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and
(u n+1
ih
A Lagrange Multiplier Based Domain Decomposition Method 139
− 2u n ih + u n−1
ih
)(un+1 ih
u n ihu n+1
ih
− u n−1
ih
)=(un+1 ih
− u n ih) 2 − (u n ih − u n−1
ih )2
= 1 4 ((un+1 ih
+ u n ih) 2 − (u n+1
ih
− u n ih) 2 ),
after several technical transformations we obtain
√
ε 1 µ −1
E n+1 −E n 1
+ S Γext ((u n+1
1h
4∆t
− un−1 1h )2 )=
1
2 S 1(f1 n (u n+1
1h
− un−1 1h )) + 1 2 S 2(f2 n (u n+1
2h
− un−1 2h )).
Therefore,
E n+1 ≤E n + 1 ( [
2 ∆tS1/2 1 (f
n
1 ) 2) S 1/2
1
+S 1/2
1
( (u
n+1
1h
− un 1h
∆t
( (u
n+1
1h
− un 1h
∆t
)2 )] + 1 ( [
2 ∆tS1/2 2 (f
n
2 ) 2) S 1/2
2
+S 1/2
2
( (u
n+1
2h
)2 ) +
( (u
n+1
2h
− un 2h
∆t
− un 2h
∆t
)2 ) +
)2 )] . (19)
Now, we will show that under the condition (14) the quadratic form E n is
positive definite; more precisely, that there exists a positive constant δ such
that
( ( (u
n+1
E n 1h
≥ δ S − )2 ) ( (u un n+1
1h
2h
1 + S − )2 ))
un 2h
2 . (20)
∆t
∆t
Obviously, it is sufficient to prove the inequality
4ε e µ e S e (vh) 2 ≥ ∆t 2 S e (|∇v h | 2 ) ∀e ∈T 1h ∪T 2h , ∀v h ∈ P 1 (e), (21)
where ε e and µ e are defined by ε e = ε 1 or ε e = ε 2 (respectively, µ e = µ 1 or
µ e = µ 2 ). It is known that for a regular triangulation
S e (|∇v h | 2 ) ≤ 1/c 2 1h −2
e S e (vh) 2 (22)
with a positive constant c 1 , universal for all triangles e,whereh e is the minimal
length of the sides of e. Combining (21) and (22), we observe that the time
step ∆t should satisfy the inequality
∆t ≤ c √ ε e µ e h e , (c = √ 2c 1 ), (23)
for all e ∈T 1h ∪T 2h . Evidently, (14) ensures the validity of (23).
Further, using the relation (20), E 1 = 0 and summing the inequalities (19),
one obtains the stability inequality (17):
n−1
∑
E n ≤ M∆t (S 1 ((f1 k ) 2 )+S 2 ((f2 k ) 2 )), ∀n.
k=1