Ta
.
p ch´ı Tin ho
.
c v`a Diˆe
`
u khiˆe
’
n ho
.
c, T.21, S.3 (2005), 216—229
NGHI
ˆ
EN C
´
U
.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ
O
.
T PHU
.
O

ˆ
EN H
ˆ
O
˜
N HO
.
.
P
TRONG MI
ˆ
E
`
N H
`
INH HO
.
C PH
´
U
.
C TA
.
P
D
˘
A
.
NG QUANG
´

´
t mˆo
.
t phu
.
o
.
ng ph´ap chia miˆe
`
n gia
’
i c´ac b`ai to´an biˆen
elliptic v´o
.
i c´ac
diˆe
`
u kiˆe
.
n biˆen hˆo
˜
n ho
.
.
p trong miˆe
`
n h`ınh ho
.
c ph´u
.

´
th´ı du
.
v´o
.
i c´ac miˆe
`
n cˆa
´
u th`anh t`u
.
hai, ba
ho˘a
.
c nhiˆe
`
u ho
.
n h`ınh ch˜u
.
nhˆa
.
t v´o
.
i c´ac cˆa
´
u h`ınh kh´ac nhau. C´ac kˆe
´
t qua
’

.
n biˆen.
1. MO
.
’
D
ˆ
A
`
U
Trong [1]
d˜a dˆe
`
xuˆa
´
t mˆo
.
t phu
.
o
.
ng ph´ap chia miˆe
`
n m´o
.
i gia
’
i phu
.
o

.
ng ph´ap c˜ung nhu
.
chı
’
ra tham sˆo
´
l˘a
.
p tˆo
´
i u
.
u cho tru
.
`o
.
ng ho
.
.
p miˆe
`
n ch˜u
.
nhˆa
.
t. Trong b`ai b´ao n`ay ch´ung tˆoi tiˆe
´
p tu
.

’
n phu
.
o
.
ng ph´ap n`ay l`a su
.
.
ca
’
i thiˆe
.
n mˆo
.
t phˆa
`
n vˆe
`
tˆo
´
c
dˆo
.
hˆo
.
i
tu
.
v`a th`o
.

n n`ay s˜e
du
.
o
.
.
c
ch´ung tˆoi chı
’
ra trˆen mˆo
.
t th´ı du
.
trong mu
.
c 3 cu
’
a b`ai b´ao.
2. M
ˆ
O TA
’
PHU
.
O
.
NG PH
´
AP
X´et b`ai to´an

217
trong d´o
Ω
l`a miˆe
`
n gi´o
.
i nˆo
.
i trong
R
2
v´o
.
i biˆen Lipschitz
∂Ω
cˆa
´
u th`anh t`u
.
c´ac phˆa
`
n biˆen tro
.
n
∂Ω =
k

j=1
S

`
ng
u = 
i
u = ϕ
i
(x), x ∈ S
i
, (i = 1, . . . , k), (3)
trong d´o

i
u = u
(diˆe
`
u kiˆe
.
n biˆen Dirichlet), ho˘a
.
c

i
u =
∂u
∂ν
i
(diˆe
`
u kiˆe
.

ph´u
.
c ta
.
p.
Dˆo
´
i v´o
.
i b`ai to´an biˆen Dirichlet, t´u
.
c l`a khi
u = u
, nhiˆe
`
u t´ac gia
’
d˜a dˆe
`
xuˆa
´
t v`a
nghiˆen c´u
.
u c´ac phu
.
o
.
ng ph´ap chia miˆe
`

i gi´a tri
.
da
.
o
h`am cu
’
a nghiˆe
.
m trˆen biˆen chung gi˜u
.
a c´ac miˆe
`
n, phu
.
o
.
ng ph´ap n`ay c´o thˆe
’
xem l`a ngu
.
o
.
.
c
dˆo
´
i
v´o
.

.
u
cu
’
a tham sˆo
´
l˘a
.
p
d˜a du
.
o
.
.
c thiˆe
´
t lˆa
.
p. Theo ch´ung tˆoi
du
.
o
.
.
c biˆe
´
t chu
.
a c´o nghiˆen c´u
.

thˆe
´
, trong b`ai b´ao n`ay ch´ung tˆoi tiˆe
´
p tu
.
c ph´at triˆe
’
n phu
.
o
.
ng ph´ap
d˜a du
.
o
.
.
c nghiˆen c´u
.
u cho
c´ac b`ai to´an v´o
.
i
diˆe
`
u kiˆe
.
n biˆen hˆo
˜

Dˆe
’
dˆe
˜
h`ınh dung ´y tu
.
o
.
’
ng cu
’
a phu
.
o
.
ng ph´ap , du
.
´o
.
i
dˆay ch´ung tˆoi mˆo ta
’
phu
.
o
.
ng ph´ap chia
miˆe
`
n gia

\ Γ, u
i
= u|
Ω
i
, (i = 1, 2)
, trong d´o
∂Ω
i
l`a biˆen cu
’
a miˆe
`
n
Ω
i
. D˘a
.
t
g =
∂u
1
∂ν
1




Γ
v´o

n nhu
.
sau: xuˆa
´
t ph´at t`u
.
xˆa
´
p xı
’
ban
dˆa
`
u
g
(0)
,
v´o
.
i
k = 0, 1, 2,
gia
’
i liˆen tiˆe
´
p 2 b`ai to´an






−∆u
(k)
2
= f trong Ω
2
,
u
(k)
2
= ϕ trˆen G
2
,
u
(k)
2
= u
(k)
1
trˆen Γ.
(4)
Xˆa
´
p xı
’
m´o
.
i cu
’
a

NG QUANG
´
A, V
˜
U VINH QUANG
trong d´o
θ
l`a tham sˆo
´
l˘a
.
p cˆa
`
n cho
.
n dˆe
’
qu´a tr`ınh l˘a
.
p hˆo
.
i tu
.
.
Trong tru
.
`o
.
ng ho
.

p (3)-(5) s˜e
du
.
o
.
.
c ´ap du
.
ng
dˆe
’
hiˆe
.
u chı
’
nh da
.
o h`am ph´ap tuyˆe
´
n
cu
’
a h`am trˆen c´ac biˆen chung. Tham sˆo
´
l˘a
.
p
θ
trˆen mˆo
˜

p xı
’
bˆa
.
c 2 v`a su
.
’
du
.
ng cˆong th´u
.
c sai phˆan c`ung bˆa
.
c
dˆe
’
t´ınh da
.
o
h`am ph´ap tuyˆe
´
n trong (5).
Khi c´ac miˆe
`
n con l`a h`ınh ch˜u
.
nhˆa
.
t ch´ung tˆoi
d˜a xˆay du

nhau.
K´y hiˆe
.
u
L
1
v`a
L
2
l`a dˆo
.
d`ai cu
’
a c´ac ca
.
nh h`ınh ch˜u
.
nhˆa
.
t,
h = L
1
/M, k = L
2
/N
l`a c´ac
bu
.
´o
.

da
.
ng phu
.
o
.
ng tr`ınh hˆe
.
vecto
.
3
diˆe
’
m.
−Y
j−1
+ CY
j
− Y
j+1
= F
j
, 1  j  N − 1, Y
0
= F
0
, Y
N
= F
N

,
(7)
dˆo
´
i v´o
.
i b`ai to´an biˆen hˆo
˜
n ho
.
.
p, trong
d´o
Y
j
l`a c´ac vecto
.
nghiˆe
.
m,
F
j
l`a c´ac vecto
.
M
chiˆe
`
u ch´u
.
a

i.
K´y hiˆe
.
u
b1, b2, b3, b4
l`a c´ac gi´a tri
.
biˆen Dirichlet ho˘a
.
c Neumann trˆen c´ac biˆen tr´ai, pha
’
i,
du
.
´o
.
i v`a trˆen cu
’
a miˆe
`
n ch˜u
.
nhˆa
.
t.
´
Ap du
.
ng c´ac thuˆa
.

diˆe
’
m lu
.
´o
.
i, ch´ung tˆoi tiˆe
´
n
h`anh xˆay du
.
.
ng c´ac thu
’
tu
.
c b˘a
`
ng ngˆon ng˜u
.
MATLAB
dˆe
’
gia
’
i c´ac b`ai to´an co
.
ba
’
n sau:

2
, M, N)
tra
’
la
.
i nghiˆe
.
m cu
’
a b`ai to´an trong tru
.
`o
.
ng ho
.
.
p
b1
l`a
gi´a tri
.
biˆen Neumann,
b2, b3, b4
l`a c´ac gi´a tri
.
biˆen Dirichlet.
+
U0100(b1, b2, b3, b4, L
1

1
, L
2
, M, N)
tra
’
la
.
i nghiˆe
.
m cu
’
a b`ai to´an trong tru
.
`o
.
ng ho
.
.
p
b3
l`a
gi´a tri
.
biˆen Neumann,
b1, b2, b4
l`a c´ac gi´a tri
.
biˆen Dirichlet.
+

C´ac thu
’
tu
.
c h`am
U0101(b1, b2, b3, b4, L
1
, L
2
, M, N), U1001(b1, b2, b3, b4, L
1
, L
2
, M, N)
,
U0110(b1, b2, b3, b4, L
1
, L
2
, M, N), U1010(b1, b2, b3, b4, L
1
, L
2
, M, N)
,
du
.
o
.
.

.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ
O
.
T PHU
.
O
.
NG PH
´
AP CHIA MI
ˆ
E
`
N
219
3. THU
.
.
C NGHI
ˆ
E

xuˆa
´
t c`ung v´o
.
i c´ac thu
’
tu
.
c h`am
d˜a xˆay du
.
.
ng, ch´ung tˆoi
tiˆe
´
n h`anh t´ınh to´an thu
.
.
c nghiˆe
.
m cho mˆo
.
t sˆo
´
tru
.
`o
.
ng ho
.

´
p dˆe
´
n. Trong c´ac
b`ai to´an n`ay ch´ung tˆoi cho
.
n tru
.
´o
.
c c´ac h`am
u
∗
l`a nghiˆe
.
m d´ung, c`on c´ac diˆe
`
u kiˆe
.
n biˆen v`a vˆe
´
pha
’
i
du
.
o
.
.
c t´ınh t`u

´
p
u
(k)
v`a
u
(k−1)
t´ınh theo chuˆa
’
n dˆe
`
u cu
’
a h`am lu
.
´o
.
i nho
’
ho
.
n
dˆo
.
ch´ınh x´ac

cho tru
.
´o
.












−∆u = f trong miˆe
`
n Ω,
u = ϕ, trˆen {−a  x  a, y = −b} ∪ {0  x  a, y = b}
∪{x = −a, −b  y  0} ∪ {x = a, −b  y  b},
∂u
∂y
= β(x) trˆen {−a  x  0, y = 0},
∂u
∂x
= g(y) trˆen {x = 0, 0  y  b}, (H`ınh 1).
Γ
b
y
0
0
0
0
0

Chia
Ω
th`anh hai miˆe
`
n
Ω
1
v`a
Ω
2
bo
.
’
i biˆen chung
Γ = {0  x  a, y = 0}
v`a k´y hiˆe
.
u
u
i
l`a
nghiˆe
.
m trong
Ω
i
, (i = 1, 2), ξ =
∂u
1
∂y

´o
.
c
ξ
(0)
= 0
. V´o
.
i
k = 0, 1, . . .
thu
.
.
c hiˆe
.
n c´ac bu
.
´o
.
c sau:
Bu
.
´o
.
c 1: Gia
’
i b`ai to´an trong miˆe
`
n
Ω

i b`ai to´an trong miˆe
`
n
Ω
2
220
D
˘
A
.
NG QUANG
´
A, V
˜
U VINH QUANG
T`ım nghiˆe
.
m
u
(k)
2
= U1000(. . . )
trong d´o
b1, b2, b4
l`a c´ac gi´a tri
.
trˆen c´ac phˆa
`
n biˆen d˜a
biˆe

cu
’
a h`am
ξ
trˆen
Γ : ξ
(k+1)
= θξ
(k)
+ (1 − θ)
∂u
(k)
2
∂y




Γ
.
Kˆe
´
t qua
’
thu
.
.
c nghiˆe
.
m kha

’
ng du
.
´o
.
i
dˆay, trong d´o cˆo
.
t “Sai sˆo
´
” chı
’
sai sˆo
´
cu
’
a nghiˆe
.
m gˆa
`
n d´ung so v´o
.
i nghiˆe
.
m
d´ung trong chuˆa
’
n dˆe
`
u. Tiˆeu chuˆa

=10x(1-x)y(1-y) u
∗
=10x(1-x)y
2
(1-y)
u
∗
= sin x sin y
Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´
lˆa
`
n
teta sˆo
´
l˘a

6
0.6 2.10
−6
10 0.6 4.10
−5
8 0.6 7.10
−6
11
0.7 2.10
−6
16 0.7 5.10
−5
12 0.7 7.10
−6
16
Kˆe
´
t luˆa
.
n: So
.
dˆo
`
l˘a
.
p gia
’
i b`ai to´an trˆen hˆo
.
i tu

p xı
’
b˘a
`
ng
0.5
.
Nhˆa
.
n x´et: V´o
.
i c´ach chia miˆe
`
n nhu
.
H`ınh 1 ch´ung tˆoi
d˜a gia
’
i b`ai to´an b˘a
`
ng c´ach l˘a
.
p cˆa
.
p
nhˆa
.
t gi´a tri
.
cu

n ngu
.
o
.
.
c la
.
i:
dˆa
`
u tiˆen trong
Ω
2
gia
’
i b`ai to´an
v´o
.
i
diˆe
`
u kiˆe
.
n biˆen Dirichlet trˆen
Γ
, sau d´o trong
Ω
1
gia
’

v`a th`o
.
i gian t´ınh to´an cu
’
a phu
.
o
.
ng ph´ap cˆa
.
p
nhˆa
.
t
da
.
o h`am du
.
o
.
.
c mˆo ta
’
o
.
’
trˆen v`a phu
.
o
.

.
.
c cho trong
c´ac ba
’
ng sau.
Ba
’
ng 1. Lu
.
´o
.
i
32 × 32
v`a
 = 10
−3
phu
.
o
.
ng ph´ap cˆa
.
p nhˆa
.
t
da
.
o h`am phu
.

9.10
−4
14 4.0
5.10
−4
0.4
5 1.6
3.10
−4
9 2.7
5.10
−4
0.5
4 1.3
3.10
−4
6 1.9
2.10
−4
0.6
6 1.9
4.10
−4
7 2.1
4.10
−4
0.7
9 2.6
5.10
−4

p nhˆa
.
t
da
.
o h`am cu
’
a ch´ung tˆoi c´o phˆa
`
n nhanh ho
.
n phu
.
o
.
ng ph´ap cˆa
.
p nhˆa
.
t h`am trong [2]. Ch´ınh
diˆe
`
u
n`ay l`a
dˆo
.
ng co
.
th´uc
dˆa

U
.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ
O
.
T PHU
.
O
.
NG PH
´
AP CHIA MI
ˆ
E
`
N
221
Ba
’
ng 2. Lu
.
´o

i gian
Sai Sˆo
´
lˆa
`
n Th`o
.
i gian
Sai
teta l˘a
.
p t´ınh (giˆay) sˆo
´
l˘a
.
p t´ınh (giˆay) sˆo
´
0.3
10 10.7
5.10
−5
14 14.9
7.10
−5
0.4
6 6.6
6.10
−4
7 7.6
5.10



















−∆u = f trong miˆe
`
n Ω,
u = ϕ, trˆen {−a  x  a, y = −b} ∪ {0  x  2a, y = b}
∪{x = −a, −b  y  0} ∪ {x = 2a, 0  y  b},
∪{a  x  2a, y = 0} ∪ {x = a, −b  y  0},
∂u
∂y
= β(x) trˆen {−a  x  0, y = 0},
∂u
∂x
= g(y) trˆen {x = 0, 0  y  b}, (H`ınh 2).

1
a
Ω
1
Ω
2
x
0
2
a
H`ınh 2
1 - Biˆen Neumann; 0 - Biˆen Dirichlet
Chia
Ω
th`anh hai miˆe
`
n
Ω
1
v`a
Ω
2
bo
.
’
i biˆen chung
Γ = {0  x  a, y = 0}
, k´y hiˆe
.
u

c thu
.
.
c hiˆe
.
n bo
.
’
i qu´a tr`ınh l˘a
.
p sau
dˆay:
Cho tru
.
´o
.
c
ξ
(0)
= 0
. V´o
.
i
k = 0, 1, . . .
thu
.
.
c hiˆe
.
n c´ac bu

D
˘
A
.
NG QUANG
´
A, V
˜
U VINH QUANG
biˆe
´
t,
b4 =

ξ
(k)
, 0  x  a, y = 0,
β(x), −a  x  0, y = 0.
Bu
.
´o
.
c 2: Gia
’
i b`ai to´an trong miˆe
`
n
Ω
2
T`ım nghiˆe

c t`u
.
b`ai to´an 1,
ϕ(x), a  x  2a, y = 0.
Bu
.
´o
.
c 3: Hiˆe
.
u chı
’
nh gi´a tri
.
cu
’
a h`am
ξ
trˆen
Γ : ξ
(k+1)
= θξ
(k)
+ (1 − θ)
∂u
(k)
2
∂y



u
∗
= sin x sin y
Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´
lˆa
`
n
teta sˆo
´
l˘a
.
p teta sˆo
´
l˘a
.

−4
4 0.5 7.10
−6
5
0.6 1.10
−6
12 0.6 8.10
−4
6 0.6 7.10
−6
8
0.7 2.10
−6
18 0.7 8.10
−4
8 0.7 7.10
−6
13
0.8 2.10
−4
20 0.8 8.10
−4
15 0.8 6.10
−6
20
0.9 0.019 20 0.9 0.0049 20 0.9 0.0015 20
Kˆe
´
t luˆa
.

’
ng
(0.1, 0.9)
, gi´a tri
.
tˆo
´
i u
.
u xˆa
´
p xı
’
b˘a
`
ng
0.5
.
B`ai to´an 3.












3
bo
.
’
i 2 biˆen chung
Γ
1
= {0  x  a, y = 0}
v`a
Γ
2
= {0  x  a, y = b}
k´y hiˆe
.
u
u
i
l`a nghiˆe
.
m trong 3 miˆe
`
n
Ω
i
, (i = 1, 2, 3), ξ
1
=
∂u
1
∂y

E
.
M M
ˆ
O
.
T PHU
.
O
.
NG PH
´
AP CHIA MI
ˆ
E
`
N
223
Γ
1
b
y
0
0
0
0
0
0
1
1

Γ
2
Ω
3
H`ınh 3
1 - Biˆen Neumann; 0 - Biˆen Dirichlet
Viˆe
.
c gia
’
i b`ai to´an
du
.
o
.
.
c thu
.
.
c hiˆe
.
n bo
.
’
i qu´a tr`ınh l˘a
.
p sau
dˆay:
Kho
.

c 1: Gia
’
i b`ai to´an 1 trong miˆe
`
n
Ω
1
T`ım nghiˆe
.
m
u
(k)
1
= U0001(. . . )
trong d´o
b1, b2, b3
l`a c´ac gi´a tri
.
trˆen c´ac phˆa
`
n biˆen d˜a
biˆe
´
t,
b4 =

ξ
(k)
1
, 0  x  a, y = 0,


ξ
(k)
2
, 0  x  a, y = b,
β(x), −a  x  0, y = 0.
Bu
.
´o
.
c 3: Gia
’
i b`ai to´an trong miˆe
`
n
Ω
3
T`ım nghiˆe
.
m
u
(k)
3
= U1000(. . . )
trong d´o
b1, b2
l`a c´ac gi´a tri
.
trˆen c´ac phˆa
`

.
c t`u
.
b`ai to´an 2.
Bu
.
´o
.
c 4:
Diˆe
`
u chı
’
nh gi´a tri
.
trˆen c´ac biˆen chung
ξ
(k+1)
1
= θ
1
ξ
(k)
1
+(1− θ
1
)
∂u
(k)
3

Kˆe
´
t qua
’
: K´ıch thu
.
´o
.
c miˆe
`
n
a = b = 1
, lu
.
´o
.
i chia
64 × 64
.
224
D
˘
A
.
NG QUANG
´
A, V
˜
U VINH QUANG
u

= θ
2
sˆo
´
l˘a
.
p
θ
1
= θ
2
sˆo
´
l˘a
.
p
θ
1
= θ
2
sˆo
´
l˘a
.
p
0.3 Khˆong 0.3 Khˆong 0.3 Khˆong
hˆo
.
i tu
.

−6
20 0.7 8.10
−4
13 0.7 8.10
−6
20
0.8 7.10
−6
30 0.8 8.10
−4
20 0.8 8.10
−6
30
0.9 0.0045 30 0.9 0.002 30 0.9 3.10
−4
30
Kˆe
´
t luˆa
.
n: So
.
dˆo
`
l˘a
.
p gia
’
i b`ai to´an trˆen hˆo
.

’
ng
(0.5, 0.6)
.
B`ai to´an 4.
MiÒn
rçng
0
Γ
4
b
y
0
0
0
0
1
1
a
Ω
1
Ω
3
x
1
Γ
3
Ω
2
Γ

Γ
1
1
Ω
4
Γ
2
H`ınh 4
1 - Biˆen Neumann; 0 - Biˆen Dirichlet
X´et b`ai to´an



















−∆u = f trˆen miˆe

2
=
{a  x  2a, y = b}, Γ
3
= {−a  x  0, y = b}
v`a
Γ
4
= {−a  x  0, y = 0}
, k´y hiˆe
.
u
u
i
l`a
NGHI
ˆ
EN C
´
U
.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ


Γ
1
, ξ
2
=
∂u
3
∂y




Γ
2
, ξ
3
=
∂u
3
∂y




Γ
3
, ξ
4
=

p sau
dˆay:
Kho
.
’
i
dˆo
.
ng
ξ
(0)
1
= 0, ξ
(0)
2
= 0, ξ
(0)
3
= 0, ξ
(0)
4
= 0
. V´o
.
i
k = 0, 1, . . .
thu
.
.
c hiˆe

n biˆen d˜a
biˆe
´
t,
b4 =





ξ
(k)
1
, a  x  2a, y = 0,
β(x), 0  x  a, y = 0,
ξ
(k)
4
, −a  x  0, y = 0.
Bu
.
´o
.
c 2: Gia
’
i b`ai to´an 2 trong miˆe
`
n
Ω
2

, a  x  2a, y = b.
Bu
.
´o
.
c 3: Gia
’
i b`ai to´an trong miˆe
`
n
Ω
2
T`ım nghiˆe
.
m
u
(k)
2
= U 1000(. . . )
trong d´o
b1, b2
l`a gi´a tri
.
trˆen c´ac phˆa
`
n biˆen d˜a biˆe
´
t,
b3 = u
(k)

.
´o
.
c 4: Gia
’
i b`ai to´an trong miˆe
`
n
Ω
4
T`ım nghiˆe
.
m
u
(k)
4
= U 0001(. . . )
trong d´o
b1, b2
l`a gi´a tri
.
trˆen c´ac phˆa
`
n biˆen d˜a biˆe
´
t,
b3 = u
(k)
1
l`a gi´a tri

.
c 5:
Diˆe
`
u chı
’
nh c´ac gi´a tri
.
trˆen c´ac biˆen chung
ξ
(k+1)
1
= θ
1
ξ
(k)
1
+ (1 − θ
1
)
∂u
(k)
2
∂y




Γ
1

− (1 − θ
3
)
∂u
(k)
4
∂y




Γ
3
, ξ
(k+1)
4
= θ
4
ξ
(k)
4
+ (1 − θ
4
)
∂u
(k)
4
∂y



´
A, V
˜
U VINH QUANG
u
∗
=10x(1-x)y(1-y) u
∗
=10x(1-x)y
2
(1-y)
u
∗
= sin x sin y
Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´
lˆa
`
n Tham sˆo
´
Sai Sˆo
´

.
p
θ
3
= θ
4
θ
3
= θ
4
θ
3
= θ
4
0.3 Khˆong 0.3 Khˆong 0.3 Khˆong
hˆo
.
i tu
.
hˆo
.
i tu
.
hˆo
.
i tu
.
0.4 0.0016 30 0.4 3.10
−4
30 0.4 2.10

0.9 0.0087 30 0.9 0.0042 30 0.9 0.0012 30
Kˆe
´
t luˆa
.
n: So
.
dˆo
`
l˘a
.
p gia
’
i b`ai to´an trˆen hˆo
.
i tu
.
v´o
.
i gi´a tri
.
tham sˆo
´
l˘a
.
p
du
.
o
.

1
1
a
Ω
1
Ω
3
x
1
Γ
6
Ω
5
Γ
2
1
Ω
4
Γ
5
MiÒn
rçng
1
1
1
0
Ω
2
Γ
3

Γ
5
MiÒn
rçng
1
1
1
0
Ω
2
Γ
3
Γ
4
H`ınh 5
1 - Biˆen Neumann; 0 - Biˆen Dirichlet















EN C
´
U
.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ
O
.
T PHU
.
O
.
NG PH
´
AP CHIA MI
ˆ
E
`
N
227
Chia
Ω
th`anh 6 miˆe

, k´y hiˆe
.
u
u
i
l`a nghiˆe
.
m trong 6 miˆe
`
n
Ω
i
, (i = 1, . . . , 6),
k´y hiˆe
.
u
ξ
1
=
∂u
1
∂y




Γ
1
, ξ
2



Γ
4
, ξ
5
=
∂u
3
∂y




Γ
5
, ξ
6
=
∂u
3
∂y




Γ
6
l`a c´ac gi´a tri
.

.
’
i qu´a tr`ınh l˘a
.
p sau
dˆay:
Kho
.
’
i
dˆo
.
ng
ξ
(0)
1
= 0, ξ
(0)
2
= 0, ξ
(0)
3
= 0, ξ
(0)
4
= 0, ξ
(0)
5
= 0, ξ
(0)

m
u
(k)
1
= U0001(. . . )
trong d´o
b1, b2, b3
l`a c´ac gi´a tri
.
trˆen c´ac phˆa
`
n biˆen d˜a
biˆe
´
t,
b4 =















(k)
3
= U0010(. . . )
trong d´o
b1, b2, b4
l`a gi´a tri
.
trˆen biˆen d˜a biˆe
´
t,
b3 =















ξ
(k)
4
, 3a  x  4a, y = b,

l`a gi´a tri
.
trˆen biˆen d˜a biˆe
´
t,
b3 = u
(k)
1
l`a gi´a tri
.
biˆen nhˆa
.
n du
.
o
.
.
c t`u
.
b`ai to´an 1,
b4 = u
(k)
3
l`a gi´a tri
.
biˆen nhˆa
.
n du
.
o

b3 = u
(k)
1
l`a gi´a tri
.
biˆen nhˆa
.
n du
.
o
.
.
c t`u
.
b`ai to´an 1,
b4 = u
(k)
3
l`a gi´a tri
.
biˆen nhˆa
.
n du
.
o
.
.
c t`u
.
b`ai to´an 2.

biˆen nhˆa
.
n du
.
o
.
.
c t`u
.
b`ai to´an 1,
b4 = u
(k)
3
l`a gi´a tri
.
biˆen nhˆa
.
n du
.
o
.
.
c t`u
.
b`ai to´an 2.
Bu
.
´o
.
c 6:

2
= θ
2
ξ
(k)
2
+ (1 − θ
2
)
∂u
(k)
5
∂y




Γ
2
,
228
D
˘
A
.
NG QUANG
´
A, V
˜
U VINH QUANG

4
)
∂u
(k)
2
∂y




Γ
4
,
ξ
(k+1)
5
= θ
5
ξ
(k)
5
− (1 − θ
5
)
∂u
(k)
5
∂y



’
: K´ıch thu
.
´o
.
c miˆe
`
n
a = b = 1
, lu
.
´o
.
i chia
64 × 64
.
u
∗
=10x(1-x)y(1-y) u
∗
=10x(1-x)y
2
(1-y)
u
∗
= sin x sin y
Tham Sai Sˆo
´
lˆa
`

´
l˘a
.
p
0.5 Khˆong 0.5 Khˆong 0.5 Khˆong
hˆo
.
i tu
.
hˆo
.
i tu
.
hˆo
.
i tu
.
0.6 0.0032 13 0.6 0.005 30 0.6 6.10
−5
30
0.7 1.10
−6
30 0.7 0.0048 18 0.7 1.10
−5
22
0.8 0.001 30 0.8 0.0048 30 0.8 2.10
−5
30
0.9 Khˆong 0.9 Khˆong 0.9 Khˆong
hˆo

.
tham sˆo
´
l˘a
.
p
du
.
o
.
.
c cho
.
n trong
khoa
’
ng
(0.6, 0.8)
, gi´a tri
.
tˆo
´
i u
.
u xˆa
´
p xı
’
0.7.
4. NH

.
.
c, ch´ung tˆoi c´o mˆo
.
t sˆo
´
kˆe
´
t luˆa
.
n v`a nhˆa
.
n x´et
sau
dˆay:
+
Dˆo
´
i v´o
.
i b`ai to´an biˆen elliptic v´o
.
i
diˆe
`
u kiˆe
.
n biˆen hˆo
˜
n ho

`
du
.
o
.
.
c mˆo
.
t sˆo
´
h˜u
.
u ha
.
n
c´ac b`ai to´an da
.
ng co
.
ba
’
n.
+
Dˆo
´
i v´o
.
i
diˆe
`

n phu
.
o
.
ng ph´ap su
.
’
du
.
ng l˘a
.
p gi´a tri
.
h`am trˆen biˆen.
+ Viˆe
.
c lu
.
.
a cho
.
n tham sˆo
´
tˆo
´
i u
.
u trong viˆe
.
c hiˆe

’
ng
di
.
nh b˘a
`
ng l´ı thuyˆe
´
t, nhu
.
ng qua
kˆe
´
t qua
’
thu
.
.
c nghiˆe
.
m cho thˆa
´
y r˘a
`
ng phu
.
o
.
ng ph´ap hˆo
.

.
i phu
.
o
.
ng ph´ap
d˜a dˆe
`
xuˆa
´
t c´o kha
’
n˘ang gia
’
i quyˆe
´
t du
.
o
.
.
c c´ac b`ai to´an biˆen elliptic v´o
.
i
diˆe
`
u kiˆe
.
n biˆen hˆo
˜

.
U THU
.
.
C NGHI
ˆ
E
.
M M
ˆ
O
.
T PHU
.
O
.
NG PH
´
AP CHIA MI
ˆ
E
`
N
229
[2] Saito N., Fujita H., Operator Theoretical Analysis to Domain Decomposition methods,
12
th
Int. Conf. on Domain Decomposition Problems, Editors: Tony chan, Takashi, Hideo,
Oliver Pinoneau, 2001, www. DDM.org, 63 - 70.
[3] Dang Q. A., Domain decomposition method for solving a strongly mixed boudary value