6 Stability and consistency
From
6
, we can state the following results.
Lemma 6.1
6
There exist two positive constants C
1
and C
2, depending only on the minimum angle of the decomposition and on the polynomial degree
C
1
∥
R
[
[
ψ
]
]
∥
0
,
Ω
2
≤
∑
e
∈
Γ
I
h
e
-
1
∥
[
[
ψ
]
]
∥
0
,
e
2
≤
C
2
∥
R
[
[
ψ
]
]
∥
0
,
Ω
.
Lemma 6.2
6
There exists two positive constants C
1
and C
2, depending only on the minimum angle of the decomposition such that
C
1
∥
R
[
[
ψ
]
]
∥
0
,
Ω
2
≤
∑
e
∈
Γ
I
h
e
-
1
∥
[
[
ψ
]
]
∥
0
,
e
2
≤
C
2
(
∥
R
[
[
ψ
]
]
∥
0
,
Ω
2
+
∥
∇
h
ψ
∥
0
2
)
,
ψ
∈
H
2
(
T
h
)
.
Lemma 6.3
6
Let
H
be a Hilbert spaces, and λ and μ positive constants. Then, for every ξ and η in
H
we have
λ
∥
ξ
+
η
∥
H
2
+
μ
∥
η
∥
H
2
≥
λ
μ
2
(
λ
+
μ
)
(
∥
ξ
∥
H
2
+
∥
η
∥
H
2
)
.
Theorem 6.1 (Stability) For δ = 1, problem (4.18) is stable for all θ ∈ (0,1).
Proof Consider first the case γ = 1, β = 1. From the definition of A
stab(·,·;·,·;·), we have
A
stab
(
u
h
,
u
h
;
p
h
,
p
h
;
c
)
=
a
(
u
h
,
u
h
;
c
)
+
e
(
p
h
,
p
h
)
+
θ
(
u
h
+
a
(
c
)
∇
h
p
h
,
-
α
(
c
)
u
h
+
∇
h
p
h
)
.
We remark that (6.4) can be rewritten as
A
stab
(
u
h
,
u
h
;
p
h
,
p
h
;
c
)
=
(
1
-
θ
)
∥
|
α
1
∕
2
(
c
)
u
∥
|
0
2
+
θ
∥
a
1
∕
2
(
c
)
∇
h
p
∥
0
2
+
∥
a
1
∕
2
(
c
)
[
[
p
]
]
∥
0
,
Γ
h
2
,
and the stability in the norm (5.2) follows from
θ
=
1
2
.
Consider now the case γ = 0, β =1. Using the equivalent expressions (4.11) and (4.12) for the first and second equation of (4.3), respectively, the problem (4.18) for γ = 0 can be rewritten as: find u
h
∈ (D
l-2(Th
))
d
, ph
∈ D
l-1(Th
) such that
(
α
(
c
)
u
h
+
∇
h
p
h
+
R
[
[
p
h
]
]
,
v
)
-
θ
(
α
(
c
)
u
h
+
∇
h
p
h
,
v
)
=
0
,
-
(
u
h
,
∇
h
ψ
+
R
[
[
ψ
]
]
)
+
δ
θ
(
u
h
+
a
(
c
)
∇
h
ψ
,
∇
h
ψ
)
=
(
q
,
ψ
)
.
From the first equation in (6.6) and (4.9) we have
α
(
c
)
u
h
=
-
(
∇
h
p
h
+
1
1
-
θ
R
[
[
p
h
]
]
)
.
Substituting the expression (6.7) in the second equation of (6.6) for δ = 1, we have
A
B
R
(
p
h
,
ψ
)
+
θ
1
-
θ
∫
Ω
a
(
c
)
R
[
[
p
h
]
]
⋅
R
[
[
ψ
]
]
d
x
=
(
q
,
ψ
)
,
∀
ψ
∈
D
l
-
1
(
T
h
)
.
Denote by B1h
(·,·) the bilinear form (6.8), we have
B
1
h
(
ψ
,
ψ
)
=
∥
a
(
c
)
(
∇
h
ψ
+
R
[
[
ψ
]
]
)
∥
0
,
Ω
+
θ
1
-
θ
∥
a
(
c
)
1
∕
2
R
[
[
ψ
]
]
∥
0
,
Ω
2
,
and the stability in the norm (5.3) follows from Lemma 6.1. This completes the proof. □
Theorem 6.2 For δ = -1, problem (4.18) is stable for all θ < 0.
Proof Consider first the case γ = 1, β = 1. The problem (4.18) for δ = -1 reads
A
stab
(
u
h
,
u
h
;
p
h
,
p
h
;
c
)
=
a
(
u
h
,
u
h
;
c
)
+
θ
(
u
h
+
a
(
c
)
∇
h
p
h
,
-
α
(
c
)
u
h
-
∇
h
p
h
)
+
e
(
p
h
,
p
h
)
.
Using the arithmetic-geometric mean inequality, we have
A
stab
(
u
h
,
u
h
;
p
h
,
p
h
;
c
)
≥
(
1
-
2
θ
)
∥
|
α
1
∕
2
(
c
)
u
∥
|
0
2
-
2
θ
∥
a
1
∕
2
(
c
)
∇
h
p
∥
0
2
+
∥
a
1
∕
2
(
c
)
[
[
p
]
]
∥
0
,
Γ
h
2
,
and since θ < 0 the result follows.
Consider now the case γ = 0, β = 1. From (6.7) the second equation of (6.6) for δ = -1 can be written as
A
B
R
(
p
h
,
ψ
)
+
2
θ
1
-
θ
(
R
[
[
p
h
]
]
,
a
(
c
)
∇
h
ψ
)
+
θ
1
-
θ
∫
Ω
a
(
c
)
R
[
[
p
h
]
]
⋅
R
[
[
ψ
]
]
d
x
=
(
q
,
ψ
)
.
We remark that formulation (6.12) can be rewritten as
1
1
-
θ
A
B
R
(
p
h
,
ψ
)
-
θ
1
-
θ
A
B
O
(
p
h
,
ψ
)
=
(
q
,
ψ
)
,
where ABO
(ph
, ψ) is introduced by Baumann and Oden
14
, and given by
A
B
O
(
p
h
,
ψ
)
:
=
∫
Ω
a
(
c
)
(
∇
h
p
h
-
R
[
[
p
h
]
]
)
⋅
(
∇
h
ψ
+
R
[
[
ψ
]
]
)
d
x
+
∫
Ω
a
(
c
)
R
[
[
p
h
]
]
⋅
R
[
[
ψ
]
]
d
x
.
Denote by B
2h
(·,·) the bilinear form (6.13), we have
B
2
h
(
ψ
,
ψ
)
=
1
1
-
θ
∥
a
1
∕
2
(
c
)
(
∇
h
ψ
+
R
[
[
ψ
]
]
)
∥
0
,
Ω
-
θ
1
-
θ
∥
a
1
∕
2
(
c
)
∇
h
ψ
∥
0
,
Ω
2
,
and since θ < 0 the result follows again from Lemma 6.3 and 6.1. □
Theorem 6.3 (Consistency) If p,c and
u
are the solution of (2.1)-(2.5) and are essentially bounded, then
(
ϕ
c
t
,
w
)
+
B
(
c
,
w
;
u
)
=
L
(
w
;
u
,
c
)
,
∀
w
∈
L
2
(
0
,
T
;
H
k
(
T
h
)
)
A
stab
(
u
,
v
;
p
,
ψ
;
c
)
=
l
stab
(
ψ
)
,
∀
(
v
×
ψ
)
∈
(
(
L
2
(
0
,
T
;
H
l
-
1
(
T
h
)
)
)
d
×
L
2
(
0
,
T
;
H
l
(
T
h
)
)
)
provided that the constant M for the cut-off operator is sufficiently large.
To summarize, for all the bilinear forms in (6.4), (6.10), (6.8) or (6.13) we have: ∃C > 0 such that
B
1
h
(
ψ
,
ψ
)
≥
C
∥
ψ
∥
1
,
h
2
,
B
2
h
(
ψ
,
ψ
)
≥
C
∥
ψ
∥
1
,
h
2
,
∀
ψ
∈
D
l
-
1
(
T
h
)
,
and ∃C > 0 such that
A
(
v
,
v
;
ψ
,
ψ
;
c
)
stab
≥
C
∥
(
v
,
ψ
)
∥
stab
2
,
∀
(
v
,
ψ
)
∈
(
D
l
-
2
(
T
h
)
)
d
×
D
l
-
1
(
T
h
)
,
where (6.17) clearly holds for every θ ∈ (0,1) for the case ((6.4), (6.8)), and for every θ < 0 for the case ((6.10), (6.13)). On the other hand, since ∇
h
D
l-1(Th
) ⊂ (D
l-2(Th
))
d
holds, boundedness of the bilinear form in (6.8) and (6.13) follows directly from the boundedness of the bilinear forms A
BR and A
BO, as proved in
13
, thanks to the equivalence of the norms (6.1) and (6.2). Thus, we have: ∃C > 0 such that
B
1
h
(
p
h
,
ψ
)
≤
C
∥
p
h
∥
1
,
h
∥
ψ
∥
1
,
h
,
B
2
h
(
p
h
,
ψ
)
≤
C
∥
p
h
∥
1
,
h
∥
ψ
∥
1
,
h
,
∀
p
h
,
ψ
∈
D
l
-
1
(
T
h
)
.
7 Error estimates
Let
(
u
̃
,
p
̃
,
c
̃
)
be an interpolation of the exact solution (u, p, c) such that
a
(
u
̃
,
v
;
c
)
-
b
(
p
̃
,
v
)
=
0
,
∀
v
∈
(
D
l
-
2
(
T
h
)
)
d
,
b
(
ψ
,
ũ
)
+
e
(
p
̃
,
ψ
)
=
(
q
,
ψ
)
,
∀
ψ
∈
D
l
-
1
(
T
h
)
,
(
c
̃
-
c
,
w
)
=
0
,
∀
w
∈
D
k
-
1
(
T
h
)
.
Let us define interpolation errors, finite element solution errors and auxiliary errors
ξ
1
=
u
̃
-
u
h
,
ξ
2
=
u
̃
-
u
,
e
u
=
u
-
u
h
=
ξ
1
-
ξ
2
;
η
1
=
p
̃
-
p
h
,
η
2
=
p
̃
-
p
,
e
p
=
p
-
p
h
=
η
1
-
η
2
;
τ
1
=
c
̃
-
c
h
,
τ
2
=
c
̃
-
c
,
e
c
=
c
-
c
h
=
τ
1
-
τ
2
.
It was proven in
18
that
∥
|
α
1
∕
2
(
c
)
ξ
2
|
∥
0
2
+
∥
a
1
∕
2
(
c
)
[
[
η
2
]
]
∥
0
,
Γ
h
2
≤
C
h
2
l
-
2
(
∥
u
∥
l
-
1
2
+
∥
p
∥
l
2
)
.
hold for all t ∈ J with the constant C independent only on bounds for the coefficient α(c), but not on c itself.
Theorem 7.1 (Error estimate for the velocity and pressure) Let (
u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L
2(0, T; Hl
(Th
)), u ∈ (L
2(0, T; H
l-1(Th
)))
d and c ∈ L
2(0, T; Hk
(Th
)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h such that
∥
(
u
-
u
h
,
p
-
p
h
)
∥
stab
2
(
t
)
≤
C
(
∥
c
-
c
h
∥
0
2
(
t
)
+
h
2
l
-
2
)
.
Proof For the sake of brevity we will assume
θ
=
1
2
,
δ
=
1
in the following content. Consider the case γ = 1, β = 1. From the second equation of (5.1) and (6.16) we have
(
α
(
c
)
u
-
α
(
M
(
c
h
)
)
u
h
,
v
)
-
b
(
p
-
p
h
,
ψ
)
+
b
(
ψ
,
u
-
u
h
)
+
e
(
p
-
p
h
,
ψ
)
-
1
2
(
u
h
+
a
(
M
(
c
h
)
)
∇
h
p
h
)
,
-
α
(
M
(
c
h
)
)
v
+
∇
h
ψ
)
+
1
2
(
u
+
a
(
c
)
∇
h
p
,
-
α
(
c
)
v
+
∇
h
ψ
)
=
0
.
That is
(
α
(
c
)
(
u
-
u
̃
)
,
v
)
+
(
α
(
M
(
c
h
)
)
(
u
̃
-
u
h
)
,
v
)
+
(
(
α
(
c
)
-
α
(
M
(
c
h
)
)
)
u
̃
,
v
)
-
b
(
p
-
p
h
,
v
)
+
b
(
ψ
,
u
-
u
h
)
+
e
(
p
-
p
h
,
ψ
)
+
1
2
(
α
(
M
(
c
h
)
)
u
h
-
α
(
c
)
u
,
v
)
+
1
2
(
u
-
u
h
,
∇
h
ψ
)
-
1
2
(
∇
h
p
-
∇
h
p
h
,
v
)
+
1
2
(
a
(
c
)
∇
h
p
-
a
(
M
(
c
h
)
)
∇
h
p
h
,
∇
h
ψ
)
=
0
.
Choosing v = ξ
1, ψ = η
1 and splitting ep
according ep
= η
1- η
2, from (7.1) and we obtain
1
2
(
α
(
M
(
c
h
)
)
ξ
1
,
ξ
1
)
+
e
(
η
1
,
η
1
)
+
1
2
(
a
(
M
(
c
h
)
)
∇
h
η
1
,
∇
h
η
1
)
=
1
2
(
(
α
(
M
(
c
h
)
)
-
α
(
c
)
)
u
̃
,
ξ
1
)
-
1
2
(
α
(
c
)
ξ
2
,
ξ
1
)
+
1
2
(
a
(
c
)
∇
h
η
2
,
∇
h
η
1
)
-
1
2
(
(
a
(
c
)
-
a
(
M
(
c
h
)
)
)
∇
h
p
̃
,
∇
h
η
1
)
+
1
2
(
ξ
2
,
∇
h
η
1
)
-
1
2
(
∇
h
η
2
,
ξ
1
)
.
Let us first consider the left side of error equation (7.5)
1
2
(
α
(
M
(
c
h
)
)
ξ
1
,
ξ
1
)
+
e
(
η
1
,
η
1
)
+
1
2
(
a
(
M
(
c
h
)
)
∇
h
η
1
,
∇
h
η
1
)
=
1
2
(
∥
|
α
1
∕
2
(
M
(
c
h
)
)
ξ
1
|
∥
0
2
+
∥
a
1
∕
2
(
M
(
c
h
)
)
∇
h
η
1
∥
0
2
)
+
∥
[
[
η
1
]
]
∥
0
,
Γ
h
2
.
We know that (7.2) and quasi-regularity that
∇
h
p
̃
,
u
̃
are bounded in L
∞(Ω). So the right side of the error equation (7.5) can be bounded from below. Noting that
|
α
(
M
(
c
h
)
)
-
α
(
c
)
|
≤
C
|
c
h
-
c
|
, we have
|
(
α
(
M
(
c
h
)
)
-
α
(
c
)
)
u
̃
,
ξ
1
)
|
≤
C
∥
c
-
c
h
∥
0
2
+
ε
∥
|
ξ
1
|
∥
0
2
.
The second and the third terms of the right side of the error equation (7.5) can be bounded using Cauchy-Schwartz inequality and approximation results,
|
(
α
(
c
)
ξ
2
,
ξ
1
)
|
≤
∥
α
(
c
)
∥
0
,
∞
∥
ξ
2
∥
0
∥
ξ
1
∥
0
≤
ε
∥
|
ξ
1
|
∥
0
2
+
C
h
2
l
-
2
,
|
(
a
(
c
)
∇
h
η
2
,
∇
h
η
1
)
|
≤
ε
∥
∇
h
η
1
∥
0
2
+
C
h
2
l
-
2
.
The fourth term can be bounded in a similar way as that for the first term
|
(
a
(
c
)
-
a
(
M
(
c
h
)
)
∇
h
p
̃
,
∇
h
η
1
)
|
≤
C
∥
c
-
c
h
∥
0
2
+
ε
∥
∇
h
η
1
∥
0
2
.
The last two terms can be bounded as follows
(
ξ
2
,
∇
h
η
1
)
≤
ε
∥
∇
h
η
1
∥
0
2
+
C
h
2
l
-
2
,
(
∇
h
η
2
,
ξ
1
)
≤
ε
∥
|
ξ
1
∥
|
0
2
+
C
h
2
l
-
2
.
Substituting all these inequalities into Equation 7.5, we have
1
2
(
∥
|
α
1
∕
2
(
M
(
c
h
)
)
ξ
1
|
∥
0
2
+
∥
a
1
∕
2
(
M
(
c
h
)
)
∇
h
η
1
∥
0
2
)
+
∥
a
1
∕
2
(
c
)
[
[
η
1
]
]
∥
0
,
Γ
h
2
≤
ε
(
∥
|
ξ
1
∥
|
0
2
+
∥
∇
h
η
1
∥
0
2
)
+
C
(
∥
c
-
c
h
∥
0
2
+
h
2
l
-
2
)
.
Using the facts
1
C
I
≤
α
(
M
(
c
h
)
)
≤
C
I
,
1
C
I
≤
α
(
c
)
≤
C
I
, and
1
C
I
≤
a
(
M
(
c
h
)
)
≤
C
I
,
1
C
I
≤
a
(
c
)
≤
C
I
we have
∥
(
ξ
1
,
η
1
)
∥
s
t
a
b
2
≤
C
(
∥
c
-
c
h
∥
0
2
+
h
2
l
-
2
)
.
The theorem follows from the triangle inequality.
Now consider the case γ = 0, β = 1. The bilinear form (6.8) from the second equation of (5.1) for
θ
=
1
2
reads
A
B
R
(
p
h
,
ψ
;
c
h
)
+
∫
Ω
a
(
M
(
c
h
)
)
R
[
[
p
h
]
]
⋅
R
[
[
ψ
]
]
d
x
=
(
q
,
ψ
)
,
where
A
B
R
(
p
h
,
ψ
;
c
h
)
=
(
a
(
M
(
c
h
)
)
(
∇
h
p
h
+
R
[
[
p
h
]
]
)
,
∇
h
ψ
+
R
[
[
ψ
]
]
)
.
Replacing (6.8) with ph
= p and subtracting it from (7.13) we finally obtain
a
(
c
)
(
∇
h
p
,
∇
h
ψ
+
R
[
[
ψ
]
]
)
-
A
B
R
(
p
h
,
ψ
;
c
h
)
=
0
.
Choosing ψ = η
1, we have
A
B
R
(
η
1
,
η
1
;
c
h
)
+
∫
Ω
a
(
M
(
c
h
)
)
R
[
[
η
1
]
]
⋅
R
[
[
η
1
]
]
d
x
=
A
B
R
(
η
2
,
η
1
)
+
∫
Ω
a
(
c
)
R
[
[
η
2
]
]
⋅
R
[
[
η
1
]
]
d
x
+
(
a
(
M
(
c
h
)
)
-
a
(
c
)
)
(
(
∇
h
p
̃
+
R
[
[
p
̃
]
]
,
∇
h
η
1
+
R
[
[
η
1
]
]
)
+
∫
Ω
a
(
c
)
R
[
[
p
̃
]
]
⋅
R
[
[
η
1
]
]
d
x
)
.
Let us first estimate the left side of (7.15). From (6.17) and using the fact
1
C
I
≤
a
(
M
(
c
h
)
)
≤
C
I
,
1
C
I
≤
a
(
c
)
≤
C
I
, we have
A
B
R
(
η
1
,
η
1
;
c
h
)
+
∫
Ω
a
(
M
(
c
h
)
)
R
[
[
η
1
]
]
⋅
R
[
[
η
1
]
]
d
x
≥
C
∥
η
∥
1
,
h
2
.
The first and the second terms of the right side of (7.15) can be bounded using Lemma 6.1 and (3.5)
A
B
R
(
η
2
,
η
1
)
+
∫
Ω
a
(
c
)
R
[
[
η
2
]
]
⋅
R
[
[
η
1
]
]
d
x
≤
C
∥
η
2
∥
1
,
h
∥
η
1
∥
1
,
h
,
≤
ε
∥
η
1
∥
1
,
h
2
+
C
h
2
l
-
2
.
Note that
∇
h
p
̃
,
p
̃
are bounded in L
∞(Ω) and
|
a
(
M
(
c
h
)
)
-
a
(
c
)
|
≤
C
∥
c
h
-
c
∥
, we have
(
a
(
M
(
c
h
)
)
-
a
(
c
)
)
(
∇
h
p
̃
+
R
[
[
p
̃
]
]
,
∇
h
η
1
+
R
[
[
η
1
]
]
)
+
∫
Ω
a
(
c
)
R
[
[
p
̃
]
]
⋅
R
[
[
η
1
]
]
d
x
≤
C
∥
c
-
c
h
∥
0
∥
η
1
∥
1
,
h
≤
C
∥
c
-
c
h
∥
0
2
+
ε
∥
η
1
∥
1
,
h
2
.
Substituting all these inequalities into the (7.15) and using the triangle inequality we have
∥
p
-
p
h
∥
1
,
h
2
≤
C
(
∥
c
-
c
h
∥
0
2
+
h
2
l
-
2
)
.
We easily deduce,using (7.19)
∥
|
α
1
∕
2
(
c
)
(
u
-
u
h
)
|
∥
0
2
≤
C
∥
p
-
p
h
∥
1
,
h
2
≤
C
(
∥
c
-
c
h
∥
0
2
+
h
2
l
-
2
)
,
which completes the proof. □.
From
7
, we state two lemmas for the properties of the dispersion-diffusion tensor, which will be used to prove error estimates for the concentration.
Lemma 7.1
7
(Uniform positive definiteness of D(u)) Let
D(u) defined as in Equation 2.6, where ϕdm
(x) ≥ 0, dl
(x) ≥ 0 and dt
(x) ≥ 0 are non-negative functions of x ∈ Ω. Then
D
(
u
)
∇
h
c
⋅
∇
h
c
≥
(
ϕ
d
m
+
min
(
d
l
,
d
t
)
|
u
|
)
|
∇
h
c
|
2
.
If, in addition, φdm
(x) ≥ d
m,* > 0 uniformly in the domain Ω, then
D(u) is uniformly positive definite in Ω:
D
(
u
)
∇
h
c
⋅
∇
h
c
≥
d
m
,
*
|
∇
h
c
|
2
.
Lemma 7.2
7
(Uniform Lipschitz continuity of D(u)) Let
D(u)
defined as in Equation 2.6, where dm
(x) ≥ 0, dl
(x) ≥ 0 and dt
(x) ≥ 0 are non-negative functions of x ∈ Ω, and the dispersivities dl and dt is uniformly bounded, i.e., dl
(x) ≤ dl
*
and dt
(x) ≤ dt
*. Then
∥
D
(
u
)
-
D
(
v
)
∥
(
L
2
(
Ω
)
)
d
×
d
≤
k
D
∥
u
-
v
∥
(
L
2
(
Ω
)
)
d
.
where
k
D
=
(
7
d
t
*
+
6
d
l
*
)
d
3
∕
2
is a fixed number (d = 2 or 3 is the dimension of domain Ω).
Theorem 7.2 (Error estimate for concentration) Let (
u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L
2(0, T; Hl
(Th
)), u ∈ (L
2(0, T; H
l-1(Th
)))
d and c ∈ L
2(0, T; Hk
(Th
)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h and Δt such that
∥
ϕ
(
c
-
c
h
)
∥
L
∞
(
0
,
T
;
L
2
(
Ω
)
)
+
(
∑
i
=
1
N
△
t
(
∥
|
D
1
∕
2
(
u
i
-
1
)
∇
h
(
c
i
-
c
h
i
)
|
∥
0
2
+
∥
[
[
c
i
-
c
h
i
]
]
∥
0
,
Γ
h
2
)
)
1
∕
2
≤
C
(
△
t
+
h
k
-
1
+
h
l
-
1
)
.
Proof The first equation of (5.1) is
(
ϕ
c
h
n
+
1
-
c
h
n
△
t
,
w
)
+
B
(
c
h
n
+
1
,
w
;
u
M
n
)
=
L
(
w
;
u
M
n
,
c
h
n
+
1
)
.
It can be written as
(
ϕ
τ
2
n
+
1
-
τ
2
n
△
t
,
w
)
-
(
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
w
)
+
(
ϕ
c
n
+
1
-
c
n
△
t
,
w
)
+
B
(
c
h
n
+
1
,
w
;
u
M
n
)
=
L
(
w
;
u
M
n
,
c
h
n
+
1
)
.
Subtracting the DG scheme equation from the weak formulation, we have for any w ∈ D
k-1(Th
)
(
ϕ
c
t
,
w
)
-
ϕ
τ
2
n
+
1
-
τ
2
n
△
t
,
w
+
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
w
-
ϕ
c
n
+
1
-
c
n
△
t
,
w
-
B
(
c
h
n
+
1
,
w
;
u
M
n
)
+
B
(
c
n
+
1
,
w
;
u
n
)
=
L
(
w
;
u
n
,
c
n
+
1
)
-
L
(
w
;
u
M
n
,
c
h
n
+
1
)
.
that is
(
ϕ
c
t
,
w
)
+
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
w
+
B
(
τ
1
n
+
1
,
w
;
u
M
n
)
=
ϕ
τ
2
n
+
1
-
τ
2
n
△
t
,
w
+
ϕ
c
n
+
1
-
c
n
△
t
,
w
+
B
(
τ
2
n
+
1
,
w
;
u
M
n
)
+
B
(
c
n
+
1
,
w
;
u
M
n
)
-
B
(
c
n
+
1
,
w
;
u
n
)
+
L
(
w
;
u
n
,
c
n
+
1
)
-
L
(
w
;
u
M
n
,
c
h
n
+
1
)
.
Choosing w = τ
1
n+1, we obtain
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
τ
1
n
+
1
+
B
(
τ
1
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
=
ϕ
c
n
+
1
-
c
n
△
t
,
τ
1
n
+
1
-
(
ϕ
c
t
,
τ
1
n
+
1
)
+
ϕ
τ
2
n
+
1
-
τ
2
n
△
t
,
τ
1
n
+
1
+
B
(
τ
2
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
+
B
(
c
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
-
B
(
c
n
+
1
,
τ
1
n
+
1
;
u
n
)
+
L
(
τ
1
n
+
1
;
u
n
,
c
n
+
1
)
-
L
(
τ
1
n
+
1
;
u
M
n
,
c
h
n
+
1
)
.
Let us first consider the left side of the error equation (7.25). The first term can be bounded as
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
τ
1
n
+
1
≥
ϕ
2
△
t
(
(
τ
1
n
+
1
,
τ
1
n
+
1
)
-
(
τ
1
n
,
τ
1
n
)
)
.
The second term of Equation 7.25 is
B
(
τ
1
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
=
(
D
(
u
M
n
)
∇
h
τ
1
n
+
1
,
∇
h
τ
1
n
+
1
)
+
(
u
M
n
⋅
∇
h
τ
1
n
+
1
,
τ
1
n
+
1
)
-
∫
Ω
q
-
(
τ
1
n
+
1
)
2
d
x
+
∫
Γ
h
C
11
[
[
τ
1
n
+
1
]
]
[
[
τ
1
n
+
1
]
]
d
s
.
The second term of B(·,·;·) can be estimated using the boundedness of u
M
and
1
C
D
(
u
)
≤
D
(
u
M
)
≤
C
D
(
u
)
:
(
u
M
n
⋅
∇
h
τ
1
n
+
1
,
τ
1
n
+
1
)
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
|
∥
0
2
+
C
∥
ϕ
τ
1
n
+
1
∥
0
2
.
Thus
ϕ
τ
1
n
+
1
-
τ
1
n
△
t
,
τ
1
n
+
1
+
B
(
τ
1
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
≥
ϕ
2
△
t
(
∥
τ
1
n
+
1
∥
0
2
-
∥
τ
1
n
∥
0
2
)
+
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
-
C
∥
ϕ
τ
1
n
+
1
∥
0
2
.
Let us bound the right side of the error equation (7.25).
L
(
τ
1
n
+
1
;
u
n
,
c
n
+
1
)
-
L
(
τ
1
n
+
1
;
u
M
n
,
c
h
n
+
1
)
≤
C
∥
ϕ
τ
1
n
+
1
∥
0
2
.
Using Taylor series expansion, we have
ϕ
c
n
+
1
-
c
n
△
t
,
τ
1
n
+
1
-
(
ϕ
c
t
,
τ
1
n
+
1
)
≤
ϕ
2
△
t
∥
D
t
2
c
∥
L
2
(
t
k
,
t
k
+
1
;
L
2
(
Ω
)
)
+
C
∥
ϕ
τ
1
n
+
1
∥
0
2
.
The fourth term in the right side of the error equation (7.25) is
B
(
τ
2
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
=
(
D
(
u
M
n
)
∇
h
τ
2
n
+
1
,
∇
h
τ
1
n
+
1
)
+
(
u
M
n
⋅
∇
h
τ
2
n
+
1
,
τ
1
n
+
1
)
-
(
q
-
τ
2
n
+
1
,
τ
1
n
+
1
)
+
∫
Γ
h
{
{
D
(
u
M
n
)
∇
h
τ
1
n
+
1
}
}
[
[
τ
2
n
+
1
]
]
d
s
-
∫
Γ
h
{
{
D
(
u
M
n
)
∇
h
τ
2
n
+
1
}
}
[
[
τ
1
n
+
1
]
]
d
s
+
∫
Γ
h
C
11
[
[
τ
1
n
+
1
]
]
[
[
τ
2
n
+
1
]
]
d
s
=
:
∑
i
=
1
6
T
i
.
Terms T
1 through T
3 can be bounded by using Cauchy-Schwartz inequality and approximation results,
|
T
1
|
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
∥
∇
h
(
c
n
+
1
-
c
̃
n
+
1
)
∥
0
2
,
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
h
2
k
-
2
,
and
|
T
2
|
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
h
2
k
-
2
,
|
T
3
|
≤
C
(
∥
ϕ
τ
1
n
+
1
∥
0
2
+
h
2
k
)
.
Terms T
4 and T
5 can be estimated using inverse inequalities,
|
T
4
|
≤
ε
h
∑
K
∈
Γ
h
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
,
∂
K
+
C
h
∑
K
∈
Γ
h
∥
τ
2
n
+
1
∥
0
,
∂
K
2
,
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
h
2
k
-
2
,
and
|
T
5
|
≤
ε
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
+
C
h
∑
K
∈
Γ
h
∥
τ
2
n
+
1
∥
0
,
∂
K
2
,
≤
ε
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
+
C
h
2
k
-
2
.
Using Cauchy-Schwartz inequality and the trace inequality, we have
|
T
6
|
≤
ε
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
+
C
h
∑
K
∈
Γ
h
∥
τ
2
n
+
1
∥
0
,
∂
K
2
≤
ε
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
+
C
h
2
k
-
2
.
Noting that [[c
n+1]] = 0, if the constant M for the cut-off operator is sufficiently large, we write the last two terms in the right side of the error equation (7.25) as
B
(
c
n
+
1
,
τ
1
n
+
1
;
u
M
n
)
-
B
(
c
n
+
1
,
τ
1
n
+
1
;
u
n
)
=
(
(
D
(
u
M
n
)
-
D
(
u
n
)
)
∇
h
c
n
+
1
,
∇
h
τ
1
n
+
1
)
+
(
(
u
M
n
-
u
n
)
⋅
∇
h
c
n
+
1
,
τ
1
n
+
1
)
-
∫
Γ
h
{
{
(
D
(
u
M
n
)
-
D
(
u
n
)
)
∇
h
c
n
+
1
}
}
[
[
τ
1
n
+
1
]
]
d
s
=
:
∑
i
=
1
3
S
i
.
Noting that
|
u
n
-
u
M
n
|
=
|
u
n
-
u
h
n
|
point-wise if the constant M for the cut-off operator is sufficiently large, we can bound term S
1 as
|
S
1
|
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
∥
D
(
u
n
)
-
D
(
u
M
n
)
∥
0
2
,
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
∥
u
n
-
u
M
n
∥
0
2
,
≤
ε
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
C
∥
u
n
-
u
h
n
∥
0
2
.
Term S
2 can be bounded in a similar way as that for S
1
|
S
2
|
≤
ε
∥
ϕ
τ
1
n
+
1
∥
0
2
+
C
∥
u
n
-
u
h
n
∥
0
2
.
Term S
3 can be bounded using the penalty term and continuity of dispersion-diffusion tensor
|
S
3
|
≤
ε
∫
Γ
h
C
11
[
[
τ
1
n
+
1
]
]
2
d
s
+
∥
D
(
u
n
)
-
D
(
u
M
n
)
∥
0
2
,
≤
ε
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
+
C
∥
u
n
-
u
h
n
∥
0
2
.
Combining all the terms in (7.25), we have
1
2
△
t
(
∥
ϕ
τ
1
n
+
1
∥
0
2
-
∥
ϕ
τ
1
n
∥
0
2
)
+
1
2
(
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
)
≤
C
(
∥
ϕ
τ
1
n
+
1
∥
0
2
+
ϕ
2
△
t
∥
D
t
2
c
∥
L
2
(
t
k
,
t
k
+
1
;
L
2
(
Ω
)
)
2
+
△
t
∥
∂
t
τ
2
∥
L
2
(
t
k
,
t
k
+
1
;
L
2
(
Ω
)
)
2
)
+
C
(
h
2
k
-
2
+
∥
u
n
-
u
h
n
∥
0
2
)
.
Suppose that m is an integer, 0 ≤ m ≤ N - 1. Multiplying by 2Δt, summing from n = 0 to n = m, we obtain
∥
ϕ
τ
1
n
+
1
∥
0
2
+
△
t
∑
n
=
1
m
+
1
(
∥
|
D
1
∕
2
(
u
n
)
∇
h
τ
1
n
+
1
∥
|
0
2
+
∥
[
[
τ
1
n
+
1
]
]
∥
0
,
Γ
h
2
)
≤
C
△
t
(
∑
n
=
1
m
∥
τ
1
n
∥
0
2
+
∑
n
=
0
m
∥
e
u
∥
0
2
)
+
C
(
△
t
2
∥
D
t
2
c
∥
L
2
(
t
k
,
t
k
+
1
;
L
2
(
Ω
)
)
2
+
∥
∂
t
τ
2
∥
L
2
(
t
k
,
t
k
+
1
;
L
2
(
Ω
)
)
2
)
.
The theorem follow from (7.3), the discrete Gronwall's lemma and the triangle inequality. □
Theorem 7.3 (Error estimate for flow in coupled system) Let (
u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L
2(0, T; Hl
(Th
)), u ∈ (L
2(0, T; H
l-1(Th
)))
d and c ∈ L
2(0, T; Hk
(Th
)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h and Δt such that
max
0
≤
t
≤
T
∥
(
u
-
u
h
,
p
-
p
h
)
∥
s
t
a
b
(
t
)
≤
C
(
Δ
t
+
h
k
-
1
+
h
l
-
1
)
.
Proof Taking L
∞ norm with time in (7.3), we have
max
0
≤
t
≤
T
∥
(
u
-
u
h
,
p
-
p
h
)
∥
s
t
a
b
2
(
t
)
≤
C
(
∥
c
-
c
h
∥
L
∞
(
0
,
T
;
L
2
)
2
+
h
2
l
-
2
)
.
Substituting (7.24) into the above inequality, we obtain (7.39). □