
BM
I
? D
TD
C
E
?
S?
?I
C
W
IA
C
S
I
I
P
YI B
K
A
LG
S
CQ
N
O
I
VH
[
X
X
H
H
H
A D
S
I
OQ
TD
PJ
I
S
V
OQ
B CG
R
P[
O
IH
O
A PB
F
N
K
T
PD
C
O
E
K
S
OQ
R
OD
?H
H
H
TD
C
E N
K
T
]
O
A
PD
F
K
A G
D
B
DH
H
P
S
I
? BM
OJ
ID
P
H
G CG
I
N
K
T
NG
C
N
O
H
KD
V
PQ
PD
I
KG
F
O
B
H
D
OB
W
IA
V
ED
TD O
O
C
N
TD
S
I
V
OB
I
P
YI B
^ E
]
HB
LG
^ N
0
FR
KG
ED
O
OB
T
P
^ N
O
IH
KG
[ h
]
J
D
D
D
A
O
N
O
B
IH
Y
KD
TD
V
NI
OB
I
A PD
C
I
A ? DH
VH
O
I
C
R
BH
RB
D OJ
B
R? D
K
AH
CG
NB
O
Ke B
^ C
U
A
A
B
OD C
I
FL
T
O
N
O
A PD
F
D
LD
O
RD
C R
I
PQ
[
AH
H
C
FM
C
O
A PD
F
ED
X
k CNk
Nmg = O(Nl ), Nmg = O(l Nl ), Nmg = O(κl Nl ).
k>0
AH
H
BM
G
O U H
H
B
B
KD
V
PQ
Y
?
C
^ N
OB
I
P
YI B
K
B
H
B
LG
N
W
IA
C
I
TD O
C
R D
YI D
KD
TV
AH
U GH
D
LD
TD O
S
OQ
OB
T
F
RB
T
CV
P
J
D QB
?
V
I
AH R? D
K
C
N
OB
I
^ E
]
B
PJ
O
C
I
%
%
$
"
$
%
! #" "
$
"
$
%
γ γ −1 −1 zγk−1 = A−1 k−1 dk−1 − Mk−1 Ak−1 dk−1 = (I − Mk−1 )Ak−1 dk−1 . H
D
H
G
H U
A
D
KD
V
NI B
O
R
]
H
U CB
E
KG
I
H
H
FM
T
KG
# U
G
D
A
G
D
H
D
H
E D
TD
IS
I
I
C
O
O
E
AH
PD
O
[
J
B
J
B
D
G
CG
ID
E
TD
SI
Ke
H
CB
?
VB
O
I
O?
WH
D
OQ B
$
3,
%
-0
,
/,
.
1,
%
*0
,
/,
-.
$B
H
D
TD
C
E
LD
O
GH RD
C
S
F
P
T
W
Q
A
CI B
E
N
K
T
]
g
]
]
I
CI J
FM
T
H
D
H
KG
O
O
P
G AH CG
T
D
VH
O
]
]
]
T B
L
F
E
H
G CG
¯ 1 (ν1 , ν2 ), M1 (ν1 , ν2 ) = M ¯ k (ν1 , ν2 ) + S ν2 p M γ (ν1 , ν2 )A−1 r Ak S ν1 Mk (ν1 , ν2 ) = M k−1 k k k−1
S
?I
R
I X A B
L
H
G
J
B
G
Ce B
?
H
G
D
]
N
BH
ID
V
^
U
G
J
U
Q
H
D KG
Ce G
F
U
A
H
D
C D
B
TD
C
E
LD
O
GH
B
H
B
RD
C
S
F
P
T
W
IA
C
IW
IC B
0E
[
B
H
I
H
D
] B
TD
C
E
LD
O
GH
B
H
RD
C
S
F
P
T
W
IA
C
WH
A
CI B
E
I
W
IA
C
I
[
[
[
UA
F
O
D
B
D
LD
[
G
J
B
D
H
D
H
H
J
WH
A
IC B
E
L
F
T
Y
E
e
PG N
Y
O
I
NB
OB O
?
CV
CG
A ID
F
VD
O
Ke
IF
RB
J TB
]
[
D
A
GH
D
G
Ce G
F
C?I
F
L
W
I
I
A ?AH
?
N
B
[
[
X
]
A
B
H
[
[
F
C
B
TD
C
H
ED
LD
O
GH
D
RD
C
LD
O
E
K h N
W
IA
C
IW
Q
A
IC B
EG
F
TB
IA
OQ
Ke B
C
[
X
D
U
D
H
A
H
G
D
G
BH
?H
V
FR
KG
ED
O
R
CI
E
E
I
P
I K
F
I
EA
O
Ce
D
D WI H O
E
TD
IS
O
B
TD
C
X
W
IA
C
S
I
I
P
B
H
BM
YI B
K
LG
[
^
O
M
IH
O
V
C
FM
P
H
CG
C
KD
GH
D
P
CD
PQ
A
G
G
I
C?I
F
L
W
I
]
A D
O
J
B
D
G
B
E U H
D
LD
O
B
H
OD
W
IA
C
I
CG
ID
E
TD
SI
P
CG
Ke
H
CB
?
T
N
K
T
D
CG
H
PJ
? j IB
S
I
P BM
YI B
K
LG
]
]
A
G
J
D
H
D
AH
C
T BH
OB
I
P
YI B
K
LG
CG
V
I
UT
V
EI B
W
A
IC B
E
N
W
I
I
V
F
V
[
^
B
B
H
BM
I[
W
IA
C
S
I
I
P
YI B
LG K
W
A
IC B
0E
D
LD
O
Ke D
PJ
I
]
]
B
J
B
A
BH
B
D
M
B
H
E
A
IH
A PD
Ke D
O
P
NG
C
W
IA
C
IF
TD O
?
X
C
B
X
^
^
[
X
B
H
O
D
RD
C
NG
C
BH
ZAH
E
]
H IB
C
ID
T
YB
N
[
K h
W
[
B
D
?B
O
?I
CG
H
GH
D
A ID
Z
C
IA
A
D
H ?H
P
G
I
C?
F
L
G
B
B
B
EB
O
?
?
?
C
P D
O
K
TI
IH
O H
H
EJ
I
PB
A
G
X
]
D
T
P
I
C
E
I
C?I
F
L
W
I
I
A
H
H
GH
D
G
A ?AH
?
[
X
A
D
B
H
B
P
B
H
B
J
NB
O
O
P
H CG
C
CD
A
CI B
E
C
W
IA
C
IW
^ W
0
D IC B
E
C
R
^ E
O
I
ED
V
H
C A
D
?H
P
T
L
[
]
G
CD
C
I
PB
H IJ
O
A J
B
D
B
W
IA
C
IW
A
B
H
B
CI B
E
^ ED
E
?I
[
+ X
]
]
] g
A
D
G
D
G
V
W
I
I
F
?
T
O
D
IM
eD
ED
V
N
IH O
Z
CD O
E
IH
PD
A
U f B
]
D
H
D
?
? B
Ce B
G
VAH I
V
O
YD
E
C
R
]
E
D
I
B
A G
$D
O
P
IF
?AH
Ce
K
IH
T
AH ]
B
H
^
U H
E
D
LD
O
GH
D
RD
C
LD
O
E
W
IA
C
F
IW
A
CI B
OB
I
P
VD D
YI B
K
LG O
YD
CG
V
OB
I
P TD
C
[
[
[
DH
H
E
E
I
N
K
T
H
IH O
A YB
PQ
O
O
FR
KD
NV
N
K
PB
CG
TD
H
H
H
U
YI B
K
AH LG
T
V
NG
C
N
O
H
KD
V
PQ
O
A PD
F
ED
O
RD
C
GH
ED
T
YB
?B
O
Ce G
F
H
A
O
A
f
O
H
IH
Y
K X
I
I
V DH
O
T[ H
B
U
BH
Q
O B
NG
C
Y
KD
A TD
V
H
$
O
IH
Y
K X
I
I
V DH
PD
O
NG
C
N
KG
R
I
U
PQ
I
Oe
H
A PD
F
[
k
KB
IC
E
E
I
PD
C
O
H
H
Ke
C
I
O
D
, %*
-
E
K
H
H
EQ
H
CG
?I
H
K
T
W
IA
H
I
K
IH
C
ID
Z
V
zγk−1
EQ
H
G CG
I
N
E
K
C
FH
T
H
KG
?
T
X
'
O
D IH
]
N
K
T
H
T
O
O
Y
K X
I
A
BH
G
M
N
P
O
N
W
I
? DH
A
C
F
I
D
A ?AH
[
]
C ] H
zν1
O
H
C l
] B
FC
FT
H
N
O
IH
D
DH
N
K
T
O
A PD
F
EQ
I
V
E
I
O
IH
ZAH
H
0V
B
?D
T
W
IA
YI D
NV
K
T
zνk1 +1
IH
S
OQ
R
PD
TD
C
H
H TD
C
D
3,
Z
O
CD
O
f
ED
I
H ZAH
O
A PD
F
ED
?
O
IH
Y
IW
A
IC B
]
H
H
O
E
zγk−1
ZAH
N
K
T
OD
?B
E
S
E
K
H LD
B TJ
O
K
PB
CG
[
^
H
f
A
K X
I
I
V
E
E
B T[ H
f
D
KG
C
I
}
C
H TD
C
OQ
GH RD
C
A
ED
B
B
T[ H O
W
IA
]
O
IH
U
B
[
0
H
¯ k (ν1 , ν2 ) M
^ NB
O
O
E
CG
ID
GH
D
RD
C
γ ek−1 − zγk−1 = Mk−1 (ek−1 − z0k−1 ),
K
H
AH
D
D
LD
O
T
DH ZAH
O
RD
C
]
OB
I
P
T
L
Oe
V
gk
T
V
D
H
AH
X
]
U
B
B
0
γ
C? G
Y
ek−1 = A−1 k−1 dk−1 . C
M GM (k, z, f )
LD
V
U
B
ID
Z
N
K
T
T
LD
+
zγk−1
]
O
TD
H
k−1
E
A
H
B
C
E
D
LD
O
RD
Ak−1 ek−1 = dk−1 . ν1
O
B
>
QF
C
S
I
C
I
W
A
CI B
0E
C
GH
D
LD
O
YI B
K
^
k−1 Ak zk = b k
O
D
J
AH
H
H
E
PD
K
?W
[
LG
T
AH
H
ν1
k = 0
k
E
^
A
R
E
RB
I
X
E
A
I
B
P
k
WH T
V
I
O
IH
K
D
D
H
OJ
O
IH
Y
R
I
V
W
IA
Ce G
F
V
M0 (ν1 , ν2 ) = 0, γ ν1 Mk (ν1 , ν2 ) = Skν2 (I − p(I − Mk−1 (ν1 , ν2 ))A−1 k−1 rAk )Sk .
W
γ ek−1 − zγk−1 = Mk−1 ek−1 , H
AH
D
A K X
I
I
^
]
N
Mk−1
T
V
J
B
X
O
H
ek−1 = A−1 k−1 dk−1
P
LB
] Z
D
A
D
ED V
E
IH
C
I
k−1 ν2
ν1
H
CG
I
z0k−1 = 0 EG
P
K
IT
H
A PB
G
B
ZAH
dk−1 = r(Ak zνk1 − bk ) zνk1
H
B
B
V
?I
C
k > 0 Sk Skν2
T
?
?
?
?I
Skν1
L
D
CB
ν2 Ml (ν1 , ν2 )
− b)
Ke
A
D
C
¯z = z −
Wk−1 (Ak z2
C
I
O
Ke
H
CB
2
YI
^
H PD
T
K
z2 = z 1 − p e 1
zνk1 +1 = zνk1 − p zγk−1 .
γ ν1 zνk1 +1 = zνk1 − p (I − Mk−1 )A−1 k−1 r(Ak zk − bk ),
γ ν1 zνk1 +1 = (I − p (I − Mk−1 )A−1 k−1 r Ak )zk + gk ,
γ I − p (I − Mk−1 )A−1 k−1 r Ak .
k>1
¯ k (ν1 , ν2 ) = S ν2 (I − pA−1 r Ak )S ν1 . M k k k−1
%
%
$
"
$
%
! #" "
$
"
$
%
^
V
]
0 ν2 6= 0 J
B
D
N
N
K
1 4C ∗
Mk (ν1 , ν2 ) A
H
P
B
CG
O
E
K
Y
E
I FR
DH
BH
H
KD
VB
J
B
D
H
>
]
2 ξk ≤ η(ν) + C ∗ ξk−1 , k = 2, 3, . . . , l.
BH
B
U
]
P
I
?
]
UT
AH
N
K
T
E
R
K
HB
M
B
J
H
?J
IA
e
VH
+
>
IE B
V
CD
OB C
CG
EI
OD
NG
PB
H IJ
O
I
\
^
?G
EJ
IA
D
J
ED
]
C
T BM
]
]
^
A
AH
D
AH
D
H
H
W
A
IC B
E
N
K
T
EQ
C
O
IH
YD
KD
VT
V
I
P
CG
O
H
D
H
PB
O
Ke
C
]
V
W
I
I
F
?
T
O
I
D
]
H
FR B
KG
C
FT
W
I
I
F
?
T
O
I
O
IB
I
PD
OG
D
[
>
X
U
P
IC G
[
H h
$
N
^
^
]
H
D
D
]
S
P
N
K
T
GH
Ce G
C
F
O?
WH
S
P
I
S
P
N
K
T
GH
GH
E
RB
I
OJ
D
X
l
]
H
U KD
TD
A
VD A
CB
A
D
VB
O
PD
I
KG
F
S
I
FM
T
H
H KG
O
IH
[
]
D
D Ke D
Z
O
RD
CB
CG
T
N
K
T
X
G
H
H
M
eD
ED
V
E
A
IH
PD
A
J
B
D
H
B
Ce
I FR
KD
0V
P
H
CG
Ke
CB
?
T
Ke
S
I
^
G
H
C
R
C
I
D
D
KB
PD
J
Ke D
V
I
EQ
P
CG
O
H
H
H
A PB
O
E
O
T
H
D KG
V >
]
H
D
] B
TD
C
E
LD
O
GH RD
C
D
LD
^
P
H
G
D
CG P
F
C
BH
YB
A
CD
D
OG
N
W
IB
I
E
G
ID
?A
VB
V
NI
OB
I
P
YI B
K
LG
S
B
CG
D [
D
D
D
D
A
D
D
H
A DH
O B
NB
O
O
CG
V
I
C
NB
C
R
F
ED
CD
V
I
O
^ N
^
A
A
X
K X H
^ E
FLV
?[
^
D
D
D
H
UA
I
Z
D
H
H
D KD
O
Ke
]
]
]
^
I
H
C
FH
T
KG
A
O H
EI B
O
OB
PQ
TD
S H
C
J
B
,
D V[
?
TD
X
K
?W
+
I `
D
] B
U
N
K
T
?
Ke D
C
PI
CG
Ke
H
CB
J
B
RB
I
OJ
Ce
CB
I
SP
C
N
KD
PJ
D
OV D
C
R
F
ED
CD
V
?
Ke D
X
U
G
J
H
A
J
H X
H
H
T
S D
A D OQ X
TD
V
E
T
P
I
V
RB
TB
S
OQ
O
Y
NA
D
D VG
EB
N
K
T
CQ B
C
FKe
J
D
H
AH
H
D
H
G
H
H
U AH
OQ
Ke
I
KD
B
^ BH
FR
KG
E
P
FM
E
O
A IH
E
I
]
X
X
H X
J
B
A ^ M V
I
C
E
A
¯ k (ν, 0). −M
ν¯ ξk ≤ η(ν) + H
E
FR
KD
P
CG
O
H
H A PB
O
I OB
O
KD
V
PQ
E
YI D
?
]
I
P
G
BH
D
H
H
O D
Ke B
C
T
Q
E
EQ
TD
SI
O
NG
C
P
J
D
H
QB
?
O
IH
R
OB
IA
LD
X
K
]
E
] S
N
O
H
D
A PD
F
S
OQ
GH
B
RD
C
O
OB
I
P
YI B
LG
D
KG
R
IB
O
F
J
DH
I
I
O
OG
BH
IH
R
O
I
OB O
T
FR
KG
]
F
K
]
,
E
]
]
I
^
]
P
EG
I
EI B
C
YI B P
D
A LD D
k=1
E
J
eD
] B
OB
?
T
FY
H
]
^
I
CD
EI
I
NG
OB
CG
K
LG
B
AH
KD
TV
k
I
A
H PB
O
I
J
T
LD
D IH O
Z
]
D
H
M
^ NG ?
C
N
O
KD
OD
CD
O
I
N
O
IH
C
= Skν
V
C
D
A
WH O?
e
VH
C
E
FH
J
,
[
V
PQ
]
]
I
D
?
?H
O
k
T
]
^
O
CD
G
DH
G Ke D
V
;
B
]
*
0
C
R
EI
H
X
?I
ν pA−1 k−1 r Ak )Sk
]
η(ν) ≤ Ce G
CI
FH
T
H
ξk−1 ≤ 2η(ν) ν ≥ ν¯ KG
O
G
D
6
5
]
W
H
H
PD
I
KG
G
^
QD A
CD
Z
Ke D
ν := kMk (ν, 0)k ξk < 2η(ν)
E
K
T
B
A
O
Ke B
C
H h
7
;
IA
C
BM
]
?I
H
PB
H IJ
O
CG
I
O
IH
YD
?H
O
p
TD
G
D
N
K
T
P
FI
?AH
135
3
5D
S
I
I
P
YI B
K
LG
R
CB
I
ED
C
PB
^
ID
O
C
O
KD
PG
−1 ν −1 ν kA−1 k−1 r Ak Sk k ≤ cp kpAk−1 r Ak Sk k = ¯ k (ν, 0)k ≤ c−1 (kS ν k + kM ¯ k (ν, 0)k) ≤ c−1 (Cs + 1). = c−1 kS ν − M k
D
P
CG
O
D
D WH O?
C
R
]
C
R
^ ED
AH
NG
S
I
M NG
C
N
H
KB
O
E
− (I −
SI
H
LD
P
K
IT
1 4C ∗
D
H
A PB
H
H
K
I
;
C
P
EQ
D
H
H
N
TD
S
G
GH
O
KD
V
PQ
Skν
Ce B
?
A PB
O
η(ν) ≤ H
^ E
FR
I
KD
V
[
7
135
3
;
C
Y
O
S
P
^ A F T K N
H
]
Y
=
T
D
D
YI D
KD
Q
1
23
OB
I
P
YI B
K
LG
5
23
38
7
7 I[
D
35 ;
J
B
D
OB O
?
;
3
67
A
G
ξk
O
G
H
C
R
TV
AH
D
CG
NV
P
IC G
FC
G
[
1
T
eFC
E
V
k
YD
D
O
T
D
B
^ DH
?
C
6
2
6
3
67
]
B CD
FR
KG
E
3
234
38
P
p
E
O
R
LD
2η(ν) C∗ 4η 2 (ν) ≤ 2η(ν) I
KG
5
7
η(ν) →
¯ k (ν, 0)k ≤ η(ν) kM
KB
3
7
3
67
]
3
5
29
6
2
3
9
7
5
9
A
CI D
EG
G
B
kSkν k ≤ Cs .
]
O
;
3
8:
7
;
;
3 ;
3
67
:
I
V
cp kxkk−1 ≤ kp xkk ≤ Cp kxkk−1 ,
3
7
3:
6
1
3
234
38
67
5
6
7
ν2 = 0
5;
3
67
5
6
2
9
7
7
k = 1, . . . , l x ∈ Rk−1
^
γ kMk (ν, 0)k ≤ η(ν). γ−1 1
:
:
k 29
38
67
35
234
ν ≥ ν¯ 1
6
2
1
A
k k
;
;
:
η(ν) Cs
6
5
34
7
3
3
7
3
6
2
5
cp , Cp Pk : R k → U k
ν→∞ γ≥2 ν¯ > 0 Skν
]
0
ν p ν p A−1 k−1 r Ak Sk
[
?C
> X
]
]
Skν2 = I, ν2 = 0, kpkRk−1 →Rk ≤ Cp
H
D
[
B
LD
] ]
A
H
G
AH
A
D
O
D
[
$[
IH O
Z
CD
D
G
eC G
P
K
IT
H
G PB
V
E
C
IH
J EB
B
KB
RB
O
U H
T
D
VD
O
H
YI D
KD
V
ID
O
R
CI
E
E
K
A
H
J
ID
O
Ke
[
D NkX
H
CB
O
^ D
O
Ke I
D
Ke D
C V
A PJ
I
V
?W
B
I a
Ce G
E
TD
SI
G
eC B
P
J
D QB
?
T
UA
U
BH NG
C
FT
W
Q
A
IC B
E
^ D
O
O
K
]
G
H
K
H
B
H
H
BM
H
BH
[
[
[
>
H
D X
B
G
AH
I
O
IH
K
PB
CG
T
V
A
ED
O
P
IF
?AH
O
NG
Ce B
PD
A
I X B J
C
FT
P
CG
Ke
H
CB
J
B
? h D
FR
KG
O
A PD
F
ED
T
YB
?B
O
W
IA
C
I
P
YI B
LG
D
TD O
FY
^
^
J
B
D
D
D
[
U
5
D AH
H
H
NB
FR
KG
N
K
T
F
E
C
E
YB
? h D
3
7
6
7
3
PD
K
?W
I a
UI `
eC G
E
TD
IS
G
Ce B
?
T
K
PB
IA
VB
?
?D
O
YD
E
O
IH
O
A PB
F
]
U
K
?W
I a
γ = 2
p
¯ k (ν, 0)k ≤ 1 kM
¯k M
ξ1 ≤ η(ν). k>1
2 ν ≤ η(ν) + kpk kMk−1 (ν, 0)k kA−1 k−1 r Ak Sk k.
¯ k (ν, 0)k + kp M 2 (ν, 0)A−1 r Ak S ν k ≤ ξk ≤ k M k−1 k k−1
2 ξk ≤ η(ν) + Cp ξk−1 c−1 p (Cs + 1).
ν1 6=
%
%
$
"
$
%
! #" "
"
$
"
$
%
"
"
&
H
GH
H
D
D
H
D
J
S
X
U
O
E
O
D
H ]
Ak−1 = r Ak p. PQ D
P
O
A B
D
CB
A
D V[ D U H
E
BH
PB
]
S
P
N GH
G
N
O
IH
RB
D
D OJ
N
K
T
NG
C
J FH
Ke D
V
X
B
C? H
GH
D
CG
R
D
G
B EB
O
[
[
UT
V
EQ
AH
Al := Ah
B
B
H
D
K
(Ax, x) > 0
V
T
%
H
H
K
?W
I `
N
K
T
N
O
IH
T
YA
P
FC
I
] B
U
H
FM
P
C
]
B
G
D
D AH
H
J
M
G
D
Y
O
I
E
T
P
I A
H
H
H
CG
ID
E
TD
IS
E
C
I
Ce
I R
K
?
I
O
YD
E
W
IA
[
H
AH
D
D
H
D
H
K
S
I
I
P BM
YI B
LG
H
CG P
R
I
KD
?
ED
Ke D Z
O
RD
CB
CG
T
I
OB
PD
C
C U H
Ce
Q
C
YD
E
X
X
X
]
]
BH
H
H
A B
A LB
B
VD
L
FM
T
KG
A
H
G
A IE B
W
IC B
E
IE
OQ
R
CI
E
E
IG
E
H
G CG
I
N
K h
] B
E
FR
KD
V
I
\
H
A
D
H
U H
A YB
PQ
EA
O
O
IH
T
H
D
PJ
I
AH VJ
R
E
BH
H
H
I
\
J
J
G
G
B
J
B
OD I
F
B
RB
TB
I K
I
I \ FI
T
N
O
IH
O
A PB
F
H
A OB
NG
I
AH PZ B
R
CG
P
J
J
B
C
N
K
P
N
M
EI B
RB
TB
EI
?I
RB
^
E A
A H
IB
V
O
^ NG
]
AH
TB
S
OQ
\
+
H \
Ke B
W
I
O
T
I
H
H
H
H
G
G
S
OQ
O
Y
NA H
D
D VG
EB
W
IB
IJ
IAH C
?
T
I
N
K
T
PD
C
O
E
K
S
OQ
R
OD
?B
TD
C
E
H
A
A
H
B
G
I
O
IH
O
E
I
V
I
NV
W
IF
C
I
D
H
H
AH
NB
O
R
I
V
C
0I
O
O
K
T
D
VD
O
Ke
C
YI D
KD
V
[
A
P
CG
O
H
J PB
CI
FH
T
H
D KG
P
H
CG
Ke
D
I
W
A B
CI B
E
C
R
^ E
YI D
KD
TV
AH
]
A
H
G
ID
O
R
IC
E
E
I
H NG
C
N
K
P
N
J
[
]
5
7
3 ;
67
7
6
5
7
5;
;
C? H
GH
]
;
B
C
C?
F
L A
H
GH
D [
6
5
3
7
68
3
X
VB
D
D
VD
ZA
S
7
5
B
C
1
9
5
3
2
4
5
7
5
3
5
:
O B
A PD
F
S
OQ
GH
GH
D
G
RD
C
S
O
IT
S
F
P
T
G
D
B
AH
D
H
H
D
] B
TD
C
E
LD
O
GH RD
C
S
F
P
T
CG
ID
E
TD
IS
E
C
D
CG
9
B
D
B
OJ
D
CG
ID
OJ
A
EQA
O
N
K
T
O?
WH
l
J
B D
CG
R
H
H
B KD
O
F
RB
TB
C
A FH
IE
G
ID
?A
[
X
C
J
]
O B
B
RB
J TB
] H
^
?I
K X
I
I
S
N
O
QD A
A CB
H
D
VD
C
R
^ C
BH
RB
I
]
K
I
H
P
E C
IH
R
I
V
C
ID
C
R
N
UP
ED
T
I P
]
H
CD
E
D CG
^ E
]
H
BM
PG
E
P
YI B
LG
OQ
NG
C
N
K
.
Ce B
TB
J
G
B
W
Q
A
IC B
E
N
K
T
P
D
B
D
X
]
]
^
H
H
P
CB
J
B
? h D
5;
3
H
9
H
H
E
C
A
X
NI
]
I
e
I
V
B
,
D CD
G
U
A
H
B
Ce G
R
NB
]
A
O
IW H
D
3
67
34
7
,
1 √ ν
PD
AH
EA ^ H
D
A
] H
C
^ W
Q
IC B
]
?
C
H
T
PQ
C
RG
B
J
H
H
A PB
E
N
O
E
I
[
1
:
29
E
[
B
C
R D
^ NB
?
C
^
1
]
H
G
C
B
OB
CG
OD
?
C
FH
CG P
F
$
5
3
3
5
3
7
^
]
CD
NB
NG
:
E
P?
U
PB
H IJ
O
D
OJ
U
IH
R
P
?
I
B
OD
EA
L
S
P
W
I
I
FO?
O
Ke
C
I
D
H
G
D
\
C l
KB
O
D
?I G
PQ
TB
K
A
H
W
IA
C
S
I
I
P
B
H
BM
IY B
K
LG
N
O
IH
O
E
I
VH
D
D
BH
KG
V
C
R
^ C
RB
[
D
H
A
P
S
?
I
B
B
OD
EA
L
S
Q
I
V
H
PD
C
O
W
I
\
I \
?
O
]
O?
J
B
O B
B
WH
A
l
m2
W
IC B
E
I
O
P
D
H
H
E[
O
O
K
T
AH
D
VD
O
H
E
BH
LB
KD
E
PD
I
KG
F
E
OQ
RD
CB
^
Z
Oe
E
V
H
VH
KB
h
CG
IB
R
7
29
OB
H NG
C
N
O
B
KD
V
PQ
K
?W
I `
CG
ID
E
Ke
C
YI D
KD
V
D
CG
T
^
CG
ID
^ N
I OB
P
IY B
K
LG
F
]
IE B
D
5
6
7
−1 A−1 k − pAk−1 r k k−1
A PD
G TD
SI
J
KB
I
V
H
B
NG
C
N
K
P
N
O
B
CG
R
P
B
KB
P
CG
[
X
A PB
FV
6
2
3
FP
]
U
DH
M
B
OJ
O
C
G
B KD
C
R
I
K
IH
X
0
PQ
D
^
B
A
A
D
D
D
H
D CD
O
TD
PJ
I
V
C
P
C
B
H
B
IN J
P
?
?
Y
?
C
?I
OD
I
EA
L
H
?I G
[
[
CD
NB
NG
PB
IJ
?
C B
C
R
V
D
^ NB
T?
NA
A
H
H
A
D
P
Ce
N
K
T
PQ
O
Y
H
D
KD
T
F
C
?H
P
?
O
IH
O
E
I
VF
ED
C
D
] B
C
B
x
CD
U
] B
D
B WH O?
C
^
D
D
PG
G ?D
?
TD
38
67
5
η(ν) = O
Ce G
C
I
A ? DH
V
^ N
O
IH
Y
KD
A
TD
V
QD A
D A ED
O
^
H
D
TD
C
E
E
O
B
TD
FL
?
B
Ak x P∞ f (x) = m=0 {am sin(mx) + bm cos(mx)} m Ak Skν x ν Skν x
D OkX V D
FT
X
H
D A CB
V
E
TB
I
%
[
CD
D
]
N
PD
I
KG
F
I
W
]
D
LD
X
NA
TD
EA
3
I
I A
^
]
H
G
N
W
I
FO?
C
FH
CG P
F
2
4
35
7
5
3
:
\
U
U
A
H
D
B
D
VD
LD
A
D
CB
V
D
LD
O
H \ AH
Ke B
W
I
O
T
I
W
IB
IJ
C
^
H
H
^
B
H
D
NG
C
N
K
P
N
K
IA PB
V
?
?
$
E
I
FM
T
KG
Ce
Q
C
YD
W
IAH
\
G IB
I
E
ID
?A
V
E
H
G
?
T
I[
WH
A
IC B
E
O
X
Ak
F
$ V
TB
CB
A
Al
N
OB
W
$
J
B
D
ED
A
G
H
X
B
FM
O
NG
TD
eC G
G IB
I
E
ID
?A
VB
V
^ N
I OB
P
YI B
A>0
I
P
IB
G
r = p∗ , I
E
E
?I
R
OD
I
[
O
Ke B
PB
H
B
IJ
O
K
?W
I a
CG
ID
E
^
U
G TD
SI
F
D
O
YD
KG
LD
O
GH RD
C
S
Z
Ke D
?B
[
D
f
Ah
YI B
K
ID
?A
H OB
?
GH
D
U AH
D
A
D
D
OD
W
I
VA
H
D KG
I
[
K
LG
B
[
F
P
T
D
V
O
I
1 η(ν) = O ν
LG
VB
V
NI
OB
I
P
IY B
C?
A
V
H
B
C
R
BH
RB
D
D OJ
C
^ C
]
O?
WH
D
P
CG
D
Al = ATl > 0
P
[
K
LG
B
F
L
N
K
T
NB
T
L
C
] H ] V
^
O
Ke B
E
C
ID
VB
H
C
FM
P
6
O
OB
LB
A
k−1
CG
P
T
P
I
ED
O
H
[
B
U H
CG
G
E I
N
^
[
[
H TD
C
B
O
A
PD
F
A
CG
ID
RD E
K
K
k
O
P
D CG
U
R
S
PQ B
A
X
GH
D
H
^ H I
C?
K?
E
D
EB
G
B
E
K
?W
ID
C
B
A OI B
L
;
CB
TD
H
TG
¯ k (ν, 0) = (I − p A−1 r Ak )S ν = A−1 − p A−1 r (Ak S ν ) . M k k k−1 k k−1
P
A
D
H
C
D
H
B
TD
IS
O
IH O
ZAH
H TD
C
E
X
O
S
κ
I
[
V
^
D
Ce G
O
YD
KG
OQ
H
CG
S
OQ
RD
GH
7
56
CG
G IS
¯ k (ν, 0)k ≤ CA η(ν). kM
O
P
D
D
NB
C
C
]
ν ≥ ν¯
I
H
N
;
;
^
WH
LD
O
TD
W
B
KG
R
I
∀ k ≥ 1.
H
B
:
X
H PJ
OI
F
H IB
IA
C
I
CA > 0
C
P
W
I
TB
IA
C
7
[
OB
IA
H CG P
R
I KD
?
≤ η(ν)kAk k ∀ k ≥ 0.
EJ
I
K
T
38
23
2
6
6
2
D
M
B
D
CG
O
H
H
κ<1
L
H
NG
C
N
K
kAk k−1
KG
[
H
H
] B
K
235
I
K
C
YI D
O
H
O
Ke B
E
C
N
O
IH
O
K
T
−1 −1 kA−1 k − pAk−1 rk ≤ CA kAk k
EQ G
G
H
G
H ID
V
B
Y
7
D
D
H
E
P
A
CI G
FC
I
6
Y
?
C
CG
ID
T
[
P
N
O
O
Ke
CB
J
2
^
EQ
kAk k
?I
R
LD
I
^
H
D NkX
O
6
ν¯ > 0
KD
TD
C
E
C
O
PI D
]
I
18
kAk Skν k
ν→∞
]
H TD
C
E
S
OQ
]
S
N
CG
ν
η(ν) → 0
]
GH RD
C
D
LD
O
E
k
k η(ν) : R+ → R+
η(ν) η(ν)
x 6= 0
k · kAl := hAl ·, ·i 2 .
1
D
B
P
V U B
NI
OB
I
P
YI B
K
LG
CG
D
PG
I
OB
PJ
O
H
IH O
OG
D
DH
\
[
[
=<X N
O
H f
Ke B
EA
%
%
$
"
$
%
! #" "
"
$
"
$
%
"
"
&
−∆ u = f P
Ω, u|∂Ω = 0, (
*
*
&
- . ]
]
E
Z
O
P
H
B
A H
D
^ NG
Ce
N
O
E
I
V
O
U
GH
D
FR B
KG
S
C
I
S
P
CG
IB
>
]
U
B
Ce B
F
ZA
O
C
H
YD
E
H
PD
I
KG
*
P
e
Z
K
I
H NG
C
N
O
KD
V
PQ
CG
ID
OA
N
FK
]
H
H
#
H NG
C
N
K
P
N
B Ce G
K X
D
KG
I
PD
C
O
E
K
O
H
H
H
G
D
D
U J
Ke D
V
IA
I
CG V
E
C
E
IH
M NG
C
N
O
B
GH
H
KD
V
PQ
T
L
P
O
D
?H
KB
T
N
O
IH
YD
L
AH
I
H
^
B CG
O
G
eD
] A ]
G
D
KB
T
FY
I
CG
ID
OA
N
KF
X
]
]
KG
D
B
D
B
H
WH O?
O
O
KD
V
PQ
B
N
K
T
C
R D
^ NB
?
C
B Ce G
K X
D
B
CD
C
NG
PB
IJ
H
H
H
H
PD
I
KG
F
O
T
O
H
IH
ZAH
TD
G
]
B
G
B
O
H
H
]
L
A D
H
P
B
CG
OB
A
CG
W
NI
FK
[
H
D
$
M
PD
I
KG
F
C
NA
P
H
AH
D
G
D
I
FM
P
H CG
C
CD P
C
R
^ E
YI D
KD
TV
[
D
S
OQ
Ke B
EI D
O
KD
I
UV
O
G RD
F
?J
CI
ND
CG
D
G
Z
PQ
O
IH
[
\
C
]
H
CV G
W
I
FO?
OB
PD K
FT
W
IJ I
IAH C
?
T
IA
C
H
A EB
V
]
A
G
H
A
UA
IC
D
D
D
AH
OA
N
FK
L
O
EA
OD
?
H
B
NG
C
N
K
P
N
C?
GH
NB
F
L
NB
EB
E
R
I
V
[
[
\
X
3
6
2
6
2
23
]
3
5
6
9
29
2
P
CG
OB
A
CG
]
6 ; 35 7
6
5
18
1
6
?
C
S
OQ
O
R
D
GH
H
S
V
OQ
C
O
E
K
O
R
UA
U
H
H
D
H
H
D
OD
S
?
OQ
O
YD
K
PB
E
H
CG
G
IB
OB
J TB
U
D
D
Ke
H
EJ
I
OA
H
ED
O
A PB
CG
H
ID
O
Ke
CB
PD
T
H
KG
D
VB
O
C
R
^ E
IY D
KD
TV
AH
U
J
B
D
J
G
D
D AH
H
CG
ID
E
TD
IS
F
E
C
FM
O
PD
D
B OG
P
CG
Ke
H
`
H X
H
CB
?
T
E
T
H ]
CD D
T
P
O
FT
H
D
D
H
J
B
D
U J
^ NA
PD
L
P
^ C
]
I
B
X
X
A
B
C A
H
EB
V
A
A
D
H
NB
I
PQ
eC
I
P
C
K
PD
B
GH
D
YD
E
FT
j
]
M
PD
L C
CG
OD
E
KQ B
E
O
RD
CB
I
KG
OB
]
D
CG
X
*
^ A
H
H
H
A
B
G
D
H
E
E
I
A ED
C
EB
V
I
?
TD
C
E
P
E
A
V f IB
X
H
H
]
T
L H
]
]
]
D
G
E
A
G
H
C G
H
IP B O
T
P
I
P
FH
AH
CD
V
W
IB
I
E
ID
?A
VB
VD
CG P
P [
X
F
P
CG
O
H
H
M
D
H
A PB
NA
P
C
K
P
[
^
B
B
I
IH O
T
YD
S
O
P
C
D
IC G
D
G
RB
I
TB
J
B
P? D
OB
D CG
NV B
K
]
X
U
FT D
u ∈ H10 (Ω)
@
?
A Ce B
P
IC B
G ZH
VF
D K X
CG
AH
S
KD
AH
U
C
YI
K
H
U
IS
hk ≤ c1 2−k , h0
+
P
C[
IB
G
B
H
H ^ H
O
H
L
H
VT
VH
Z
H
O
X
5
6
8
6
'
9
A
GH
∀ v ∈ H10 (Ω),
D
A
H
EG
E
FT
E
PD
OQ
R
Ω
+
#*
O
X
H
]
U
H
IH O
YD
K
P
^ B
H
OB
I
A
TB
O
f ∈ L2 (Ω)
#
(
.
O
YD
E
A
D
J
D
,
IB
RD
K
FL
H
PD
I
OD
S?
OQ
EA
PQ
B PJ
O
C
H
D
C
FH
A
3
3
P
I
eC
f v dx
'
X
[
S
I
D KG
F
C
^ ED
+
OQ
E
A
B
IO B C
L
;
] B
K
Ω
%*
'
#
S
O
H
C
N
TD
S
P
N
?
H
H OB
K
T
I
>
]
B
D
AH
H
D
]
1
6
8
:
75
Z
-
.
O
E
C
B U D
?
GH
\
]
Uk−1 ⊂ Uk
*
C
Q
^ C
H
YD
E
?
P
C?I
D
OD
?
C
IB
?A
V
U RD
CB
CG
]
I
OB
I
P
YI B
K
LG
E
C
C
R
^ H
I OB
27
2
3
(f, v) :=
.
[
H2
#
B
NG
H
H TD
C
E
?
O
I
?H
O
G
D
H
D
B
D
AH
U CG
V
I
UT
V
ED
7
2
5
3
5
T
a(u, v) = (f, v)
]
C
N
O
YD
KG
F
J
P
CI G
G
Ke D
FL
D
LD
B
I
O
S
OQ
O
T
N
K
T
K
?W
I a
Ω
]
IFR
KG
H
B
I
FC
CD
BH
I
OB
B
[
D
Ce G
E
O
E
O
c0 2−k ≤
'
! & %
.
+
OQ
YD
KG
OQ
O
IH
F
LG
A
FR
K
D LG
Y
NA
D
D
VG
EB
G
U
G
B
H
TD
IS
O
TD
A
PI
X
Uk d
*
* #
KB
I
RD
C
GH
D
LD
O
I
V
O I
P
A TD O
I
[ K h N
W
B
H
BM
K
PD
OB
I
E
VH
IA
D
J
B
CB
B
A
FY
K
W?
H
P
I
Z X 2 ∂u ∂v a(u, v) := dx, ∂x i ∂xi Ω i=1
$
.
,
H
D
DH
B
OB
L
?
E
S
O
E
U
B VH
Y
?
C
]
IA
C
S
I
I
P
IY B
CG
F
B
X
k H2 (Ω) H2
'
&
D OQ X
TD
A V GH
F
KG
H
J
B
D
D
Ce B
?
T
O
K
LG
K
Q
EI B
I `
;
?W
I
]
D
?
O
I
Ke D
FL
LD
O
E
^
D
AH
H
E
K
H H
CB
P
K
PI B
KG
F
V
X
L
H
NG
C
J HX Ce B
Y
I
N
O
D
KD
V
PQ
A LD
D
CD
?
N
K
T
U
B
B
H
I T
P
C
E H
H
IH
O
A PD
F
ED
O
RD
C
GH
ED
O
W
IA
C
I
[
T
L H
G
H
H
^
]
H
R
D
LD
C
H
H
KD
B Ce G
Ω 0
A
X
D K X
NB
O
O
R
OB
IA
LD
QD
X
0
C
C
R
Ce B
C
RG
I
E
FT
[
^
D
^
^
H
G
H
H
KB
C
T
S
?I
R
SO
I
C
S
D T?I J
A ED
Ω
&
*
NV G
OQ
CG
H
D
B
PJ
I
?
O
TD
B YD
E
K
?W
I a
B
KG
R
ID
LD
U
#
6
Tk
N
UA
I
V
I
A
PD O ^ H
F
U
D
Ce G
E
TD
O
4
5
∂2 i=1 ∂x2i
P2
*
.
O
TD
D
T?I J
[
ED
O
RD
C
GH
ED
T
YB
?B
O
W
[
Z
BH
D
Ce G
4
∆ :=
+
?
'
^
A
B
G
H
B
H
ED
L
E
e
P
W
IA
C
IW
Q
A
H IB
IA
C
I
G
Ke D
FR
c0 , c1 u
'
CG
ID
"
+$
$
D
IC B
E
EQA
O
D
SI
RB
X
E
w
*
H
E
TD
SI
.
X
I
O?
WH
H
BM
D
KG
(Ak ) w
+
+
!$ # '
*
UO
Y
K X
I
A
G
B
3,
H
TB
O
,
Ω ∈ R2 , I
P
CG
]
FC
TF
O
D
EB
A
P
J
(CA + ν1 )(CA + ν2 )
NV
O
%
H
Ke
,
NG
Ce
C
YI
CA
1 w
$B
IH
$
/,
.
1,
%
*0
,
-0
J
B
D
.,
*
*
$
4
/,
-.
'
CB
?
T
N
K
)
'
1$
3$
4
7$
.
TD
PB
Wk =
+
T
G
B YD
OD
O
N
S G
K
T
B
F
RB
I
B
T
CQ B
J
AH
H
H
0
-0 "
I
P
w
J
H
C
FKe
OQ
O
FR
KD
V
E
O
kMl (ν1 , ν2 )kAl ≤ p
TD
C
E
D
LD
O
RD
GH
A IH
E
I
CA
TB
N
O
IH
ZAH
D
C
D
LD
O
E
PD
K
?W
I a
I `
−1 −1 kA−1 . k − pAk−1 rk ≤ CA kWk k
]
H
H
D
LD
O
C
O
E
K
O
R
U
U
Wk = WkT ≥ Ak .
U
H
OD
?D
LD
O
CA
Wk
Sk = I − Wk−1 Ak
∀ u, v ∈ H10 (Ω).
U0 ⊂ U1 ⊂ · · · ⊂ Uk ⊂ · · · ⊂ Ul ⊂ H10 (Ω),
%
"
$
$
"
"
%
$
"
$
%
! #" "
"
%
&"
"
&
%"
$
"
$
%
"
&"
%!
J
B
H
F
\
Pk : R k → U k
H
S
O
TI
GH
D
G
B
B
C
H
?
CG
I
PG
D
O X
D
H
F
T
Y
E H
EJ
A D A U OB
CG
V
I
\
]
0
A D
CI
O
E
K
OQ
J
H
H
Ke D
PJ
I
[
]
kzk − p zk−1 k ≤ CA kAk k−1 kfk k,
?[
OQ
O
[
FT
A
D
Rk
Y
O
]
F
?
IA
T
P
I
P
E
H
A
H
D
B
B
U D
PG
^ A
D
A CB
V
?
?
W
Q
IC B
E
D
[
]
D
AH
H
P
TD
S
V
A CB OQ
E
IA
Z
H
PD
G
D
LD
C
H
D KG
RB
[
X
A
G
H
AH
O D [
H \
H
Ke B
W
O I
I T
O
P
H
D K U H
O
R
OD
?
ED
CG
D
PG
NG
J FH
Ke D
E
U
G
H VD
P
T
L
IH
\
^
W
I
FO?
S
OQ
C
O
H
ZAH
Ce B
PD
IE
ID
?A
VB
V
N
O
IH
ZAH
D
LD
O
E
K
[
[
\
AH
H
A D
J
D
B
H
U
A D
O
R D
H
D OD
?
P
CG
A OB
CG
PV
PD
C
O
W
I
\
I \
?
P
CG
OB
A
CG
V
T[ H EI
V
]
A
A
H
P
G
H
B
KB
RB
O
I
FM
T
KG
W
IB
I
E
ID
?A
VB
VB
CG
D
PG
?I
PD
V
OB
K
V
[
[
Z f B
]
N
]
U
A
B
H
H
H
PD
C
O
E
K
S
OQ [
O
ID K
O
G
BH
RD
F
?
FR
KG
P
FM
F
L
O
X
]
H
D
A
A B
D
N
OB
I
A PD
C
I
? DH
V
A CB
V
N
K
T
OD
i B K X
$
] G
I
]
C?I
K?
E
GH
D
H
G
[
r
NA
D
VG
A
O
IH
ZAH
S
OQ
A PD
H
EQA
O
E
H
]
H
]
Rk
D
CB
A
H
D
H
Z
O
IH
H
AH
D
G
C
O
E
CG U H
H
D IH O
Z
]
U
H
H
U
C
BH
RB
D
H
D
OJ
EA
O
I
P
PD
C
O
E
K
S
H
]
g
C?
K?
E
H
GH
D
H
[
H
?
D
Ke D
C
H
IB
Y
K
^
A
H
U TB
O
I
V
OQ
IA
Z
P
hk−1
F
V
C
V
H VAH I
V
K
T
V
e
FN
J
Ke D
I
TB
J
I
0
O
OQ
R
OD
?B
]
A D
P
$
H
CG
A OB
CG
PV
C
A
G
FH
EI
ID
?A
VB
V
^
I
U
U
0
H
H
B
J
D
H
H
D
TB
O
C
O
E
K
O
R
OD
?
O
IH
ZAH
?
C
[ ]
C?
K?
E
H
GH
D
H
IB
[
H
A
B
^ D
F
L
?
?
H
IB
Y
K
[
X
0
O
J ED
Ak zk = fk Ak−1 zk−1 = r fk fk ^
]
W
IB
I
E
G
ID
?A
VB
VD
P
CG
IH
PD
H
\
D CD
G
J
]
]
E
D
TD
Ce G
F
1 8
E
VD
[
IH O
C
O
W
I
I \
H
H
AH
D
NI G
C
N
K
T
V
H
GH
BH
?
H
D
NB
A
CD
1 8
I
H
C
I
]
Y
KD
A TD
B
D
D
N
C?
[
FR
O
C
E
Ce G
D
?
R
I
1 8
RB
^
H
P
N
FR BH
O
KG
IH
T
H
D
V
?
C
R
^ E
D IH O
Z
D
I RB
KD
WH
D
H
E
^
1 4
E
D
Uk K
O
OQ
O
K
I
I
F
O
A PJ
I
[
ED
P
CG
A OB
O
I
ED
V f B
D
H
J
TB
V
O
IH
O
TD
IS
1 8
D
U
H
P
CG PD
C
O
B
U
D
+
D
^
D
]
O
CD
K?
B
H
]
B
KD
V
PQ
1 8
OJ
X
Pk∗ D
R
OD
?
H
H
CG
H
OB
^
^
P
P
P
? ^ B
V
LD
OA
W
A
H
CG
FV
T
Y
E
G
E
O
IH
ZAH
NA
TD
A
1 8
eC G
H
J TB
F
H
O
]
N
KB
H
H
D
EJ
J
O
B
0
LB
K
0
]
J
AH
H
E
?
Ke D
T
L
G
N
O
N
O
IH
RB
OJ
N
O
I
FO?
S
OQ
C
O
E
K
O
R
Uh
EJ
IA
H
G
?D
F
K
hk
CG
ID
S K
B
OQ
R
OD
?
OJ
A
D
H
ED
JH
EI
RB
OB IH
OJ J
D
Ce B
A PD
B OQ
?
V
G
\
I
W
A
CI B
H
hk
] S
N
O
EQA
O
P
K
F
^
EI
FE
G
D
\
D
B
Ce G
R
NB
T
L
D
J
A PD
F
S
OQ
O?
WH
CG
H
X
G
H
G
EI B
P
?
EI B
P
H
CG
O
IA
B A PB
V
Uk
E
I
F
I
ED
O
O
P
CG
RD
C
D
O
ED
LB
ZG
H
IH O
J
IB
A
DH
IH
KG
H
G
D
U
H
OD
J
ED
I
Rk
GH
B
E
p u2h I
P
%
OA
D
r uh
N
KG
I
H
N
X
A
O
Ke
C
I
A PB
NG
C
BM
J
TB
?
ED
P
CG
[
H
hAl z, yiRl = a(Pl z, Pl y) ∀ z, y ∈ Rl ,
?
I
J
O
[
E
R
I
PQ
N
K
T
O
H
CD
?
O
Y
r = p∗ .
KG
$
ED
B
X
IH
RB
O
D
H
X
H
H
PD
C
O
E
K
S
NA
H
OB
I
PD
H
T
L
ku − uk k0 + hk ku − uk k1 ≤ c h2k kf k0 ,
O
IH
H
NG
H
A
D
]
A
NV G
[
X
D
VG
A OB
OQ
O
P
CG
ku − uk k1 ≤ c hk kukH2 .
I
RB
OJ
C
N
A OB
E
I
F
O
AH FH C
C
GH
D
H
OQ [
O
A
D
CG
A
a(uk , vk ) = (f, vk ) ∀ vk ∈ Uk .
O
Ke D
FL
D
D
H
FM
T
ED
IG
S
GH
D
A
A
D
CB
C
I
? DH
A
V
H1
FE
P
T
S
G
H
H
L
O
D PB
A
O
TI
G
C?I
F
L
]
A
r p hp x, yiRk = hx, r yiRk−1 ∀ x ∈ Rk−1 , y ∈ Rk
AH
X
NV
W
E
O
P
KG
N
K
T
OB
PD
J
IH
KJ
F
ED
H
KJ
F
K
ID
C ] H
H
l
V
hfl , yiRl = (f, Pl y) ∀ y ∈ Rl , P nk Pk : R k → U k i=0 zi yi
Uk
AH
AH
H
NI
KD
A
O
M NG
C
G Ke D
V
% O
OB
T
U
O
RD
G
V
L2
C
Y
ED
B
H
IH
RB
OJ
N
H IG
e
T
E
H
F
^
p = Pk−1 Pk−1 ,
C
V
C
O
NI B
D
KG
R
I
PQ
PD
C
O
hk−1 = 2hk
O
H
C
[
X
I
C?I
]
]
P
O
AH
?
ku − uk k0 ≤ c h2k kukH2 ,
S
OQ
GH
D
^ H
C?
F
H
H
GH
[
H
H
O
C
u k ∈ Uk
O
H O[
O
K
I
AH NG
C
HX
A
E
D
H
G
QC
NJ
P
fl
YD
D
L C
K
K?
FO?
A
f ∈ L2 (Ω)
KD
G
[
C
^ C
CG
ID
E
N
I
hz, yiRk = h2k
A VB
GH
D
X
YI
H
K
TB
YI
H
K
TB
H O[
O
K
T
^
g
O
I
V
OQ
A
IA
Z
P H
U TB
kv − vk k1 ≤ c hk kvkH2 .
H
^
C?I
A
B
O
A
K
I
Al
^
NG
C
B T[ H
O
B
C
R D
NB
?
C
v k ∈ Uk ^
]
]
A D
D
H
O D
Ke D
PJ
I
NV
K
T
O
O
E
I
P
EI B
O
ED
O
OB
A PD
IG
A ZB
K ^ H
[
\
^
A D
D
V
P
B
CG
A OB
CG
V
C
R
eC B
C
I
H RG
E
TF
A
U G
EQ G
P
C
FM
EI
G
ID
?A
VB
A
H10
CG
H
C
2h ]
F
L
S
O
A
I
V I
V
u
N
O
X
KJ
F
P
Rk
IH
IB
A
H10 ∩ H2 (Ω)
Uk v∈
Rk
−1 k(A−1 k − pAk−1 r)fk k ≤ CA kAk k−1 kfk k
uh
Rk
Fk = (Pk∗ )−1 fk ,
%
"
$
$
"
"
%
$
"
$
%
! #" "
"
%
&"
"
&
%"
$
"
$
%
"
&"
%!
hAk y, yiRk = a(Pk y, Pk y) = k
kPk yk21 ≤ 2 ch−2 k kPk yk0
≤ 2 c0 h−2 k kykRk .
y ∈ Rk
1 1+ν . N
C
R
D
^ NG
Ce
f (x0 ) ≤ 1 1+ν .
D
H
P
N
O
IH
RB
λ
kB(I − B)ν k = max{|λ(1 − λ)ν | : λ ∈
OJ
L
D
1 , 1+ν
Z
Ke D
X
C
B = BT ,
B
BH
kB(I − B)ν k ≤
OI
BH
E
FR
B
]
ν ≥ 0.
1
3
7
8
7
7
7
B
C
L
λ(B) ∈ [0, 1] KD
?
B
+
5
9
34
K
A
H
H NG
C
N
K
P
B
A H
H ?I A
A PD
V
N
K
T
ED
C
O
FE
CG
O
I
E
OQ
CA
T
OB
CG
BH
FR
N N
KG
ED
O
R
CI
E
E
OB
I
P
YI B
LG
B
[
G
T
H
B
H
T
] B
A
H
G
H
G
G
K
U X F
N
T[ H
H
gB
F
B
ED
TD
A
\
C
LB B
E
OQ
R
IC
E A
D
H
E
E
I[
OB
I
P
YI B
K
LG
N
K
%
]
GH
P
K
TI
D
H
G PB
A
VB
T
L
P
F
ED
C
D
]
H
A
B
O
W
H
IB
B
AH
A
H
IG
?
K
C
EB
V
KG
I
f
A
D
B
H
I
WH
O?
?
A PD
^
G
D
CD
C
I
PB
IJ
^
^
]
D
C
R
eC
A
D
IH
D A PD
V
O
RD
CB
CG
T
J
AH
H
B
D
I
Ce G
J
C
K X
I
R
]
A
H
B
G
D
A EB
V
I
?
TD
C
E
OB
PQ
W
I
[
X
X
K
I
I
D
?I
A
A PD
H
K h
NV
C
A ^ ED
C
H
Wk = w1 Dk kWk k ≤ ckAk k kDk k ≤ ckAk k w k Dk = (Ak ) kDk k ≤ kAk k kWk k ≤ B
V
L
FM
T
H
KG
WH
OD
?
P
O
T
H
A PD
CV
V
BH NG
C
P
O
IW H
D
]
]
AH
H
H
K
C
S
I
I
P
B
H
BM
YI B
LG
H
CG P
RB
?
P
KG
I
T
I
#
U
G
B
B
P
E B
G
ID
?A
VB
VD
CG
D
PG
K
?W
I
CG
ID
E
TD
IS
P
CG
H
J
[
`
G
J
G
G
B
J
B
D
] D
OB
?
T
B
OD
F
RB
I
TB
N
K
T
W
IB
I
E
ID
?A
VB Ke
U
D CG
P
P
J
D
CB
IA
P
?
T
U
J
D
B
ED
I
EQA
O
O
D
VD
S
K h N
J IB
Z
O
CI
FH
T
H
KG
]
?
P
X
CD
C
OB
CG
O
D WH
J
B
H
OD
S?
C
I
IG
e
K
I
P
IN D
D
O
D
G
eC G
E
I
IJ PB
V
NG
FH C
C
W
I
I
UA
]
H
IH
R
Ke
EJ
H f
\
[
]
S
N
O
A
PD
F
U
^
]
A B
D
H
H
H
D
S
OQ
GH
GH
D
G
B
RD
C
S
O
TI
O
O
IH
ZAH
S
OQ
C
O
E
K
O
R
OD
?
CG
ID
OJ
F
O?
WH
[
A
D
H
D
?
? B
eC B
G
VAH I
V
O
YD
E
P
CG
O
H
H A PB
O
e
H
VH
+
]
^
H
J
B
D
D
D
B
A
D
Ce
A
IH
PD
V
O
RD
CB
CG
T
P
H
CG
Ke
CB
?
T
N
K
T
O
Ke
H
CB
PD
T
K
]
kAk k ≤ c h−2 k .
;
0 ≤ B = B T ≤ I, 7
P
CG
D
I >
[
]
j
V
P
F
E
GH
$
OQ
Ke
C H
YI D
KD
^
A H
A
H
P
B
CG
O
PB
PQ
K
TI
PB
V
O
F
O?
D
H
H
B
WH
C
R
K
P
H
D
H
U D A
O
FM
O
A Ke B
C
?H
G
VB
O
I
O
Ke
CB
PD
T
KG
B
λmax (Ak )
CG
V
?
?
PG
$
RB
I
J
TB
D
H
GH
B
IH
F
I
C?
O
[
I
CQ
OB
CG
OD
∀ φ ∈ Uk ∀ φ ∈ Uk−1 .
ID
^
3
7
CG
ID
IH O
YD
KD
I
PD
E
λ(Ak )
^
I
B
0
B
B
RB
I
W
A B
CI B
E
?
?
?
J TB
N
O
IH
ZAH
+
^
A
P
D
CG
O
H
H
D
B
H
G
A PB
O
C
O
R
IC
E
E
Ak
T
f (x) = x(1 − x)ν T
5
6
7
3
$
ckAk k
M
PD
I
]
≤ c h2k−1 kFk k0 ,
KG
F
D
NG
E
≤ c h2k kFk k0 ,
W
J
FH
Ke D
P
∀ φ ∈ H10 (Ω).
I
G
VD
P
D
IB
?
C
kuk − uk−1 k0 ≤ ku − uk k0 + ku − uk−1 k0 ,
FO?
]
$[
W
\
LD
C
^
P
O
Ke B
W
AH
H \
\
B
B
D
OJ
A
EA
uk−1
>
O
∀ vk ∈ U k .
IH
I
FO?
S
N
kuk − uk−1 k0 ≤ c h2k kFk k0 .
$
VH
CG
CD
PD
C
H
OQ
C
K h
B
[
I
O
T
I
M
uk
IH X
D
LD
O
Ke D
O
E
H
]
?
?
D
I RB
J
H TB
O
IH
ZAH
O
f
AH
C kuk − uk−1 k0 ≤ c CA kAk k−1 kFk k0 .
D
D
K
S
H
^
]
O?
D
WH
0
IH
VT
p zk−1 = Pk−1 uk−1 .
F
A
O
E
H
D
D
Uk
k
PJ
I
OQ
K
O
R
OD
?
N
h2k ≤ c kAk k−1
NV
kvk k1 ≤ c h−1 k kvk k0
K
H
U
D
C
NG
E
FH
J
Ke D
a(u, φ) = (Fk , φ)
R
OD
C
R
^
G
VD
P
A
ZAH
V
= (Fk , φ) = (Fk , φ)
T
E
BH
?B
H
^
J
Ce B
B
D
ku − uk−1 k0
T
K
T
ED
P
H
CG
O
?
ku − uk k0
FR
P
I
PB
H
VG
eD
B
e
H
H
?
zk = Pk−1 uk ,
KD
G CD
C
I
PB
IJ
A
C
?
?
B
T
L
uk − uk−1
]
]
O
KB
CG
l
?
C
+
a(uk , φ) a(uk−1 , φ)
CD
G
B
E
X
hk−1 ≤ c hk
C
I
PB
IJ
P
OQ
CB
A
D
VH
c C Pk−1
uk−1 ∈ Uk−1
H
H
c
OD
O
j
B
u
PD
C
OB
CG
OD
B
Pk
uk ∈ U k
?L
^
ED
O
I KD
V
Fk ∈ U k hAk y, yiRk ≤ c0 h−2 k kyk2Rk
kuk − uk−1 k0 ≤ c kAk k−1 kFk k0 .
Uk−1
(B)}.
kB(I − B)ν k = max x(1 − x)ν . x∈[0,1]
x0 =
%
"
$
$
"
"
%
$
"
$
%
! #" "
"
%
&"
"
&
%"
$
"
$
%
"
&"
%!
^
kWk k = kAk + Lk Dk−1 LTk k ≤ kAk k + kLk k2 kDk k−1 ≤ ckAk k. B
H
Sk = I − Wk−1 Ak . W
IA B
H
[
[
P
A
H
@
? ]
ν = 1, 2, 3, . . . . UT
K
P
H
6
15
23
6
;
23
6
2
3
9
kBk ≤ 1 3
6
I
O
?I
E
E
K
?
?
B
G
B
H
B
NB
U
A H
H
F
KG
C
YD
E
ED
C
O
FE
CG
O
EI
I
N
TD
S
TD
V BH
]
G
A O
D
Ke B
C
?H
[
[
^
U
A
H
D
H
P
K
AH
P
Ce
A
IH
PD
A
NV G
FH C
C
N
OB
I
P
YI B
LG
D
CG
D
PG
KG
I
O
Ke
C
O
IFT
CB
J
B
D
H
[
X
]
^
]
U
\
G
D
C
P
Q
EQ
E
K
BH
H
H
OB
I
PD
J
Ke D
V
I
LD
C
H
A ED
J
j
X
I \ FI
I T
W
I
I
?H
P
OD
?
]
O
O
Y
NA H
D
D VG
EB
G
H
OB
RB
J TB
NB
O
TD
S
G
^
O
H
IH
O
A
A
H
PB
F
E
A
IB
V
O
? ^ B
^
G
G
B
K
A J
H
H
Y
?
C
H
B
B
TD
C
E
K
I
I[
OQ
OB
PD
IC
E
E
G
IH
O
N
T[ H
H
gB
F
TD
KG I
C
H I
^ NB ?
%
^
]
J
B
D
B
B
H
H
B
H
G
NG
C
FH
J
Ke D
V
I
OB
I
P
YI B
K
LG
N
K
T
KG
I
K
T
C
ND
CG
?
W
IA
C
I
E U H
I
K
?W
]
[
[
+ X
]
H
B
P
P
EI D
CG
D
PG
E
I
BM
IY B
K
LG
NG
C
N
K
P
N
I `
CG
ID
E
O?
D
WH
D
B
A
H A
D
H
BM
P
J
S
PQ D
B
B
CG
D
PG
L
P
YI B
KG
?I
PD
V
N
K
T
C
TD
IS
TD
VH
O
E
[
[
X
]
]
A
H
G
BH
J
G
G
FR
KG
H
K
I
I
KG
I
^ BH
N
T[ H K
H %
gB
F
B
D
LD
O
R
CI
E
E
I
N
K
T
C
P
>
A
H
H
G
UE
K
C
Q
A
H
D
CI B
E
OQ
R
IC
E
E
IH
O
^ W
0
]
J
B
G
G
G
B
J
B
D
OQ B
?
T
B
OD
F
RB
I
TB
W
I
I
E
ID
?A
VB
V
D
D A LD
CD
?H
O
E
I
Z
Ke D
I
OB
P
YI B
K
LG
[ X
U
O
C
O
E
K
O
R
H
D
H
H
D
OD
?
N
K
T
[
G
D
[
B
V
D U CG
V
K
I
AH
ED
KG
IG R
K
?W
I a
Ce G
E
TD
IS
I
OB
I
P
YI B
K
LG
U TI
U
D CG
V
I
^
]
]
G
D
H
D
B
B
V
AH
ED
KG
R
I
E
PQ D
?
O
TD
I
K
?W
I `
Ce G
E
TD
IS
O
Ke
CB
UT
U
H PD
T
K
A
H
H
G
G
K
H
K
[
%
?I
B
A J
B
D
D
H
H
] I
E
C
N
K
T
N
D AH
H
PD
I
KG
F
EQ
TD
IS
O
OQ
AH
H
A PD
^ ED V
E
+
OB
I
P
YI B
LG
H
CG P
RB
?
P
N
T[ H
H
gB
F
B
B
TD
C
E
D
LD
O
R
CI
X
X
]
H
H
I
? D
TD
C
E
P
E
IY D
X
^
N
K
T
T
B
LD
C
A
D
H
PD U H
V
L?
K
]
kAk Skν k = kWk2 B(I − B)ν Wk2 k ≤ kWk kkB(I − B)ν k ≤ c 1 kWk k ≤ kAk k. ≤ 1+ν 1+ν
H
]
^
E
:
;
9
O
CG
H
PJ
FR
KG
D CD V
O
R
K
CI D
C
V
L
F
T
TQ
AH
]
]
]
U
G
B TD
SI
P
C
R
^ E
E
E
G
G
FN
J
Ke D
V
,
^
]
^
N
K
T
E
FR BH
KD
V
I
?
?
?
C
D
B
B
+
^
E
RB
I
OJ
D
X
l
]
]
T
B
LD
C
W
Q
I
E
A
^
G
H
AH
N
O
IH
K
T
D
VJ
NI
TD
S
I
Ce
D
D WI H O
O
YD
^
]
c kAk k. ν +1
H
BM
YI B
K
LG
NB
A
IB
V
O
A
ED
O
R
CI
H
G
^ H
D
H
FH
T
KG
N
VW H
OD
D
CG
Ke
H
E
KD
(B) =
C
BM
3
7
34
6
7
B ∈ Rn×n
I
P
K
2 , πν
WH
Ce G ^
H IP B
OB
1
1
H
IB
E
E
K
NB
E
E
>
EA
O
%
C
J
CB
ν¯ > 0
A
CI B
E
A RD
r CG
F
D
k(I − B)(I + B) k ≤ 2
ν+1
V
ν
B
;
H
H
H FM
T
IH
O
E U H
K
T[ H
H
?
H PD
O
X
X
R
A
D QB
?
Wk
O
YD
:
]
O
J
IH
KG
Ce
I
]
^ E
C
IH
%
A
1
G
E
E
23
23
3
7
5
9
9
P
H IC G
T
CD
E
I
NG
P
\
G
G
gB
F
B
PD B K
OB
C TD
D CG
K
Q
F
D
]
H
E
LD
A
H
D
B
B
D
EB
I
N
eC
I
kAk Skν k ≤
PD
^
D AH
H
H
D
7
6
3
:
KD
I R U H
KG
YI
?
C
BH
N
W
PB
IJ
FO?
NB
OG
TB
H
j
I X B J
NB
H
O
EI
OQ
R
OD
?G
[
O
R
CI
E
I
?
IH
A PD
?
T
H
H
K h
− 12
TD
H
C
EQ
E
K
P
H
CG
Ke
3
W
Q
A
[
T
YB
I
PV D
G
D
KG
R
ID
C
[
H
H
G
]
O
IH
T
G
G
B
E
G
H TD
C
D
E
I
B = W k 2 A k Wk Sk = I − Wk−1 Ak
−1
C
E
S
H NG
C
N
K
P
N
EQ
J
B CB
3
7
3
6
2
]
H
D
IC B
E
?A
[
]
K
C
D
H
A
I
N
K h
\
E
T
C
R
Ak
OQ
OD
W
IA
C
NI
H
j=1
|(Ak )ij | ≤
E
i
i−1 X
D
max Ak
K
i,j
≤ c max(Ak )2ij ≤ ckAk k2 . Uk
? h D
|(Ak )ij | CG
H
FM
P
H
\
I
O
O
FY
S
B
] B
A
N
K
T
Wk = (Lk + Dk )Dk−1 (Dk + LTk ) = Ak + Lk Dk−1 LTk ≥ Ak ,
H
i=j+1 CG
C
W
I
FO?
S
OQ G
IH
I X B J R
GH
AH
H
I
FM
T
H
N
K h
kWk k ≤ c kAk k.
D
G CD P
P
H V DH
P
FK
S
FL
IA
T
ED
H
LB
A
0 ≤ Wk−1 Ak ≤ I.
K
j nk X
1 w Dk
E
max
Wk =
H
H
H
H
O
J KG
I
kAk k ≤ c2 kDk−1 k−1 .
T
[
P
IC G
1 c2
]
kLk k2 ≤ kLk k1 kLk k∞ = PD
C
O
E
K
S
PQ
FK
O
O
KG
O
H
w=
≤ 2 c−1 1 hk
F
ED
ED
R
C
= maxi (Dk )−1 ii
P
D
^
D
P
I
−2 −2 k k k (Dk )ii = (Ak )ii = h−2 k a(φi , φi ) = hk kφi k1 ≥ c1 hk .
CG
O
Ke
H
CB
C
CG
c2 = c0 c−1 1 kDk−1 k
NI
W
I
FO?
J
NB
OG
H
D
G
G
B
H
H
H
U CB
E
CQ
O
E
K
OQ
Ke B
eC G
F
N
K OD
H
gB
F
B
TD
C
E
LB ]
A
H
G
LD
O
R
CI
E
E
I
A EI D
T[ H
T
IB
T
LD
C
\
I X B J
NB
O
Ke B
?
%
D
K
?W
I a
B
J
KB
I
OB
N
K
T
NG
FH C
K
NB
D
D
H
C
R D
H
YB
?
T
^ E
^
O
O
E
] B I
U
U
0 U
X
i
D
PG
H
k
PD
T
K
Ak
Uk
φki
(Wk−1 Ak ) ⊂ [0, 1]
1
P
D
H
D
D
H
H
H
H
BM
P
A DH U AH
C
D
CD
?D
CG
D
PG
P
YI B
K
LG
NG
C
N
O
D
KD
V
PQ
C
R
H
YB
?
UC
B
G
D
B
A EB
V
I
?
TD
C
E
N
K
T
CG
^ E
^
H
X
[
"
$
$
"
"
%
$
"
$
%
! #" "
"
%
&"
"
&
%"
$
"
% H
$
%
"
&"
%!
[
BM
PG
P
IY B
K
A
LG
E
IH
A PD
W
Q
A
IC B
E
!
Ax
UN
B U CG
]
zi+1 = M zi + N b, R
P
$B ]
k = 1, 2, . . . .
K
P
N
T
P
I
W
B
H
]
G
M
I
EA
O
P
H
D
I K
k·kB −1 G
P
G
D
EB
A
l
H
Y
NA
]
I K
I
A
D
H
H
?[
OQ
O
O
E
l
^
G
H
A
O I
IH
T
H
D PJ
I
V
ED
OA
KB N
?
P
O
Y
NA
D
VG
D
H
(B·, ·) U AH
V
BH
J
NG
C
P
I A
QB
O
C
H A CB
V
^ PD V
O
H
TB
IA
L
S
OQ
O
H
D
VG
TD
C
E
H
I
I K
]
AB ¯z = b, z = B ¯z.
N
O
IH
O
PB
G
F U B
@ ?
[
]
R
I
B
L
P
D
H
BM
YI B
LG K
P
CG
^
I
?
C
B
]
]
H
B
F
E
H
G CG
I
E
BH
ZAH
D
CG
E
P
T
LD
Ce
I
KG
+
A D PJ
I
NV
K
T
C
R
D
^ D
[
D
E
C
A B
D
?H
P
LD
O
Ke D
U
D \
PQ
O
P
C
I
D
?H \
O
YD
B
C
R
^ E
O
D
ED I
V
O
AH
G
H
B
H
H
A
H
D
A
D
CB
V
NG
C
E
I
O I
D
IH
Y
NA
A VB
E
Z
O
P
C
R D
IY D
KD
TV
^ E
BH
H
H
] D
O
O
K
T
E
C
P
QF X
^
?
I # KG
3
:
^
]
A G
D
A
S
KQ B
E
I
V
P
C
TD
IS
I
VB
?
B
D
D
?I
ID
Z
EA
O
C
X
]
]
]
$D
P
K
IT
H
G A PB
V
?I
ID
Z
X
23
6
2
3
9
6
1
23
3
3
7
3
6
2
3
7
3
3
6
18
:
9
;
A
B
Q
C
I
A
H
EB
V
]
]
M
A D
B
H
[
K h N
W
IA
C
I GH
G
WH
V
P
NG
C
N
KG
R
I
PQ
H
BH
] F
E
C D AH
H
FM
FM
T
KG
E
FR
KD
]
AH
D
$ F
D
AH
A H
X
]
I
E
E
K
P
H
H
NB
LB
KD
A
B
H
H
H
AH
B
H
TD
W
IB
IA
C
ID
T
YB
?
O
FH C
C
TD
C
PD
C
O
IA TB
SL
OQ
O
Y
NA
D
VG
ED
?
C
I
D
LD
?
]
H
VD
C
R
^ N
NA
O
D
A
PD
V
B
A
CB
A
V[ D H
O
IH
O
E
I
V
W
IB
IJ
KB
I
CD
CG
G ID
E
I
C A
IJ PB
P
S
F
?H
P
F
ED
A
P
T CD D
K
I
I ?H
O
PQ
A
[
A ^ HD
D
CD
I
IA
C
O
IH
D eC G
E
K
A
H CB
V
AH
A
B
H
7
5
3
6
65
;
k = 1, 2, . . . .
U RD
D
^ PD
TD
C
H
H
G
D A PB O
V
G TD
IS
]
H
CB
P
A
D
PI B
A CB
H
D
K
X
V
KG
F
D
]
P
eC
CD
F
W
I
Ke
CB E
G
J
D B
?
T
N
K
T
[
4
35
6
V
zi+1 = zi + αi (zi − zi−1 ) + β i (Azi − b),
C
Y
A−1 b = M A−1 b + N b.
O
z = M z + N b.
E
?
^ E
CG
ID
E
eC B
P
R
H
D VH X G IH
C
FT
X
k·kB
P
IT D
S
OQ
OD
I
OQ B
TB
J
H
G
H
TD
IS
E
K
H
^
]
$
7*
B
VH
O
NG
X
]
EQ
J
B
D
CB
?
V
E
I
FRZ
K
,
0
D
D EG
V
D
AH
H
D
ID
?A
VB
V
N
K
T
EQ
C
C
5;
h
/
&
OB
CG
G
k √ η¯ − 1 kz − z kB ≤ 2 √ kz − z0 kB , η¯ + 1
I
PD
AH
[
I
N
O
IH
N
O
IH
T
"
0
$
) 6
&+
O
YD
E
IE B
H
H
B
TD
H
B
O
x
W
B
H
C
J
B
B
H
B
ZAH
N
K
T
k
H
IA
C
S
E
FE I
X
]
OQ
J
B
GH
W
IA
PQ D
S
P
N
K
T
C
IW
A
IC B
0E
Ce G
F
B
B
YD
S
O
RB
TB
J
&
3,
H
C
E
C H
H
TD
A
IH
O
E H
E
B
√ η−1 k kz − z k ≤ 2 √ kz − z0 k, η+1
C
H NG
C
N
O
KD
V
PQ
H
[
C
FH
G
AH
H
%
C
O
E
K
IE
OQ
R
OD
?
EI
OQ [
P
I
34
k
'
N
K
P
N
z = A−1 b
EQ
H
G
H
CG
I
O
IH
^
T
L
O
IH
O
C
O
-0
(
H
[
(BA) = η¯ η
$
D
D
ZAH
C
R
C
ID CG
O
D
]
G
A PB
F
S
?I
H
G
IB
O
2
(A·, ·)
*
^
NG
H
DH
KG
R
I
O
A Ke B
C
X
R
I X A B
H
H
^ H
K
.
1,
%
*0
,
/,
.
O
¯ b ¯ = Bb BAz = b,
Bx
B
G
[
$
D
C
G
?H
V
E
RB
I
D
OJ
D
L
KB
B
7
0
%$
AB (B −1·, ·)
+
?
?
Ce B
]
N
K
T
?
A TJ
E
H
AH
'
7
CG
D
βi
*
H
D VAH I
V
O
B
N
O
J
AH
H
S
OQ [
H
F
T
TQ
V
P
K
IB
OD
7
αi
D YD
E
C
R
$
W
Q
O
I K
EQ
H
G CG
I
N
BA
$
U
IH
O
PB
]
E
YI D
KD
PG
BT > 0 (BA) Bx
+
M
A
A
CI B
E
CG
H
ID
O
O
K
PD
O
IH
? j IB
G
η1
(
AH
]
B
F
D
LD
OA
[
M
$
C
K
?J
EQ
I
H
ED
O
D
H
TD
N
K
T
O
O
FR
λmax (A) . λmin (A)
H
H
CG OQ
PJ
NI
O
IH
RB
A
KD
V
^ W
Q
CI B
A
H
A z = b.
TD
^
H
H
OJ
OQ
O
X
E
N
K
T
E
TV
AH
2 kAk k. πν
C
G
?H
V
IAH
F
C
F
?A
A
h → 0.
E
D CG
I
P
CG
D
G
^
KG
X
ZAH
ν+1 1 ν kAk Sk k = kWk (I − B)(I + B)ν k, 2
H NG
C
N
K
P
N
PD
TD
H
W
IB
^
] A
CG
J
(A) := kAkkA−1 k =
C
E
S
V
= O(h−2 )
I
EA
D
\
IF
A
F
D
η
OQ
O
I \
\
V f IB
r
OD
B
T
I
kAk Skν k ≤ C
W
[
π h)
IA
H
π h)
ID
O
H
]
J
H
A = AT > 0 A
C
S
Ke
C
O
KD I
D
cos2 ( 12 sin2 ( 12
I
I
FRZ
J
V
T
IP
η=
K
EI
O
G CI G
FC
EI
kI − 2Wk−1 Ak k ≤ 1 kWk k ≤ CkAk k,
A TI D
A CD
I
η=
[
X
B = I − 2Wk−1 Ak
B= (AB) =
%
%
$
"
$
%
! #" "
!$
"
#
#!&"
!
$
" $
$
"
$
%
≤ sup
y6=0
$
?
O
H
IH
Y
K X
I
^
T
D
QF A
WH
A
VH
T
T
L H
U
0 O
B
zold
¯z = AM GM (k, z, f ) { k=0
I P
M
B
H
^]
AH
EQ
H
G CG
I
N
O
IH
ZAH
N
K
T
B
B
B
B
U J
A
LB
A
VJ
B K ] B
H
TJ
A
WH
OD
?
P
NG
eC B
T
NI B
H
B
GH
G FY
C
FT
S
C?
O
\
BM
NG
S
I
F
TD
IH
D LG
O
D
H
X
D Ke B
RB C
E
OJ
D
D D
NI
O
IH
YD
KD
TV
V
FN
J
LD
O
P
C
I
TI B T
W
IA
C
F
O I
,
?
X
X
]
D
D
H
BM
GH
H
BH
H
A G
D
V > D
C?I
EQ
R
Ke
EJ
NG I
I
F
LG
O
Ke B
?
K
S
I
J FM
Ke D
H
OQ
P
CB
I
?
K
I
V
C
I
FKe
E
E
R
W
IB
IJ
J
U G
^ IR B
TB
N
K
T
^ H
^
AH KB
I
N
O
J IAH
I
G R? D
C
^
]
H
H
T
H
H
H
[
A
X
D
X
OQ X
FT
H
KD
eC
Q
FC
LD
E
TQ D
C
E
OQ
P
C
I
TI B
D
LD
C
H
KD
TD
C
E
OQ R
S
A B
X
;
;
V D
BH
\
29
7
GH
D
U RD
C
LD
O
E
E
OQ
P
C
I
:
?H \
H
H
KD
NG
C
P
J
D QB
?
D
CG
B
H
A D
D
M C U H
e
EV D
S?
OQA
D
G
G
WH
V
LD
O
EB
O
O
IH
KG
I R
PQ
N
[
K h
] D
O
Ke
H
CB
PD
T
KG
UA B
U KD
C
V
PQ
P
C
D
B
A
A B
B
A D
NG
C
N
TD
PJ
I
V S
N
O
A PD
F
S
OQ
GH
B
RD
C
O
CQ
H
RG
?
?
B
H
M
^ B S C ?
C
e
EV D
?
H
BH
K D
PB
IA
NV G
C
TB
J
B
^ NG
C
N
H
D
KG
I R
PQ
O
O
P
]
G
H
N
O
IH
KG
R
I
PQ
D
LD
?
R
C
? \ IB
N
K
T
H
\
UO
G
K
A D
D
S
OQA
D
G
G
WH
V
B
LD
O
E
O
N
OB
I
P
K
IH
A KB
B
A VB
N
K
T
OQ
P
C
I
?H \
Oe
RD
A
H
KG
R
I
I
]
]
G
B
D
H X
?W
O H
I a
^ K
U
O
O
D
G
D
^ TD
C
E[ H
OQ
P
CB
I
?
K
I
V
C
I
FKe
ED
?
O
G
G
H
KG
S
I
?I
R
C
I
KB
I
OB
N
K
T
C
FYI
U H
T l
CG
ID
E
TD
IS
D
KG
P
H
CG
O
A D
D
G
H
Ce G
?
O
A
PD
F
S
OQ
GH
B
RD
C
KG
R
I
CD
FM
NG
PB
H
BH
IJ
O
C
P
R
D
[
V HX G IH
OD
]
H
H
G
B
G
J
G
H
H
RB
TB
S
?I
R
C
I
V
K
K I
S
OQ
O
Y
NA
D
D VG
EB
W
I
E
ID
?A
VB
V
S
OQ
O
CG P
C
[
C
]
U
29
7
7
H
P
G
H
B
CG
O
I
Z
Ke D
N
K h
ED
TD
C
E
E
OQ
RD
C
GH
D
LD
O
1E
1
B
X
^
]
AH
H
$
E
FR BH
KD
V
E
O
IH
K
PB
CG
T
H
A
B
J
B
C
P BM
QB
O
T
H
LD
O
I
K
B
[
D
H
TJ
L
F
T
TQ
AH
VJ
TI D
C
E
OQ
GH RD
C
D
LD
O
A
E
O H
IH
Y
K X
I
I
V
E
OQ
]
H
H
H
D
H
BH
E
TD
SI
O
P
FR
KG
P
CG
O
I
Z
B
H
Ke D
P
W
IA
C
S
I
I O
ZH
O
P
CG P
RB
?
P
PD
C
[
X
X
B
H
H
1
9
6
2
G
B
TD
C
E
D
LD
O
RD
C
GH
D
LD
O
E
PD
K
?W
I
W
IA
C
I
N
O
IH
O
KD
H
G
B
B
AH
B K
H
TJ
A
ED
C
PD
LD
O
O
CD
EG
A
TD
C
E
H
H
D
UO
H
TB
IA
SL
OQ
O
Y
NA
VG
TD
C
E
^ H
^
[
^
H
A
H
EQ
V
H
G
M
CG
I
O
IH
ZAH
?
E
O
IH
Y
K X
I
I
NV G
C
D
H
H
KG
V
E
OQ
O
FR
KD
]
H
D
D
LD
O
GH RD
C
D
LD
O
E
LD
O
P
C
I
T
TI B
N
B
H
K h
EI
N
W
IA
C
I1
2
2
6
7
2
TD
C
E
]
N
O
IH
D
KG
R
I
PQ
N
K
T
K
PB
IA
V
E
I TB
T
^
P
N
D
H
C A
D
?H
P
O
O
E
I
W
Q
A
IC B
E
N
K
T
H
A
J
B
D
A D
D
LD
O
Ke D
PJ
I
NV
K
T
ED
$
E
I
FM
T
KG
X
UN
K
^
J
2
6
2
B
B
H
BM
OQ
GH RD
C
D
LD
O
E
IE
N
W
IA
C
I1
C
P
QB
O
PD
C
O
H
TB
IA
L
S
OQ
[
]
H
B
U J
]
H
B
H
H
H
BH
UO
Y
NA H
D
VG
TD
C
E
FR
KG
ED
C
W
IA
C
S
I
OQ
Ke
C
ND
D CG
G EB
H
CG P
RB
?
[ >
]
H
D
D
AH
A
H Ke
CB
P
K
PI B
KG
F
V
E
I
D CG
AH
X
D
H
H
B
X
X
P
E
R H
D OkX H ^ H
FT
I
T
B
LD
O
I
^ H
O
TD
L
PQ
Ce
Q
C
H
YD
E
N
K
CB
P
K
PI B
KG
F
OQ
GH
D
RD
C
LD
O
E
C
R D
YI D
KD
TV
^ E
]
[
C
D
G
H
H
D
J
B
H U AH
V
N
OB
I
TB
N
K
T
TD
C
E
LD
O
GH
D
RD
C
LD
O
E
I
OB
PD
J
Ke D
V
F
I
ED
B
GH
D
U RD
C
LD
O
E
K
?W
ID
LD
O
B
H
B
TD
W
I
IA
C
IW
N
A
CI B
0E
Ce G
F
IH O
ZAH
N
K
T
C NI G
B
TD
TD
H
D C
E
LD
O
G IS
TD
C
E
]
K
I
I
q p 1+ξ k √ −1 η¯ − 1 1 − 1 − ξ 2k 1−ξ k √ < ξk . =q = k 1+ξ k η¯ + 1 ξ +1 1−ξ k
A
Ke D TD P
znew = AM GM (l, zold , fl )
I
V DH
G
V E
I
D
G
DH
D
IC G
A
CD
FH
H
A
A
J VG
e
FN
Ke B
RB
V
PQ
B
d = r(Ak z − f)
¯z = A−1 0 f
Al zl = f l
kI − M k k ≤ kIk + kM k k ≤ kIk + kM kk ≤ (1 + ξ k ) K
(1 + ξ k ) . (1 − ξ k )
kyk kyk ≤ sup ≤ (1 − ξ k )−1 . kyk − kM k yk y6=0 (1 − kM k k)kyk k(I − M k )−1 xk kyk = sup ≤ k kxk y6=0 k(I − M )yk T
KG
?
D
[
N
W
A PB H
B
O
(BA) = kI − M k kk(I − M k )−1 k ≤ ]
B
^
C
?H
P
OQ
Ke D
x
J
] B
D
D
C B
k = 1, 2, . . . .
TB
V
N
K
H
D
TD
C
E
LD
A PJ
O
R
B = N = (I − M k )A−1 ,
D
I
H
CB
P
K
PI B
KG
F
X
O
RD
C
GH
D
LD
CI B
E
^ H
^ E
C
IH
E
C l
y = M k y0 + N x = N x.
T
B
D
AH
H
D
B
W
Y
O
I[
OB
I
Ke D
O
y0 = 0 :
J
O?
V
LD
I
V E O
O
IB
IJ
KB
I
[
PD
T
A¯ y=x
CI
FH
D
WH
BA = (I − M k )
T
H
I
H
N
PD
K
IH O
AH
D
B
y = Bx
x ∈ Rl y
x6=0 D
O
O
YD
AH
?W
I
O
kM k ≤ ξ < 1
k(I − M k )−1 k = sup KG
P
K
T
NV
YD
O
B
CG
D
K h
I
O
FE
k
O
P
CG
O
IH
O
M = Ml
Ke B
H
H
H
N = (I − M )A−1 .
R
A
H
H
PB
O
O
T
H
KG
D
P
]
A−1 = M A−1 + N
I
k A
]
G
H
D
AH
H H
B EG
?
?
Ce
D
D WI H O
O
YD
E
E
K
CB
P
K
IP B
KG
F
V
E
OQ
GH RD
C
D
LD
O
E
PD
C
X
A D
%
^
H
H
H
H
H
A D
D
G
D
H
G
H
H
D
UO
H
TB
IA
L
S
OQ
O
Y
NA
B VG
TD
C
E
CG
ID
E
TD
SI
Ce G
?
O
Ke
CB
PD
T
K
C
R
K
P
%
$
"
$
%
! #" "
!$
"
#
#!&"
!
$
$
" O
T
H
KG
V
D
$
$
%
"
ID CG
O
Ke D
PJ
I
V F
K
G
I >
b
zl
H
KG
kSh k < 1 U
K
IH
* U
C
I
FKe
E
]
A D
]
D
P
CG
OB
A
CG
A
TV D
V
Vi v∈V C
D
H
D
N
O
A B
I
B
A
J
B
D
ED
$
E
I
UA
V
B
H
D U
NV
K
T
S
P
N
W
I
EA
O
I
A BH
B
D
NG
C
D
IG
D
LD
C
H
D KG
D
GH
D
G
\
C
X
B
D
U G
^
V
U
D
A
H
B
D
H
J
H
H
A U B
Z
Oe
E
EQ
H
G CG
I
BM NG
C
ZAH
O
Y
[
K X
I
I
V
T
L
P
CG
OB
A
CG
A
TV D
Ke D
X
]
V
] S
P
H
B
CG
OB
A
CG
[
X
l
C
H
YD
E
H
H
B
J OB
A
]
H
O
IH
ZAH
TD
C
E
OQ [
U
VA
PQ
B
B
D
D
LD
X
B
O
H
IH
YD
KJ
A
]
C ] H ]
M
F
D
LD
K
N
K
T
G
X
E U H
H
B
O B
NG
C
A
FH
W
A I DH
V
E
H
G CG
$
I[
I
FM
T
KG
OB
K
B
UA D
TV D
V
O
O
IH
O
A PB
F
E
H
G
M
CG
I
O
IH
ZAH
?
N
[
H
W
IA
C
I
B
H
CG
I
PG
U OD
0
Ke
WH
B
Z f B
^
P
H
H
O
O
P
DH
H
CG
O
IH T
O
eC
Q
C
YD
E
C
FH
CG
H
OQ
P
C
I
T
X
[
GH
B
B
U GH
S
P
O
W
I
H
B
NG
Ce
N
O
KD
V
PQ
C
H
YD
E
I
LB
]
Vi ⊂ V,
K
B
]
O
M
]
CG
H
ID
OA
E
H
TD
ED
M
IA
C
IH
I
BM
P
O
I
G
H
I
T
L H
D
H
D
H
$
FM
E
NA
V
O
Ke
CB
J
D N X k
O
P
CG
A OB
CG
A
VT D
F
V
E
FE
]
H
D
F
P
CG
A OB
CG
A
CV B
Y
K
A TB
O
I
V
B
U H
YD
KJ
A
D
OB
TB
J
Ce G
F
]
A z = b,
T
A
D
X
N
O
IH
ZAH
PQ
Vi V Pi , Q i : V → V i
A PD
H
D
M
A CB
V
W
I
? DH
FM
T
H
I
OB
PD
A
?
N
O
IH
U
G
C
H
O
C
KD
IT
] D
O
Ke
H
O
I
[
A
H
H
G
H
A PB
F
S
?I
I X A B R
L
KB
S
OQ [
O
K
IF
E
H
G CG
I
A FM
D
CD
?H
O
E
IC D
EG
G
B
O U H
H
D
TD
C
E
LD
O
P
C
I
T
IT B
K
?
I
$
] B
]
] G
I
D
] S
P
B
CG
OB
A
CG
A
TV D
V
^
H
D
D
H
B
H
B
B
H
D
O
W
I
I
B
A ?AH
?
PD
TD
C
E
H
PD
OG
P
Y
K
W
A
[
@
?
P i
B
TD
C
E
LD
O
U
A
O
H
IH
Y
KD
TD
V
]
A
N
O
IH
T
H
D PJ
I
V
CG
I
BH
A
D
H
UP
C
I
IT B T
I
I
C
S
B
A E? B
P
NG
C
L
CG
D ID
T
C
^
U
W
IA
C
I
B
H
A
N
W
I
? DH
V
[
U
G
B
A
O
O
K
D
H
PD
KG
F
D
B
KG
R
ID
LD
O
Ke B
C
?H
V
N
K
T
S
D
B
B
WH O?
O
X
]
H
T
I
T
D
H
B
H
D
W
Q
A
CI B
E
N
K
T
?
D WH O
TD
P
PQ
O
O
NG
C
I
G
U PB
KG
F
D
AH
H
VB
?
?
NG
C
FH
J
Ke D
V
IT D
C
E[ H
OQ
RD
GH
D
LD
O
E[
OQ
P
I
C
A
H
C
D
O
N
OB
P
YI B
K
LG
IH
U
I
DH
ZAH
O
RD
C
CG
L
H
J
KB
I
O
Ke
CB
P
K
I
U
X
^
I
I
V
A
H
D
KG
I
H
^ BH
FR
KG
P
?
Ke D
C
E
OQ
H
RD
C
C
FT
X
O l
] B
TD
C
E
CG
ID
E
G
CD
C
I
PB
IJ
θ
D
A CB
V
N
O
IH
H
A
B
KG
C
I
A ? DH
V
Y
A
D
QD A
I
I
K X
A
H
D
V
LD
PD
O
N
O
IH
FR
K
CB
V
E
K
l
D
G
?I
H
E
P
K
KB
B
V = Rn
K
T
D AH
V
H
Ai : Vi → V i
?
?
eC B
A
P
CG
OB
A
AH IH
T
V
US
H
OB
I
R
CB
I
O
Ke
IA PB
A
Z
G
CD
C
I
PB
H IJ
O
Vi ,
VJ
O ^ I
P
D CG
A
D
H
PD
T
G
X
H
]
]
H
v ∈ V Pl i=0 vi , vi ∈ Vi
FH
H
B
A
H
[
X
H
B
O
X
P
YI B
K
LG
i=0
l X
T
H
v i ∈ Vi Ke
A
D
TV D
V
O
C
I
I
C
H
D
Ke
P
CV
FH
G
V
NG
Ce
N
O
KD
X
LB
ZD
V=
C
f
ED
C
CB
P
K
IH
RB
OJ
DH
[
J
B
+
BAh
IC B
EG
G
B
H
B
KG
G
H PB
C
H TD
C
[
FO?
A
IB
D
A
D
B
^ ED
E
?I
C
R
^ E
C
IH
ν → ∞.
KG
P
A
D
(Qi v, vi ) = (v, vi ),
IC G
O
YD
E
EQ
Y
I
] S
H
IAH
F
PI B
KG
F
E
AH
B
O
V
PQ
E
H
] D
O
Ke
%
}
[
D
l
T
IT B
D
H
X
N
KD
PJ
X
E
C
TD
H
v =
H h
A
U
A
P
CG
KD
S
CB
H
D
OQ
X
C
E
OB G
A PD
E
FC
A
SH
LD
S
N
O
CI G
F
S A PD
D M
A
EB
Ma
I
H
A CB
V
D
[
OB
A
CG
B
A
D
AH
TD
IS
D
H
D
LD
O
P
C
I
T
TI B
eC G
D
K
K
K
KB
θ(ν) → 0
D
]
Ai S
]
I
I
H
D
D
E? B
C
D
P H
C
I
O
LB ^ H
Z
ED
C
N
VB
O
E
TD
IS
I
B
b z V
OQ
Ke D
LD
O
RD
C
GH
C
C
A
A
V
C
R
T
IT B
F
B
A
ν
PJ
I
V
N
−1 ν Ma = I − (θ pA−1 2h r + (I − Sh )Ah )Ah ,
P
ν→∞ ]
O
IH
K
S
[
EI B
C
TV D
K
I >
A
D ED
C
] D
VB
?
?
C
J
FM
O
H
CG
A
θ
PB
CG
F
P
T
B
H
P
]
G
BH
Ke D
V
PI
FR
PJ
IH
C
V
z0 = z zi+1 = zi − Wk−1 (Ak zi − f) i = 0, . . . , ν − 1 ν
CD
I
I
AH
, W
IA
O
VB
ED
NG
C
^
A
H KG
P
O
TV D OQ
RD
C
C
znew
R
T
J
[
H
B
O
E
Q
C
H
EB
A
I
PV D
C
O
H
TB
IA
L
S
OQ
O
CG
H
EB
A
zν
K
V
B
A
TD
I
PD
A
B
VH
OQ
OD
W
B
B IG
?
K
AH
H
X
-
IC D
]
W
C
IW
W
I
I
B
H
B
V[
W
IA
C
S
I
?I
C
Ce B
Ce
Q
P
[
H
A
D
?AH
?
PD
IE
OQ
PD
D
G
H TD
C
E
S
OQ
OD
W
CG
ID
O
I
Z
I
TD
H
IA
C
S
IC B
E
D
H
OG
N
K
CB
C
R
I
E
IE D
B
Y
NA
V
¯e = AM GM (k − 1, e, d)
C
E
OQ
RD
I
I
BM
R T C
^ NG
I
eC
T
I HX
D
P
C
H
H
B
A
IA
A
X
θ=1
I
H
C
GH
D
P
YI B
E
F
O
FT
C
B
E
I
U
^ E
OQ
LD
[
B LG K
W
I
TD
H
D
?
H
T
I
U
RD
C
O
E
OQ
P
C
I
A
C
E
H f
D
OJ
C
I
U
NG
C
N
H
T
TI B
CI B
0E
LD
O
P
V
¯z = zν − θp ¯e
K
P
N
D
D
H
C
I
e=0
O
C
R
D
T
L
θ
LD
O
^
O
T
P
I
Sh
P
CB
I
?
K
I
V
zold
(Pi v, vi )A = (v, vi )A ∀ v ∈ V, vi ∈ Vi , (Ai zi , vi ) = (Azi , vi ) ∀ zi , vi ∈ Vi .
(Ai Pi v, vi ) = (A Pi v, vi ) = (Pi v, vi )A = (v, vi )A = (A v, vi ) = (Qi A v, vi ).
U
H
B
H
[
G
M
TD
IS
O
H
H
D
H
IH
FRZ
FK
?
C
T
P
O
Ke
J CB
D NkX
O
OB
I
P
YI B
LG K
KG
R
I
O
IH
R
%
%
$
"
$
%
! #" "
!$
"
#
#!&"
!
$
" $
$
"
$
%
]
EQ
^
AH
A H
G
O D
ri
Qi = ri+1 . . . rl−1 rl , Ql = I.
M
Qi BH
X H
A
H
]
pl pl−1 . . . pi+1 vi Ri−1 Ri
(Qi v, vi )Vi = (v, pl pl−1 . . . pi+1 vi )V = (p∗i+1 . . . p∗l−1 p∗l v, vi )Vi = = (ri+1 . . . rl−1 rl v, vi )Vi .
H
OB
U
K
P D
H
KG
U
D
ED
O
RD
C
GH
ED
D
?B
P
CG
A OB
A CG
B U
0V
U
H
A D
H
H
D
P
CG
A OB
CG
S
V
OQ
C
O
E
K
O
R
OD
S?
OQ
O
H
]
G
H
H
I
? D
ED
TD
C
E
Ce B
TB
VP D
C
FT
X
X
^
D
UA D
TV D
VB
O
W
I
I
A ?AH
?
H
D
TD
C
E
C
^
V
J
B
X
U0
T
L
?
H
B
?
?
B
NG
C
TD
IS
O
H
F
ED
C
D
]
C A B
D
?H
PB
C
O
H
OD
EV D
NB
V
W
Q
A
IC B
E
N
K h
E
FR BH
KD
B
B
J
H
0
I
A ED
D
CD
?
P
M
H
[
^
OQA
E
T
B
LD P
CG
OB
A
CG
^ D A C
D
TV D A
[
S
V
OQ
Ke B
OD
LD
D
H
C
?H
P
OQ G
I X B J
OQ
O
CA
D
R
A
X
S
C
?H >
] B
?
ID
Z
NB
I
I
T
0H
O
CG
H
H PJ
I
0O
A
H
O
M
IH
ZAH
?
IH O
Y
K X
I
I
V DH
O
CG
PJ
D
A B
H
KJ
E
EJ
P
I
E
YI D
KD
D
LD V
C
N
K h
]
e G
H
TJ U
O
G
H
D
TD
F
E
FE
P
O
IH
YD
AH
IY D
KD
TV
A
B
D
H
BH
UW
IA
C
B
H
NI G
C
ZAH
O
Y
H
]
X
J
H
1
C
P
2
6
2
6
2
BM
QB
O
O
IH
O
PB
A
*
1
2
6
3
5
B
D
P
CG
OB
A
CG
A
VT D
V
O
W
I
I
G
B
H
B
H
H
I
? D
TD
C
E
K
?
P
L
G
G
B
D
H
BM
TB
A E
D
V
VB
TD
C
A
A
D
IC D
EG
A
E
E
I
V
LD
O
Ke B
P
IA
IC
6
6
7
5
63
6
3
27
6
4
]
]
H
^
H
?AH
P
H
? > RB
^
T
29
6
;
L
CG A
]
7
2
5
7
35
3
D
?
PD
TD
C
E
^ B S
K X
I
I
V
E
H
CG
G
ID
C
R
^ E
n PnR R = i=1 span{¯ ei }
^ E
FT
N
EQ G
O
J
H
Ce
]
A
D
ED
E
I
O
IH
ZAH
$
O
IH
O
^
?
N
P
0O
H TD
C
E
E
OQ
OD
^ ED
n
E
FR
P
P
Ce B
E
[
T
L
B
H
YD
K
P
E
H
FM
T
H
KG
H
A PB
F
C
D
H
X
^
;
]
]
AH
H
1
eC
N
O
E
I
V
E
OQ
FE
3
67
A
A
H
J
B
NG
C
N
K
PB
CG
T
V
EQ
H
G CG
I
N
O
IH
Z
;
A J
B
D
B
^
I
KG
I
H
^ ED
E
?I
+
]
]
X
D
U AH
N
K
T
C
Ri = RiT > 0
KD
I
C
H
OD
I
I
W
^
i
OB
J
O
E
H
H
I
FO?
Ui
V
NI
K
V
E
X
\
^
]
#
$
P
CG
OB
A
H
DH X
D
D
?
O
C
C
A
A PD
C
I
A ? DH
O
$
FT
H
H
Ri
O
IH
K
H
IH
B
P
CG
A OB
S
OQ
C
O
E
K
D
Ce G
F
CG
G
IB
YI D
KD
V
I
KG
D CG
A
^
O
B T[ H
RD
H
D
ID
Z
?
O
IH
Y
O
C
R
NA
P
C
K
PD
FT
?I
ID
Z
A = AT > 0
AH
?
V
T
VD A
Y
KD CB
?
D
U
O
R
H
OD
V = R l Vi = R i
D
A TD
P
A
Vi ⊂ V CG
V
B
B U J
S
P
CG
OB
A
Ri = A−1 = a−1 i ii
A
A Ce B
P
H
^
B
D CG
A
B TV D
V
A K X
I
I
^
Ce
YI D
KD
AH
v
VD A
CB
IC B
O
IH
YD
K >
?
S
OB
CG
OB P
A
CG
J TB
[
B TV D
H
O
X
G
H
Ce B V
?
EI
FT
A
VT
#
H
H
H
P
H
D
CG
O
A PB
NG
C
N
K
P
N
I
E
P
CI G
T
KG
E
EQ
NA
5
1
29
2
6
3
135
1
29
2
6
6
6
3
23
6
7
]
;
3
7
6
3
3
7
9
3
7
6
3
6
29
2
6
6
;
]
3
3
5
67
A D
%
E
FR
3
23
6
7
3
3
6
29
2
6
3
135
5
6
5
5
BH
'
9
$
%
$
"
$
%
! #" "
!$
"
#
#!&"
!
$
" KD
V
I
$
$
%
"
ID CG
O
Ke D
PJ
I
V F
K
G
I >
6
B
VD
V[ D H
G
A
S
V
OQ
H
]
]
H
PD
C
O
\
Ce G
F
D U B
O
O
YD
E
C
R D
YI D
V
O
N
O
T
L
?I
ID
Z
N
K
T
H
Vi
J
H
G
B
C
O
E
H
D
]
W
I
I \
AH
KD
TV
IH
O
A
D PB
F
C
R
^ B
?
C
NG
O
IH
O
A PB
F
E
]
V
I KG
A e = r0 .
?I
R
A
H
EG
E
K
H
H EG
O
A
PD
F
B VD
?
^ E
^
O
T l
] D
O
RD
Ce
FR
I
KG
C
H
A
IC D
V = Rn
AH
H
G V[ D H
?I
H
X
U
O
R
OD
?[
O
IH
Y
^
J
B
G
D
eC B
?
B
C
H
YD
E
EG
G
B
Ri
R
OD
I
OB
FT
ˆei ,
eC B
?
R
OD
I
i=0
RB
^
NG
OB
0?
H
H
vi ∈ Vi OJ
X
G
D
KB
?F
C
O
E
K
N
O
IH
Y
O
YD
B
H
U0 ⊂ · · · ⊂ U l
D
G
H
%
B
M
A K X
I
I
C
E
S
O
EIA
C
S
?
?
Ce B
ZAH
D
H
ei = Pi e.
D
D
H
V
G
e
T
U
O
O
IH
V
NB
X
B
B EQ D
?
OJ
>
H
O
Ke
C
>
vi i aij (¯ ei , e¯j ) = aij δji
]
^
O
O
P
H CG
C
?H
T
O
I
pi i
Ke
H
CB
PD
P
CD
D
G
[
H
ZAH
?
A
D
IG
B
H
?
O
IH
Y
K X
I
O
KD
V
e¯i
T
H
OQ
OD
[
O
IH
Y
^
T
LD
+
]
?
O
IH
Y
ˆei = Ri Qi r0 ,
K
]
z1 = z0 − B(A z0 − b)
B
H
H
A K X
I
I
H
A K X
I
I
V
O
A
Ai zi = b i ,
U H
]
Ri Qi .
W
IA
C
I
OQ
O
K
X
]
?
Ke
CB
P
X
0
A
IFT
C
Ai Pi = Qi A.
KG
CG
ID
P
K
TI
PB
V
P
PD
KG
F
l X
NG
BH
z1 = z 0 − D
V DH
PD
^ B S O
P
CG
A OB
K
IP B
KG
D
AH
H
Ai
E
i=0
l X AH
D
CG
A
F
0V
I
V
C
I
z0 e = z0 − z
T
B= H
B
B VT D
V
X
H
A i e i = Q i r0 ,
Y
V
?
?D
G
OB
H
H
D
T
L
A−1 i
HX
[
I
VAH
T
O
?
C
R
e
D
H
H
X
I P
P
E
ID
Z
?
vi
F
A
E
K
V
Ce
Q
C
H
YD
D
TD
IS
zi = P i z bi = Q i b
?
[
H
CB
P
K
PI B
X
E
C
R
ei
H
D
H
D KG
F
$
H TD
C
E
ˆei
A PD
V
O
O
P
CG
T
G
D
AH
H
G
z = z0 −e
E
H
[
VH f
V
Ri Ri
K
H
FM
T
z r 0 = A z0 − b
v
B
Qi v = vi e¯i , (A¯ ei , e¯j ) =
Ai = aii .
Vi = Ri ∼ Ui ⊂ Ul ∼ R = V,
l X i=0
6
v = 7
P i
(Ri−1 vi , vi ) ≤ K0 (Av, v). i,j=0
l X B
B
(vi , vj )A ≤ K1 l X i=0
1/2 !1/2 l X (Rj−1 vj , vj ) (Ri−1 vi , vi ) j=0
I
O?
I
A PD
U
E
A
H
BH
[
B
D
^
(BA) Ti
P
B
CG
O
R
I
K
IH >
]
B
U
J
H
H
BM
Y
E H
ED
K
FL
C
P
QB
O
F
^ A
A
D
H
S
OQ
Ke B
OD
LD
CA
N
K
T
E
V f B
I
T
LD
?
B
]
]
A
B
B
GH
O
G
H
IH
O
KD
V
PQ
C
A FH
OI B C
L
T
L
? D
B
T l O
γ = 1
O
H
H A PB
O
NB
P
J
D QB
?
T
eC
FR
I
]
]
^ N
W
FI
C
P
]
C
D
OB
CG
OD
[
A D ?[
D
CD
D
UO
H
H
DH
A PB
O
O
R
QD
C
R
^ E
C
IH
EB
X
%
S
OQ
O
Y
NA H
D
VG
B
]
∀ u1 ∈ U 1 , u2 ∈ U 2
I
P
CG
A OB
CG
A
VT D
A
D KD
P
G
IB
O
Z j ID
U AH
H
VH
H TD
C
E
CG
ID
E
H
D
S
P
B
CG
OB
A
CG
A
TV D
VB
O
W
I
I
?AH
A
D
D ?[
O
Ke
K
A KB
B
]
]
A G
B
A B
B
K H
H
TJ
P
O
AH
CD
EG
U
D
H
B
B
KG
I
A
P i
Z j ID
P
CG
^ W
^
* AH
B
H
H
B
S
OQ
P
C
IB
TB V
O
KG
R
I
ED
C
P
B
TD
C
E
D
LD
C
N
W
IJ
KB
I
NB
^
H
H
H
H
A D
D
V
PQ
P
D
CG
OB
A
CG
V
H
PD
T
K
I
P?
I K
D
D
I
P
CG
A OB
CG
A
VT D
NV
K
T
C
R
NA ^ C
D
PD
H
D
UO
TB
IA
L
S
OQ
O
Y
NA
D
VG
TD
C
E
P
Ke
CB
P
K
PI B
X
PD I
CA
HX
D KG
F
AH
H
VB
?
?
FM
^ C
O
O
UO
H
H
DH
H
G
A PB
O
O
O
K
F
I
D
H
KD
^
T
L H
U
Ke
0 IL
U
D
PB
PV
EQ
LB
KG
NV
K
W
FH
X B J
D
B
P i @
T
A
AH
D
V
O
YD
E
J
EI B
P
CG
OB
CG
A
D
A OB
A CG
YD
EJ
P
CG
G
B
TD
IS
P
CG
Ke
H
V
PD
TD
C
H
D
O X
T
]
^
^
U
H bc
A
D
D
I
O
R
G
D
QD
L
PD
$
C
PB
O
D IH
E
I
]
#
d
c
b
V d
O
I
X
C
T
I
U
A J
B
D
B
DH
D
H
B
H
H
B
PJ
O
C
E
I
Ke
CB
P
K
PI B
KG
F
D
AH
H
ED V
E
?I
C
OQ
O
A
CG
X
X
A
G B
O
I
E
OG
I
ND
EB
C
ZB
E
E
OQ
R
KJ
IB
E
I
FM
P
CG
US
I
V
C
?
A
D
G
D
D
B
Y ^ H ?
]
[
X
A
H
B
B
H
H
D
D
P
D
UC
H
CD
G
CG
OB
A
CG
A
TV D
S
V
OQ
C
O
E
K
O
R
OD
?
N
K
T
W
^ P
U
B P i
P
H CG
O
P
]
H
H
A
H
G
B
UA B
H
OD
LD
O
O
K
F
I
?
H
PD
A
NV G
C
N
K
P
N
^ PD
TD
C
SE
OQ
GH RD
C
D
LD
O
SE
OQ
P
C
I
H
P
H
D
CG
O
PB
A
DH
#
EI
O
H
IH
Y
A
]
^
H
H
W
A
G
B
B
B
U
A
A
D
H
H
H
IT B T
I
I
C
P
E
EQ
N
O
E
I
]
A B K H
H
TJ
P
NB
O
O
AH
CD
@
?
P i @
TD
C
E
ED
O
P
C
I
I T
TB
EG
UT
H
G A PB
V
ED
C
I
A
]
C
H
A K X
I
I
V
E
OQ
Ke B
^ ED V
O
FE
A
CG
O
EI
D OQ X k
FT
E
TI D O
H
RB
E
PQ
O
\
FK
G
O
D
D
O
Ke B
EA
P
K
I
^
E
?
H
H
H
N
K
H
CB
P
K
PI B
KG
F
D
AH
H
VH
CG P
RB
?
PB
TD
C
]
C
A
A
J
B
D
D
H
AH
D
H
E
C
D
B
AH
D
H
NG
C
N
K
P
N
EQ
C
ED
V
C
R D
IH
X
X
C l
^ E
]
@
A D
H
G
ED
K
?W
I
E
IT D O
W
IB
IA
C
ID
CG
F
V
TD
C
H
H
H
E
O
C
O
ED
LD
O
GH RD
C
KB
P
I
?
C
FT
D
X
]
P
;
;
LD
O
E
3
234
7
[
^
^
]
X
J
H
D AH
H
D
A
D
H
H
D
P
E A
A H
IB
V
O
O
T
P
I
V
I
O
YD
KG
O
EQ
C
P
H
CG
Ke
CB
B
? h D
^
]
H
H
B
H
W
I
I
A ?AH
D
D ?[
O
Ke
K
A KB
B
TV D
H
B
F
W
A
IC B
E
O
EQ
H
G CG
I
CG
IB
R
D
A
VH
O
IH
YD
O
FE
?
?D
C
^
G
OB
B
PB
H
+
D
ED
C
R
Ce
T
P
A
ID
L?
K
e
VH
I
VJ
O
Ke B
E
[
34
B
H
G
WH
A
IC B
E
EQ
H
G CG
I
N
O
IH
ZAH
N
K
T
W
IA
C
I
[
[
;
H
U D A
eC
Q
C
YD
E
\
X
]
A
D
J
IB S
C
^
]
AH
A H O
H
IH
O
E
I
V
N
O
IH
K
PB
CG
T
VJ
CI
H FH
T
KG
? j B
]
D
U
H
B ?H
P
O
C
FH
CG P
T
S
N
O
IH
RB
D
D OJ
S
PQ D
O
I
?
I
OB
P
YI B
K
LG
N
K
T
[
>
X
X
[
(Ti v, v)A
]
1/2
$
LD V
C
Z j ID
P
CG
]
I
N
K
T
F
D CG
A
D
G ?H
O
H
RB
?
P
ED
J
B
D
CB
?
T
N
K
H
FT
X
U2 ⊂ U
D
CB
A
N
K h
O
H
PB
A
H
H
VT D
D
P
CG
H TD
C
E
E
OQ
GH RD
C
D
LD
O
E
E
OQ
P
C
I
C
D
J
U1 ⊂ U
WH
H
D
O
S
OQ
O
l
P
P
T
IT B
T
A
^ B S
Ke D
V
j=0
D
V
]
]
N
O
IH
YD
KJ
O?
WH
TV D
V
N
G
E
N
I
+ ]
PD
C
l X
J
W
K
G
B
V F
T
H
NG
C
J
FH
H
C?
GH
!1/2
I
A
IC B
E
IE
N
O
B
F
I A
J
C
I
G
U H
V
PD
γ < 1 arccos γ
eC B
H
AH
D
G
IH
K
T
V
e
FN
J
Ke D
V
I
V
B
D U X F
B
C
O
H
]
K
CB
PD
C
O
Ke D
G
|(u1 , u2 )U | ≤ γku1 kU ku2 kU
PD
^
]
CG
IB
R
O
W
I
KG
H
TB
IA
L
S
OQ
O
Y
NA
D
B
VG
TD
C
H
P
K
PI B
H
TB
IA
V
I
(Ti y, y)A
T
I
D I[ R
X
E
CG
ID
E
KG
F
γ ∈ [0, 1)
H
?AH
A
D
D
B
H
H
AH
D
WH O?
O
K
T
V
Ce
Q
C
H
G TD
SI
X
L
i=0
KG
H
X
67
D
H
?[
O
Ke
B
B
A KB K
V
H
CG
D
G
B
A ID
?
l
γ=0
1
1
5
I
O?
D
WH
N
K
T
H TD
C
E
N
K h
H
YD
E
E
K
CB
P
X
O?
WH
D
λmax (BA) λmin (BA)
C
FT
Ti y ⊂ V i , ]
H
ED
C
O
FE
A
CG
O
]
W
Q
A
IC B
E
K
IP B
KG
F
D
K1
D
3
9
7
EI
I
] B
TD
S
TD
N
D
H AH E
C
S
VB
P
CG
CG
H
ID
O
O
K
PD
(BA) :=
C
7
6
3
34
;
1
3
7
;
NB
FM
T
OB
KG H
A
D
CG
A
NG
H
TV D
AH
Vj
?
?
?
5
3
7
68
3
+
vi , vi ∈ Vi ,
7
5
6
;
18
3
3
6
9
Ri
29
2
6
6
6
A
i,j=0
(Ti y, Tj v)A ≤ K1 l X
23
6
2
3
2
6
3
23
6
7
3
3
C
N
K
KG
Ri = (I − Siν )A−1 i .
ν l X
]
K0 9
3
67
7
6
5
6
V
Si = I − wDi−1 Ai
]
H
E A
A H
V f IB
O
]
Ce G
F
N
OB
I
P
YI
B
B
H
BM
3
3
7
3
6
2
A U PD
F
ED
U B
O
W
IA
C
S
I
I
P
YI B
LG K
W
A
IC B
0E
[
:
U
H
H
H
H
D
M
K
7
5
B
LG
NG
C
N
K
P
N
B
TD
C
E
LD
O
RD
C
GH
D
LD
O
E
ED
C
O
E
K
E
OQ
Ke
J CB
N X k
l
5
67
2
3
9
3
27
7
2
5
3
:
7
5
3
3
6
9
;
Ti = R i Q i A = R i A i P i K1
:
:
P
N
BA θ=1
5
29
8
B
Ai
]
3
27
7
2
5
3
3
3
7
3
v ∈ V Vi
Si
i
6
2
(I−Siν )A−1 i
Si
y, v ∈ V
≤ K 0 K1
Vi Qi
(Ti y, y)A = (ATi y, y) = (ARi Qi Ay, y) = (Ri Qi Ay, Ay) = = (Ri Qi Ay, Qi Ay) = (Ri−1 Ri Qi Ay, Ri Qi Ay) = (Ri−1 Ti y, Ti y).
%
%
$
"
$
%
! #" "
!$
"
#
#!&"
!
$
" $
$
"
$
%
U
0
?H
B
B
K1 ≤ ρ(G), K1 hl
H ]
B
G
E
FR BH
]
I
GH
D
A U H
E
I
V
hm kvk kA h−1 m kvm k ∀ vk ∈ Vk , vm ∈ Vm , k ≤ m. hk
X
G
]
E
P
^
U
D
H
O
O
E
I
W
I
FO?
S
OQ
C
O
E
K
O
R
H
D
H
B
H
H
D
OD
?
H
PD
C
O
W
I
[
\
\
I \
?
P
CG
OB
A
CG
^
]
W
Q
A
IC B
E
]
D
G
H
H
B
D
UA
V
P
F
O?
D
WH
D
FM
FM
P
H CG
C
CD P
C
R
K
P
O?
WH
I
J
B
D
F
T
Y
E H
EB
EJ
IA
ED
NIA
TD
LB
K
D
H
AH
O
M
IH
K
T
V
I
A
\
]
5
67
2
3
9
H NG
C
N
O
KD
V
7
5
3
7
2
56
6
2
29
3
3
34
7
]
]
O
O
D
G
D
H
KD
V
PQ
P
9
;
;
^
P
H h
]
D
[
CG
G
A
H
G
D
D
PB
V
EQA
O
O
A Ke B
C
?H
V
I
W
Q
A
IC B
E
[
Ce G
F
]
G
A
H
G
TB
IA
F
W
A
B
A OQ
E
FM
O
R
CI
E
E
I
E
PB
CI B
[
I
CG
D
G
A Ke B
C
?H
0V
C
OB
T
L
H
UA
NV
K
T
O
Ke
C
D
D
H
CI G
[
O
]
^
^ P
E
I
V
O
A
A H
B
^ NB
P
K
TI
D
@
?
^
N
D
H
G
H
J
B
D
B
H
U OB
?
T
E
E
K
NB
O
Ke
CB
LD
ED
P V
NB
FM
T
KG
NG
C
I
TB
OD
V
EB
G
E
E
I
H
G
f
X
K h
UC
F
]
I
CD
EI
I
NG
PB
IJ
CG
H
ID
OA
EJ
]
A
J
B
D
A D
^
IE B
C
OB
CG
OD
?G
I
S
OQ
Ke D
PJ
I
NV
K
T
U OD
?J
I
I
W
I
FO?
PQ
CD
C
NG
PB
H IJ
O
CQ
OB
CG
OD
?
[
O H
c 0 , c1 , c2
H
H
J
A
A
KD
V
I
FN
O
I
I
ED
J
B
D
eC B
?
T
V
hm k∇uk h−1 m kvk. hk
AH
W
B
P i
D
^
]
]
]
U RD
C
GH
B
KG
R
I
CD
I
]
E
H
OB
I
A PD
EI
FE
G
PQ
S
A
,
\
X
]
]
c3 kvk k2A ≤ (Rk−1 vk , vk )
] B
R
Ce
P
CG
j
O
CB
BH NG
C
FR
KD
V
H
CG
O
O
R
I
]
−1 −2 2 2 c1 h−2 k kvk k ≤ (Rk vk , vk ) ≤ c2 hk kvk k ,
OB
J
B
D
l ?
H
hm −1 −1 (R vk , vk )1/2 (Rm vm , vm )1/2 , hk k
T
A
H
O
O
]
D
CG
T
D
G
B
EB
O
O
I
O?
WH
RD
C
I
PB
H
D
IJ
O
C
R
EI
D
CD
^
N
O
]
Rk = V k
O
WI H
[
.
D
D
]
≤ c( 12 )i−j G O
D
DH
H
G
PB
C
O
O
CG OB
K
OD
F
I
OB D
B
D
?G
?
J
I
T
D IE B
O
H
)
I
hi hj
hj hi
CD
r
YD
^
O
Ke
CB
CG P
A OB
H PD
T
K
7
0
%$
'
7
$
7*
I
Uk
NB
H
H
hi , hj
E
$
E
D CG
A
TV D
NV
K
T
,
0
0
B
]
OQ
PB
A
H f
vk ∈ Vk , k = 0, . . . , l vk
NG
W
Q
A
H
(s
ρ(G) ≤ kGk∞
IC B
Ti Tj |ρ(G)| E
γij = c min
r
PB
[
H
UO
Y
NA
D
VG
B
$
.,
"
) 6
+
?I
C
IH
FR
KD
Ke D
PJ
I
0 ≤ k ≤ m ≤ l
IJ
I
W
NI
FK
P
CG
H TD
C
E
*
*
$
Vm
H
^
A OB L
IC
[
B
H
RB
?
P
ED
D
Ce G
E
)
7
^
EI B
B
|(∇u, ∇v)| ≤ c
O
G
[
P
CG
D
PG
K
LG
E
A
H TD
C
G
H
TD
SI
E
'
1$
3$
4
0
Vk
E
F
TB
IA
OQ
,
GH
CD
Ce
PB
D
B
YB
?
T
r
B
A
Ke B
C
?H
J
U
IC D
EG
IJ
C
G
B
G
G
B OD
F
(vk , vm )A ≤
O?
D
U H
]
E
E
OQ
RD
C
GH
D
LD
O
E
E
OQ
P
C
I
$
[
(vk , vm )A ≤ c r
D
$ F
EA
V
B
X H
TF
H
Ke
X
T
3
&
]
H
V
N
H
γij xi yj = (Gx, y) ≤ kGkkxkkyk = ρ(G)kxkkyk =
WH
L2 L2
A
k
D
O
D
A
B
TJ
P
?
^ H ?
H
O
H
CB
P
K
PI B
KG
D
-0 "
'
C
OB
E
Uk , v ∈ U m
C
U
∞
?
+
J
U G
B
U B
A
P
CG
OB
A
CG
D
O
H
G
D
Uk
?
B
U
I
O
Ke
C
I
Y
B
CG
ID
E
TD
G
H
d
bc
IT B
RB
I
.
50
O
A PD
K h
j=0
?
U
BH
X
D
?I
C
eC B
LB
IS
N
CB K
AH
F
UV H J
D
B
B
?
V
N
K
G
PD
"
+$
.
CG
O?
WH
l
1/2 !1/2 l X x2i yj2 .
NG
IH
RB
D
OJ
D
A
AH
KD
VT
O?
WH
l
H
CB
J
TB
AH
H
F
O
IH
O
i, j
C
X
O
CD
H
H
D
VH
OQ
C
O
E
K
O
V
H
X
]
H
E
S
OD
?
yj = (Ri−1 vj , vj )1/2
RB
OJ
D
D
H
H
PD
C
O
E
K
S
U
H
E
GH
B
P
C
O
TB
IA
A
OQ
hl
A ] B
D
0 k·k C
]
H
R
OD
FT
F
C?
O
X
K
PI B
X
L
I
V f B
Vi Vj G = {γij } G kGk = ρ(G)
O
O
D
H
KD
V
PQ
H
AH
A
I
V
N
K
T
C
R
^ E
A
D
D
YI D
KD
TV
V
e
VH
+
I
S
OQ
Ke D
PJ
^
Ke B
OD
LB
0 IT
]
]
^
A
D
B
^
A D
A B
D
B
D A PD
C
?H
P
O
D
LD
O
Ke D
PJ
I
NV
K
T
X
]
]
H
O?
O
F B
D WH
H
KB
RB
O
P
E
YB
? h D
H
I
? D
B
TD
C
E
N
K h
X
Rk = Dk−1 , $
K
H
TJ
A
EG
P
K
TI
H
G
D
H
PB
V
O
Ke
CB
PD
T
H
KG
Ak
K1
^
P
G
?H
T
O
[
OQ
R
OD
S?
?D
C
R
^ E
YI D
D
KG
F
D
S
OQ
(vi , vj )A ≤ γij (Ri−1 vi , vi )1/2 (Rj−1 vj , vj )1/2
Dk
^
J
AH
H
B
I
H OQ [
O
ID K
O
I
?
N
OB
I
AH
H
K1 = ρ(G)
P
EA
R
T
LD
?I
O
I
U
N
PD
I
KG
*
]
V
i=0
l X
D
B
O
I
Y
?D
AH
B
Ke D
FL
C
O
CG
IB
P
YI
= ρ(G)
O
^
O
I
X
AH
KD
TV
I
xi = (Ri−1 vi , vi )1/2 K1 γij ρ(G) l
PD
T
K
G
D
Ce B
?F
OG
N
X
H
G
RD
F
]
]
H
ρ(G)
I
H
C
H
B
V
E
FT
D K X
Ω
B
TF
CG
?H
C
OD
EQ
?J
CI
K
x, y ∈ Rl+1
P?
T
LD
?B
W
, Tk
^
?J
H
H
KB
N
ND
CG
l
I
]
h
O
IH
X
i,j=0
I
\
EQA
D
I
hl
B
D
O
l X
FO?
EA
O
γij γij (l + 1) × (l + 1).
vi ∈ Vi , vj ∈ Vj i, j = 0, . . . , l vi ∈ Vi vj ∈ Vj
u ∈
k ≤ m.
√ 2 2 = max |γij | ≤ c √ . i 2−1 j=1 l X
!"
$
"
$
$
"
%
$
"
$
%
! #" "
"
" %
&"
"
&
%
!$
"
#
#!&"
!
$
%$
*
.
# * .
D
unew = uold − Br = uold − l X i=1
Ri Qi r,
B
D A U OB
CG
A
E U H
D
LD
O
RD
C
GH
D
LD
D
0
3,
%
-0
'
'
.
V D
IB
?
C
^ E
N
O
A
PD
F
E
OQ
K
G A U KB
B
A VB
$
W
IB
IJ
K
IH
KB
H
N
IH O
ZAH
N
K
T
PD
C
O
TB
IA
L
S
U
O?
R
G FM
E
I
PB
H IJ
O
C
P BH
D VH X G IH
B
^
U
Z
C?I
GH
B
LB
CD
H
NG
PB
H IJ
O
[
^
D
H
C
D
B
TD
C
ED
LD
O
GH RD
C
D
LD
O
E
LD
O
?
D
(BA) ≤ C I
^
B
OD
G
G
B
^
D
AH
H
H
OQ
O
FR H
KD
V
Ke
CB
P
K
IP B
KG
F
0V
[
X
J
B
D
H KG
I
^ ED
A
+ X
U
U
^
CQ
OB
CG
OD
?B
O?
H
H
H
D
H
G IAH C
?
D I[ T
O
C
O
E
K
O
R
OD
?B
W
A
NG
IC B
^ V
I
CD
NB
PB
0E
IJ
O
H ] B
O
FR
KD
]
C
R
K
P
I
O?
D
H
H
H
B
WH
N
N
O
T
KG
D
H
H
H
M
D
CG
E
C
E
I
W
NI
FK
]
U
D
H
H
PD
I
KG
F
O
O
KD
V
PQ D
C
R
H
E
N
O
E
I A
H
H
VH
KB
h
^
]
AH
CG
ID
OA
N
FK
L
]
U
0
CA
D
V
B
^
P
N
W
I
A ? DH
NV B
O
Ke B
OD
LD
]
E
R BH
C ] H
]
A D
^
V
T
P
I
C
E
O
I
I
V
I
S
P
B
CG
OB
A
CG
V
A
BH
H
H
X
^
^
U F
KD
K H
B
H
Ce G
F
PD
C
O
W
I
\
I \
D
?B
P
CG
CG
^
]
G
D
H
P
P
A DH
D
CD
?
CG
ID
O
Ke B
OD
LD
CA
D
B
CG
D
PG
F
K
I
P
O
Ke
CB
PD
T
[
A B
H
H
h
H
H
I
PD
C
O
E
K
S
OQ
R
H
OD
?D
P
CG
D
G
H
N
K
T
O
FM
P
CG
C
H
A
IH
P
E
IC D
YD
U
G
B EG
A
CD
KJ
N
TD
S
TD
V
E
O
^ H
A OB
^
H
KB
T[ H
A
D
]
FM
D
PV
O
D
CB
A
NG
E
OA
P
H
D
KG
(vk , vm )A = 0 k 6= m
ED U H
LD
O
GH
D
RD
C
LD
O
ED
LD
O
P
C
I
H
WH
D
B
[
h
TV D
V
O
O
E
0
A
B
OQ
O E
H
K
C
H
[
F
N
O
IH
O
E
B
l
W
I
LD
GH RD
C
T
TI B
B.
I
1
7
M
]
O
D
P
C
I
T
IT B
E
H
A PB
F
?I
l
hl
l X
?AH
*
,
V
?
H
Y
NA
D
VG
B
TD
C
PD
A
C
OB
CG
OD
O
IH
H
WH
k=0
A
O
D
VB
ED
CB
P
K
IP B
X
H
F
S
D
D ?[
A
O
E
B
2 h−2 k kvk k ≥ c
P
C
-0
4
P
H CG
KG
F
D
E
CG
ID
E
OQ
RD
C
GH
B
CD
C = K 0 K1
l X
D
$4
O
IH
D
AH
AH
H
G
H
W
I
kvk k2A ≥ c
I
'
?
Z
F
E
I
EV
V
TD
IS
G
P
CG
A
IJ
]
A
B
OB L
IC
O
S
N
O
IH
YD
B
^ NB
?
C
k=0
?B
T
TI B
'
D CD O
V
X
H
D
I
V
[
\
AH
KD
TV
V S
OQ
O
B
Ce G
l X
TD
6
?D
O
YD
[
Ce G
D
OQ
Ke
C
A Ce G
G ?H
O
D
X
]
H
D
D K X
kvk2A =
C
H
K
,0
EJ
P D
IH O
?
L
I
NG
O
I
KD
H
R
K
I
B
E
]
E
B
KG
R
I
C
H
EQ
D
G
A
?W
.
-0 "
,
N
CG
D K X
TD
PQ
N
K
V h D
?
k=0
I
&'
*
0
.
$
H
B
CG
G
]
l X
O
1
'
"$
"
0
V
E
P
N
TD
I
A
B
[
k=0
S
B
D
D
TD
I
C
R
"
0
I
OB
P
K
C
]
kvk2A ≥ c
l X
O
H
O
CG
'
OQ
RD
C
IH
K
W
I
FO?
O
I
FM
T
kvk2A =
EIA
^
H
PJ
IH
GH
D
]
K0
C
P
G
B
Ce B
I
VJ
O
FY
k¯ uk k ≤ chk k¯ u k k1 .
]
kvk k ≤ chk kvk kA .
B YD
E
f
U
CA
Az = b
Vk
A B
O
B
V
PQ
D
O
O
H
KD
E
A D A U OB
CG
PV
T[ H
AH
H
] D
]
C
R
^ DH
?
C
^ H
IH
YD
KJ
C
I
]
A D
W
I
FO?
M
FM
O
Ke D
PJ
I
V
E
A
IC D
EG
\
N
K h
]
E
CQ
OB
CG
OD
?
N
K
T
I
D
M
WH O?
O
IH
FR
KD
V
?
T[ H
AH
H
^
AH
U B
O
NG
FH C
C
[
X
0
I
D
$
H NG
C
N
O
D
KD
V
PQ
LD
A
CD
?
N
K
T
I
D
F
T
Y
E H
EJ
IA
ED
J
G
D U B
A
LD
C
N
K h
^
]
]
^
A B
BH
O
H
IH
YD
KJ
E
FR
KD
V
EG
\
Ul
B
EB
D
GH
u ¯k = uk − uk−1 , k = 1, . . . , l,
Vm
V
]
$
TD
C
J
k¯ u0 k ≤
U D
]
E
S
P
N
H
H
H
H
KG
A
u ¯k ∈ U k :
S
P
CG
vk = Pl−1 u ¯k H
>
(∇¯ uk , ∇¯ um ) = 0 ∀ k 6= m.
BH
kuk − uk−1 k ≤ chk−1 kuk − uk−1 k1 ≤ c1 hk kuk − uk−1 k1 .
T
FR
KD
(∇uk−1 , ∇ψ) = (∇uk , ∇ψ) ∀ ψ ∈ Uk−1 .
L
] B
B
V
H
H
K
T
C
R
k = 1, . . . , l
Pl Pl v = k=0 vi
^
G
O?
WH
D
^
E
E
K
P
u ¯k .
&
&
>
A
I
]
D
e
F
V
T
A
∀ ψ ∈ Uk .
A
A
B
I
V
m
S
IT
I
]
D
C
H
H
]
B EG
I
^
K0 v
(
NB
P
B
EQ
O
R
LD
I
KB
O
IH
CV
E
ku − uk k ≤ chk ku − uk k1 .
R0 = A−1 0
H
u ¯ 0 ∈ U0 CG
?
?
P
D OB
B
YD
KJ
A
IC D
O?
D
I # KG
H2
(
O
B
G
eC B
]
^
O
Ke
A
EG
G
B
WH
V
C
(∇uk , ∇ψ) = (∇u, ∇ψ)
Uk
LB
N
K
T
T
LD
+
N
N
K
PB
CG
TD
VD
B
H
H
, D
u
KD
V
H
J
D
A PB
O
I
O
I
VAH
H
D
^
H
CB
PD
E
E
Ul ⊂ H10 (Ω)
!
H
\
H
V
O
YD
Ce G
LD
C
?
?
B
+
k=0
l X
PD
C
O
W
I
O
P
CG
AH
h0 ∼ O(1)
]
D
E
D
ψ=u ¯m
*
I \
T
G
D
C
R
]
F
O
H
T
H
?
u=
D
?B
P
CG
OB
A
O VH
C
^ E
YI D
O?
WH
Ke B
H
KD
ul = u.
D
H
FH
T
KG
KD
l
OD
K
u ¯0 = u0 ,
!
A CG
PV
G
TV
AH
A
Uk−1 ⊂ Uk ,
(
$
/
e
B
NB
A
L
Ω
+
)
P
h0 k∇¯ u0 k
%
$
,
'
u ∈ Ul u = Pl v u uk ∈ U k H1
v ∈ V := Rl
kvk k2A .
2 h−2 k kvk k .
(Rk−1 vk , vk ).
k=0
!"
$
"
$
$
"
%
$
"
$
%
! #" "
"
" %
&"
"
&
%
!$
"
#
#!&"
!
$
%$
Ω J
B G
^
P B
O B
F
C
F
A
?A
B
CG
PD
]
A
H
I
eC B
PD
FE
A
K U H
B
G
B
A KB
V
ED
C
] S
B
KJ
H
>
A G
D
H
C
TD
IS
I
CV
RG
AH
H
P
U
H
H
A
D ?AH
?
TD
C
E
]
BH
A
3
H
I
O
H
IH
O
E
I
V
5
6
5
8
;
]
UA
F
V
D
J
B
]
H
H
B
D
D
J
D
CD
A
OA
P
[
C[
D IB
O
O
YD
E
PD
TD
C
E
D
LD
O
X
X
@
?
U
B
C?
S
OQ
P
C
H
K
I
V
C
I
KeF
E
I
D
LD
O
P
C
I
IT B T
O
IH
O
PB
A
B
GH
IB
TB V
O
PD
C
H
RG
] S
B
^
A
B
H
A
D
G
D
G
G
H
A U B
N
K
T
OQ
K X H O
V
I
VB
TD
C
E
D
LD
O
P
I
CB
?
K
I
V
C
I
KeF
E
I
OB
CQ
I
P
]
ED
A
]
H
H
D
H
C
FM
CG P
F
X
Ce G
O
YD
KG
FM
O
Ke B
E
C
ID
V
Ce
E
CI
FT
V
I
A
J
]
N
K
T
CG
ID
O
YD
KG
J
NI G
C
N
D
M
[
X
C?I
N GH
IH O
R
Ke
H
EJ
I
LI
]
U
B
H
D
CB
A
D
VG
T
P
I
H
H
TD
A CB IH
C
E
[
Y
OQ
P
C
I
T
TI
CG
D
O X
D
TD
V
K
I
[
A
B
H AH
BH
D
O
Ke
C
D
H
KG
I
R
I
PQ
V
E
OQ
Ke B
E
C
ID
V
eC
Q
C
CG
V
H
PD
K
TJ
S
IF
T
UO
E
K
S
OQ
H
H
]
I
V A
G
F U B
RB
I
J
B
H
TB
A
A
0 IR
T
L H
H
EA B
D
NB
O[
O
I K
0I
[
X
G
G
H
H
U
VJ
TI D
C
E
OQ
GH
D
RD
C
LD
O
E
?I
R
G
IB
K
?
^ C
I
RB
O
[
[
%
]
]
A
H
GH
D
I
C
J
B
OB
CG
OD
C?
F
L
D
G EB
PD
KJ
F
D
KG
[
[
j
X
\
X
ED
B
[
A
PD O ^ H
F
ED
O
RD
C
GH
TQ
^
B
O
O
A
H
H
D
PB
C
FT
O
A PD
F
ED
B
X
^
A D
B
J
H
H
H
B
D EJ
P
QD
C
R
^ NG
C
A
FH
C
IB
TB V
?
A
D
I J
eC B
C O
H
IH
ZAH
ID
X
X
K X
I
I
V
D
D
RG
H
B
H
ZA
S
D
Ce
A
A
A
D
K
Q
D
D
V
O
FT
C
O
E
I
V
LD
O
O
O
\
YD
H
H
D
H
FO?
S
OQ
C
O
E
K
O
R
OD
?
P
CG
A OB
CG
V
P
?
N
P
O
YD
K
T
AH
NV
] S
N
CG
K h
KJ
F
D
B
D K X
S
?I
C
P
U
U
0
PD
U
H CG P
R
KD
I
?D
C
R
2i−1 ) n0 Cn → 2 AH
H
]
^
H
D [
A
A
D
H
D
D
UC
F GH
ED
C
W
Q
OB
IA
L
KD
?
E
IB
V
O
CG
IB
K X
D
TD
VD
A
CD
?H
O
P
NG
C
]
?
C
IB
?A
V
B
AH
D
G
O UN
H
EJ
I
O
Ke
I
RB
I
J
D TB
O
\
[
H \
H
Ke B
W
I
O
T
I
O
IH
ZAH
T
LD
^ H ?
U
G
H
TD
S
U
Q
O
N
O
A
B
PD
F
D
LD
] B
B
B
BH
AH
O B
NG
C
R
CG
P
D
CG
R
N
W
IF
C
G
NB I
?
[
]
D
eC B
G
TB
I
K
V
H
D VAH I
V
O
YD
E
P
+ ^
D
H
B
GH
U AH
VA
I
V
C?
G
eB
KB
R
Ke
H
EJ
IH
T
L
CG
IB
K X
D
TD
V
TB
I
K
CV D
^
^
U
U
Q
A
H
GH
D
B
B
D
J
BH
DH
D
D
O
E
B
DH
BH
LB
KG
C
O
RD
C
F
RB
TB
E
ZAH
C?
F
L
O
I
?
IE
[
X
X
^
A
H
BH
[
D
VD
O
A PB
CG
IB
[
K X
D
TD
PV
NG
C
R
Ke
H
EJ
I
Oe
PD
F
OQ
GH RD
C
[
^
A H
H
D
G
A
bc
#
D
D
H
B
[
d
X
H
G
H
AH
A
B
GH
TD
S
V
I
V
C?
CG
IB
K X
OA
FE
P
T
P
T
H
LD
?A
E
I
V
E
I A
OB
I
P
YI B
LG K
Ke
CB
P
K
PI D
UN
U
D
AH
H
H
KG
F
UV
E
N
O
E
I
V
EQ
I
KG
D
LD
k
.
B
U
H
H
F
^ E
O
PD
PD
C
O
E
A
VH
O
A
PD
F
(k) φi ,
FK
K
PI B
X
N
K
T
B
D OkX
F
E
ED
A
6
6
3
67
P
CB
I
?
O
$D
OA
B TD
S
E
I
i=1
D
KG
D
AH O
VD
FT
B
Q
$
T
P
I
C
E
O
I
I
V
1
>
X
H
ED
U
O
W
I
FO?
\
Pk
EA
\
F
C
!
O
U
(k)
PD
H
(rh , φˆi ) ˆ φi a(φˆi , φˆi )
YD
I
D
CG
A
G
ED
PD
a(φi , φi )
K
S
H
H
E
V
Oe
H IH
KB
T
C
U PB
B
AH
H
U
A
(k)
O
E
K
S
OQ
R
OD
]
J
M
B
F
X
BH NG
C
D
KG
F
D
O
H
NV
K h
O ^ H
A PD
F
ED
S
(1+ 32
G
H
?S
I
O
f
D
(rh , φi )
H
OQ
R
]
]
H OQ [
]
H
PQ
C
FH
Vk = Pk−1 (Uk − Uk−1 ) k k−1
I X B J
B
hl V
OD
?B
TD
^]
B
G
O
IH
P
CG
OB
A
P
CG
C
H
(k)
S
G
H
W
IB
H
S
FO?
i=n0 +1
nl X
?I
R
C
E
CG
ID
H
RD
O
A PB
G
D
D CG
A
P
D
G CD
k=1 i=nk−1 +1
nk X
S
IJ
A
IB A
G H C
?
T
H PD
C
VS
PQ D
B VT D
V
D
l X
IAH
F
I
O
+ Pl−1
W
I
I
FO?
\
^
O
D
C
R
U
CD
H AH
R
OD
?
]
RB
I
U
T
L
k
TD
I
I K
A−1 0 Q0 r
C
A
IB
$H
B
\
P
O
W
I
I
C? G
H
NI B
A
B
RG
FK
EA
D
BH
?
Ke D
C
ED
Y
B
H
OQ G
I X B J
B
FR
KG
BH ^ H ?
EQ
H
H
D
TD
C
E
P
C
R
J TB
P?I D
OB
B
OQ G
I X B J
D
KG
O
\
k
H
LB
O
T
P
I
P
^ N
OB
I
P
K
I
Pl−1
D
B
FR
KG
]
G
B
Ce B
I
VJ
B
$[
Q
[
^ E
O
I
ED
V f B
[
W
I
I
FO?
R
0I
[
(k)
E
W
OB
PD
U NG
]
+
B
A
^
U
O
EQ
H
G CG
I
CG
IB
R
D
γ≥2
OG
H
D
B
Br =
ZG
A
LD
OA
H
ED
O
TD
U
D
A−1 0 Q0 r
K
IC B
A
IC G
M
D
AH
O
CD
W
I
H PB
A
B Cn ≥ q > 1
LD
GH
E
A 0
A
F
^
Br =
C?
[
K
K
I
N
K h
ED
IE
O
IG
E
O
EI B
D
D
D
H
H
H
CG
V
O
C
O
E
K
OQ G
D
n0 l → ∞
NG
D
V
H
X
NV
W
O
A
A PB
V
P
C
U
I X B J
^
W
I
k −1
1 2
BH
K?
[
D
P
K
PI B
KG
D
LB
KG
A
NG
Ce B
B
?H
P
I
W
A
O
R
0
k+1
C
TB
H
U
φˆi
E
# KG I
+
T
D
WH
A
F
.
D
EB
B
TD
C
H
[
O
N
B
H
ED
O
R
Ke
H
(l)
Ke
U
I
FM
E
D
\
I
? \
I
i = nk−1 +
G
D
EV D
P
CG
V
I
FO?
H
G
H
EJ
CI
hb, e¯i iRl = (f, φi ).
H
U
]
H
[
IT
GH
D
G
O
C
GH
YD
E
B
GH
D
B
H
OD
?
φi
C
H
D
θ=1
C
PV
?
H
H AH
(k) φi ,
k
YD
?
ν =1
D
C
O
P
D A PD
F
TD
I
E
C?I
KQ J
F
C?
C
R
[
IC B
E
G
OD
,
E
OQ
I OB
P
]
NB
M NG
C
N
X
^ E
IY D
KD
D IH O
Z
b
Ke D
C
P
D
G
G
H
EB
i=nk−1 +1
nk X
CD
A
V
OQ G
KJ
F
(k)
!
C
D
IF
G
H
(rh , φi )
Ke B
CG
YI B
K
?I
R
LD
P
H
FM
T
KG
KG
D PB K
T
H
AH
O
PD
C
(k) φi (k) (k) a(φ , φ ) i i k=1 i=1
]
I
B
?AH
U
H
H
B
EQ G
C
E
I
O
TV
CD
D
G
nk l X (k) X (rh , φi )
NG
C
^ A
LG
a(φi , φi ) (k)
CG
N
Br = AM GM (l, 0, r)
TD
]
D
+
G
C
(k) (k−1) φi 6=φi
W
I
I
D I X B J
?
(k)
IB
D
>
C IE
EI B
KJ
F
nk−1
X
PJ
]
A
D
EQ
FE
G
J
,
T
LD
C
A PD
F
ED
xi
K X
[
O I
I
I
L
G
EI
M
G
^
U
k a(·, ·)
D
A
?H
J
B
ED
A
Ke D
C
i
P
KD
V
X
I
\
H
H
TD
S
AH
H
R
I
K VA
I
k−1
GH
FO?
CQ
O
K
C
] ?
M
]
N
K
NV G
C
N
K
PD
H
CB
P
K
I
(l)
NV
H
OQ G
X
N
O
PD
H
H
D O X k
A
hAl e¯i , e¯j iRl = a(φj , φi ),
P
D
O
IH
D
]
:=
H
E
N
TD
X
i
P
O
A
A
S
$H
O
IH
RB
D
OJ
D
l
PQ
I
F
P
i=1;
OD
EV D
W
I
I
D
LD
]
H
K
IH
P
P
O
Q
(l)
KG
I X B J
V
H
FO?
\
U
P
O
A PD
F
H
T
L H
]
^
?
N
P
0O
B
J
H
]
H
G
H
O
G
A G
B
U CD
EG
N
O
IH
PB
A
H
CG P
RB
? >
N
O
IH
KG
R
I
PQ
Ce
CB
IA
D
D
H
?G
O
O
CG P
F
r = A uold − b
B
I
?D
I
H
N
K
T
1
9
23
ED
(k) (k−1) φi 6=φi
R
G
LB
FM
T
H
KG
OQ G
I X B J
?
NG
EQ
T
L
k=1 φ(k) 6=φ(k−1)
X −1 Br = A−1 0 Q0 r + P l
B
e
?
NG
C
J
FH
H
PQ D
O
GH
U
B
f
Pl−1
^
PQ
B
I
ZB
O Ke D
C
I KB
AH
P
H
k
K
O
IH
EJ
]
I
C?
+
] A
CG
EG
Ke
k
GH
T
LD
NG
M
]
C ] H
X
H
]
C
N
KG
R
I
PQ
X
H
a(ψ, φ) = (∇ψ, ∇φ). Al
O
A PD
F
ED
S
I M
QD A FK
O
CD
?
P
D
C
A−1 0 Q0 r +
NG
C
BH
U
B
k
R
H
O
OQ
A PB
I
?
rh = P l r nk
Ke
J
^ ED
X
^ EB
KJ
F
Br =
EJ
IH
B
H
B
T
A
D
CQ
O
O
k 1, . . . , nk
L
CG
IB
E
?I
k
^
D
K X
T
r
$
%
$
"
$
%
! #" "
#
&"
&
$
&"
&
$
"
&$
%&$
!$
"
&$
%$
"
$ #%"
UW
B
G O
J
H
IH
X U
K
P?
N
Ce
C
M
AH
B
^
C?I
GH
U
C
P D
Ql−1
I
A IB
C
BH
PD
KJ
F
]
GH
H
UC
GH
KQ J
F
P
E
<
D
J
B
TD
C
X
^
W [
\
VH
GH
CG
C?I
]
H
D
?H
R
N
KG
0 IR
H
P
TQ D
D
PG
O
H
[
X
I
C?
E
A
B
GH
H
]
H
H
LB
Z
C
E
Ce G
F
hk = 2−k
E
ID
T
^
Ql−1 ?A
V[ VB
D
J
D
I
FO?
Qk mk
YD
A
CI
E[ H
H
D
E
I
RB
D
D
D
GH
H
OJ
?
C
S
OQ
O
YD
K
P
eC G
O
Ke
H
CB
PD
T
KG
X
H
H
H
C?
O
H
GH
D
A CB
TB
P
?B
O
EI B
C
O
E
K
EI
OQ
R
OD
?
[
A D
D
D
H
G
G
B D
U G
F
N
O
IH
O
A PB
F
W
IB
I
E
ID
?A
VB
V
NB
FR
KG
L
Z
H
CG
NV
K
T
L
[
]
A G
B
H
D
GH
G
H
U CD
EG
EQ
Y
I
?
C
S
?I
R
F
I
S?
OQ
A CB
TB
P
?
IB
?
C
^ S
f
X
]
A
AH
B
UT
C
C
S
OQ
A
B
H
Ke D
FL
C
N
K
T
?
?
E
O
IH
E
I
V
H
J
TD
C
Q
J
B AH
A
^
A B
BH
D [
H
M
H
KQ
FL
I
N
H
TD
S
P
E
I
CG
IB
[
D K X
O
C
H
RG
FR
? @
KG
P
K
I
I ^
A
H
BH
\
B
D
UW
I
\
I \
?
EI
OQ
P
QJ A
A
G
J
I \ FI
T
I
N
O
IH
O
A PB
F
FR
KG
P
E
A G
D
TD
IS
I
VB
?
?
EI B
^ C
^
P
CG
D
PG
EI
OQ
H OB
K
Y
C
BH
TB
D
H
B
[
K X
O
RB
TB
J
NB
^
G
D
I
UW
O I
T
[
\ AH
H \
NB
E
A FH
IE
G
ID
?A
VB
VB
T
LD
?
W
IB
IF
C
I
P
O
Ke
H
CB
?H
D
H X
D
G
D
B
B
H
G
H
D
NG
C
I
PD
OB
CG
B
TD
C
E
LD
?
H
I X A B R
L
KB
Ce G
O
O
C
] S
O?
D
D
WH
S
OQA
D
H
H
J
B
P
UA
IB
V
NI
O
IH
ZAH
CG
ID
T? B
K
L
S
B
CG
D
PG
O
NG
C
A FH
[
I X B J
O
KB
I
OB
L
^
G
J
P
G
B
B
RB
I
J
D TB
O
\
[
H \ AH
Ke B
W
I
O
T
[
D I[
O
TD
S
I
CG
D
PG
C
FH
Ke D
V
I
O
IH
CG VH
]
H
D
GH
H
H
H
Z
Oe
E
H
H
PB
TD
C
E
O
A I DH
D
D
B
CG
V
?
[
T l O
?
C
S
OQ
O
YD
K
P
CG
ID
O
Ke
CB
P
]
U
H
H
B
G
A
H
J
D
D T D
H
KG
V
eC G
O
R
IB A S
CI
FH
Ke D
V
IH
Y
?
C
TD
C
E
CD
C
E
PQ D
W
O
@
?
H
H
H
H
G
H
AH
UF
OQ
O
J
H
YD
K
T
^ TD V
C
E
?I
[
[
j
I X A B R
L
KB
NG
C
N
K
P
N
N
K
CB
P
K
PI B
KG
H
D
M
D U X F
AH
H
VD
LD
O
GH
D
RD
C
LD
O
E
I DH O
A
CG
V
?
ED
TD
S
TD
V
E
OQ
P
CB
I
OA
C
Ke
*
0
*0
3,
%
-0
2
.
1,
%
,
/,
-.
&'
1
(
(
" 6
0
'$
/
7
A
H
G
H
[ I X B J
?I
R
H f
IB A S
I
$
] G
]
] G
I
A B
D
H
I
D
H
C
C
R D
B D
CG
R
CG
ID
O
K
PD
KG
F
D
B
KG
R
CG
A
O
X
C
G
H
IH
[
?I
R
C l E
PD
C
RG
^ E
]
D
B
CG P
TF
J
D
B
[
X
IE \
A
B
]
D
AH
H
H
C
I G
PB
B
IJ
O?
D
H
WH
E
K
CB
P
K
IP B
KG
F
UV
G
X
d
bc
T
L H
]
(2)
D
V
D
G IC G
N
K
T
PD
KJ
F
R
I
D
X
A
U H
R U X B
IB A
A
H
IH
H TD
C
E
P
S
G
H
CD
S
?I
R
D
G
Ce G
E
I
PB
O
[
I
D LD
K
M
φ5
O
C
H
GH
X
S
H
^
IE
OQ [
H
K
P
I
V
S
A
IB A
H
IH
H TD
C
E
D
A D OkX H H
TD
IJ
[
D
H
H
WH
A
O B
?
(2)
O
C?I
S
B
OQ
OQ
C
O
V
]
O
K
I
I
O
Ke B
D
]
H
B CG
E
K
E
I
φ4
E
Q
VF
VA
A
B
GH RD
C
S
F
S
AH
Qk
H
D
B
VF
VA
L
H
D
O
O I
FC
]
A
IB
V
O
]
?
D
D WH O
P
H
CG
Ke
CB
J
B
? h D
@
?
X
D
H
K X
S
OQA
(2)
K
L
AH
B
H
E
K
H
ED
H IE B
C
O
]
AH
D
J
D
BH
G
NG
C
P
K
I
F
I
G
PD
φ3
U
O
R
O
O
P
T
E
U
O
R
OD
O
P
D
J
OQ
K
I
I
@
?
P
Ce G
H
RD
A
D
IH
O
(2)
H
H
X
H
NB
PA
A
CI D
EG
H
^
H
Ke B
A
H
B
V
O
G
PB
TD O
A
D
φ2
OD
O
J
IB
H
$ V
Q
VF
G
B
?
P
KG
R
0I
E
A
]
P
Y
?
C
Ce G
H
A RD
CD
(2)
IH
IB
Y
K
KQ J
A
B
CG
A OB
B
I
]
]
]
]
]
]
V
NG
C
BH
^
φ1
?
[
]
^
VA
L
A
KB
RB
h = 2−l Qk k = 1, . . . , l
H
M
KQ J
F
F
C
I
CB
O
LB
O
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
AH
C
RG
I
(2)
6 φ0
NG
N
K
P
N
A ≡ A(l) ^
H NG
I
YIA
H
G
TD
Ql
PZ B
D
CG
D CG
Nk
W
φˆ6 AH
ZG
6
Q
O
A
H
φˆ2
A
NB
R
-
BH
V
]
]
]
]
]
]
6
C
?I
]
]
]
]
]
]
]
]
]
]
]
-
IC B
A
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
φˆ0
R
φˆ5 ]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
SE
C
N
K
T
K0 = O(| ln hl |2 ), K1 = O(1).
E
[
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
φˆ1
I X B J
G
H
φˆ4
?I
R
S
IB A
A
,
φˆ3
H
$
]
]
]
]
]
]
]
k=0 hl
] G
]
]
]
]
]
]
]
]
]
]
]
]
]
k=1
] G
k=2
I
G
I X B J
S
?I
hl
YD
E D
PD
G
J
V
I
O
YD
E
cond(B A) ≤ K0 K1 , φ6
(2)
k=2 -
6
^
]
C D
$
%
$
"
$
%
! #" "
#
&"
&
$
&"
&
$
"
&$
%&$
!$
"
&$
%$
"
$ #%"
H
$B
^
A
D
0 I22 −A−1 33 A32
C
AH
˜ w) = hl (Kv, F
(l) Wh
G∈Ql−1
X
˜ K 1 2 ∂G
I dv h dwh ds, ds ds
D
∀v, w ∈ RNl −ml−1 , $
I
O
v N
O
B
[
^
D U X D
W
H
OD
? f B
AH
AH
ED
T
YB
?B
X
D
A B
CG
ID
OA
H
EJ
BH
V
J
NI G
C
FR
KD
]
J
A
w
PD
KJ
F
S
OQ
HX
C? B
GH
NB
OA
H
X
HX
F
C?
GH
FM
OA
AH
Ke
H
C
I O
K E
AH
D
IH D
T
V
F
W
CI B
E
Ce B
J TB
QD
C
R
D
LD
C
KD
X
^
^
N
K h
N
V
O
IH T
O N
A
A
HX
AH
IB
H
H
PI D
KJ
F
IH
T
P
P
E
FC
D
D
Ce G
O
F
V
G
B
PD ?
?
B
U J
J
]
A
W
Q
A
IC B
E
C
?H
G
VB
O
I
O?
D WH
E
FR
I
KD
V
I
S U HG
F
P
T
N
^
Y
O
I
G
H
H
OB
I
V
PQ
C
FT
Ke
C
D
G
B
A D NG
C
I
CG
?
?
^ E
H
YB
?
V
EQ
B
R[
GH
B
^ H
]
X
O
DH
OB
PD
T?
NA
D
h
]
U
H
B
D
H
D
D
AH
A
P
K
B
PI B
KG
F
V
E
C
I
L
KB
OQ
GH
D
RD
C
LD
O
E
O
Ke B
EA
D
H
KB
T
?
C
I
[
>
X
K H
A D
O
R D
LD
I
KB
OB
NG
C
I
CG
QD A
CD
?
^ E
^
H
CB
P
[
^
D
C?
O
e A
H
GH
B
H
VH
C
O
U
U
B
D
AH
H
G
K
IP B
KG
F
V
X
?
W
IA
C
I
B
H
D
Ke D
G
H ?
O G B
C
H
G
D
N
O
NG
A
D
KD
V
PQ
N
O
IH
WH
V
LD
O
B OD
W
IA
H
D ZAH
D C
I
LD
H
LD
O
Y
[
K X
I
A
O
H
X
I
VH
WH
A
IC B
E
G
B
E
H
G
N
I
CG H
H T K
T
O[
O
K
I
FTe
O
ID
LD
?
?
[
BH
U
C
E
D
H
ED
O
GH RD
C
D
LD
O
E
] D
O
RD
C
NG
C
ZAH
>
]
D
GH
H
GH
D
H
H TD
C
E
ED
O
A
RD
C
S
F
P
T
C?
F
L
H
H
B KD
O
RB
>
X
EQ
G
H
CG
G
IT D
VH
O
IH
ZH
] W
A
IC B
E
S
OQ
Ke B
OD
I
J
LB
I
N
TB
[
B
T
X
A
D
IH O
H
O
IH
ZAH
C
X
]
0
1 (l−1) 2A
B
O
O
IH
G
Nl − m l FY
[
P
CG
(l)
IH
A PD
D
]
O
S
OQ [
OB
A H
CG O
I
K
D
A
Wh
D
C
?H
W
I
FO?
\
]
J
?
Ql
Z
O
CD
G
D
D
(l) Wh
P
CG
ID
∂Ω
H
P
CG
B
I
(l)
L
EI
N
OA
^
A
H ?H
R
^
CG
IN
[
A I DH O
S
B
FM
H
B
O
S
OQ
P
Q
AH
N
OD
D
[
PG
B22
O
PB
H
O
FKe
D
Γh
IH
A
EJ
B
D
O
VH
A
D
CG
V
U
IB
R
D
PB
D
D CG
^
V
WH
A
IC B
E
[
G
AH
H
FH C
C
E
CG
G
TI D
S
V
OQ
Ke B
D
H CG
O
IH
ZAH
F
Ke D ?
?D
X
D
G
H
AH
H
D
EQ
G
H
CG
G
IT D
VH
O
IH
ZAH
?
Ke D
C
Ce
N
K
PB
CG
T
CV
YD
E
e
TJ
Ce G
O
FT
A
+
H
]
]
A D
AH
H
H
D
D
W
A B
CI B
E
C
R
^ NG
eC
T
H X I
F
O
YD
E
E
O
IH
YD
O
E
V
E
QC G
Ce
KG
I
R
I
Q
LB
>
i
M
H
H
UC
O
EI
N
T
GH
B
GH
D
G
NG
C
N
K
P
N
O
Q
VF
J AH
VA
L
Ce
C
I
I
[
D
D
K X ?
C
R
^ E
H H
C
IH
EB B
R? D
C
P
T
?I
?
O
I
%
^
H
D
?
?
?
C
B
B
WH
A
CI B
E
O
Ke B
OD
LB
I T
NG
C
N
K
P
N
[
[
0 B22 0
O
T
H
Y
KG
D
A
O
H
Ql−1
KD
TD
P
CG
P
CG
B
B
O
FE
Ce
N
C
>
$
O
IH
K
PB
H
0 0
P
K
H
P
NG
S
X
S
O
J
l
]
AH CG
T
VH
H
FM
T
]
?
1 (l−1) 2A
M
A
TI
PB
OB
A
CG
I
BM
B
D
GH IAH T
P
U
P
QB
?
(l−1) Γh
eD
G∈Ql NV G
A
]
D
A
YD
O
E
Ce G
]
A
B
V
I
FM
P
A(l−1)
C
G
VD
A
H
U
Ke B
OD
LB
V
P
W
NV
K
[
H
CG
C
P
CD
H
BH
G
D
KG
C
?F
V
T
ED
G
wh M
(Av, w) = hl
N
V
(l−1/2)
K
J
[
I
NG
C
I PD
OB
CG
KG
I R
D
W
T
D
[
D
G
H
V
C
R
I
W
C?I
GH
H
H
I[
T
O
PQ
A
CI B
Q
A
CI B
D
R? D
C
O
PB
W
Q
A
CI B
T
L
1 (l−1) 2A
P
N
vh W
^
Ce
A
IH
PD
I
FO?
B
J
Wh
A
IC B
I
A
E
[
]
C?
(l) Γh
E
W
\
(l−1/2) Γh
E
D
I
RB
OJ
D
H
H
GH
D
?
H
Γh (l−1/2) Γh
K
V
H
B
O
C
O
E
K
[
X
Nl
I
L?
U
D
A
L?
(l)
I
\
U X F
E
O
IP
P
D
F
FM
J
E
E
N
K
T
A
CV D
H
0 0 . 0
FO?
D
AH
[
H
?
H
H
E
OQ
R
O
I
I
T
0H
0 0 N = 0 A23 A−1 33 A32 0 0
IH
]
H
H
U
O
R
OD
B
K
Ce
B
C? B
N
K
CB
C
IFY
KG
D
G
H
%
A
T
l
T
L
]
V
NB
A
L
O
B
K X
I
FE
D
CG
N
O
P
K
PI B
X
VH
O
IH
K
PB
e
T
0 0 FAT − N, A33
U H
G
D
x ∈ R Nl F
−1 0 0 FBT , A33
V
0 0 . I33 OQ
E
O
I
?
NG
RB
−1 A11 − A12 B22 A21 I
O
Ce
−1 (A11 − A12 A−1 22 A21 )
GH
I
D
KG
I
H
B
M
G
IH
R
K
?
D KG
F
A
RD
C
S
]
Ql−1
F
P
T
TB
J
?
K X
^
X
D
IH
KG
AH
H
VD
CG
T
AH
]
U
G
i
D
WH
A
IC B
E
O
Ke B
OD
LB
T
ID
O
RD
[
[
K X
F
E
H
G CG
I
Ce
I
ZH
LB
0 A23 . A33
B
0 B22 0 NB
M
N
I
E
CB
IA
D
D
H
X
N
E
TB
I
N
K
P
D
O
IH
EB
W
A
IC B
A−1 22
H
GH
C
R
J
D
E
TD
L?
K
VD
C
R
LD
O
V
0 , A23 A−1 33 I33
H
A B
D
CB
A
V[ D H
^
C?
O
H
GH
B
]
B
$
Ce
KG
I
R
I
Q
LB
D
>
i
K X ?I
O
Ql
?H
$
C?
K
T
CG
O
YD
E
B
D
Q
B
X
^ E
C
RD
C
0 A22 0
D
J
N
B
ID
^ B
?
C
H
VA
C
K
Q
IH
E
A12 A22 A32
P
0 0 TB
B
^ W
H C? G
A
F
W
B ?H
P
O
H
]
LD
1 (l−1) 2A J
^ N
IH
−∆ O
A
CI B
0E E
R
I
V
A
IC B
E
TD
O
IH
Y
B
B
X
B
D
Y
WH
A
^
F
W
A
H
^ NG V
C
A11 − A12 A−1 22 A21 A = FA 0 0
ED
A
D
O
A
H
IC B
E
O
C
O
IC B
E
FM
O
O
E
YB
X
A CB
F eC G
H
KB
P
A11 A = A21 0
Ke D
B = FB
∂2 ∂y 2
PJ
I
A
D
D
H
B
V[ D H
W
I
,
A(l−1)
I
G
H
H
H
? ^ B V
C
P
I
?
A22
FM
T
H
KG
Z
PQ
OQ
O
E
G
ID
?A
VB
V
A12 A−1 22 I22 0
V
H
YD
KJ
IN
O
I
A33
N
D
x ˜ = Bx A
I11 −1 A21 FB = −B22 0
K
N
K
T
Ke
CB
P
K
PI B
A
T
H
KG
B22
LB
A
H
T
L
Z
IC
B
>
I11 FA = 0 0
Ijj
J
O
IH
KG
I R
Q
Ql
NG
C
N
K
P
CG
2
∂ −∆ ≡ − ∂x 2 −
y = FBT x
0 0 z = y; A33
x ˜ = FB z
Ql−1
BA
B22
X 1 I dv h dwh ds ∀v, w ∈ RNl , 2 ds ds ∂G
$
%
%
$
"
$
%
! #" "
$
"
$
%
"
&
$
&"
!
D
G
ρ(BA) = max
v∈H
ρ(BA)
(Av, v) . ˆ v) (Bv, D
max v(i)
v∈H
(A(i) v(i) , v(i) ) ≤ 3. ˜ (i) v(i) , v(i) ) (K
U
^
H ?H O
K D
UA
O
B
H
NG
J
ds
H
E
FH
h h dv(i) dw(i)
IH
R
N
ds
F
h h dv(i) dw(i)
KG
ds Ke D
ds
R
∂G(l)
I
VF
G
ml−1
i=1
X
VA
i=1
VD
P
˜ (i) v(i) , v(i) ) = (K
I
v ∈ H B
i=1
L
ml−1
i=1
X
PQ
DH
(l−1) Gi E
OQ ^
(A(i) v(i) , v(i) ) =
F
O
i=1,...,ml−1 v(i)
max
H
∂Gi
(l−1)
I
NA
max
1 2
O?
Ke
(l−1)
WH
]
max ai bi
CB
G(l) ⊂Gi
X
D
ai i=1,...,ml−1
W
hl 2
C
≤
B
˜ (i) v(i) , v(i) ) = bi := (K
A
A
?
?
N
O
B
IH
T
H
D PJ
I
V A D U
[
I
?
R
N
H
H
D
[
[
B
$H
T
P
I
ED
O
RD
K X
P
E
PB
I
AH CG
T
V
I
?I
R
N H
V
^
J
[
N
OQA H
KB
U
D
CD
?H
O
CG
H
ID
O Ce
G
^
J
P
D
G
A
PB
PD
I
VF
VA
L
] D
O
H
CG
C
H
CD
O Y
?
I
I
H
B
?H
R
N
S
W
P
PB
B
OB
A
IA
L
O
W
I
TB
A
D
BM
M
O
I
VS
K
AH
H
CG I
T
V
E
YD
I
E
B OG
K
EQ
G
AH
H
LD
CQ
R
D
J
GH
D [
HX
KQ J
F
C
R
K
P?
C?I
OA
AH
KQ J
F
F
Ql−1
BH
bi
D
i=1
i=1 mP l−1
A
ai := (A(i) v(i) , v(i) ) = hl
T
bi H
ml−1
X
IC D
D
H U G
O
Q
VF
VA
L
KQ J
F
Ql−1
C
mP l−1
[
O
N
O
B
[
WH
A
B
?
Ke D
G
?D
Ql
H
$ F
I
OQ
ai
?H
H
H
ˆ v) = (Bv,
P
O
A
I
A
B22
H
[
ρ(BA) =
D
S
X
W
CI B
B
D
H
,
[
A
CI B
E
W
CI B
E
TD
VD
O
ml−1
X
?
R
N
G IH
FY
IH
RB
OJ
D
D
H
H
H
]
E
C
O
O
E
(Av, v) =
ED
A
D
CD
I R
K
IH
FP
OQ
T
P
P
T
L
7
3
]
H
H
GH
Ke B
^
Q
VF
VA
L
Ql (l−1) Gi
C
H
G
ρ(BA) PG
?
Ke D
U D A
\
7
6
;
]
C?
P
OD
LB
T
I
NG
C
H
]
\
$
D
B
A
BM
GH
G
D
H
U CB
SE
OQ
Ke B
?
K
E
OB
I
A PD
K X
I
E
B
NG
C
T
D A YD
V
I
W
Q
IC B
O
I
A PD
C
?H
P
EI
N
IH O
Y
KD
TD
A
%
$
"
$
%
! #" "
$
"
%
$
C NV G
%
"
&
$
&"
!
M
N
K
P
N
I
W
I
I
H
˜ K
FR
v(i) Ce G
ˆ = Au, λBu
H
$
G
?D
[
[1, 3]
?H
O
D
ED
P
H CG
O
P
]
D
K
X
A
G
ED
FL
A
N
K
FO?
T
L
A
PD
S
A
I
O?
B
H CB
Y
O
RD
H
CB
P
K
PI B
[
T
FL
T
P
I
Q
VF
VA
L
G(l)
KB
A
H
IB
Y
H
C
D WH
.
L
C
O
S
K
?H
P
N
U
OJ
$
N
2
D
AH
H
KG
F
V
ED
O
GH RD
C
S
F
P
T
P
D
H
M
H
i
I
TB
O
H
K
T
H = {v : A32 v2 + A33 v3 = 0}.
OQ
A
T
L
UC
]
K
TB
O
A
K X
P
9
7
P
R
K
N
T
L
,
I
V
.
w
Ke B
?
H
H
IH
O
RB
OJ
F
W
1
5
68
29
2
6
7
5
3
9
C
FM
P
GH CG
T
v
O
˜ = Sw, ˜ λKw
EI
OQ
O
B
λB −1 v = Av.
KB
P
P
I
VA
D
H
0 A23 A33
I
?
P
X
C
OQ
A
IC B
E
FM
X
B −1
P
CG
D
G
?H
P
H
D
H
BA
K
NG
C
O
X
B
A CB O
E
Z
I % V
3
7
?
H NG
C
N
K
P
N
(l−1/2) Γh
IT
H
]
A12 A22 − A23 A−1 33 A32 B
H
D
G
B
CG P
O
F
RB
TB
J
FM
A12 B22 + A23 A−1 33 A32 A32
A PB
RB
M
OQ
O
H
X
FM
T
5
6
(l−1/2) Γh
N
K
P
D
G
H
CG P
I
FM
P
H CG
C
H
KG
A
G
0 0 TB
N
λ
J
NB
P
CD
D
G
D
E
6
A12 B22
V
˜ K 0 B
D
F
P
C
A11 A21
O
C
R
eC
A
IH
^
IC D
7
3
G
Ke B
A
^
A
D
PD
V
H
D
CG
A OB
G
B
$H
E
˜ K
F
TB
ˆ= B C
?H
A11 A21
D
λ 6= 1 G
A EG
I
˜ = K
IA
V
A
L?
K
C
D
CG
A
TV D
I # KG
A11 = A21 0
LD
^
I
V
S˜ =
O
^
^
NB
P
CQ
*
R @
V
wh
Ke B
A
H
[
J
RB
I
TB
EI
N
O
I
B −1
RB
J
TB
D
Te
H
AH
RD
DM
PG
IH
R
vh
C
?H
V
N
O
A Ke B
C
?H
G
VB
>
λ 6= 1
K
T
F
H
O
B22
ED
C
T
L
(l−1/2) Wh
ai ,
bi ,
ds,
ds.
(A(i) v(i) , v(i) ) , ˜ (i) v(i) , v(i) ) (K G(l)
k>0
O
P
IF A
U
D
G
[
ED
K
?W BH
J
NG
C
P
QB
I
O
H
G
E
K
D
H
H U CG
Y
W
Q
A
IC B
E[
O
O
K
PD
KG
F
X
O
IB
I
PD
D
B
B
H
H
B
OG
O
P
Z
Q
O
X
?
?
EQ
N
B
H
H
KG
R
I
PQ
C Q ^ A
H
B
A EB
−1 0 T 0 Fk . (k) A33
C
TF
O
e
H
H
G
H
TJ
NG
Ce
I
A TD
P
I
F
V
ED
D
AH
D
H
+
X
^
B
D
G
B
D
D
B
[
I
E
TB
I
%
X
H
A
B
G
D
AH
D
H
D
D
R
I
C A
?A
S
?I
H
B
B
G
H
[
I X A B
H
KB L
CG
R
I
C
I
O
EQ
C
P
H
CG
Ke
CB
J
B
? h D
N
K
T
G
?H
T
O
I
O
H
PD
F
O
N
K
CB
P
7
:
6
2
3
5
8
7
3
7
5
6
2
2
6
3
D
AH
H
D
P
E
T
P
P
N
K
H
H
H
CB
P
K
PI B
KG
F
V
LD
O
RD
C
X
2
6
3
1
9
2
6
7
7
235
2
6
3
5
3
5
8
9
−1 BAM GA
U J
?
T
O
R
LD
I
KB
OB
H
D
G
CG
E
I
PB
A
IJ
S
OQ
P
IF
?AH
JH
KB
I
OB
O
OB
PD
W
Q
A
TB
K
IP B
KG
F
D
AH
H
GH
D
LD D
V
LD
C D
¯z = Fk (d1 , d2 , d3 )T
C
V
EQ
F
P
D OG
]
^
H
R
^ E
d1 = e 2
]
^]
H
D
CI B
E
O
Ke
H
CB
PD
T
H
O
E
N
O
IH
K
AH
BH
ai , bi
CG
Ke
C
D
KG
O
RD
GH
k ˆ (1) = B (1) B
H
W
?AH
NG
C
BH
CB
J
B
ˆ (l) BAM G = B
P
b TB
W
VH
OQ
OD
W
B
H
V
E
C
S
F
P
T
D
P
J
QB
D
(A(i) v(i) , v(i) ) ≥ 1. ˜ (i) v(i) , v(i) ) (K
IP B
KG
F
H
D
K
RD
AH
O
E LD
B H
VA
IA
C
0 (k) B22 0
D
X
A
IC B
@
E
D
IH
K
^
T
N
V
N
K h
?
min
min
K
A
D
AH
H
]
C
N
OA
OQ
CG
AH
D
IH
T
V
X
V
bi i=1,...,ml−1
V
E
G
?H
V
H
PJ
I
O
IH
RB
OJ
D
D
ai
C
?H
P
B
H
?
0
$
N
O
A PD
FB → F k
O
^
D
?
?
OQ
R
I
K
IH
P
ˆ (k) B 11 = Fk 0 0
(k)
OQ
RD
C
?
O
D
WH
S
OQ
Ke B
−1 2 1 (k−1) (k−1) Y (k−1) (k−1) ˆ (k−1) −1 (k−1) = A − (I − τj [B ] A ) , I 2 j=1
(k)
[
H
GH
O
A PD D
LD
F
ED
T
YB
G
H
(k)
P
IC G
H
H
G
H
e
B →B
T
Re
O
E
E
?I
D
H
X
F
i=1
i=1 mP l−1
≥
^
W
Q
R
?
SK
OQ
CB
A
%
v(i)
¯z = AlgM GM (k, b) { k=0
D
$[
A
?B
O
OQ
T
ˆ (k) B
¯z = [A(0) ]−1 b
A
IC B
I X A B
ID CG H
L
D
(k)
F
E
N
A(l)
T
D
WH
A
P
N
A→A k = 2, . . . , l
e0 = 0
?
(k)
KB
τj
K
T
ˆ (k) B 11
d2 = [B22 ]−1 z2 d3 = [A33 ]−1 z3
(z1 , z2 , z3 )T = FkT b
V
mP l−1 (k)
gi−1 = AlgM GM (k − 1, A(k−1) ei−1 − z1 ), ei = ei−1 − τi gi−1 , i = 1, 2;
}
√ −1 cond BAM G A ≤ 3 + 2 3. l
G
D
P
D
CG
O
H
H
J
A PB
O
FN
Ke D
V
I
O
^
R
LD
I
KB
O
$
%
%
$
"
$
%
! #" "
$
"
$
%
"
&
$
&"
!
(
A H
BH
H \
G
B
N
O
J
IAH
I
R? D
C
?
?
E
OQ
P
I C
?H \
O
NG
C
P
J
D QB
?
^ AH
E
I
V
ED
O
OB
T
P
*
J
B
H
AH
A
S
I
? BM
OJ
ID
P
W
A
CI B
E
S
OQ
O
Y
S
I
Z
B
G
G
B
Ke D
N
K
T
F
N
O
X
O(N 2.2 )
*
$%
(
,
,
'
'
*
-
$
U
G
G
B
A
B
H
H
GH
G
TI D
C
C?
D
H
ED OA
[
HX
PB O
O
F
?
ED
O
R
O
I
T
B IH
P
OD
F
N
O
IH
O
*
*
*
#
+
*
'
-
$
*
$
'
$
*
&
^
]
H
D
H
H
D
F
W
A
IB
I
E
G
ID
?A
VB
V
O
C
O
E
K
O
R
U PB
U
A
H
BM
OD
?
I
V
?
OJ
ID
P
[
[
]
A B
A G
B
H
A
A G
D
D
H
EQ
H
CG
G
B
DH
H
D
H
E
N
PD
I
KG
F
Ce
NA
P
C
K
H
PD
FT
C
FT
H
IH O
ZAH
O
O
FR
KD
V
H
PD
I
KG
F
E
OQ
^ E
I[
D
O[
O
K
I
N
O
IH
ZAH
V
P
NG
KB
P
CI B
EG
K
H
TJ
>
X
G
B
E
I
A
A
VA
IC
E
A
H
CI D
EG
A
CG
IB
] B
^
J
H
D
G
D
G
H
OB
IA
L
E
QD
O
IH
O
A
PB
F
DH
PG
Ce B
ZAH
C
FT
O
FY
O
CG
IB
X
X
D K X
TD
V
UR
D K X
W
I
I
I
U
D
A
AH
H
B
H
D
EV D
?H
T
ED
TD
C
E
?G
C
O
I
Z
O
Ce B
PD
P
I
CD
I
E
QD
C
R
LD
C
N
^
K h
X
D
B
H
T D
YB
?
P
W
I
IA
C
ID
T
YB
?B
O
C
R
RB
I
J
D
TB
O
GH
H
D
RD
C
N
O
IH
Z
[
[
[
[
X
]
H
W
I
I
I J
D
EV D
?
TQ D
C
E
I
J
D
A
B
D
A
D
U AH
E
C
I
L
KB
OQ
B
H
OD
W
IA
C
ID
?
C
Ce
I
D
D
CG
V
O
YD
E
W
I
I
I
EV D
?H
T
@
?
[
[
]
AH
BM
AH
H
H
H
B
B
CG
IB
D
B
B
D
J
H
H
H
H
PD
TD
C
E
PD
OG
D
B
IH O
Y
A
NA
V
O
O
C
CQ
O
OD
EV D
[
D ?[
O
Ke B
EA
O
?@
D K X
N
O
IH
K
H
TJ
A
TQ D
C
E
OQ
EIA
C
Y
?
C
NG
C
R
CG
P
F
CB
A
f
]
AH
A G
U H
C
K
D
J
I[
O
R
ND Q
?G
F
N
O
IH
O
A
D
PB
F
LD
O
\
>
H \
H
Ke B
W
O I
T
I
O
IH
ZAH
NG
H
I
CG
B
D
ID
T?
YI
?
CD
V
CG
A
ID
O
P
Q
AH
VH
O
PD
I
KG
F
NG
C
BH
LB
KB
O
S
QD A
CD
?B
K
^
A B
B
H
D
D [
H
IH
T
L
CG
IB
A
D
B
B U H
TJ
A
WH
OB
IA
L
O
F
T
G AH
FM
CG
I
B eC G
D K X
TD
V
KI B
OB
?
Ce G
UV
U
B
D K X
TD
V
K X
C
CG
IB
D K X
TD
V
O
O
IH
K
H
TJ
C
J
FM
UC
^
G Ke D
V
IH
RB
O
I
^
H
J
B
G
D
A
D
H
X
H
A
J
BH
H
J
B
B
D
H
[
h
X
O B
CG
IB
D K X
O
C
RG
A
H
IH O
IB
C
P
FE
A
TD
IB A
E
O
IH
O
A
PB
F
^ G V
K
I
IB
?
C
QD A
CD
?
IE B
E
C
I
L
KB
IE
PQ D
?
Ke D
C
N
K
CB
C
R
I
E
EI D
?
D
H
D
O
E
O
E
C
EQ
H
H
H
H
PB
K
L
C
H
KG
EQ G
S
OQ
TD
U
[
?
C
>
D
^
H
A
D
BH
G
D
B
H
D
LD
E
P
Q
I
V
W
I
IA
C
Ke
I
T
AH
G
D
CG
I
V
P
N
O
IH
R
C
^ G IB
B
D
OJ
^ H V
^
]
A PJ
I
S
V
OQ
B
CG
R
[
\
H
H
A
D
H
[
H
H
U
BH
G
D
A
H
Pe B
IE
N
O
IH
PB
F
D
LD
E
P
Q
I
V
KB
OB
?
P
N
O
IH
R
C
H
IC G P
TD
E
I
^ H
P
O
IH
O
A PB
F
I
I
C
N
K
T
ED
O
OD
W
I
TB
IA
C
P
CG
IB
D K X
C
NI
D
OD
^ NA V
PD
L
[
f
A H
B
D
B
D
G
D
^
]
U D
P
FM
^ C
J
Ke D
V
IH
O
S
QD A
CD
?J
IH
LD
I
O
E
^ PD
TD
C
E
O
LD
Z
Ke D
H
H
B
B
[
A
A
B
H
U J
P
E
V f IB
H
B CG
D K X
TD
VB
K
H
TJ
A
WH
OB
IA
L
O
N
PD
I
KG
F
OQ
Ke
C
X
X
X
G
B
H
J
D
B
U
A
C
B
N
CG
D
H
O
D KD I
V
T
NG
C
BM
J
H
TB
K
TD
E
N
O
IH
O
T
IH
[
=<X
N
K h
B
K X
D
TD
S
V
OQ J
] S
N
K
T
CG
IB
D K X
W
I
I
I
EV D
?H
T
TQ D
C
E
N
OB
I
PJ
O
H
OB
I
PD
J
Ke D
V
I
eC
O
I
]
]
^
U
0
^
B
D
G
B
H
G
D
D
BH
H
G
D
G
A
P
K
IH
H
TD
E
OQ
R
KJ
IB
FN
J
Ke D
V
I
Ce B
P
Q
I
VD
O
CB
P
?H
TB
NG
C
TF
QD A
C
=<X N
O
YD
E
E
C
I
K
L
OB
CG
H IB
R
OQ
PB
CG
O
O
IH
YD
KJ
A
U
^
]
A
A
B
D
A B
D
D
H
B
H
CG
R
C?I D
O
Ke
CD T
I
LD
K
WH
N
O
IH
K
H
TJ
VW
I
O
I
V
U
[
[
H
P
D
G
H
D
?[
O
IH
K
P
N
S
?I
R
J
I \
I
CG
H
YD
O
E
C
FH
CG P
F
RB
I
TB
J
A P?I D
I KF
E
X
X
^
J
B
G
D
D
H
H
G
]
A
BH
G
B
H
H
D
O
U D A
I
O D
IH
ZAH
B
TD
C
E
?
H
Ke D
C
O
NG
C
?A
E
A
J
B
GH
D
D
H
H
AH
CG
Q
Ee
A
OA
E
I
V
P
CG
IB
I
\
D
J
D
K X
W
I
I
I
EV D
?H
T
P
+
0
H TQ D
C
V[
C
IAH
^ N
S
I
FM
J
Ke D
V
I[
LD
I
KD
SO
C
PI D
TD
C
E
Ce G
O
F
V
?
BM
PD
C
P
QB
O
?
C
X
^
AH
D
H
B
NG
Ce
N
O
H
B
H
KD
V
PQ
C
FT
N
W
IA
C
NI B
T
YB
?D
?
O
TD
W
IB
IJ
KB
I
O
Ke
C
[
X
,
,4
G
D
U
D
D
H
B
PD
T
KG
V
N
K
T
I
C
R
G
WH
A
V[
OQ
B
H
OD
W
IA
C
IH
Y
CD
C
Ce B
PD
J
Ke D
V
I
^ G
'
%*
$
7 6
'
5'
'
+
]
B
GH
D
B
X
A
O D
YD
E
ED
C
I
H
B CG [
D K X
TD
V
Ee
P
O
M
IH O
J
IB
A
D
OG
K
LD
G
EB
A
D
G
"
0
,
-
&
3
,
0
,
3,
%
-0
'
.
7
'
+
20
.
,
.*
U
A D
D
B
A D
D
G
WH
V
V
TQ D
D
PG
O
IH
VH
CG
Ce
K
IH
T
AH
G
D A VB
O
RD
CB
CG
ST
A
D
G
G
WH
V
IE
X ]
^ A
A H
A G
B
U G
eD
PB
O
EQ
H
G
H
CG
I
Z
D PQ [
O
O
AH
CD
EG
N
O
IH
ZAH
N
K
T
E
I
V f B
]
G
A B
D
H
BM
B
D
D
H
EB
RB
TB
J
TD
V
V
O
Ke
TJ
A
NG
C
P
CQ B
OQ
OB
T
S
O
I
P
F
Ke D ?
?D
V
X
X
AH
J
D
P
K
^ N
OB
I
P
K
IH
KB
A
B
G
A VB
D
G
D
V[ G
OQ
O
H
CG
H
H CG
C
BM
LB
K
T
V
W
I
I
I
EV D
?H
T
X
]
A
A
A
H
D
B
H TQ D
C
O
IH
H
KG
I R
PQ
W
IJ
K
IH
K
B A KB
VG
OB
J
PG N
E
I
V
FL
[
[
[
h
]
H
G
A G
G
AH
PD
F
S
OQ
O
IB
Z
E
M
EI D
OD
?
FM
O
O
H
CG P
F
^
H
J
H
M
G
B
E
FR
I
KD
V
EQ
N
O
IH
R
K
?
I
H TD
C
E
A CB
CB
S
OQ
Ke
C
I
KG
R
I
PQ
D Ce G
O[
O
]
H
B
G
G
B
G
RB
TB
J
TD
NV
K
T
F
TD
C
E
Ce B
J PD
Ke D
V
U
K
IH
O
NB
P
CQ
* R @
I
h = 25−1
.
*
'
#
N
,
'
'
]
O(N 2 ) !
# ! * $
*
^
M IH
R
K
?
U
X
X
I
I
G
H
B CG [
K X
D
TD
S
V
OQ
GH
D
B
A PB
Ee
P
O
F
?
B eC G
K X
D
J
H
G
G
^ A
GH
H
ED
LB
ZG
C?I
N
K
T
E
A
IB
V
O
^ B
?
] B
F
B
B
TD
C
E
A IB eC
CG
ID D
O
YD O
P
C
I E
+
U
\
B
B
D
J
D
G
H
[
?H \
?
R
C
IB
?A
V[
WH
H
G
G
U H
R
I j KD
] B
F
B
ED
TD
C
E
RB
TB
IA
OB
TD
S
V
OQ
Ke B
?
K
O I
IH
ZAH
O
OB
L
NG
Ce B
PD
C
?
T
I
C
H
YD
E
RB
TB
J
TD
PD
OG VD
P
D
CG
^
A B
D
AH
D
AH
H
B
H N
K
CB
P
K
PI B
KG
F
V
N
W
IJ
KB
I
A ED
D
CD
?
P
CG
H
ID
OA
EJ
Z
Ke D
[
X
X
G
D
D
H
D
E
H
G CG
I
N
O
IH
ZAH
B
G
G
WH
A
VD
LD
O
B
H
OD
W
IA
C
NI G
L
N
TD
S
A
CG
Q
N
O
IH
X
]
J
D
G
D UA
D
CG
V
N
K
T
OQ B
J PD
Ke D
V
I
Ce
Q
FC
LD
E
W
I
I
I
EV D
?H
T
TQ D
C
H
X
104
A
H
H
J
B
AH
H
T?
NA
V
B
D
D
LD
O
O
FE
E
H
G
H
AH
A
M
CG
IS
OQ
O
Y
O
IH
ZAH
?
O
O
E
I
V
∼
h = 50−1
^
]
D
H
M
H
D
B
B
G
G
B
D
H
B
H
D
D
LD
E
TD
SI
O
N
O
J IAH
I
G R? D
C
E D
D
G
H
IB
?
C
O
X
B
WH OQ
?
R
<
G
H
^
A
[
X
X
E \ IH C
I
ZH
F
V
eC
Q
C
YD
E
F
TD
C
E
?
O
T l
C
IN
EB
V[
OA
C
e
EV D
?
$
&"
!
&&
&
"
$
! #" "
Λ = {η ∈ H (Γ) : η = v|Γ , v ∈ V }. = {v ∈ H (Ωi ) : v|∂Ωi = 0},
29
]
D
∀ µ ∈ Λ.
]
i = 1, 2,
C
R D
U
V
D
NB
B
KG X
^
D
Ce
C
IH
CD E
O
RD
CB
D
H
U PD
A
IEA
O
OQ
P
C
IF
?A
1/2
]
^ H
B
D CG
CG
^
H
FR BH
KG
E
Z
H [
EA
O
G
AH
X
O
O
K
F
I
O
D
H
G
D
H
[
B YkX
OG
I
$H
O
IH
RB
D
D
DH
OJ
O
Ke B
W
IH
V
F
A CB [
D
H
C
D
K
I ]
C
I A
V
D
O
I
W
I
FO?
J
PD
T
H
D KG
P
CG
OB
A
CG
A
ED
[
\
I
B
P
C
E H
H
Γ
U KD
TD
A
?I
[
OD
]
D
^
FT
H
X
I
FR BH
KG
>
A
G
B
F
W
IA OB
L
O
C
[
H \
H
TD
SI
PQ
C
O
S
B
R? D
C
P
]
Ce B
RB
X D
]
G
D
G
?H
T
O
eC B I
?F
V
E
B
A
H
H
NG
P
eC G
N
I
OB
A PD
LI
C
O
I
Ce G
B
D K X
T
LD
^ HB ?
FR
KG
>
G
A
A D
H
P
H
CG
A OB
CG
PV
O
IH
T
H
D PJ
I
V DH
OA
KB N
?
CG
IB
K X
T
E
FT H
X
g
Ce G
F
^
P
B Ce G
K X
D
]
U
H
V
N
D
K h
N
O
IH
O
A
H
PB
F
O
IH
ZAH
NG
C
IH
T
L
'
5'
'
+
,4
KF
EA
V
H
D
D \
A
A D
B
D
D
P
H
CG
A OB
CG
PV
EA
O
C
Q ^ W
O
I
T
B
D
H
IH
NA T?
V
C
EB
T
I
BH NG
C
J TB
,
0
*
0
-
&
3
,
0
,
3,
%
-0
2
,
.*
&'
1
'
%$
-
0
%$
1
H 2 (Γ)
EG
I
a1 (u1 , R1 µ) + a2 (u2 , R2 µ) = (f, R1 µ)Ω1 + (f, R2 µ)Ω2
C
ui ∈ V i
OB
g
T
H
η (s) ds |s − xb |2
I
]
]
B
K
2
A
PD
O?
WH
?
eC B
B
K X
A CG
V
P
∂Ω H10 (Ω)
V
A CB
CG P
H
D H
F
Γ
V
O
B
Γ
G
D
∀ vi ∈ Vi0 ,
C[
IB
O
WH
S
J PD
Γ
X
D
B
IB
U
]
H
1
2 H00 (Γ)
27
H
RB
D OJ
OA
η (s) ds + |s − xa |2
Z
6
L
O
P?I
Z
Ke D
X
2
H
D
J
X
H
O
N
G
H
H
D
TD
Γ Γ
$
B
P
K h
V
EI
FT
A
[
H
BM
NV G
I
? GH
B
P
Z Z
D
ai (ui , vi ) = (f, vi )Ωi
ED
A
IG
Z
Λ
J
D
H
]
PD
T
H
[
U
C
O
ID
E
H AH
O
= kηk2L2 (Γ) +
Ri (Ri η)Γ = η AH
E
P
CG
H
O B KG
S
H
H
^ H
T?I B
K
D
[
H
B
VH
O
P
T
O
B
X
$
Γ Γ
H
R
EI
N
OB
A
CG
D
EQ CG P
E
A OB
Z
FE
B
L
O
H
H
H
FH
O
E
EI
Z
O
Ke B
T
P
C
J
A IB
D
I
FM
PD
I
KG
A
D
O
Ke B
T
>
U
B
D
O
IH
O
A
PB
F
E
^
OJ
D
D
= kηk2L2 (Γ) +
Z Z
^
H
G
A CG
PV
?I
H
D
^
H
C
^
B
Ce G
D K X
]
^
M
E A
A
G
H
IB
V
O
^ E
I
I
KG
#
A
H
D
D
H
H
ID
EA
OA
N
KF
[
\
BH
[
QFC
J O
H
U
B
TB
G PV D
^ ED
CG P
YD
O
E
E
D
PD
T
H
B KG
P
CG
TB
AH
A CG
FV
TD
PD
VD
[ G
E? B
L
K ] B
H
A OB
D K X
C
B TJ
A
WH
H \
H
NG
C
N
K
P
N
C
O
I
B OB
IA
L
O
P?I
IG
Z
]
H
FR BH
KG
H
H
FM
T
KG
E
C
IH
E
CD
>
Λ
C
R
T
F
PV
D
D
Γ
Z
[
KG
RB
TB
EQA
O
N
K
T
C
R
]
Γ
Ω1
]
B
^ D
?
ai (u, v) = (∇u, ∇v)Ωi Λ Vi C
C
T
NI
P
T
O
C
O
KB
G RD
F
0?
P
P
^
+
^
J
TD
V
P
I
?
^
C
R
K
[
R
^ E
IY D
KD
A
IC D
Γ ∩ ∂Ω 6= ∅
]
K
H
K
L
N
K h
]
B
]
B
RB
TB
J
B
+
B
TJ
A
WH
OB
IA
L
G
AH
1 2 (Γ) H00
B
^
PQ
D
\
O
B
T
LD
B
Y ^ H
?
C
E
YI D
TV
kηk
TJ
P
B
B
P? D
OB
CG
UA
I KF
EA
D
G
H
B
$
O
P?I
IG
Z
IE
N
AH
KD
TV
ED
EG
1
∂Ω1
]
]
[
O
B
CG
D
X
Y
B
D
O
]
]
G
B
H 2 (Γ)
A
J
G
V
FM
B
KG
E
?
C
O
YD
E
B
P
H
^
CG
I
kηk
8;
NI
U
u1 = u 2
I
FM
H
%
]
D
O
]
Γ
e
T
FM
C
R
I
ED
V f B
LD
KB
OB
]
O
B
PD
I
]
O
IH
G
[
H \
xb
O
IH
Y
U
RB
I
TB
?
O
G KG
F
B ZAH
a1,1 = a2,2 = 1
T
H
T
H
∀ v ∈ V,
]
i = 1, 2, KG
D
H
B
^ DH
?
C
]
]
J
D
Ke B
EA
O
C
BH
RB
EB
RB
TB
J
TD
V
(·, ·) = (·, ·)Ω
i = 1, 2, O
Ke
C
I O
D
g
P?I D
OB
[
X
X
Ω2 , ∂Ω2 \ Γ,
1 KD
V
T
E
a(u, v) = (f, v),
H
C[
IB
O
D
D
CG
V
B
OJ
D
D
−∆u2 = f u2 = 0 ∂u1 ∂u2 Γ, = ∂n ∂n
'
1 2
u∈V
KG
= {v ∈ H (Ωi ) : v|∂Ωi \Γ = 0},
1
Γ,
T
]
[
$
RB
I
I
FM
P
H CG
C
KG
N
u1 = u 2
P > H
]
J
TB
P
D
G CD
Ω1 , ∂Ω1 \ Γ,
$B
N
K
T
= f = 0 ∂Ω ∂Ω1 ∂Ω2
Vi0 ¯ u ∈ C (Ω)
Vi P
CG
A OB
]
K h
H
−∆u1 u1 1
Γ Ω, ∂Ω.
]
D
Ce B
T
L
Ω1 Ω2 Ω
L2 (ω)
]
A
V = H10 (Ω), a(u, v) = (∇u, ∇v). CG
F
P? D
P
−∆u = f u = 0
H
O
n ω⊂Ω (·, ·)ω Ω Rd
T
L
Ω
VH
I
Γ ∩ ∂Ω = ∅
1/2 (η(s) − η(t))2 dsdt . |s − t|2 xa
a1,2 = a2,1 = 0 Γ1 = ∂Ω (η(s) − η(t))2 dsdt |s − t|2
.
kηkΛ ≥ kηkL2 (Γ) .
$
$
$
&"
!
&&
&
"
$
! #" "
&&
&
"
#
$
"
#
%
&
$
&"
"$
"
27
:
]
∀u2 ∈ V2 ,
Ω1 Ω2
−∆(Hi ϕ) Hi ϕ Hi ϕ
N
]
^
CG
IB
J
[
X
= 0 = ϕ = 0 Ωi , Γ, ∂Ωi \ Γ.
D
M
[
X
C D
CG ]
P
UP
H
B
U J
D ^
A
GH
W
I
NI
FK
OB L
CI
GH
>
D
CQ
OB
CG
OD
^ ED ?
T?
NA
V
E
QD A
D
A
UN
FK
OB L
CI
S
D
AH
OQA
N
FK
L
S
OQ
EA
B
U
?
T
CG
H
ID
OA
EJ
A
CD
C
OB
CG
B
D
Vih
KD
*
]
C?I
C
P
]
PV
D
P
CG
A OB
CG
A
CG
[
UO
K
T
V
N
H
H
AH
D
K h
^ A
A
UO
P
G
D AH
H
CG
T
O
VH
C
FH
T
H
H
D
H
KG
O
C
E
V f IB
] B
^
D
G
P
CD
Ce G
E
I
PB
H IJ
O
C
BH
RB
D
D
D
GH
OJ
C
?
C
S
OQA
H
ED
C
OB
CG
OD
^
B
A D
P
CG
OB
A
CG
V
CG
H
ID
OA
EJ
A
CD
U D
D X
N
K
T
Uhi h
CG
X
C
NB
P
CQ
I
G
G
B
R @
] F
K
F?
PD
KJ
OD
?
D
PG
S
O
PB
?
]
M
A
H
D
H
N
O
IH
Y
KD
A TD
V
QD A
CB
V
OQ
GH
B
RD
C
O
NG
C
N
O
U G
I
PB
IJ
H f
^
A D
V
P
B
CG
OB
A
CG
V
A
A
H
D
A
N
O
IH
Y
KD
TD
V
QD A
CB
V
ED
C
I
D
H
H
D
VH
OQ
C
O
E
K
O
R
U TD
U
H
B
OD
S?
I
O
P
CG
A OB
CG
A
VD
O
H
J
D EB
O
Ke B
E
[
[
]
^]
H
H
H
D
AH
D
H
BM NG
C
FR
KD
V
E
C
LD
I
KB
OB
OQ
C
O
E
K
O
R
U D A
U
H
OD
j
\
6
7
5
:
3;
3
7
3
3
3
7
7
5
9
27
7
2
5
3
6
4 ;
Ω1 Ω2
]
∀µ ∈ Λ,
IB
D
G
B
ED
G
\
D
G
Ce G
E
I
O
T
P
IH
RD
H
D
J
H
EB
J
D
H
3
7
8
F
5
2
6
3;
3
3
7
6
3
67
7
6
5
5
3
7
:
H
D
H NG
C
N
O
KD
V
PQ
eC G
C
P
P
]
1
6
8
5
3
:
9
'
6
65
6
67
7
6
5
;
2
u i |Γ
R
OB
A
N
K
T
?
?
AH
F
OD
?
N
K
T
CG
OQ
B OD
W
I
E
G
O
^ NA
A
B
D eC G
E
h R12 , Rih
K X
D CG
A
VT D
S
]
^]
EQ
^
PB
H
D
IJ
O
?
C
GH
ID
?A
I
P
?
OA
P
^
Ω i Ri : Λ → V i
D
TD
VH
B
H
H
D
C
B
KG
I
S
B
ID
O
CG
>
G
G
B PD
H
PD
L
N
K
T
D
Ce G
F
] F
A OD
D
CG
FM
X
D ED
C
V
H
kui |Γ kΛ ≤ C˘i kui kVi ,
^ E
O
I
ED
V f B
P
T
V
OQ
C
O
E
K
D
H
H CB
Y
R
I
CD
C
NG
I
A FM
R
NG
BH
VB
VJ
D
Uhi
K
K h
U
O
R
C
B
K
A TB
O
PB
IJ
H
]
C
K
I
O
^
LD
C
H
A ED
j
A
D
H
CB O
P
O
Γ
O
D
H
O
NG
Ce B
K
I
Y ^ H V
O
IE
G
ID
?A
TF
S?
OQ
B
kRi µkVi ≤ Cˆi kµkΛ
C
IB
OD
S?
G
QG
E
H
O
I
H
AH
∂Ω2
H
A
Ω ∂Ω1 6= Γ ∂Ω2 6= Γ 1 2 Λ = H00 (Γ) Hi ϕ ∈ V i O
NI
K
IB
FT
X
OQ
O
VB
V
∂Ω1
IH
T
?
C
CG ^ P
A OB
EQ
CD
[
Ce B
J
H
]
O
H
IH
T
YA
P
FC
E
H
^
C
YI
K
TB
^
H
N
K
T
O
Ke
PQ
D
H
CB
PD
T
K
]
]
H
N
W
I
FO?
NG
C
N
O
KD
V
>
]
2 a(R12 u2 , R12 u2 ) ≤ C12 a2 (u2 , u2 ).
]
]
I
D
D
K X
P
F
RB
D CG
OB ^ H
^
W
I
]
O
IH
Y
KD
A
D
TD
V
D
H
A
ui ∈ V i
^
^
D
H
B CG [
D K X
T
C
^
C
R D
]
]
]
3
7
D
H
E
YI D
KD
OA
P
Ce G
F
5
6
7
3
^
F
6
B
B
+
H
B
C
B
OB
CG
OD
NB ?
?
C
C
FH
CG P
2
6
18
?
?
?
U
V
D
CD
NB
NG
PB
IJ
C12
$
HB
C
]
J
A
G
B EG
A
$
I
V, Vi , Λ
B
FR
KG
[
A
TB
E
S
H
Cˆ12 , C12 , C˘i , Cˆi
P
Y
KD
TD
R
C ? ^ B
ED
G
H \
Γ A
ED
C
P
PD
T
A
H
G
I
R
E
]
]
L
Uh , Uhi , Λh
O
G
H
C
O
IC D
EG
V
G
J
U D
PV
CG
OB
A
H
F
P
C˘i Cˆi
V DH
?
R
OD
H
D KG
P
CG
OB
I
?I
D
Cˆ12 , C12 , C˘i , Cˆi
I
EA
L
E
A
CG
Ω1 Ω2 B
IH
G
H
A
H
CG
S
V
OQ
C
O
J
I
P
Ω1 Ω2 .
K
D
]
O
A PD
F
ED
O
GH RD
C
EI
?
Ke D
E
A
YB
?
T
e
O
∀ vi ∈ Vi0 .
B
Ω R12 : B
Ω1
T
A
Γ ^
E
H
{(f, v|Ωi − Ri µ)Ωi + (f, Ri µ)Ωi } = (f, v).
AH
V
NG
C
P
N
A TD
P
I
V
NG
eC
IE D
B
I
EI
G
H
B
K
D
{ai (u, v|Ωi − Ri µ) + ai (ui , Ri µ)}
D
?
OJ
D
D
?I
R
O
O
BH
Rµ ∈ V
V
^
CG
IB
K X
D
TD
NV G
O
YD
E
U
H
G
H
?D
V
C
YI
K
IY D
KD
]
+
I
Ω1 Ω2
Ω1 Ω2
O
Ω2
]
H
BM
SO
I
H
P?I
R
OD
FR
KD
V
E
K
IH
T
AH
TB
Γ
I
EI
C
NG
IG
^
A
]
]
VH
]
ui = u|Ωi
G
[
H \
ED
A
? GH
O
IG
C
M
?
D
O
I
V
u V u1 = u 2 v ∈ V µ = v|Γ ∈ Λ v|Ωi − Ri µ ∈ Vi0
6
2
6
3;
H
AH
D
D OkX
Z
N
^
V
N
W
g
V
a(u, Rµ) = (f, Rµ)
7
5
3;
3
kR12 u2 kV ≤ Cˆ12 ku2 kV2 3
C
O
H
VH
O
F
D
I
]
R1 µ R2 µ
3
I
I
N
N
K
T
H
H
?
Ke D
LD
>
u1 u2
7
3
7
K A
P
TD
PD
I
]
EQ
E
O
i=1
H
KG
F
H
2 X
3
P
T
B
eC B
A
IH
O
E
X
FO?
EI B
W
N
K
H
\
I
B
TJ
A
A
D
G
PD
FK
I
EA
C
G
?D
u=
6
B
H
A
VH
KD
D
I KD
M
Rµ =
O
B
X
] B
LD
C
N
O
IH
T
YA
H
*
K
ui = u|Ωi
3
OB
C
J
IB
A
P
H
CG
Ke
CB
J
B
D
P
C
i=1
2 X
3
67 IA
L
EI
Ce G
F
@
?
T
J
P
a(u, v) = a(u, v ± Ri µ) =
7
7
7
6
D
T?I B
K
L
O
]
=
5
U
G
RD
F
?G
^
HX
a(ui , vi ) = a(u, vi ) = (f, vi )
Cˆ12
5
'
?
V
3
Ω
7
3
7
8:
V2 → V
7
2
5
3
;
Ω2
6
4
Vi0 ⊂ V
Γ
R12 , Ri
ϕ ∈ Λ
$
$
$
&"
!
&&
&
"
$
! #" "
&&
&
"
#
$
"
#
%
&
$
&"
"$
"
3
7
6
3
29 :
5
9
'
D
uk+1 1 E
O
H
IH F
O
R
I
7
"
0
H
W
IA
C
B
H
TI D
C
E
[
B
B
H
H
U OD
W
IA
C
CI
FT
D X
A G
B
B
O
TD
I
E
A
IC D
EG
EQ
J
B
H
U OD
W
IA
C
IM
O
IH
FR
I
?
C[
I
H AH
]
D
H
H
AH
D
Ce G
O
O
K
T
V
]
H
N
H
H
NG
C
N
K
P
N
EQ
E
K
E
C
H
D
H
M
Ke B
EA
O
O
ZH
O
P
NG
N
K H
P IC G P
T
KG
B
D
E A
A CB
D
H
D CQ G
A
[
^
J
M
W
Q
OB
IA
L
CG
B IB
R
O
e
Z
K
I
NG
S
I
N
FK
OB
W [
\
^
]
A
B
G
B
A DH
D
CD
?
?A
TI
I
P
CG
O
H
H
J
B
A PB
O
I
C
OB
CG
>
+ 1)C˘i2
Y
K
IH
P
'
'
C
X
H
NB
O
Ke
V
I
FO?
1
uk+1 2 ?
A K X
I
H
]
KB
RB
O
V
QD
C
R
F
H
OD
D
D
H
H
AH
H
D
A
D
$B
A CB
V
Ce G
O
O
K
T
V
FM
]
^
S
OQ
V
V
CI
E
E
I
E
FR
A
A
H
D
M
H
G
BH
KD
J
B
A
O
O
D
I
IH I
Z
CD
D
G
ED
O
OB
T
P
NB
E
I O
I
X
D
$B
A
CB
T
U
A D
Ke D
PJ
I
NV
K
v = Hi λ
I
FM
E
OB
I
]
CG
IB
K X
^
B
I
O
IH
K
N
)2
I
V
E
N
O
J IAH
?H
RD
C
S
D
V
T
LD
C
AH
O
H
µ
N
B PJ
O
TD
VAH
F
OQ
C
IY D
KD
T
0?
Ke
C
YI D
Γ
Z
O
Ke OQ
H
.$
-
(
H
ED
C
?
^
K
G
(C(Ωi
O
Ke B
A
CB
C
W
I
J
D
V
D
T
D
H
H
KD
V
I
∇Hi µ · ∇Hi λ dx =
IH
C
P
[
( 0 _
$
R
KJ
I
I
PQ D
CD
?
V
P
OA
P
X
]
A
D
ED
$
E
I
u = Hi µ
Y
H
5'
G IB
A
EV D
I
M
S
CI G
T
E
K
I
P
[
D
L
D eC G
O
Ce G
F
≥
RB
O
G
G
G
J TB
I
TB
"
OQ
CG
?H
EQ
D IH O
Z
H
Ωi
K X
A
D WH
A
]
C
I
T
PD
O
Y
e
%
∇Hi µ · ∇Hi µ dx ≥
I
I
V
E
TD
B
V[
0
H
H
PJ
D O X k
A
A
H
]
B
T
U D
H FM
T
KG
Ce
PB
I
AH
D
CG
T
V
O
YD
E
^
]
0
N
O
B
IH
O
A
H
PB
F
O
IH
ZAH
0
]
O
T
H
KG
B
T D
YB
[
? > @
O B
P
]
G
G
B
P
B
OD
F
N
O
IH
O
A PB
F
A D
O D
Ke D
PJ
I
NV
[
K h
]
$
R
E
J
AH
H
RB
I
OJ
D
D X
P
W
I
I
A B
D
E
FR BH
KD
V
O
I
KQF
EA
M
eD
ED
V
\
g
5 3
7
6
7
3
H
Ce G
F
O
IH
ZAH
U
A D
FO?
?I
R
I
B
H
H
G
H
OD
EA
L
OQ
Ke D
PJ
I
V
\
u, v
IS
D
OQ
OD
%
]
SK
A
R
IC
CD
D
?
C
FO?
Ωi
Z
O
O
'
3,
A
+
V
U
Z
CG
N
O
IH
0
e
H
E
G
D
H
∂ λ Hi µ ds = ∂ni
^
H
λk
B
%
H TD
C
E
VH
E
I
Ω1
W
IA
O
A PB
TD
S
V
S
OQ
LD
\
O
IH
Y
KD
TD
A
∂ Hi µ ds = ∂ni
PJ
H
F
N
H ^ TD _ I
C
E
[
O
IJ
I
n
C
D
I h
U OB I
E[
H f
OQ
H
]
C(Ω)
I
I
E
O
O
GH
G
H
V DH
?
R
Γ
µ, λ Z
Ce G
A
S
[
B
D
OD
I
EA
L
C
R
D
Ωi
F
0 λ0 ∈ Λ I
]
Γ
IH
H
C
N
K
T
P
G IAH T
D
D
H
FH
T
KG
^ C
µ
CG
C
FT
B
v|∂Ω\Γ = 0
Z
]
B
D
^
O
EA
FO
[
[
^
CG
IB
Ce G
F
Γ
Z
ZAH
N
D
A
S=
∂ (H1 − H2 ) : Λ → Λ0 ∂n 1 Λ0 = H− 2 (Γ) IH
∂u2 ∂u1 = . ∂n ∂n
V
K h
Ωi
O
*
[
H
]
D
G
H
KD
V
?
R
C
IH
LA
H
D
]
Hi λ
I
2 U
Sλ = χ
6
X
O
A PB
∂ ∂ (H1 − H2 )λ = (T2 − T1 )f. ∂n ∂n
CB
H
]
GH
H \
O
IH
O
A PB
F
DH
D
D
O
O
(∇Ti f )2 dx.
CD
O
IH
O
PB
$
A
H
B
B
OA
I
C
R
S
B
B
O
ui = u|Ωi = Hi λ + Ti f.
;
A
H
D
VB
A
D
C
O
I
O
D
H CB
A
D
^
?I
Ke B
P
Ke
H
CB
PD
D K X
TD
Γ
6
[
P
F
H
B
PD
?
M
CD
V
V
Ce
FR
I
KD
0V
λ
D
H
[
C
T
L
D
eC G
O
A
C
E
H
CG
O
H
V
Ωi , ∂Ωi .
6
V
O
CG
D
PG
^
K
BH
P
Q
AH
VH
Ke
WH
I O
EI
A PB
T
Ti f ∈ H 1 (Ωi )
3
23
ZH
O
P
NG
PD ] B
T
BH
?H
C
U AH
E
FR
O
NG
E
B
BH
, K
u
6
7
J
P
I
C
A
B
FH
J
C
J
Ωi Ωi
Z
E
FR
I
E
I O
I
VH
O? B
F
A
KD
V
B
Z f B
TB
(∇Hi λ) dx +
FM
M
\
D
]
NA
TD
Ke D
[
K
2
3
K
?H
C
B
U AH
O? B
F
[
H
D A CB
V
E
K
IH
G
VD
W
I
λ
3
F
I
[
EA
D
D
D
∂ (T2 − T1 )f ∈ Λ0 ∂n
N
K
P
Γ ED
D
H
E
H
OA
B
Ce G
R
FM
T
LB
P
v|Γ = λ
N
C
V
^ E
Z
H
O
H CB
A
D
A PB
l
A
−∆(Ti f ) = f Ti f = 0
29
^
?
Ke B
D
O
Ke B
IH
O
A PB
F
V
P
AH
K
Ωi
:
∂ Hi ∂ni L2 (Γ) EA
O
BH NG
C
P
V
(∇ui ) dx = Z
8
J
T
P
D
OJ
H
B KD
V
J
QB
]
V
I
2
H AH
ni
67
R
E
l
O
χ=
1
D
RB
I
OJ
X
G
I
v ∈ H 1 (Ω)
1
M
E
e
H
Z
]
P
u
]
?
O
IH
Z
S u
f ∈ L2 (Ωi ) Ωi Z ∂v ∂u u ds−(∇u,∇v)Ωi =(u, ∆v)Ωi = 0 = (∆u, v)Ωi = v ds−(∇u, ∇v)Ωi. ∂ni ∂ni Γ
∂ Hi λ ds ∂ni
1 kHi µk2Vi C(Ωi )2 + 1 kµk2L2 (Γ) .
∂Ωi
S
θ>
$
$
$
&"
!
&&
&
"
$
! #" "
&&
&
"
#
$
"
#
%
&
$
&"
"$
"
= (I − θ(I + ( D
SK
A H
B
G
T
P
I
P
H
Ce B
J
CG
IB
D K X
TD
V
I
I
O
O
SK
A
B
H
I h
TD
S
H
D VAH I
V
H
I h
I
E
I
PD
∂ H2 µ, µ ∂n2
U OB I
AH
H
B
D
∂ H2 µ, µ ∂n2
E[
H f
[
VA
D
B
G KG
F
B
RB
J TB
H
D
H
H
AH
D
D
G
J
D IH O
Z
CD O
CI
BH
?H
C
PQ
Ce G
O
O
K
T
V
NB
O
Ke
C
IY D
KD
]
[
NB
P
CQ
$
W
B
IH
?
f
OB
E[
BH
AH
H
RG
I
V
∂ ∂ H2 µ + H1 µ, λ ∂n2 ∂n1
IA
Y
KD
A TD
V
O
WH
D
J
B
H
H f
A
NB
P
?
BH NG
C
O
YD
D
CI
FH
T
H
KG
]
A
E
H
PD
I
KG
F
G
B
RB
TB
J
≥
A CB
E
F
]
2 C12
H
H
]
H
D
H
Ce G
O
O
R
IA OB
L l
U H
Y
?I
SK
I h
U OB I
BH NB
P
A
H
ZAH
LB
C
kI − θKk ≤
D
EB
D
(∇H2 µ) dx =
FV
D
O?
WH
D AH
2
I
EA
B
E
H
Ω2
Ce G
E
O
D IH O
Z
C
X
A
D
K X
I
ID
Z
I
H f
B
H
E[
W
I
IA
C
IH
I OB
?
BH NG
C
ZAH
H
LB
TZ D
TZ D
V
ED
PA
H
V f B
∂ ∂ H2 µ + H1 µ, µ ∂n2 ∂n1
O
B
O
B
D
H
Ce G
O
O
Y
NA
D
D VG
EB
CI
J
FH
T
H
KG
]
A H
H
G
H
]
]
EQ
E
K
F
K
I
P
P
O
K
T
H
H
AH
D
VD
O
Ke
C
YI D
KD
V
OI
R
IC
E
E
I
H
G
L2 (Γ)
]
G
H
O
CD
I
KB
A
H
B
E
V
LD K
B
G
D
PJ
O
C
I
O
TD
H
A
D
B
ED V
C
P
O
P
O
B
^
G
A
H
AH
G
H
D
A
D
A CB
VF
Ke D ?
?D
V
Ce
K
IH
T
D
VD
O
YD
E
O
IH
T
H
D PJ
I
V DH
OA
KB N
?
]
hµ, λi = (
TD
IS
O
F
u|Γ = θu|Γ + (1 − θ)u|Γ , Z
PQ
D
G
Ω2
Z
FM
O?
WH
E
J PD
U
C
O
I
O
B
H
H
OD
W
IA
C
CI
RG
AH
H
IH
GH
[
H \
H NV G
C
N
K
O
]
A
H
P
CG
H
G
D
F C
F
D ED
C
LB
AH T
D
VD
LD
C
O
B
V
B
K
G
D
uk+2 |Γ = θuk+1 |Γ + (1 − θ)uk+1 |Γ . 1 2 1
H
^ ED
$
E
N
O
IH
O
A
PB
F
E
I
FM
T
C
N
K h
NG
C
N
I
O
N
O
IH
^
E H
TZ D
V
E
H
AH
I
hKµ, λi =
?
A
OQ
OB
2 C12
R
E
H
≤
H
J
B
S
N
B
CG
]
G TD
S
H
ZAH
F
T
KG
?
eC
^
IC
A
Ke D
G
G
Γ
J
X
V
I
EQ
Ω2 ∂Ω2 \ Γ
B
D
B
+
H
%
I
[
K X
D
TD
PV
W
IB
IA
H
D
H
H
D
E
?I
e
T
]
B
B
P
H
M
D
H
KG
C
NA
P
C
K
PD
FT
O
QD A H
IH
Y
C
I
C
R
^ E
hKµ, µi =
E
?
T
B
]
P
]
O
O
I K X
CD
?
A
D YB
?
N
O
hKµ, µi =
ED
H
2 2 1+C12 I
NA
TD
LB
A
B
P
^
I
S
V
OQ
B
OD
W
IA
H
C
k+1
L
BH
θ=
C
RB
A
K
Ω
D
D
I
V
D
U H
V
$
P
Γ.
N
O
IH
K
CG
H
B
∂εk+1 1 − ∂n 1
OJ
∂ ∂ H2 )−1 ( H1 )εk+1 + (1 − θ)εk+1 1 1 ∂n2 ∂n1 H
T
YD
C
S
F
O
B
O
= 0 = 0
C
∂ ∂ H2 )−1 ( H1 )))εk+1 . 1 ∂n2 ∂n1 T
D
P
T
N
O
=
AH
I OB
C
R
I
J
∂n2
V
I
J
[
H \
PQ
CI
FH
T
εk+2 = θεk+1 + (1 − θ)εk+1 1 2 1 k+1 −∆ε2 εk+1 2 ∂εk+1 2
]
A
H
B
I GH
C
O
I
H
D KG
P
= 0 Ω1 = 0 ∂Ω1 \ Γ = ξk Γ εki = u − uki
R
εk+2 = −θ( 1 O
? X D
k+1 −∆ε1 k+1 ε1 εk+1 1
CG
ξ = u|Γ −λ k
u|Γ
BH
ID
Z
O
H
A PB
GH
[
H \
k
E
FR
H
H
H
O
I
λk+1 = θuk+1 |Γ + (1 − θ)λk . 2
Hi PD
T
O
T
H
KG
D
C
I
λk
KD
V
KG
N
K h
I
Γ λ
]
D
PD
A
CB
A
Γ O
B
P
k+1 = f Ω1 −∆u1 uk+1 = 0 ∂Ω1 \ Γ 1 k+1 k u1 = λ Γ k+1 = f Ω2 −∆u2 uk+1 = 0 ∂Ω2 \ Γ 2 ∂uk+1 ∂uk+1 2 1 = − ∂n1 Γ ∂n2
D
H
D
H
H
LD
OI
K
T
H
A
D
CB
A
D AH
VD D
H
C
R
^ E
YB
?
$
N
O
IH
O
A PB
F
O
Ke
CB
PD
T
H
D KG
NV B
VD
H
B
ZAH
S
N
CG
K X
D
O
Ke
C
YI D
KD
V
^
M
TD
PV
O
IH
ZAH
G
H
A
I
O
IH
T
H
D PJ
I
V
ED
OA
KB
N
?
P
O
R
IA OB
^ O
Y
NA H
D
D VG
EB
G
∂ ∂ K = I + ( ∂n H2 )−1 ( ∂n H1 ) 2 1
∂ H2 µ, λ)L2 (Γ) ∂n2 ∂ ∂n2 H2
= hµ, Kλi.
= hµ, µi.
Z Z ∂ ∂ H2 µ + H1 µ, µ = (∇H2 µ)2 dx+ (∇H1 µ)2 dx ∂n2 ∂n1 Ω2 Ω1 Z Z ≤ (∇H2 µ)2 dx + (∇R12 H2 µ)2 dx Ω1 2 = C12 hµ, µi.
Ω2
εk+2 = (I − θK)εk+1 . 1 1
2 hµ, µi ≤ hKµ, µi ≤ C12 hµ, µi,
2 C12 −1 2 + 3, C12
$ $
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
%"
"
&"
$ &
B
uk+1 1 k+1 a1 (u1 , v1 ) uk+1 = λk 1 O
?
C
B
∈ V1 , = (f, v1 )Ω1 Γ Ωh1 CQ
B
]
EQ
H
CG U
T B
YD
A
D
D
U
FO?
S
OQ G
\
^
H
I
H
IB
Y
K
TB
D
J
B U< X
A
^ WH
OB
IA
L
[
UN
KG
R
I
PQ
CG
D
IB
R
[
]
D
F
G
I X B J
V
H
P
IE B
C
O
W
I
\
I \
A D
$
P
B
CG
OB
A
CG
V
U
O
R
H
D
OD
?
S
?I G
H
I X A B R
L
H I
IH O
KB
S YD
B KJ
A
H
H
OQ [
O
I K
E
H
G CG
I
?
C
]
H
H
D
$
P
D
CG
OB
A
CG
A
TV D
V DH
O
C
O
E
K
]
BH
M
J H
D
P
B
CG
A OB
CG
A
VH
O
EB
P
NG
C
R
K
B
?J
RB
I
J
TB
U
H
B
O
R D
H
OD
?D
L
O
]
I
U
O
C
O
E
K
O
R
H
H
D
H
OD
j
^
^
]
D
G
D
Ce G
F
V
EI B
PD
KG
EI
FL
IA
ED
T?
NA
V
E
QD A
C
P
h
^
G
A
[
G
A
G
N
W
IJ
B
IAH
C
?
T
I
C
O
CI
A
G
[
H \
H
HB
Γ Ωh = Ωh1 ∪ Ωh2 Γh = NB
Y
K X
I
I
V[
QD A
[
H \
H
AH
CD
?
C
O
I
OQ
C
?
T
I
E
I
RB
OJ
]
D
A
W
IN
FK
M
OB L
CI
FM
O
EA
OD
?
\
J
AH
H
U X D
D
%
IB CG
D K X
P
E
TB
I
TB
Γh
P
?
G
H
I X B J
S
H
Au = f.
O
?
I
V
C
R
B
A
O
A
PB
V
I
fi
CB
Y
O
P
O
I
A
D
TD
I
P
A
∀v h ∈ Uh .
H
H
H
^
OQ [
ZH
O
P
H
$
K
W
Q
?
IE
I
VB
\
B
H
H
^
A
H
IE B
P
CG
O
PB
IE
OQ
T
P
IH
G RD
e
I
PZ B
PD
J
>
P
I
I
J
FM
P
CG
[
]
C
R
^ E
O
I
D
ED
V f B
O
T H
IH
Y
KD
A TD
VD A
A CB
[
V[ D H
QD A
CD
^ N
U
BH ?H
O
C
RB
D
D OJ
X
T
P
I
H
]
]
]
P
Ce
PB
I
AH
D
CG
T
V
O
YD
E
RB
I
J TB
TD
VH
IH O
ZAH
G
LI D
I
KB
OB
D
]
C
R
?
C
D
DH
B
C[
IB
f
uk+1 ∈ V2 , 2 a2 (uk+1 , v2 ) = (f, v2 )Ω2 ∀v2 ∈ V20 2 a2 (uk+1 , R2 µ) = (f, R2 µ)Ω2 + (f, R1 µ)Ω1 − a1 (uk+1 , R1 µ) ∀µ ∈ Λ 2 1 λk+1 = θuk+1 |Γ + (1 − θ)λk . 2
E
K
K
IH
K
PB
CG
H
V
KQ J
B
]
B TJ
A
IC B
OQ
CG
A
D
B
P? D
FK
I
EA
C
O
Ωh
P
TB
Q
VF
CG
T
AH
H
^
F
O
S
N
H
PJ
H
G
V
K
VA
L
[
O
K
^
]
EG
H
H
O
IH
IO
[
ui ϕi
P?
C
NG
YD
O
E
ID
O
RD
Ωh1
V DH
TD
U
G
IB
Y
fi = Z
D
D
O
RD
F
KQ J
F
H
H
∇ϕi · ∇ϕj dx,
D
A
[
A
?D
K
E
CQ
O
E
Aij
O
C
P
W
V
B
H
X
D
ZAH
D
O
IH
O
A
PB
F
a(uh , v h ) = (f, v h )
P
CG
A K X
IC
O
P
H CG
O
J
B
D
ED
A
K
C
R
H
H
LD
O
C
O
E
K
ui ϕ i
IB
I
I
ED
O
O I
AH
^
E
I
FM
^ E
T
I O H
KG
ED
V f B
]
H
(Γ, Γh ) = O(h2 )
D K X
\
u1 f1 0 AΓ2 uΓ = fΓ . A22 u2 f2 FO?
W
PD
KJ
F
D
M NG
C
P
TD
S
A1Γ AΓ A2Γ Q
C
YD
O
FC
O
P
P
CG
H
YD
O
[
uh =
VH
A
A
]
$
Q
VF
VA
B
Ωh
OQ G
I
V
BH NG
C
KQ J
F
Ωh2
Ω
OQ
X
X
A
Aij = Z
IC B
E
H
IH O
J
L
IC
U H
C
ID
E
P
Ω
RD
I X B J
K
IH
P
IB
H
AH
H
D
CG
A
Z
B
T
P
I
P
E
Z
Ke D
G
VD
Ri
GH
A11 AΓ1 0 C
J
IB
A
A
O
I
O
FC
O
E
U X F
D
AH
H
G
H VAH I
V
O
]
H VAH I
V
Λ
C
Ωh2 HX
G
D
DH
^
[
O
P
S2−1
D
I
O
F
O
RD
W
I
ϕi
^
?
Ke D
H K X
C
]
V
TV D
H
^
¯ h2 ¯ h1 ∩ Ω Ω
P
IB
]
C
E
Ce G
H
H
θ(S2−1 (−S1 λk + χ) − λk ) θS2−1 (−(S1 + S2 )λk + χ) θS2−1 (−Sλk + χ).
?
G
]
D
[
B
PD
SK
A
^
!
C
S
U
H
H
VD
O
C
O
E
K
D
K
?H
C
B
U AH
I h
U OB I
B
Γ,
H
X
O
R
λk+1 = λk + θS2−1 (−Sλk + χ).
OQ [
O
K
I
B
U
H
E[
H f
TD
−
?D
∀v1 ∈ V10 IS
H
OD
?
H
O? B
F
N
O
IH
O
H
∂ ∂ ∂ T1 f − T2 f S i = Hi ∂n1 ∂n2 ∂ni
O
EIA
C
P
?
SK
C
E
X
FT
^
−∆T2 f = f T2 f |∂Ω2 =
^
$
F
Ce B
G
A
I h
J
B
D
^ ED
A
∂uk+1 ∂T2 f 1 − ∂n1 ∂n2 ∂ ∂ ∂ ∂ = ( H2 )−1 − H1 λk − T1 f − T2 f . ∂n2 ∂n1 ∂n1 ∂n2 ∂ H2 )−1 ∂n2
D
]
H
I
VAH
A
PB
F
N
O
IH
ZAH
P
RB
I
OJ
C? D
k+1 −∆(u2 − T2 f ) = 0 Ω2 uk+1 − T2 f = 0 ∂Ω2 \ Γ 2 ∂(uk+1 −T2 f ) ∂uk+1 2f 2 1 = − ∂n1 − ∂T ∂n2 ∂n2
]
C
E
E
K
P
H
D
$
H
E
= λ +
O
U OB I
E[
H f
B
E
K
N
K
T
k
V
O
Ke
H TD
C
H
CB
P
+
= λ + = λk +
R
A
CB H
H
D
E
N
O
B
k
IH
E
I
V
K
Y
H
?I
D
X
l
= (uk+1 − T2 f )|Γ = ( 2
DH
^
Ce B
H
H
A
IH
O
E
K
W
IB
IA
k+1
C[
K
L
I RB
TB
J
H
PI B
KG
λ
TG
D
H
D
χ=−
f
L?
K
C
I
NV
K h
uk+1 |Γ 2
IB
]
P? D
OB
CG
0
EA
D
\
uk+1 = H1 λk + T1 f. 1
Vi
Uh
Uh
Ω
f ϕi dx.
Γh
$ $
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
%"
"
&"
$ &
K
T
TF
H
X
F
W
UE
C
M
H ]
T Ni (AΓ − AΓi A−1 ii AiΓ )Ni
GH
G
E
(i)
B
I
D
LD
O
RD
K X
U
J
B
O
I
K
I
OB
PD
A
O
C
B
B
J
A IB X
U
VS D
LD
I
O
E
B [
H
PD
C
O
H
TB
IA
L
S
OQ
O
Y
^
H
H
O
M
IH
ZAH
?
OQ
K
T
AH
D
G
U CB
E
F
Ke D ?
?D
$V H
IH O
D
DH
X
X
1
1
3
5
29
3
23
63
2
5
;
;
6 19
7
5
(
7
3
1
6
2
6
23
3
7
8
'
7
3
6
2
D
P? D
OB
CG
TD
V
G
J
E G
H
]
H
OB
I
P
K
I
?I
N
P
O
O
J
H
H
IH
KG
R
I
PQ
X
IP B
D
EI B
W
A
KG
F
H AH
]
G
H
H
B
TD
C
E
W
IB
IA
C
ID
O
TD
A QC B
J
H
H
CB
OQ
Ke
C CG
H
KG
I
R
I
PQ
OQ
PD
OG
I[
O
IH
ZAH
CI
ND
CG
D
G
IC B
E V
WH
A
CI B
E
EQ
H
G CG
I
N
IH O
ZAH
[
[
]
[
l
]
]
λk+1 = λk + θS2−1 (−Sλk + χ).
A
I
N
K
X
FR
KG
NA
D
VD
H
5
5
F
]
F
F
V
A
D i
]
]
]
H
H
G
B
D
G
E
O
H
IH
KD O
V
T
EQ
H
G
B
CG
I
N
K
T
OD
TA
R
I
TD
C
E
NG
C
N
K
P
N
H
^
W
I
V
A B
D
H
G
V
H
A
D
B
TD
C
E
ED
LD
KB
OB
E
?I
H
I X A B R
L
KB
F
ED
C
H
D
W
IA
C
B
H
IH
[
?I
A
B
H
W
IB
IA
C
ID
CG
[
^
J
B
D
H
+ X
G
H
H
D
N
A B
D
A CB
V
ED
LD
KB
OB
E
?I
I X A B R
L
KB
^ ED
E
O D
H
U CB
E
NG
C
N
K
P
H
AH
D
A
D PD
?
K X
S
OQ
R
IC B
E
N
IH
K
V
I
G
\
LD
I
KB
T
[
^
H OB
H
A P? D
FK
I
EA
?
I X A B R
L
KB
?
C
N
A TD
P
I
V
OQ
C
H
H
^
A D
G
]
]
U
U
H
B
O
R D
OD
S?
I
O
P
P
CG
OB
A
CG
S
V
?I
P
H
D
G
B
KD
O
H
EB
X
%
]
]
Sλ = χ S = S1 + S2 χ = χ1 + χ2 (i) Si = AΓ − AΓi A−1 ii AiΓ (i) −1 χi = fΓ − AΓi Aii fi
IC B
M
{0, 1} C
S T
D
B Ce G
Ω
E
N
[
B
D
VG
O
Ke
D
6
2
2
6
6
23
27
6
U
KD
V
N
O
IH
O
J
NI G B
BH C
FR
KD
V
H
D
A PB
F
?A
CG
]
O
IH
O
$
SK
A
Si
K
A
G
Γ
P
F
i
O
H TD
C
A CG
V
BM
6
2
FM
PD
F
H
AH
CG
I
O
C
IY D
KD
C
G
]
D
S2−1
CG
C
O
D
D K X
C
R
E
Ce
O
V
I
:
ξΓ 0
H
G
D
Ni F
H \
H
^
H
VD
O
A IH
E
I
6
J
[
\
A
NB
C
P
D A PB
F
O
^
O
E
K
I
R
J
AH
H
]
I
I T
P
P
E
H
PB
I
AH CG
T
V
P
C A B
D
?H
P
N
K
T
T
P
I
H
]
C A B
D
?H
P
I
W
Q
A
IC B
E
X
I
P
CG
IB
D K X
TD
TV B
K
P?
E
RB
I
D OJ
l
AH
I
O
H
IH
K
PB
CG
T
V DH
O
P
C
I
T
TI B
NB
P
CQ
I
* R @
]
^
H
E
Z H
VAH I
V
fΓ
O
X
A
B
CQ D
]
?F
D
I
W
I
FO?
S
OQ G
J
[
^
I h
U OB I
^
AΓ
Ke B
K
H
D
C
O
I
D
YD
E
OQ
R
V
T
D
D
Ωi
E
V
H
^
A CG
NV
K
T
D
]
1
5
=
H
J
E
O
IH
G
D
KD
[
V
O
A
H
2
3
?IA
[
[
I X B J
D
P? D
OB
^
I
W
\
Aii
L
K
D
G
KD O
V
T
UV
H
YD
IC
E
G
W
1
:
D
D
P
i=1
W
CI
ND
m X A
H Ce G
B
E
K
E
E
I
IB
H
,
A
J
CG
C
P
CG
TD
V
ηΓ η2
Q
TD
C
H
H
CB
]
N
IA
C
6
GH U RD
C
N
O
IH
Y
KD
BH NI G
C
FR
KD
V
]
D
CG
CG
S≡
Ωi
IB
Ni D
i=1 G
Ni AΓi A−1 ii fi ,
CI B
H
D
]
[
P
K
I
IH O
A
KQ J
H
A TD
A PV D
D
AΓ2 A22
K X
]
?A
W
Q
H
E
E
OQ
O
PI B
AH
X
Q W
A
I
PB O
F
F
OQ
H
D
]
N
O
IH
O
PB
A
?IA
CG
[
I
FO?
H
S2
D
[
CG
E
B CG
IY D
KD
AH
KG
F
D CD
EG
AH
H
G
A
CB
VS
OQ
Ke B
P
IA
F
B
D
(2)
TD
D
H
CG
K X
D
TD
TV
[
G
B
V
B
PD
K
?H
C
U AH
D
Ke B
CG
P
[
H \
[
H
AΓ A2Γ
V
T
A
CI B
G
IB
T
LD
S
V
]
H
B CG
H
U GH
T
KG
BM
B
O? B
F
A
H
^ GH
C
O
B
D
IC
E
D
H
Ke
C
S
P
W
EQ
H
CG
G
+
H
D
K X
T
]
FR
KG
F
N
O
IH
O
]
I
O
O
OB
TB
PA
J
S
P
CI G
E[
H f
λ ≡ uΓ
YB
Q
E
H
I
N
] B
OQ
Ke
K
B A KB
V
U AH
VH f
H
H
PD
C
O
E
K
S
ED
O
Ke B
W
^
?
O
OQ
Ke D
m X I
[
O
H K IH
K
\
H
Ωhi
PD
T
D
Sλ = fΓ −
IH
FL
D
PB
B
A TJ
CG
T
EI
N
U H
]
H
PD
C
O
W
I
I \
OQ
R
S2
TB
O
E
NA
AH
H
O
A PB
F
H
KG
EQ G
P
Ce B
D
A
G
B
PB
F
e
I
VJ
FM
O
R
A
CI B
S
K
H
TD
H
H
V
O
IH
O
E
OD
I
OD
AH O I
?B
\ H \
TD
C
T
ID
O
R
LD
I
KB
OB
?A
CG
NB
PA
V
OQ G
[
H h
Si
P?
OB
VH
[
A
DH
PB
ID
H
F
O
H RD
C
PA
NV
O
]
?
A ED
D
H
B
H
D
I OB
J
PD
Ke D
H
B
S2−1 ξ
I
PD
X
S = T
]
D
C
E
S
O
IEA
C
P
J
xk ∈
L
]
−1 −1 −1 (AΓ −AΓ1 A−1 11 A1Γ −AΓ2 A22 A2Γ )uΓ = fΓ −AΓ1 A11 f1 −AΓ2 A22 f2 .
Aii
KD
H
F
M IH
R
K
G
H
V
O
H
?H
P
O
K
PB
uΓ
fΓ
A
O
A
i
W
A
IC B
X
?
EI
FC
D
AH CG
T
^
W
G
NI G
I X B J
λk+1 = θuk+1 + (1 − θ)λk . Γ
F
D
H
E
^ AH
H
[
F
?A
GH
[
H \
H
YD
A
B
V
V
BH
A11 uk+1 = f1 − A1Γ λk , 1 k+1 (1) uΓ fΓ − AΓ1 uk+1 − A Γ λk 1 = , uk+1 f2 2
F
V
E
O
IH
O
L
KB
D
NV G
CG
E
C
O
I
I
I
FO?
C
R
AΓ
K X
D KD
V
T
Ce B
P
H O[
BH
FR C
FM
O
RD
K X KD
NV
FM
A
D
$
I
^
0 f1 u1 (1) (2) AΓ2 uΓ = fΓ + fΓ . u2 A22 f2 V10 V20
Ωi
I
J
QB
O
I K
$
O
IH
C
E
Ce
FR
I
\
U0,h 1
O
H
]
P
D
ZAH
D KD
V
[
]
U0,h 2
E
A
?A
CG
G
NB
T
FY
G
B
D
fΓ
P
CQ
N
O
I
V
^
O
H
A
EI
N
O
U0,h 1
D
H
^
NV
Ke
C
[
O
FK
O
A1Γ (1) (2) AΓ + A Γ A2Γ
O
Ke
C
G
D
B
D
O
IH
P
IC G
µ Rih µh (xk ) = 0 i = 1, 2 h
I
O
?
CD
ZAH
D
^
YD
AΓ2 A22
O
D
H h
]
(2)
PD
SP
I
V
Ce G
LD
O
RD
C
AΓ A2Γ
D
N
TD
S
O
A
AΓi AiΓ
LD
O
RD
GH
[
P
Q
A11 AΓ1 0
(i)
S
OQ
RD
C
T
P
O
fΓ
OQ
H
[ K X
^
I
IEA
C
J
H
B KG
O
AΓ
CD
W
Q
FN
H
AH
U0,h 2
A
Ke D
eC
C
IC B
E
G
, (i)
(i)
]
−1 AΓ −AΓ1 A−1 11 A1Γ −AΓ2 A22 A2Γ A uΓ N
V
AΓ
AΓ
η=
,
S2 m
$ $
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
%"
"
&"
$ &
A11 Bs = AΓ1 0
F
]
H
A1Γ (1) AΓ + AΓ2 A−1 22 A2Γ A2Γ
G ]
Bu
0 AΓ2 . A22
CV
]
M
I
PQ
I
V
D AH
A
X
B
D
B1
^ E
R
K
BH
M
?J
W
I
I
F
?
T
O
I
I
C A B
D
?H
P
E
E H
H
gk = Bu−1 A
U H
X
D
^
F
P
CG
]
D
C
NA
P
C
H
D
H
]
uk+1 = uk + θBu−1 (f − Auk ).
KG
D
FH
A
D
G
Ce G
?
D
G
NG
A OB
N
K
T
C
K
PD
W
[
V
? B
ID
Z
NB
O
Ke B
X
]
C
IY
K
A
H
TB
O
I
G
E
A
IC D
EG
B
U
I
AH
C
N
TD
D
CG
A
U
W
C
W
IB
VT D
^
FT
H IB
IA
C
ID
CG
RB
B
D
OA
P
D
H
D
LD
A
CD
H f
J
D
]
]
KS
A H
I h
U OB I
E[
W
I
I
I
EV D
?H
TB
G
AH
Ke B
H
D
H N
K
CB
P
B
H
D
CG
A OB
CG
A
VT D
CV
YI
K
A
B
TB
O
I
V
?
ID
Z
G
G
X
A J
B
D
B
^
U
J
B
D
Z
ED
T
YB
B
H
D
O
W
IA
C
LB IH
OD F
LD
KB
OB
E
OQ
GH RD
C
EQ
D AH
H
C
W
A
CI B
ED
C
BH
DH
D
NG
C
T
A
CD
ED
X
?
D
+
?B
O
C
R
Ce B
?
V
KG
I
H
^ ED
E
?I
] D
PI B K
D
^
] ^
A
D
WH
V
L
]
]
F
EQ
H
G CG
I
N
O
IH
ZAH
N
K
P
T
PD
C
O
H
X
H
TB
IA
L
S
H
AH
D
KG
F
V
LD
P
C
I
O
A
\ ?H \
H
CG P
RB
?
P
eC B
PD
J
C
[
G
D
Ke D
V
I
O
?H
P
OQ
>
H X
H
RB
O
E
Q
EQ
CG
I
YD
E N
K
T
OQ
O
Y
NA
VG D
A
H
G
DH
W
H
D
H
^
UA
I
EV
H
YD
E
EQ
O
Ke
CB
PD
T
K
H
EQ
H
CG P
RB
?
P
S
QD A
U > D P C
G
J
P
O
H
IH
Y
KD
A TD
V DH
O
Ke B
W
IH
V
K
IN
P
UR
CI
E
E
I
H
H
G
DH
A
H
G
NG
C
N
K
P
N
E
FE
P
E
BH
KG
T
YB
?F
A ] B
^
?
Ke D
G
?D
LB
]
N
K
T
CI B
E
NB
O
R
IC
E
E
0I
D
TD
C
U
IE B
A
CI B
E
RB
TB
G
H
CV B
Y
]
B
TD
C
D
O
T
?
N
K
T
H
H TD
C
E
C H
E
Ce
^ E
OQ
H
]
D K X
TD
VH
OQ
C
R
H
S
W
K
A
TB
O
I
I
] H
C
A
TV D
+
EQ
CG
G
D I[
O
[
B
O
IH
E
H
C
O
I
O
ED
T
KG
E
OQ
OB
T
A
GH
[
H \
H
B
H
LA
I
O
O
H
H
D
Ke B
E ]
[
B
G
GH
U J
W
I
FO?
S
P
G IAH T
[
\
^
D
G
G
B
H
D
H
KD
V
O
Ke
CB
PD
T
D KG
PV B
H
D
J
B
B
O?
O
T
D
H
H
D KG
U
B
D
G
H
B
D K X
N
W
I
FO?
NB
?
R
OD
I
G AH
EA
L
O
C
?
T
I
S D
G
D
H
H
I H
FM
T
KG
KB
I
PD
J
Ke D
V
I
O
Ke
CB
PD
T
H
KG
V
EQ
[
V >
O
CG
IB
H PQA
X
D K X
TD
V
E
<
A
WH
\
H
H
X
\
A
PD
FE
OD
A
QD ^ G
]
UV > D
N
O
IH
A
U PB
T O
H AH
KG
F
V
E
D
G
FH
A
EIA
FY
[
IB
G
?H
T
O
I
CG
H
D
ID
O
C
R
I
BH
B OG
K
G
H
LD
Q
VF
VA
L
P
T
IE
C
T
Ke
CB
P
K
I
(B2 x2Γ , x2Γ )
E
IA
H
^
H
B
X
LD
e0 = u 0 − u
J
C
[
]
X
E
?
H
G
H
TD
S
NI
K
T
K
A
H
OB CQ
I
P
x ˜2 xΓ
K
TD
S
ID
VF
H
IB
IA
C
I
V
?I
ID
Z
GH
C
E
H
CB
P
K
IP B
X
FL
IA
T
E
A
IC D
EG
G
B
CB
P
K
O
TD T P
N
W
B
IJ
21
ek ∈ U A2Γ ekΓ +A22 ek2 = 0 k + A22 g2 = 0 ek+1 = ek − θgk ∈ U ek 2 0 θ = C 2 +1 u = Bu−1 f P
I l
Z
]
]
D
AH
H
KG
F
S
H
VB
O
IEA
C
P
x ˜2Γ =
W
PI B
KG
N
O
AH
D
KB
I
C
R
^ E
I
#
D
ek = u k − u
A
X
IH
ZAH
D
LD
O
RD
C
D
LD
C
I
KG
U
C
x ˜2
IC B
D
E
H
G
AH
F
V
FM
=
E
H
G CG
I
O
A2Γ gΓk
O
A
=
I
IH O
IH
D
D
G
H
?H
P
S
?I
R
I
B
D
OD
EA
L
O
C
=
=
xΓ
R
H
G
E
C l
C
R
U
V
^ E
H
D
U
AH
G
H
F
W
A
CI B
E
E
A
IC D
EG
G
B
(i)
T Ni (AΓ − AΓi A−1 ii AiΓ )Ni
CI
E
E
I
E
A
IC D
H
. Ke
CB
PD
T
i = 1, 2
ZAH
A
Ce G
OQ
O
Y
NA
U = {A2Γ xΓ + A22 x2 = 0}
I
F
A
Ωh2
E
OQ
C
O
VG
D
G
EB
NG
C
H
?
T
I
P
CG
A OB
0 0 . A22
]
H
D
KB
P
I P
?
N
K
P
N
D
CG
A
EG
G
(Bu x, y) = (B1 x1Γ , y1Γ ) = (B1 y1Γ , x1Γ ) = (Bu y, x), (Bu x, x) = (B1 x1Γ , x1Γ ) ≤ (Ax, x),
0 < (Ax, x) = (B1 x1Γ , x1Γ ) + (B2 x2Γ , x2Γ ) ≤ (B1 x1Γ , x1Γ ) + (B2 x ˜2Γ , x ˜2Γ )
2 2 (B1 x1Γ , x1Γ ) = C21 (Bu x, x). ≤ C21
yi yΓ
B
V
yiΓ =
,
NG
H
xi xΓ
x, y ∈ U O
^
[
B
D
H
TV D
V
O
C
R
^ E
A1Γ (1) AΓ A2Γ
C
xiΓ =
N
A
WH
A
CI B
Bu Ke B
C
BH
E[
EQ
D YB
?
A11 Bu = AΓ1 0
O
G
H
H
?H
V
EI
OQ
O
K
T
PB
TB
J
^ D A
A PD
KD
,
V
PQ
AiΓ (i) AΓ S
Aii AΓi
OQ
Ke D
AH
D
VD
O
H
H
D A CB
V
S2 = X
i
]
D
^
Ke
C
YI D
K
S2
PJ
A
Bi = I
B
N
K
T
S
NV
K
T
T
LD
+
S
. xΓ
Bu
Bu u0 = f,
A2Γ u0Γ + A22 u02 = f2 .
ek+1 = ek − θBu−1 Aek = (I − Bu−1 A)ek .
2 C21 −1 2 +3 C21
A22
$ $
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
%"
"
&"
$ &
B
D
H
H H
1 −1 2 S1 GH
J
G
E
H
PD
O
UA
C
B
H
ID [
U
H
G Ke D
V
NI
K
CB
]
1 1 λk+1 = λk + θ( S1−1 + S2−1 )(−Sλk + χ). 2 2
I
KG
F
TD
B
P
K
H
D U KD
V
T
N
K
T
Ke
CB
] ^
C
O
A
GH
[
H \
H
B
H f
[
UT
V
AH
OB
E[
H f
U OB
E[
W
B
H
.$
-
0
.$
-
(
(
0 _
D
H
I
H
PD
KG
F
O
O
H
KD
V
PQ
O
KG
D
AH
H
D
[
U X F
V
LD
O
J
D
OD
W
I
I
EV D
?H
T
CG
ID
O
C
O
KB H
D
O
B
D
D
AH
H
B
H
UA F
O
?
C
I
A
B
TD
P
I
V
Ke
CB
P
K
IP B
KG
F
V
X
A˜22
RB
O
WH
I
O
N
O
IH
O
IA
C
(
I
H
P
P
I
?
O
?
K X
B
C?
F
L
O
A
GH
D
B
RB
I
J TB
TD
V
]
O
B
[
[
X
X
CG
IB
K X
D
TD
V
O
E
H
IH
J
D
H
A IB
O
O
T
A YD
D
^ H V
B
B
H
H
H
O
D
O
Ke
C
D
H
KD I
V
T
N
O
IH
T
P
P
C
RG
B
J
NG
C
N
A OB
CG
F
H
CG
B
[
[
K X
D
TD
V
KG
R
I
CD
H TD
C
E
E
OQ
GH
G
RD
C
S
F
P
T
IL D
I
KB
OB
D
H
CG
B
D
G
Ce G
E
I
PB
IJ
[
K X
D
^ ED
]
B TD
V
K
^
D
CG
H
ID
O
O
K
PD
KG
F
D
D
KG
IB R
H
B CG [
X
K X
T
U
GH
R
I
G
CD
eC
PB
H
IJ
C
TF
X
U
B H
A B
D
H
H
V
D
ED
CG P
R
I
KD
?
E
I
Ke D Z
C
Q
?
V
Ce
Q
C
YD
E
B Ce G
X
X
D K X
NB
O
C
RG
A
]
AH
A D
H
H
B
A
J
D
ED
C
I
W
I
I
I
EV D
?H
T
TD
C
E
PD
D
B OG
O
NG
C
I
CG
Te
H
RD
DM
PG
P
^
D
J
D
D
AH
H
H
N
K
T
Ke
CB
P
K
IP B
KG
F
V S
QD A
CD
?
P
CG
IB
X
K X
W
I
I
I
EV D
?H
T
TQ
U
J
D
BH
D
A D
C
E
PQ
O
A
D
D
H
H
H
PD
F
LD
O
E
Ce
I
CG
eC G
O
YD
EJ
P
C
T
CB
C
KeF
AH
CD
C
1
6
7
7
6
5
3
2
6
27
6
6
7
A
P
K
IP B
X
A
PB
F
H
TI D
C
^
W
Q
A
Ke B
C
O
%
H
H
OJ
OQ
O
X
]
D
D
C
R
^ E
D
H
YB
?
A˜1Γ (1) A˜Γ + AΓ2 A−1 22 A2Γ A2Γ
IP B
X
KG
F
D
AH
O
IH
5'
$
0
A
IC B
G ?H
H
EB
D
G
B
CG P
FO
RB
]
C
6
7
5
63
6
3
7
!
TB
NB
KG
F
D
AH
$
H
[
H
[
"
H
M
J
^
A˜1Γ (1) A˜Γ
S
G
H
H
VH
NG
V
P
E
H
%
'
E
NI
K
CB
V
O
IH
Z
A
TB
E
IC D
Bs−1 A
OQ
?
R
P
CG
BH
DH ZAH
O
B CG
D K X
TD
3,
%
P
K
PI B
KG
U
G
B
EG
A
H KG
I
D
D
A˜11 A˜Γ1
RD
C
C
B OD
W
IA
VS
F
P
0
H
]
D
C
R
O
∼
S C
E \ IH F
P
A
H
RB
H
B CG
K X
D
TD
V
TB
H
[
X
BH
IE
OQ
O
C
D
^
T
P
T
I
S2−1 + 12 S2−1 ]
P
I
ZH
PD
O
A
PB
F
C
P? K
ID
ED
T
X
A
H
?
P
D
F
A
i
G
FR
KG
>
D
P
CG
J
O
H
A1Γ (1) AΓ
E
H CG
E
P
F
D
H
D
A
C
N
K
PB
CG
^
]
?
D
G
P
F
I
OG
^
Ke
C
R l IH
˜ A11 ˜s = A˜ B Γ1 0
O
J
B ^ ED
D
S
A11 AΓ1
IH
B
X
A
CD
?
I
V
T U H
AH
D
O
WH
D
W
Q
O
I
T
IH
CD
EI
OQ
R
A22
ZAH
E
G
D
D
PJ
V
D
H
A2Γ xΓ + A22 x2 = 0,
NG
H
H
FE
F
A
NV
O
IH
O
U
D
D
O
A
TD
P
PQ
D
K
G IC D
]
^
F
P
CI G
[
H h
C
N
K
T
AH
D
KD
H
H
2 2 (Ax, x) ≤ C21 (B1 x1Γ , x1Γ ) = C21 (Bs x, x).
V
N
K
V
C
Ke
C
YI D
KD
I
V
C
R
D
A PD
C
?H
P
S
I
O?
?
Ke D
(Bs x, x) = (Ax, x) − (S2 xΓ , xΓ ).
E
B
H
]
WH
[
D V[
A
^ N
O
H
H
D
WH
N
(2)
NV G
+
UO
CB
P
K
PI B
X
A
CI B
E
D
O
R
H
E
H
OQ
O
X
K h
]
C
?I
C U AH
TB
IA
D KG
F
AH
H
D
H
CI
E
G
G
B
IH
T
FY
A
H
H
J
E
I
P
D CG
Ax = λBs x.
FH
J
H NG
C
L
S
H
V
O
C
O
KB
P
E
G
Y
C
FN
^
EI
N
N
S2 = AΓ − AΓ2 A−1 22 A2Γ ,
W
BH
^ NG
C
P
N
K
P
OQ
O
Y
NA
D
VG
LD
O
P
C
I
?H \
\
I
NG
H
G Ke D
V
F
G
G
?D
I
B1 Ce
Q
H
J
QB
H
H TD
C
E
H
P
I
A
C X
B1 YD
N
] D ? l
H
N
K
O
CB
P
K
PI B
RD
B
X
A
H
?D
O
C
^ I
N
O
I
−2 C21 A ≤ Bs ≤ A.
I
[
C
S
N
CG
KG
F
^
A
P
CG
A Ke B
N
K
P
N
BH
O
IH
RB
K h
IH
R
^
AH
G
EI B
W
A
IC B
E
RB
J TB
TD
S
V
C
N
O
I
^
^
P
H
7
6
35
29
27
6
$D
O
O
P
CG
C
H
CD
D
G
A22 B1 A22 y1 x1 = , xΓ yΓ − AΓ2 A−1 22 y2
]
E
WH
A
IC B
E
G
D K X
D
Bs
N
K
T
E
K
CB
X
TD
AH
E
H
RB
C
[
X
E
ED
D
OJ
Bs = BsT
P
K
PI B
KG
D
H
H
IB
V
G
?H
V
J
B
^ ED
A
D
FR
C
V
A1Γ 0 A1Γ 0 A11 A11 AΓ1 A(1) + AΓ2 A−1 A2Γ AΓ2 = AΓ1 A(1) + A(2) − S2 AΓ2 , 22 Γ Γ Γ 0 A2Γ A22 0 A2Γ A22
F
EQ
CG
PV
EV D
O
S
?
^
WH
A
[
E
KD
V
AH
G
I RB
TB
C? B
Ce B
J
PD
CI B
E
D
B
+
x2 = A−1 22 (y2 − A2Γ xΓ ). A = AT
H
G
H
J
H
B
P
G Ke D
V
?I
Bs ≤ A
I
IH O
H TD
C
E
S
OQ
B
OD
W
IA
H
O
O
H
V
H
D
H
H
]
H
K
A1Γ (1) AΓ
EQ
CG
Ω1 CG
T
O
E
OQ
O
K
H
AH
l
IC
C
N
K PI
Bs
ZAH
OQ
RD
C
GH
Ce B
T
V
PD
C
Bs
G
CG
IB
D K X
ZAH
H
OB
I
PD
X
S2 ≥ 0
H
D
TD
V
P
C
R
A11 AΓ1
I
O
IH
ZAH
O
O
O
H
EB
J
D H
H
7
3
(
U H
ZAH
D
LD
O
D RD
C
LD
O
Ke
CB
PD
T
KG
1
5
9
7
6
6
6
6
5
3
7
8
AH FH CV
C
EQ
H
G
H
CG
I
O
IH
ZH
9
'
X
;
;
Bs x = y
,
0 AΓ2 A22
˜s ∼ A. B B1
A22 ≤ A˜22 ≤ (1 + ch2 )A22 .
$ $
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
%"
"
&"
$ &
*
U
]
]
f v2h dx ∀v2h ∈ Uh2 ,
Z
Ω2
f v1h dx ∀v1h ∈ Uh1 ,
D
Ω1
Z
]
(uh1 − uh2 )µh ds = 0 µh ∈ Λ0h .
^
AH
S
B
KJ
F
S
O
O I
B
0
O
C
?
T
DH
AH
G
IB
O
O
P
D
CG
OB
A
[
B
U CA
T
B
OB
CG
]
]
B
E
FR BH
KG
NG
E
R
I
OB
IA
LD
Q
P
H CG
O
A PB
H
A
D
O H
U
B
KG
P
e
Z
K
I
BH NG
C
E
OD
I
VG
e
H
C
O
I
O
A
[
GH
[
H \
H
B
NG
S
I
FM
?
^
^
W
NI
FK
CQ G
B
\
A
BM
OB L
CI
FR
KG
?B
E
TJ
O
W
I
FO I
eC G
O
EQ G KG
P
Q
AH
VH
FO
P
?
Ke D
O
^
O
H
AH
J
H
D
D
UA
I
V B
OD
C
R
C
NG
C
N
K
P
N
P?I D
OB
CG
D
VD
O
OB
T
CG P
F
E
I
V
EI
[
]
(u1 − u2 )µ ds = 0 µ ∈ Λ0 .
PG
H
G
D
D CG ^
A
V
E
O
IH
D
P
CG
A OB
J EB
I
W
NI
FK
]
] H
C
G
?D
TI l
[
AH
D
OB
I
?
EQ B
J
D
KG
R
I
E
K
IH
T
V
[
]
]
A
O
IH
VH
D CG
T
YB
?
[
i
K h N
F
F
D
H
OQ
S
N
V
CG
B
D K X
TD
V
P
S
CG
H
H PJ
OI
S
O
O I
AH
H
U
P
I A D
]
∀v2 ∈ V2 ,
[
]
?
D
FC
O
P
P
S
OQ
OB
J TB
D CG
A
VT D
A OB L
IC
NG
S
KG ^ H
EQ G
ED
IH
E
CG
IB
D
B
V
E
O
IH
FC
O
P
O
IH
^
B
D
X
H
CG
B
[
D K X
TD
V
KG
R
I
DH
Ke D Z
O
X
]
U
P
Ke
C H
IC G
[
H h
[
BH
D
D
D
B
J
B A U B
FR
KG
O
NG
C
L?
H
OB
[
E[
H f
U OB
K
H f
B
H
E[
W
IA
C
TI D
C
H
X
X
]
]
∀v1 ∈ V1 ,
W
H
I
h
H
H
V DH
O
C
O
E
K
D
I
FM
Vi
I
FO?
\
O
PV
NG
[
\
H
Λ0
IH
O
PB
A
B
B
D
U
O
R
OD
?
V
J P U H
R
[
H \
^
[
D KD O
V
T
G
H
H
M
G
f v2 dx
S
F
^
?A
I
W
I
FO?
P
CG
DH
X
W
A
A
H
OG
C
O
NI
K
T
JH
O
F
KJ
F
H
FM
P
H CG
C
E
R
K
N
IH O
I
D
AH
H
BH
E
K H
A BH
D
UA D
D
^ B VS
RB
J TB
TD
PV
N
PD
I
KG
F
LD
P
F?
V
C
ID CB H
P
FR
I
KD
KV
X
H
H
TV D
C
E
I OB
PJ
IP B K
KG
F
S
V
EQ
T
Y
H f
H
O B
B
OB
E[
E
PD
I
KG
F
G
RB
T
^
]
H
D
AH
A
U f B
I
OB
I
P
K
PI B
KG
F
V
I
V
X
^
^
C
R
D
V
O
SK
A
B
H
I h
I
!
&
&
$
$
!
&
&
A
H
D
E
RB
I
OJ
D
P? D
OB
CG
V
FM
O
Ke B
L
C
O
?I
C
E
H
B
B T[ H
O
I
$
&"
!
&&
&
"
$
! #" "
l
$
I
X
&"
$ &
B
U J
S
OQ
GH
J
H
RD
C
S
F
P
T
I
?I
] F
^
N
P
O
I
O
IH
A
KG
R
I
PQ
I
∂u ∂n |Γ
-
[
C
B
G
[
H \
D
OB
A
CG
?
CQ G
Uh
OQ
uh1 (xk ) − uh2 (xk ) = 0.
C
Γ
O
Γ
BH
M
B
\
$
A
H
A
O
X
Γ
R
K
?J
B
A P? D
I KF
EA
D
NB
O
H
H
D
C
O
V
B
D
I
F
W
A
CI B
E
FM
O
Ke B
I
^ K
F
D
G
Ω2
Z
W
D
C
O
E
K
H
j
I
UA
H
CI B
E
OB
I
PD
A
K X
I
E
GH
G
B
OD
LB
T
I
CD
C
S
IB
YA
H
TD
G
P
CD
CG
G
?
ID
O
λ=
H
NI
λh v2h ds =
FK
∇uh2 · ∇v2h dx − Z
ds =
H
L
C
R
S
B
U
O
R
OD
A
H
ED
\
[
^
Γ
E
K
D
A OB
^
?I
C
U
E
A
ED
C
I
E
D
FH
A
IEA
G
^
H
B
CG
I
^
λv2 ds =
! +
U
H
Z
O
R
CI
NG
D
B
Λ0h
'
OD
?B
V
O
A PB
i=1
S
I
U
[
I
FM
]
EB
KG
R
CG
IB
D K X
TD
[
X
?
C
T
I
Z
P
H
D
O
B U J
I
T
H
KG
NG
C
BH
TB
J
$ F
W
A
IC B
E
I
E
C
I
S?
X
VD
TQ D
D
PG
Ω2
Γ
Z ∇u2 · ∇v2 dx −
CG
O
FM
?
C
NA
H
X
W
Q
A
IC B
E
N
K
T
FM
O
R
OQ
A CB
D
D K X
TD
Λ ui ∈ Vi i = 1, 2 Z Z Z ∇u1 · ∇v1 dx + λv1 ds = f v1 dx Ω1
+
BH CQ G
Γ
Z
H
A PB
M
FR
KG
Ω1
∇v1h
[
OB
I
PD
X
H
P
C
O
IH
PD K
J
FT
P
?
N
OB
I
J
PD
H
I O
T
BH IH
C
T
A YD
Γ
Z
,
$B
G
H \
AH
Ke
CB
P
A
IB
S
OQA
H
ED
G Ke D
@
Ke
CB
P
K
IP B
X
^ O
T
YB
V
0
A
H
λh v1h
·
(
+
C
Ω2
dx + Z
∇uh1
#
H
Z
ED
Ke
C
K
PI B
KG
F
X
CG
ID
Uhi
C
B
U
P
IC G
[
B
(BS S)
O
S
B
B
O
NA
CD
H
[
d d C (BS S) ≤ C(1 + log ) max{ 2 ; log }. h d h
*
O
CG
H
BH
TD
C
E
T
D
AH
H
H
O
O
A PB
∈
?
R
>
X
H EG
O
^
WH
A
IC B
V
CG
B
H
uhi
NG
C
T
I
F U CG
Ce
FR
KG
^ H
H
H
H
E
E
O
K
V
I BH
FR
KG
PD
C
R
xk
IB
KJ
F
P
T U BH
D
A YD
Q
C
H
YD
EI
O
E
O
E
+
D K X
B
U J
I
P
^
?
Ni Di Si−1 Di NiT .
0
^ H V
P
KQ B
E
Oe
H
OQ
C
PD
KG
X
?
I
ED
[
J
B
D
D
Ce B
?
D KG
F
Ni Di NiT = I.
&
B
A P? D
"
E[
? j IB
]
O
TD
F
D
D
KG
I
R
LB
ZG
W
NI
FK
V
O
YD
D
I R
Di
FM
7
'
C?
GH
B
H
B
CG
]
H
RD
H
KB
S
V
LD
I
BH
H A ED
C
L
A OB
V
?
Ωi
&
I
FM
KF
T
EA
>
O
D K X
CD
S
KG
D
'
0
RB
I
TB
TD
B
NG
C
BM
P
P
I
O
E
FR
KG
EB
T
IG
IC
K
I
Ωi i = 1, . . . , m
H
]
7
[
[
?D
O
A Ke B
C
P
CG
BS S
C
^
Ω 1 Ω2
BH
J
D
h
G
D
"
Ce G
D
P
J
QB
H
B
CG
G
?H
X
$ IB
AH
H
S
?F
V
T
CG
'
V
KG
R
I
G
D
m ∼ d−2
IB
K X
H
E
?
K X
D
TD
B
J
B
^ ED
A
D
H
ED
J
Γh
O
Ke
C
CD
V
V
H NG
C
E
S
OQ
Si−1
]
g
D
]
@
O
KD I
I
PB
IJ
KG
R
I
N
K
P
N
H
D
K X
T
h
'
?
D
Ce G
E
CD
C
G
O?
WH
D
i=1
m X
TD
(
CG
D
G
G
H
FM
?
R
N
O
IH
RB
OJ
O
B
+
Γh
]
5'
$
T V
N
A
B
EI \
I
LD
K
I
?I
BS =
VH
P
Ω IB
K X
D
TD
I
PB
IJ
O
IH
A SK ^ H
H
I h
U OB I
ZAH
C
RG
B
J
B
E[
H
PB
IJ
d
T
"
0
B
H
X
O
H
DM
OQ
(BS S)
B
%
'
V
O
E
O
[
H f
PG
m X
P?I
M
IH
J
IB
A
D
OB
A
BS
IG
O
O
X
0
3,
%
E
¯i Ω
Z
EI
N
I
IH
J
IB
A
¯i Ω
PD
KG
F
G
RB
TB
J
?
H
NG
E
OA
H >
BS
TD
S
VF
P
T
Di
Γh A−1 ii λ∈
Ω1
Γ Λ0h = {δ(xk )}. λh ∈ Λ0h
uh1 , uh2
[
@
?
[
A U H
SK
I h
B
P
V D
G
Ri : V → Vi0
a(Ri u, v) = a(u, v) ∀v ∈ Vi0
D
U OB
i = 1, . . . , m.
∀v ∈ Vi0 ]
W
A
D
H
H
F
P? B
V
V
E
T
P
P
^
]
U
B
VS D
QB
KG
S
O
IEA
X ]
Ωi
A
H IB
IA
C
EI
O
]
A H
PD
LB
TZ D
V
N
O
IH
O
E
I
VH
KG
D
NV G
m
CG
A
, v),
PV
H
H
IH
E
H
B
C
TD
IS
B
]
O
O
M
IH
ZAH
U
E
B
H
U H
K
] D
OB
TB
J
LB
Z
A
A
H
[
A B
B
]
W
P i
TD
C
E
E
IH O
E
I
V
I
RB
J
D
BH
TB
P
?
N
]
E
H
]
P
D
CG
H
YD
O
E
UO
I
O
B
B
VAH Z
A
B
B
D
^
B Ce G
D K X
T
^
?
N
K
T
[
^
^
K
N
K
T
C
R
S
?I
C
F
D
M
D
B
G
[
X
0
D
C
Q A
DH
C? D
eC G
B
K X
D
TD
V
A
D
CD
U
H
V
O
D
B
TD
Q
N
CD
S
C
FH
P
CG
]
X
OQ
H
CG
]
[
A
J
H
C A B
D
?H
P
N
K
T
E
H
G CG
I
?
C
TD
I
P
I
V
I
I
S
CG
H
H PJ
IO
H
H
H
H
H
H
I
W
A
IC B
E
CQ
O
E
K
PQ
FK
O
O
T
L
J
X
V
O
CG
IB
D K X
H
IH O
A IB
E
A
IC D
EG
G
B
]
U
B
U
A BM
B
D K X
TD
NV G
S
I
P
Q
?AH
H
K
?
W
Q
O
I
T
D
M
G
IH
?
Ke D
0C
,
0
5
$
"$
3,
%
]
]
A
B
AH
D
B
D [
P
S
I
N
O
ED
@
?
VH f
EI
OQ
?IW
O
O
H
YD
K
T
VB
^ W
P i
]
D
J
B
I
A
D
H
B
TD
C
E
D
LD
O
P
C
I
IT B T
I
I
H
B
^
A
B
D
A
H
G
W
IB
IJ
IAH C
?
I T
PD
C
EB
VS O
I
K S
C
H CD V
I
NG
PB
H IJ
O
IB
?
C
^ H
X
NG
C
FH
A
I X B J
S
IJ O
I
P
CG
O
I
Z
Ke D
A
A
D
ED
$
E
I
FM
T
H
KG
]
] U
H
G
D
J
C
T
NI
K
L
PQ
I
RB
TB
N
K
T
B
A P? D
FK
I
EA
NB
?
H
\
I X A B R
L
K
H
D
AH
H
D
H
F
F
NV
O
A
D
IH
KD O
V
T
N
K
T
K
CB
P
K
PI B
KG
F
V
[
^
U NG
PB
B IJ
?
?
@
?
i
X
S
E
T
B
i−1 m
P
C
P
Ce B
A PD
O
¯ \ Ωi , Ω
n
P
A
RB
I
TB
Vi0 = H01 (Ωi ) ⊂ V i = 1, . . . , m J
D
?
H
u
W
+ zi ,
I
B
I FK
EA
H
O
IH
Y
O
IH
ZAH
H f
H
E[
E
I
PD
KG
OB
K h N
F
G
RB
O
C
?H
P
H
IH O
YD
O
Ωi i = 1, . . . , m Ωi ∩ Ωj 6= ∅
A
= (f, v) − a(un+ i−1 m
i−1 m
V
zi = un+ m − un+
B
[
P
CG
B
O
O
IH
Y
A K X
I
I
V
J
BH
H I
A
TB
S
PQ
S?
F
P
T
FE
B
m
P i
= un+
IH
i
LB
D
X
\
O
A K X
I
I
H VH
V
*
N
X
i=1
m [
B
O
H
G
D
n+ i−1 m
TZ D
O
OB
A
O
B
Ce G
F
W
A
IC B
E
T
I P
NB
P
CQ
R @
I
J
B
^ ED
A
D
H
O
T
R IH
P
I
Ω
TD
i
V
D
B
B
YD
E
W
A
P i
FM
T
^
]
O
IH
ZAH
E
B
+
Ω=
C
un+ m H
zi : a(zi , v)
i E
SD
E
X
A CG
V
E
K
l
H TD
C
= u
D
D
AH IH
T
V
D
T
P
I
un un+1
CG
N
O
IH
ZAH
]
?
PD
u
U B
M
T C2Γ S2 C2Γ , O
O
IH
Y
N
O
IH
T
YD
S
B
OB
D
H
D
= f
ID
A K X
I
I
F
V
E
H
O
N
K h
CG
OA
P
−∆un+ m i n+ m
E
C[
IB
F
?H
C
?
U H
H
LD
A
D
CD
?
N
i
TD
IS
N
OB
I
PD
OG
X
O
U BH
V
E
Z
I
V
PQ
^ OB
K
T
X
C
X
?I
Ωj j 6= i
QD A
C
A ? DH
D
D
]
A
H f E[
i = 1, 2.
CA
B
W
IB
IJ
F?
V
C
ID
VH
SK
A
I h
T T SD = C1Γ S1−1 C1Γ + C2Γ S2−1 C2Γ .
V
$
?
C
AH
D
H
KB
I
L
A
^
$
J
B
ED
A
D
[
FH
AH
d
D
N
C2 B2−1 f2 ). K h
BH
G
NG
C
P
Q
B
I
U OB
W
Q
A
CI B
E
D
E
I
FM
H
GH
H \
C
SD ≡ C1 B1−1 C1T + C2 B2−1 C2T
H
CG
I
VJ
U H I
[ K X ?
OQ G
[
H \
T
A
H
(C1 B1−1 C1T + C2 B2−1 C2T )λ = C1 B1−1 f1 − C2 B2−1 f2 .
]
H
I
W
H
V
Ce
H f
H
A
H
KG
C
T CiT = (0 CiΓ )
CG
B
D K X
IA
C
]
H
E[
E
K
]
C
O
Bi−1
TD
S
V
−
TI D
H
SK
A
I h
I
H
CB
P
(i)
C
GH
H
Si = AΓ − AΓi A−1 ii AiΓ ,
LD
SF (C1 B1−1 f1
E
D
[ I
ZAH
H \
E
E
K
CB
+
^ OB
H
X
K
PI B
X
H
0
I
F
W
A
IC B
E
C2
I
[
A
TD
SI
D
$
KG
F
D
AH
H
G
O
IH
O
A YB
H
]
]
G
X
^
U
E
DH
O
H
IH
ZAH
O
GH
H
D
D
RD
C
Y
C
I
O
BM
TD
C
D
AH
T
NA
KQ
WH
O
H
YD
K
T
V
K
Q
N
C
IH
KD
CG
D
LD
K
ZA
PV D
TD
L
S
G
NB
O
[
]
]
J
B
D
D
D
NB
O
R
QD
I
C
R
Ce B
?
V
O
I f RB
H
H
H
H
^ H
O
E
O
E
T
]
YD
D
ED
P
CG
A YD
V
N
W
IJ B
IAH C
?
T
I
X
U
V
?
A D
U
H
O
NG
C
N
K
P
N
H
H
D
D
W
I
E
H
B
G
ID
?A
\
[
D V[ VB
O
EA
A U OB
CG
A
TV D
NV G
C
N
K
P
N
OD
]
]
D
?H
O
P
CG
A OB
CG
F
V
Ke D
G
?D
C
R
^ E
C
IH
E
C l
D
^
+ ]
]
Λ0h
O
B
C
B
P
K
N
K
T
Ke
V
LI D
I
KB
F
V
T
L
u1 f1 C1T −C2T u2 = f2 , 0 λ 0 C1
E
E[
H f
O
I
H
H
PI B
X
T C1Γ S1 C1Γ
FR
KG
B
SF SD λ =
U OB
B
O
D KG
F
H
CB
P
K
IP B
KG
OB
u2
H
O
E
QD A
CD
AH
H
[
SF =
E[
H f
H
CG
G
?
V
BH
X
]
EG
0 B2 −C2
O
X
ED
I
NB
^
EQ
F
D
u1
O
X
TD
H
RB
TB
P
D
AH
B1 0 C1
D
D
O
O
K
PD
X
J
J
QB
O
IH
C
H
D KG
F
S
PQ
A−1 ii
C
E
G
IL D
I
KB
Bi−1
BH
G
D
H
AH
H
Si−1 SD
?F
V
T
OB
D
λ
SK
A
I h
I
Λ
0
h
Ωi
i = 1, . . . , m.
∈ Vi0
$
$
$
&"
!
&&
&
"
$
! #" "
!
&
&
$
!
&
&
&"
$ &
&"
!
&&
&
$
"
$
B ]
]
T
LD
D
H O
O
KD V
PQ
F
D O ^
Ke
H
CB
PD
H
eC G
3
7
5
6
7
I − T = Q1 + R1 − Qm · · · Q1 = R1 + (R2 + Q2 )Q1 − Qm · · · Q1 = R1 + R2 Q1 + (R3 + Q3 )Q2 Q1 − Qm · · · Q1
T
+
?@
3
D
D
A LD
CD ?
N
K
^
T
D C
A
A ? DH V
E
a(Qi u, v) = 0 ∀v ∈ Vi0 . K
! #" "
"
#
"
$ K
T
IH
AH
D V
I
Qi = I − R i
(f, v) = a(u, v) = a(Qi u, v) + a(Ri u, v) = a(Ri u, v) ∀v ∈ Vi0 , H
OQ
I[
CB
C
A
H
H E
I
vi = Ri Qi−1 · · · Q1 (I − T )−1 v ∈ Vi0 ,
U H
D V
EQA
D O
D
D O? ]
;
;
2
6
6
7
6
3
D A
A CB
H
D
G
U B
A
B V
PA
H
^
C
A IE B
D
? DH
] A
NV G C
M N
K
P
^
D
LD
H C
H
KD
A
]
^
OQ
B
H
V
O
O
IH
A YB
D
DH
CD
O
RB
X [
X A CB
D OQ
O
H R
OB
IA
LD C
H E
H
[
kR−1 k ≤ mγ 2 .
OJ
D O
TD
D O
B E
J I
P
C
H K
P
CG
H
G F
D
A
D
A CB
H
D I
I
3
3
46
5
67
6
8
3
:
3
27
7
2
5
3
6
2
6
;
6
7
]
U ] N
GH P
?
B
+ B ?
? I
P
O
D O
H Ke
PD T
H
K V
I
q γ = (1−q)−1
7
3: 6
3
3
7
3
6
2
5
67
6
8
:
6
2
6
5
H OQ
O
KD V
PQ
D C H
Y
NA
D
D
VG
G
EB
] ]
R
1 a(u, u), mγ 2
;
6
|[Ri u]|
2
6
3
3
7
6
15
67
6 1
3
6
6
3
7
6
3
19
23
V
7
3
i=1
]
;
a(Ru, u) ≥
kT k ≤ q < 1.
m X
i=1
|[v]|
v6=0 i=1
≤γ
1 2
m X |[Ri u]||[vi ]|
i=1
N
: 3
27
|[v]|
≤ sup
= γ m a(Ru, u) .
7
8
6
7
67
7
6
5
5
7
2
5
3
m X |a(Ri u, vi )|
√
CB
6
15
7
13
5
5
' : 3
vi ∈ Vi0 i = 1, . . . , m
kRk ≤ m
V
m m √ X √ X 1 1 a(Ri u, Ri u)) 2 = γ m( a(Ri u, u)) 2 ≤ γ m(
;
WH
NG
BH C
P
R E
RB
[
D VH X G IH
B W
A
B I
]
OJ
D
X
l
|[u]| = a(u, u)
|[vi ]| ≤ γ|[v]|
m γ
Ri
m X |a(u, v)| |a(u, v1 + · · · + vm )| |a(u, vi )| = sup ≤ sup |[v]| |[v]| |[v]| v6=0 v6=0 i=1
v6=0 i=1
1 2
'
3
34
3
7
D ]
= sup
3
6
2
A
v6=0
T = Qm Qm−1 . . . Q1 .
v = v 1 + v2 + · · · + v m , γ
]
D A
CB
A
H
D V CB
H A
D
VD C H
|[u]| = sup
E
IH O
D Z
O
CD
D
G
J
OQ B
N
PG
B W
A
B
P i @
B
TD C
H E P i @
B
TD C
H B S
TD
E
D
eC G AH
E
TD IS H
B
5
R
.
D VA
CB
A
v∈V
R
^ E
YI D
TV
KD
AH A V
e A E
EG
BH
i−1 m
X ?I
ID Z
J
B E
?I
A
^ ED
D
B
+
X
− u) + Ri u − Ri u = Qi εn+
εn+1 = T εn , kT k
Ri
i=1
E
FR
V
?I
ID Z
N
K
KD
i−1 m
i
εn+ m = Qi (un+
m P
R=
]
T
J
X
IC D
i
εn+ m = un+ m − u.
,
H
VH
+
H
i
] V
D C
^
F
?
G
i = 1, . . . , m.
|[vi ]| ≤ k(I − T )−1 k|[v]| ≤ (1 − kT k)−1|[v]| = γ|[v]|.
LB
TZ D
H V
E
F
?H
B C
O
F
?
ID Z
E
T
H
P > H
Ke D
?D
^ H V
H
+ Ri u.
KB
h
kRi k = kQi k = 1
+ zi = Qi u
n+ i−1 m
+ zi = (Ri + Qi )u
n+ i−1 m
]
=u
n+ i−1 m
] H
H T
L
,
X
u
i n+ m
C
^
= (R1 + R2 Q1 + · · · + Rm Qm−1 · · · Q1 )(I − T )−1 v = v1 + · · · vm ,
F
ED
C
V
D
zi = R i zi = R i u − R i u
I −T
v = (I − T )(I − T )−1 v
∀w ∈ V,
, w)
n+ i−1 m
k(I − T )−1k ≤ (1 − kT k)−1
]
T
C
^
a(Ri u, v) =
I
a(Ri zi , w) = a(Ri u, w) − a(Ri u
n+ i−1 m
D
X H X
AH
E
PQ
D
LD
O
Ke D
D
PJ
A
I
NV
B
, v) ∀v ∈
w∈V v = Ri w ∈ Vi0 a(Ri u, Ri w) = a(Ri Ri u, w) = a(Ri u, w) K h
Vi0 . LD
a(zi , v) = a(Ri u, v) − a(u
n+ i−1 m
^
C
D
kT k < 1 ^
F
?
Ke D
G
?D
]
E
H E
I
I
I
J
= R1 + R2 Q1 + R3 Q2 Q1 + · · · + Rm Qm−1 · · · Q1 .
N
OB
I
A
X
^
V
E
FR BH
KD
A
]
D
γ → +∞ O H
T = (I − Pm ) · · · (I − P1 ) un+1 − u = T (un − u),
en
D
U G
en+1
U H
]
i−1 m
RB
en+1 = (I − Pm ) · · · (I − P1 )en .
]
) U F
?H
C
D
V
G
G
IE B
LB
TZ D
A B
EQ
H
G CG
I
N
O
IH
ZAH
N
K
T
W
B
P i @
B
H
A
]
EQ
H
G CG
I
W
Q
CI B
E
O
IH
FY
G
B
]
T
D
M
D
W
Q
A
IC B
E
e
ED
V
W
Q
O
I
IH
[
X
^
W
A B
^
H
J
ED
T
YB
?
P
W
O
I
IH T
I
KF
O
NI B
ND
CG
D
G
[
U
AH
H
D
H
H
D Ke D
FL
E
NA
V
Ce G
N
O
G
B
D
IH
FY
A
A CB
V
O
IH
K
PB
CG
T
V DH
O
R
AH
GH
B
]
A
W
I
IN
FK
OB L
CI
S
KJ
F
S
O
I
O
FC
O
P
S
N
O
I
^
^
U P
AH
D
J
H
D
FY
G
QD A
A CB
V
E
I
TB
TB
I
U H
P
PD
KJ
F
S
O
O I
FC
O
P
CG
H
H
H
D
LD
I
KB
OB
OQ
C
O
E
K
O
R H
R
H
OD
? KD
I
?
^
A
B
CG
H
ID
OA
EJ
H
D
A D
CV
N
D
TD
S
V
W
Q
OB
IA
L
C
R
PQ
]
A EB
AH X
^
I
O
E
O
B
IH
EB
%
S
V Vi n(ni ) 0
T
L
^
T
B
LD
+
O
X
CG
IB
K X
D
H
IH
J
IB
A
Ce G
F
A
H
H
N
W
IN
FK
OB L
IC
I
U
?
C
D
B
Ωh
OJ
en ^
= (I − Pi )en+
OB
J TB
KG
I
m
D
H
LB
TZ D
G H
D
un+1
X
[
VB
A
V[
OQ
WH
^
N
O
IH
OD
W
H TD
C
[
IC B
E
A
IC B
D
D
n, ni
D
I
BM
O
N
O
IH
Y
Y
K X
B
?
O
TD
?
H
H F
H
D
KD
O
A
D
CD
W
A
CI B
A
I
X
I
D
A
IA
E
E
X
P
P
C
H
H
Ωh
?I
P
$A
i−1 m
ID
J
n+ − RiT A−1 i Ri Ae
Q
C
? DH
I
K X
RiT zi ,
Z
N
B PG
A
i−1 m
H
$ F
TD
TD
S
A
H
C
A OB
B
[
$
CG
IB
?
WH
NB
O
^
E
E
I
]
Ωi
O
^
I
C
[
X
I
P
G
U
E
TB
KD
CG
X
F
G
U Uhi Ωh Ωhi
IH
H
H
+
C
E
= Ri (f − Aun+ n+ i−1 m
I
VD
D ID
G
H
?I
H
K X
D
TD
I
TB
J
+
?
Ke D
G
?D
F
K
I
P
h
K
D
V
I
VD
LD
O
T
AH
C
N
R
I X A B
H
B
J
B
D
^ ED
A
B
l
i=1
^
^
m S
AH
0
H
RD
K
PB
CG
T
AH
L
K
B
O?
D WH
D
E
?I
]
E
O
IW H
Ω =
T
H AH
Pi = I − RiT A−1 i Ri
F
LD
= u
ED
A
H
N
O
IH
T
V
V
O
H
H
H
CG
E
C
E
]
xk
V
V
O
B OD
W
$
O
IH
Y
B YD
S
O
]
B
Ai
FN
O
P
i−1 m
H
H
X
i n+ m
O
IA
C
I
?I
A K X
I
I
N
K
T
I
P
Ri
J
D
A CB CB
I
IH
O
D
PB
A
H
H VH
H
CG
O
H
rih : Uh → Uhi
Ke D
G
H
D
i
en+ m = en+
?
K
V
F
OA
P
Ai zi
V
J ^ ED
B
ID
Z
H
A
G
PB
F
K
I
P
ni × n i Ωi
I
V
W
N
K h
u
C
I
B
X
A
IC B
3
εi .
I
FKe
A
+
H
−u
E
D
B
E
T
L
3
7
$
O
R
OB
IA
]
N
K
T
D
|[R2 vε ]|2 ≤ ε2 .
E
C
FH
E
?I
A
3
2
5
i n+ m
H
P
CG
C
E
7
7
6
u
H
D
CI D
35
]
D
LD
IA
√ εi−1 )2 .
Z
P
CD
U
CD
G
EG
7
35
^
H
H
C
R
BH
i = 3, . . . , m
ni × n
∂Ωi
$
H
∪ Ω2 ) I
U
\
?
N
K
3
3
X
E
E
^ E
P
J
D
D
QB
?
V
O
|[R1 vε ]|2 ≤ ε ≡ ε1 .
VAH
[
FO?
FM
1
23
3
A
D
D
R
LD
I
KB
O
I
M
D
LD
K
N
K
T
X
H
C
R
?
C
C
D
DH
B
FH
CG P
F
G
]
^
]
C
R D
D
LD
O
P
CD
IA
VC D
E
H
YB
? h D
]
C ] H
^
Ce G
F
kT k < 1 kT k = 1 vε ∈ V |[vε ]| = 1 |[T vε ]|2 ≥
^
]
v2 = (1 − ξ)v. I
^ D
A
B
5
67 5
2
D
LD
D
H
EI
C
R
^ E
C
NG
CB
i=1
m X √
U AH
FL
T
B
B
H
3
7
KB
D
ε>0
H
G
A
A
A
6
E
O
2
V
T
Ω1 Ω2 ^
E
]
9
4
εi ≤ mγ
QD
T? B
K
^
PD
I
V
]
I
I
V
C
R
O
T
1 − εi−1 −
]
O
L
E
N
K h
1
1
RD
CB
i=1
KD
B
IG
QD
A CB
TB
D
CG
m X √
V
A
I TB
TB
%
]
6
6
5
3
7
T
E
FH
^
p
O
I
KD
]
P
Ce
5
8
N
K
T
H
k
C
I
Z
VH
OD
B
S?
O
IW H
3
2
3
7
R
J
Ke D
|[Ri vε ]|≤kR −1
εi → 0 i = 2, . . . , m R R−1 εi = 1 − (
^
#
O
PB
d
?
KG
|[ξ]| ∼ d−1 Ω I
A
D
CG
OQ
1 . 1 − kT k
NG
H10 (Ω1
C
v∈V =
AH
v1 = ξv, A
γ= D
7
6
3
G
H
+ P
2
EI
[
D
5
3
7
9
D
IAH
P
CD
A VD
P
e
VH
R l IH
1 − ε2 ≤ |[Qm · · · Q3 Q2 vε ]|2 ≤ |[Q2 vε ]|2 , R2 = I − Q 2
D
O
YD
1
'
'
i=1
R
O
I
T
IH
S
E
2
IA
V
I K
m X
N
I OB
A
F
P
k
TD
B
P
Q
?AH
]
;
23
6
3
−1
O
O
T
N
6
6
2
6
;
FR
I
KD
T
L
|[R1 vε ]| + |[Q1 vε ]| = 1
2
CG
[
H
1
(
|[Ri vε ]| ≤ εi ,
ε→0
F
?
Ke D
FM
O
OB
I
P
F
:
I
1−ε ≤ |[T vε ]|2 = |[Qm · · · Q2 Q1 vε ]|2 ≤ kQm · · · Q2 k2 ·|[Q1 vε ]|2 ≤ |[Q1 vε ]|2 2
FCI
D
X
G
A
C
OB
CG
OD
3
6
6
Rvε ]|≤kR
O
D
V
QD
P
Q
?AH
[
23
2
|[Qi vε ]| ≥ 1 − εi ,
P
G
?D
B
H
j
2
C
ξ(x) A PB
A
KD
V
O
I
VB
FR
KG
γ
T U H
M
^
J
AH
SZI
A
−1
K
N
K
?I
G
M
Q
T = Qm · · · Q2 (I − R1 ) √ √ √ 2 2 |[Qm · · · Q2 vε ]| ≥ |[T vε ]| −|[Qm · · · Q2 R1 vε ]|2 ≥ 1 − ε1 − ε1 = 1 − ε2 √ √ ε 2 = 1 − ( 1 − ε 1 − ε 1 )2 |[Ri vε ]|2
L
Q
T
E
K
W
I
1=|[vε ]|=|[R
N
CD
γ ∼ d−1 S
AH
D
IH
T
V
I
V
P
Q
?
1−ε
ε>0
rih uh (xk ) = uh (xk ) Ωhi Ri
Ai = Ri ARiT .
un
i = 1, . . . , m. i
en+ m =
,
−1 2 T −1 T −1 T −1 Pi2 = (RiT A−1 i Ri A) = Ri Ai Ri ARi Ai Ri A = Ri Ai Ai Ai Ri A = Pi .
"
"
$
&"
!
&&
&
"
$
! #" "
#
$
W
H
^
H
KQ J
F
O
I
]
^
TV
K
I
D
X
∂Ω1
A
IC B
E
A12 A22 , ]
H
KQ J
F
O
R1 = (I1 O) , R2 = (O I2 ) . ^
H
√ d>C τ
I
ln εh2 τ −1 , O?
]
S
^
H
D
LD
C
A
G
H
EA
O
O
B
D
IH
O
A PB
F
S
?I
R
[
I
H A ED
C
EB
A
B
UR
I
V
C
τ >0
OQA
N
H
KD
AH FK
L
D
CG
B
D
P
A
τ →0 WH
H
I
V
U H
O
A
H
GH
D
PB
F
C?
O
OB
PD
A
G
[
D
\
S
OQ
P
C
I
?H \
N
O
I DH
A
CG
V
CG
ID
D
H
BH
D
U H
V S
I
Z
Ke D
P
eC G
E
TD
IS
O
C
TB
VC D
J
X
X
U
OQ
O
F
D
D
G
H
EJ
P
OA
N
FK
L
O
IG
O
IH
O
A PB
F
S
QD A
CD
[
P
D
B
H
D
[
B ] B
W
A
P i
B
TD
C
E
LD
O
P
C
I
IT B T
CG
G
H
H
TB
A
B
BH
D
U KB
CV
FTZ
S
F
C
R
^ RB
J
TD
S
V
Ke D ?I
?
O
K
I
S
IF
P
T
O
IH
ZAH
P
H
A H
D
B
OQ
US
P
S
D
CG
H
H PJ
IO
eC G
R
NB
Ke D Z
C
^ DH
Ke D Z
C
I
Q
?AH
V
KG
I
#
X
X
]
N
K
T
OA
P
I
J
D
H
H
OB
R
H
?
Y H
K X
I
E
C
^ H
U H
O
A
G
H
PB
F
S
?I
R
C
I
V
K
I
K
S
FL
IA
T
I
RB
I
TB
]
A D
D
G
H
G
C D
CG
ID
E
TD
SI
eC G
?
FY
S
IW H
I O
T
IH
A
H
$
H
B CG [
D K X
TD
NV
OB
I
L
KB
O
G
QD
KD
V
OQ
I
I
Z
CD
C
G
H
H
U AH
H
VH
Z
Oe
E
E
R
d
B
YD
B
q<1 O
E
O
I
? \
I
B
O
YD
FI
I \
\
H PG
OQ
C
A
I
D
VD
]
^
U
D
H
A D
D
G
D
H
H
PB
B
H
IJ
W
IA
C
I
CG
ID
E
TD
IS
G
Ce G
?
O
R
CD E
K
Q
FY
[
X
? j B
P
NG
C
U
H
B
B
A
H
D
IAH C
V
C
O
I
O
YD
E
W
A
P i @
TD
C
[
]
N
K
T H
]
AH
D
D
G
H
M
B
H
H
F G
G
B
B
TD
C
E
D
LD
O
RD
H
U B
U
[
X
[
K X
W
IA
C
F
I
TD O
C
N
K
PB
CG
T
C ] H V
]
%
H
D
N
E
O
H
IH
O
A PB
F
OQ B
J
PG
B
TD
C
E
−1
P
K
H
V
E
O
D
BH
EJ
A
?H
O
O I
IH
H
I
Q
?
^
EQ
H
G CG
I
N
O
IH
ZAH
N
K
T
W
E
OQ
P
C
I
K
I
I
]
T
P
I
P
H
NG
]
G
D
H
[
H
B CG [
K X
D
TD
PV D
TB
K
P?
E
I
OB
A PD
EI
FE
O
B
H
B
H
D
UC
P
Q
I
B
B
HB
G
B
VJ
W
A
P i @
B
TD
C
E
LD
O
P
CB
I
?
K
I
V
C
I
FKe
E
N
W
IA
C
I
O
T l
A
H
G
H
B
B
BH
A G
B
H
UO
AH
CD
EG
Z
PQ
CD
NG
C
R
IC D K
W
A
P i
B
TD
C
E
N
I
P
NB
]
]
AH
D
D
G
H
M
B
H
[
X
H
D
I
? D
B
TD
C
E
LD
O
RD
X
K X
W
IA
C
F
I
TD O
C
N
K
PB
CG
T
C ] H V
−1
IT
PB
B I X KD
G KQ B
E
W
IB
P
C
P
Q
I T
D A EI B
A ZAH
I
[
EB
J
B
G
BH
[
LD
O
0 A22
A
CG
A
B
]
D C? D
C
R
A CB
H
D
]
O
I
UA
L
G
R
M
H IB
IA
C
ID
T
IT B
A11 A12
IB
S
V
N
W
IJ
C
PD
IAH
D E
A G
C
?
L
KB I
A
S
I T
I
VA
S
A
C
T
I
X
D
E
I
K
B
B
B
?J
W
A
P i
B
]
D
P
CB
I
?
K
I
O
P
C
I
T
IT
0 A22
G
VD
?
NG
AH
B
−τ ∆ + I IJ
^
^
N
C
I
H
U D
A
H PB
IJ
H
D
TD
C
E
LD
O
P
A CG
P
C
IE
CG
W
I
IB
?
AH
J
H
OQ
Ke
H
Ke
K
D
Ce
I
IE
OQ
OB
O
H
un+1 = un +
D K X
X
G IAH C
?
T
IF
E
A
K
KB
D
H
CG
D
D O X
BM NG
C
ZAH
RB
I
J
TB
TD
C
H
C
I
T
TI B
V
C
I
KeF
[
A11 0
C
I
O
IH
U
I \
\
I
V f B
Q
V
IE
U
LD
O
QD A
CD
?B
FT
T
NG
Ce
N
D
D E X H O
E
X
AH V$
I
T
TI B
E
K
J
B
A ^ ED
TV D
C
X
X
E
D
H
J
]
B
H
T
A
B
?AH
B
TV D
P
CB
I
?
I K
O
CG
IB
kT k
OB
A
A
IS
OQ
P
N
O
A
V
S
P
CG
C
H
B
[
K X
D
TD
D
H
TD
SI
O
F
C
Ce G
D
CB
P
D H
?I
I l
1
IB
I
TD
S
AH
H
B
D
I
I
N
ED V
O
I O
OB
A
E
?
^
H
B
CG
H
BM
NV G
A
W
F
?A
CG
O
H
E
+
O
= un +
L
V
D
VA
X
CG
D
C V
I
FKe
E
M
G
K X
D
TD
I
P
H IB
IA
C
[
Ke
K
PI B
B
un+ 2 un+1
KB
[
X
ε>0 O
B
G
LD
A CB
H
D
D
A U OB
A
?
eC
AH
CD
A
H
V
NG
S
I
Q
D
CB
B
E
?I
Z
E
?I
C
H
CG
M
V
I K
CI
N
FE
B TV D
V
O
I
P
] D
BM
?AH
H
[
O
H
?
?
Ce B
Au = f
B
K
D K X
AH
H
P
T
F
G
B
EG
A
D
D
FM
OA
C
?H
P
P
]
ID
T
Ke
X
Pi )
N
K
T
UO
FC
C
YI
K
TB
T
H
W
I
O
YD
E
B
B
D
O
Ke B
A
[
YB
KG
F
^
i=1
TD
N
K
BH
O
P
OQ B
O
A
Y
E
I
D
A ?AH
?[
W
A
O
R
Q
H
B
CG
A ?H
K
P
D
A PD
Pi )en .
T
FR
KG
ED
A
H
I
V
N
I C
A
D
H
U
RB
?AH
H
VH
O
ˆi Ω
PD
FE
O
AH
Q
?AH
H
D
O
Ke
P i
B
A
IC B
B OJ
I
C
H
K
B
A KB
AH
H
i=1
= Ri (f − Aun ), i = 1, . . . , m, m X = un + RiT zi .
V
Ωh2
m X
C
J
JH
∩ Ωh2
H
B
KI
V
LD
K
A KB
TD
C
E
D
E
?
O IB
I
I
V
n RiT A−1 i Ri Ae = (I −
>
V
PA
H
G
VD
F
Ωh1
f.
R
H
C
O
B
H
C
X
K X
D
TD
H VD
L
(I − T )u = (I − T )A −1
^ E
O
I
O
[
X
E
D
LD
J
IB
A
IAH O
R
OD
D
i=1
m P
IH
K
AH
D
]
]
]
[
R
^ E
C
IH
%
?
B=(
R
^ E
Ke B
E
V
K
I
ID
O
P
C
I
T
EB
W
V
O
I
i=1
FC
M
B
O
I
O
Ke
H
CB
TI B
ZB IG
B
eC G
A
m X
PB
CG
O
K
T
LD
g
KQ
H
PD
H
K
en+1 = en −
YI D
KD
Ωh2
P
H
]
]
B
TJ
A
T
K
I
I
un+1
T
0
E
B IE [
]
EG
KG
Ai zi
AH
D
?
FR
Ωi
V
OA
J
CB
AH
∂Ω2 Ωh1 A, R1 , R2 A11 A= A21 P
Ωh1 ^
KG
E
A
IC D
EG
G
B
Ω
B
^
N IE
W
NI
FK
L
D
H
A H
B
B
H
D
H
EQ
H
G CG
I
N
K
T
W
IB
IA
C
ID
CG
A
TV D
C
E
Ce G
W
[
P i @
TD
C
E
F
ED
C
H
D
B
B
BH
G
B
I T
P
P
H
NG
C
P
Q
I
A VJ
W
P i @
B
TD
C
E
LD
O
P
C
I
T
B
H
B
IT B
N
W
IA
C
I
O
T l
un+1 = un + (I − T )u − (I − T )un = un + (I − T )A−1 f − (I − T )un . n T −1 n un+1 = un + R1T A−1 11 R1 (f − Au ) + R2 A22 R2 (f − Au )
(f − Aun ),
n = un + R1T A−1 11 R1 (f − Au ) 1 n+ 12 = un+ 2 + R2T A−1 ) 22 R2 (f − Au
(f − Aun ),
"
"
$
&"
!
&&
&
"
$
! #" "
#
$
]
H
D
@
? G
BH
A
D
U D A
^
[ IC
I
LD A
Q H
CG K
I
P
CB
I
?
K
I
V
C
I
KeF
E
O B
~b
IB
D
U
\ J
I \ FI
T
I
W
I
I
?H
P
OD
?
D A ED
U H
D
−τ ∆ + E
P
O
B
^
J
G
S
N
V
B CG
D K X
TD
PV
RB
TB
S
EQ
I
PB
H IJ
O
ED
D
X
A
G
H
U B
S
OQ
O
FR
KD
@
?
P
U
A D
VH
C
N
O
IH
O
PB
A
D
H
H
F
L
Z
CG
NV
K
T
F
E
SH
G
FM
O
P
N
O H
[ ]
Ce
D M
Ce G
O
G RD
C
EQ
H
G
M
CG
I
O
IH
ZAH
F
O
ED
O
H
RD
C
?
O
IH
Y
IH
EB
B
O
CG
ID
C J
H TD
N
KD
A PD O
D PJ
V D
V
K
∂Ω
?AH
H
D K X
D
A
D
G
Ce G
?
A CB
$
H
A
B
un+1 = 0
G
ED V
O
Ke B
E
A
[
H
D
VG
N
O
^
E
O
IAH
O
Ωi
TD
VB
A
X
O
I
IE
I
VH
YB
T
D
H A ED
C
IH
O
PB
A
u ˆi = 0
AH
H
D
D
AH
H
H
D
?G
PQ
B
A EB
F
I
E
H
OB
I
TB
J
I
Z
ˆ i i = 1, . . . , m Ω
O
V
LD
V
O
IH
HB NG
C
V
E
O
IH
I
Ωi ,
O
A
B PB K
V
O
D
B
A ZAH
I
Ω,
P
C
I
I KG
H
A
CG
W
KQ B
B
G E
P i
B
TD
C
(−τ ∆ + I)ˆ u i = τ f + un
?H
P
^
EI
N
B
CG
D K X
TD
FV
H
τ >0
OD
?
ED
C
?H
P
C
TB
PV D
G
i = 1, . . . , m
F
E
O
T
Y
E
(~b·∇)
Ke B
OD
LD
C
(−τ ∆ + I)un+1 = τ f + un
D
O
H
IH
T
B K X M
O
C
ε
∂Ωi ,
un+1 |Ωˆ i = u ˆi |Ωˆ i .
Ωi
U
H
D
A
A
BH
K X
I
I
V
C
P
D
B
B
V HX G IH R
W
P i
B
TD
C
E
LD
O
P
C
I
T
B
H
B
TI B
N
W
IA
C
I
O
TD
"
"
$
&"
!
&&
&
"
$
! #" "
#
$
H
?
CB
I
E
CB
]
]
] ^
]
]
]
^
]
]
g
g
] B
]
O
I
[
c
A B
D
H
H
PJ
P
CG
OB
A
CG
A
VT D
PV
W
c
`
]
]
]
]
]
]
] H
G
J
T U H
H
PB
S
5
7
OQ X k
FR
Z
PQ
N
CG
I
,
B
P i
4
2
H
H
1
N
@
/
,
%
%
?
3
.
)
-
OB
I
A PD
OA
I
C
Ke B
TD
C
1
3
6
2
3
@
?
]
^
^
]
]
U
0
$
1
"
b
6
^
]
]
0
&
H
F B
?B
%
%
@
?
f
$
?I
CB
I
E
CB
D
H
E[
O
Ke
C
KG
I
R
I
PQ
TQ D
H
C
8
?@
]
^
]
^
]
]
0
$
H
J
D
(
f
>
$
W
I
I
I
EV D
?
TQ D
C
6
6
6
l
;
@
?
^
]
U
0
b
@
?
]
^
]
]
0
A
JH
KB
I
OB
P
N
O
IH
YD
K
I
S
V
I
A
H
O? B
F
QD A
CB
V
B
U AH
l
3
3
4
l
;
]
^
^
I
B
PD
K
?H
C
#
c
f
>
6
6
6
?@
]
0
.
c
@
?
]
^
F B
?B
f
$
]
]
^
]
g
]
0
c
`
A
2
56
H
j
?I J
I \
D
G
H
I[
?
R
CB
I
E
CB
E
I RB
J
H
BH
TB
PQ
9;
@
?
]
]
]
]
]
U
5
4
%
H
P
U D
]
]
]
X
/
,
c
/
3
1 2
) . 0
.
)
-
+
@
? )*
U
] B
]
E
CB
E
]
]
]
G
H
G
?D
A
EI B
R? D
C
EI
PQ D
K
FL
I
EI
?I
R
I OD
?G
S
N
CG
B
+
D K X
P[
O
I
]
]
^
]
]
]
]
]
g
]
A
G
H
c
`
PB O U H
F
S
?I
R
C
I
V
K
I
K
N
K
T
I RB
J
H
BH
TB
PQ
A
6
7
2
3
j
;
?@
U
&
!
? @
]
]
]
]
]
]
]
]
]
]
g
H
I
E
H
J
CB
E
I
E
CB
E
KG
R
I
PQ
IE
\
+
]
]
]
^
g
0
c
A
H
D
D
P H
O
IH
R
OB
IA
LD
EI
OQ
O
CG
H
H
G
B
CG
A
D
CB
V
UN
G
H
LD
?
R
C
I
V
K
I
K
N
K
T
]
]
]
]
0
$
&
b
`
c
#
$
G
H
D
B
TD
C
E
LD
O
OD
W
B
B
AH
IG
?
K
D
LD
O
TD
CG
ID
E
TD
IS
3
l
?@
]
]
(
'
]
]
]
]
]
^]
^
]
]
]
g
&
c
%
+
>
I >
]
U
H
H
?
"
!
$
%
%
#
@
TD
C
E
ED
O
B
H
OD
W
IA
C
I
ED
O
TD
6
2
7
5
X
l
@
?
]
]
^
]
]
]
]
]
^
]
]
]
]
g
0
g
K
c
`
+
f
? h D
B
B
A
B
?
@
S
O
IB
Z
E
S
PQ D
W
IS
OQ
A OD
C
?H
K
O
W
A
B
P i
B
H
D
H
M
TD
C
E
L
\
K
B
A H
UA F
I
OA
C H
Ke B
N
O
IH
O
E
I
V
CG
ID
OJ
A
D
D
G
H
WH
6
56
X
l
;
]
^
?@
F
B
?B
f
$
]
0
]
]
^
^
]
]
]
H
H
H
H
H
PD
C
O
E
K
S
OQ
R
OD
?
TQ D
C
E
OQ
GH RD
C
D
LD
O
3
g
0
:
A D
>
;
@
B
?
F
?B
f
$
EQ
H
G CG
I
IG
QG
WH
RV
Q
H
B H
B
J
S
B
RB
TB
S
OQ
A
B
OD
W
I
I
P
P
CG
IB
A D
G
G
WH
V
]
]
^
]
]
]
g
+
>
I >
] B
D K X
N
O
IH
K
TJ
A
TQ D
C
E
I
H
B
PD
K
?H
C
B U AH
O? B
F
QD A
A CB
V
6
6
6
3
3
4
l
;
?@
D
LD
O
B
H
D
OD
W
IA
C
I
LD
O
TD
CG
ID
E
TD
IS
G
CG
D
G A ID
?
3
2
6
l
3;
6
@
?
]
]
^
]
]
]
]
g
G
H
[
A B
UK
S
OQ
CG
D
OJ
N
O
IH
ZAH
TD
C
E[ H
OQ
OD
W
B
B
H
IG
?
K
3
2
6
3;
6
+
>
I >
O
IH
O
A PB
F
S
?I
R
C
I
V
K
I
@
?
]
^
]
0 I
A
C
6
H
H
H
H
B
G
H
U J
S
?I
R
C
I
V
K
I
K
N
K
T
PD
C
O
E
K
S
OQ
R
OD
?
TD
@
?
$
RB
T
^
]
]
]
+
f
]
] g
XIII
]
]
A
D
K
? h D
CG
A
A
D
H
B
B
ID
FL
F
V
I
I
C
P
W
A
P i
E
C
I
L
K
6
3
3
@
?
0
K
CI B
EJ
I
]
>
$
^ C
]
EB
TD
C
D
E U H
E
OQ
GH
D
RD
C
LD
O
E
NV
O
IH
O
A YB
F
V
I
W
I
I
?H
2
5
:
@
?
$
"
"
&
]
]
]
1
1
1 ]
@
?
1
$
1
1
$
? @
]
]
^
]
]
]
]
g
1
c
`
&
@
?
]
]
]^
g
c
`
]
]
]
]
]
]
]
5
.
)
-
/
,
/
? @ ]
]
^
]
]
]
]
]
]
]
]
g
c
`
5
.
)
-
/
,
@
?
]
]
] ^
]
]
g
c
`
1
)
%
%
@
? ) /
]
^
&
c
]
0
"
4
%
]
c
7
@
?
]
]
^
]
]
]
g
c
`
]
]
1
-
5
%
@
?
]
]
^
]
]
]
g
&
c
`
c
]
]
"
5
#
7
4
%
%
%
@
? - 0 )
]
]
^
]
]
]
]
]
g
c
`
]
]
1
5
%
^
&
$
'
$
'
%
)
)-
$
+
)
0
@
?
]
^
]
0
"
c
%
4
)
7
%
0
0
!
!
@
?
]
]
^
]
]
]
]
]
]
]
]
g
c
`
5
3
%
%
^
*
)
%
@
?
]
]
^]
]
]
]
g
c
`
]
]
)
b
+
0
@
?
$
"
"
&