RESEARCH Open Access
A stabilized mixed discontinuous Galerkin
method for the incompressible miscible
displacement problem
Yan Luo
1
, Minfu Feng
2
and Youcai Xu
2*
* Correspondence: [email protected]
2
School of Mathematics, Sichuan
University, Chengdu, Sichuan
610064, PR China
Full list of author information is
available at the end of the article
Abstract
A new fully discrete stabilized discontinuous Galerkin method is proposed to solve
the incompressible miscible displacement problem. For the pressure equation, we
develop a mixed, stabilized, discontinuous Galerkin formulation. We can obtain the
optimal priori estimates for both concentration and pressure.
Keywords: Discontinuous Galerkin methods, a priori error estimates, incompressible
miscible displacement
1 Introduction
We consider the problem of miscible displacement which has co nsiderable and practi-
cal importance in petroleum engineering. This problem can be considered as the result
of advective-diffusive equation for concentrations and the Darcy flow equation. The
more popular approach in application so far has been based on the mixed formulation.
In a previous work, Douglas and Roberts [1] presented a mixed finite element (MFE)
method for the compressible miscible displacement problem. For the Darcy flow,

nation of discontinuous discrete concentration, velocity and pressure spaces. Based on
our results, we can assert that the mixed stabilized discontinuous Galerkin formulation
of the incompressible miscible displacement problem is mathematically viable, and we
also believe it may be practically useful. It generalizes and encompasses all the success-
ful elements described in [2,6] and [5]. Optimal error estimate are obtained for the
concentration, velocity and pressure.
An outline of the remainder of the paper follows: In Section 2, we describe the mod-
eling equations. The DG schemes for the concentration and some of their properties
are introduced in Section 3. Stabilized mixed DG methods are introduced for the velo-
city and pressure in Section 4. In Section 5, we propose the numerical approximation
scheme of incompressible miscible displacement problems with a fully discrete in time,
comb ined with a mixed, stabilized and discontinuous Galerkin method. The bounded-
ness and stab ility of the finite element formulation are studied in Section 6. Error esti-
mates for the incompressible miscible displacement problem are obtained in Section 7.
Throughout the paper, we denote by C a generic positive constant that is indepen-
dent of h and Δt, but might depend on the partial differential equation solution; we
denote by ε a fixed positive constant that can be chosen arbitrarily small.
2 Governing equations
Miscible displacement of one incompressible fluid by another in a porous medium Ω
Î R
d
( d =2,3)overtimeintervalJ =(0,T] is modeled by the system concentration
equation:
φ
∂c
∂t
+ u ·∇c −∇·(D(u)∇c)=qc
∗
,(x, t) ∈  × J
.

)
, x ∈ 
.
(2:4)
The no-flow boundary conditions

u ·n =0, x ∈ ∂
,
(D(u)∇c − cu) · n =0, x ∈ ∂
.
(2:5)
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 2 of 17
Dispersion/diffusion tensor
D
(
u
)
= φ
d
m
I + |u|
(d
l
E
(
u
)
+

and d
t
are the longitudinal and the
transverse dispersivities, respectively, and are assumed to be nonnegative . The impos ed
external total flow rate q is sum of sources (injection) and sinks (extraction) and is
assumed to be bounded. Concentration c* in the source term is the injected concentra-
tion c
w
if q ≥ 0 and is the resident concentration c if q < 0. Here, we assume that the a
(c) is a globally Lipschitz continuous function of c, and is uniformly symmetric positive
definite and bounded.
3 Discontinuous Galerkin method for the concentration
3.1 Notation
Let T
h
=(K)beasequenceoffiniteelementpartitionsofΩ.LetΓ
I
denote the set of
all interior edges, Γ
B
the set of the edges e on ∂Ω,andΓ
h
= Γ
B
+ Γ
I
. K
+
, K
-

+
+ u
−
·n
−
, {{u}} =
1
2
(u
+
+ u
−
), {{p}} =
1
2
(p
+
+ p
−
), [[p]] = p
+
n
+
+ p
−
n
−
.
For e Î Γ
B

s
(
K
)
.K ∈ T
h
}
.
(3:1)
The usual Sobolev norm on Ω is denoted by ||·||
m, Ω
[19]. The broken norms are
defined, for a positive number m,as
|v|
2
m
=

K∈T
h
v
2
m,K
.
(3:2)
The discontinuous finite element space is taken to be
D
r
(
T

(the tensor product of the polynomial spaces
of degree less than or equal to r in each spatial dimension) because P
r
(K) ⊂ Q
r
(K).
The cut-off operator
M
is defined as
M
(c)(x) = min(c(x), M),
M(u)(x)=

u(x)if|u(x)|≤M,
M
u(x)/|u(x)| if |u(x)| > M
,
(3:4)
where M is a large positive constant. By a straightforward argument, we can show
that the cut-off operator
M
is uniformly Lipschitz continuous in the following sense.
Lemma 3.1 [7] (Property of operator
M
) The cut-off operator
M
defined as in Equa-
tion 3.4 is uniformly Lipschitz continuous with a Lipschitz constant one, that is
M(c) −M(w)
L

, v ∈ (L
∞
())
d
.
We shall also use the following inverse inequalities, which can be derived using the
method in [20]. Let K Î T
h
, v Î P
r
(K) and h
K
is the diameter of K. Then there exists a
constant C independent of v and h
K
, such that

D
q
v
0,∂K
≤ Ch
−1/2
K
D
q
v
K
, q ≥ 0
.



h
{{D(u)∇
h
w}}[[c]]ds −


h
{{D(u)∇
h
c}}[[w]]d
s
+


h
C
11
[[c]][[w]]ds +(u ·∇
h
c, w) −


cq
−
wdx,
L(w;
u, c)=


K
+
x ∈ ∂K
+
∩ ∂
,
(3:6)
here c
11
> 0 is a constant independent of the meshsize.
We now defi ne the weak form ulatio n on which our mixed discontinuous method is
based
(
φc
t
, w
)
+ B
(
c, w; u
)
= L
(
w; u, c
)
, ∀w ∈ H
k
(
T
h

n+
1
h
− c
n
h
t
, w
h
)+B(c
n+1
h
, w
h
; u
n
M
)=L(w
h
; u
n
M
, c
n+1
h
),
∀w
h
∈ W
1,∞

.
4 A stabilized mixed DG method for the velocity and pressure
4.1 Elimination for the flux variable u
Letting a(c)=a(c)
-1
. For the velocity and pressure, we define the following forms
a
(
u, v; c
)
=
(
α
(
c
)
u, v
)
,
(4:1)
b
(p, v )=(p, ∇
h
· v) −


I
{{p}}[[v]]ds −



,
b(ψ, u
h
)=(ψ, q), ∀ψ ∈ D
l−1
(T
h
).
(4:3)
In order to eliminate the flux variable, we first recall a useful identity, that holds for
vectors u and scalars ψ piecewise smooth on T
h
:

K∈T
h

∂K
v · nψds =


h
{{v}} · [[ψ]]ds +


I
[[v]]{{ψ}}ds
.
(4:4)
Using (4.4) we have

, v) −


I
[[p
h
]] ·{{v}}ds =0
.
(4:6)
We introduce the lift operator R:L
1
(∪∂K) ® (D
l-2
(T
h
))
d
defined by


R[[ψ]] · vdx = −


I
[[ψ]] ·{{v}}ds, ∀v ∈ (D
l−2
(T
h
))
d

πw, v
)
=
(
w, v
)
, ∀v ∈
(
D
l−2
(
T
h
))
d
.
(4:9)
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 5 of 17
Equation 4.8 gives now
α(
c
)
u
h
= −
(
π∇
h

Î D
l-1
(T
h
). The
Equation 4.10 gives
α(
c
)
u
h
= −
(
∇
h
p
h
+ R[[p
h
]]
).
(4:11)
Using (4.5) and the lifting operator R defined in (4.7) we have
b
(ψ, u
h
)=−(u
h
, ∇
h

, ∇
h
ψ + R[[ψ]]
)
=
(
q, ψ
).
(4:13)
For future reference, it is convenient to rewrite (4.13) as follows
A
BR
(
p
h
, ψ
)
=
(
q, ψ
)
, ∀ψ ∈ D
l−1
(
T
h
),
(4:14)
where
A

A
(
u
h
, v; p
h
, ψ;c
)
= l
(
ψ
)
, ∀
(
v, ψ
)
∈
(
D
l−2
(
T
h
))
d
× D
l−1
(
T
h

ψ
)
=
(
q, ψ
).
(4:17)
In a sense, (4.16) can be seen as a Darcy problem. The usual way to stabilized it is to
introduce penalty terms on the jumps of p and/or on the j umps of u.In[2],Masud
and Hughes introduced a stabilized finite element formulation in which an appropri-
ately weighted residual of the Darcy law is added to the standard mixed formulation.
In Hughes-Masu d-Wan [5], the method was extend within the discontinuous Galerkin
framework. A family of mixed finite element discretizations of the Darcy flow equa-
tions using totally discontinuous elements was introduced in [6]. In this paper, we con-
sider the following stabilized formulation which includes the methods of [2,6] and [5].
The stabilized formulation of (4.16) is
A
stab
(
u
h
, v; p
h
, ψ;c
)
= l
stab
(
ψ
)

ψ

,
l
stab
(ψ)=l(ψ),
e(p, ψ)=a(c)


h
C
11
[[p]][[ψ]]ds,
(4:19)
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 6 of 17
where g and b are chosen as the following (i) g =1,b = 1. (ii) g =0,b =1,δ could
assume either the value +1 or the value -1. The definition of θ will be given in the fol-
lowing content.
5 A mixed stabilized DG method for the incompressible miscible
displacement problem
By combining (3.8) with (4.18), we have the stabil ized DG for the approximating (2.1)-
(2.5): seek c
h
Î W
1,∞
(0, T; D
k-1
(T

(φ
c
n+1
h
− c
n
h
t
, w)+B(c
n+1
h
, w; u
n
M
)=L(w; u
n
M
, c
n+1
h
), ∀w ∈ W
h
,
A
stab
(u
n
h
, v; p
n

,
(5:2)
where
p
2
1,h
=
1
2
a
1/2
(c)∇
h
p
2
0
+ a
1/2
(c)[[p]]
2
0,
h
,
a
1/2
(c)[[p]]
2
0,
h
=

0,
≤

e∈
I
h
−1
e
[[ψ]]
2
0,e
≤ C
2
R[[ψ]]
0,
.
(6:1)
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]]
2
0,
≤


we have
λξ + η
2
H
+ μη
2
H
≥
λμ
2
(
λ + μ
)
(ξ
2
H
+ η
2
H
)
.
(6:3)
Theorem 6.1 (Stability) For δ =1,problem (4.18) is stable for all θ Î (0,1).
Proof Consider first the case g =1,b = 1. From the definition of A
stab
(·,·;·,·;·), we have
A
stab
(
u

c
)
∇
h
p
h
, −α
(
c
)
u
h
+ ∇
h
p
h
).
(6:4)
We remark that (6.4) can be rewritten as
A
stab
(u
h
, u
h
; p
h
, p
h
; c)=(1−θ )|α

h
Î (D
l-2
(T
h
))
d
, p
h
Î D
l-1
(T
h
) such that

(α(c)u
h
+ ∇
h
p
h
+ R[[p
h
]], v) − θ (α(c)u
h
+ ∇
h
p
h
, v)=0,

(6:7)
Substituting the expression (6.7) in the second equation of (6.6) for δ = 1, we have
A
BR
(p
h
, ψ)+
θ
1 − θ


a(c)R[[p
h
]] ·R[[ψ]]dx =(q, ψ), ∀ψ ∈ D
l−1
(T
h
)
.
(6:8)
Denote by B
1h
(·,·) the bilinear form (6.8), we have
B
1h
(ψ, ψ)=a(c)(∇
h
ψ + R[[ψ]])
0,
+

+ a(c)∇
h
p
h
, −α(c)u
h
−∇
h
p
h
)
+e
(
p
h
, p
h
)
.
(6:10)
Using the arithmetic-geometric mean inequality, we have
A
stab
(u
h
, u
h
; p
h
, p

θ
1 − θ
(R[[p
h
]], a(c)∇
h
ψ)+
θ
1 − θ


a(c)R[[p
h
]] · R[[ψ]]dx =(q, ψ )
.
(6:12)
We remark that formulation (6.12) can be rewritten as
1
1 −
θ
A
BR
(p
h
, ψ) −
θ
1 −
θ
A
BO

]] ·R[[ψ]]dx
.
(6:14)
Denote by B
2h
(·,·) the bilinear form (6.13), we have
B
2h
(ψ, ψ)=
1
1 −
θ
a
1/2
(c)(∇
h
ψ + R[[ψ]])
0,
−
θ
1 −
θ
a
1/2
(c)∇
h
ψ
2
0,
,

2
(0, T; H
l
(T
h
))
)
(6:16)
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
1h
(ψ, ψ) ≥ Cψ
2
1
,
h
, B
2h
(ψ, ψ) ≥ Cψ
2
1
,
h
, ∀ψ ∈ D
l−1
(T
h
)

(T
h
))
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
1h
(
p
h
, ψ
)
≤ Cp
h

1,h
ψ
1,h
, B
2h
(
p
h
, ψ

˜
p, v)=0, ∀v ∈ (D
l−2
(T
h
))
d
,
b(ψ,
˜
u)+e(
˜
p, ψ)=(q, ψ), ∀ψ ∈ D
l−1
(T
h
),
(
˜
c − c, w)=0, ∀w ∈ D
k−1
(T
h
).
(7:1)
Let us define interpolation errors, finite element solution errors and auxiliary errors
ξ
1
= ˜u − u
h

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
|
2
0
+ a
1/2
(c)[[η

h
)))
d
and c Î
L
2
(0, T; H
k
(T
h
)). We further assume that p, ∇p, cand∇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
)
2
stab
(t ) ≤ C(c −c
h

2
0
(t )+h
2l−2
)
.
(7:3)

h
))∇
h
p
h
), −α(M(c
h
))v + ∇
h
ψ)
+
1
2
(u + a(c)∇
h
p, −α(c)v + ∇
h
ψ)=0.
(7:4)
That is
(α(c)(
u
−˜
u
),
v
)+(α(
M
(c
h

(α(
M
(c
h
))
u
h
− α(c)
u
,
v
)+
1
2
(
u
−
u
h
, ∇
h
ψ)
−
1
2
(∇
h
p −∇
h
p

2
,from(7.1)andwe
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
((a(c) − a(
M
(c
h
)))∇
h
˜
p, ∇
h
η
1
)
+
1
2
(ξ
2
, ∇
h
η
1
) −
1
2
(∇

1
, ∇
h
η
1
)
=
1
2
(|α
1/2
(M(c
h
))ξ
1
|
2
0
+ a
1/2
(M(c
h
))∇
h
η
1

2
0
)+[[η

− c
|
, we have
|
(α(M(c
h
)) − α(c)) ˜u, ξ
1
)|≤Cc −c
h

2
0
+ ε|ξ
1
|
2
0
.
(7:6)
The second and the third terms of t he 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
+ Ch
2l−2
.
(7:8)
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

2
0
+ ε∇
h
η
1

2
0
.
(7:9)

(7:10)
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 10 of 17
Substituting all these inequalities into Equation 7.5, we have
1
2
(|α
1/2
(M(c
h
))ξ
1
|
2
0
+ a
1/2
(M(c
h
))∇
h
η
1

2
0
)+a
1/2
(c)[[η

h
)) ≤ CI,
1
C
I ≤ α(c) ≤ C
I
,and
1
C
I ≤ a(M(c
h
)) ≤ CI,
1
C
I ≤ a(c) ≤ C
I
we have
(ξ
1
, η
1
)
2
stab
≤ C(c −c
h

2
0
+ h

(
p
h
, ψ;c
h
)
=
(
a
(M(
c
h
))(
∇
h
p
h
+ R[[p
h
]]
)
, ∇
h
ψ + R[[ψ]]
).
Replacing (6.8) with p
h
= p and subtracting it from (7.13) we finally obtain
a
(

)+


a(M(c
h
))R[[η
1
]] ·R[[η
1
]]dx = A
BR
(η
2
, η
1
)
+


a(c)R[[η
2
]] ·R[[η
1
]]dx +(a(M(c
h
)) − a(c))((∇
h
˜
p + R[[
˜

, η
1
; c
h
)+


a(M(c
h
))R[[η
1
]] ·R[[η
1
]]dx ≥ Cη
2
1,h
.
(7:16)
The first and the second terms of the right side of (7.15) can be bounded using
Lemma 6.1 and (3.5)
A
BR
(η
2
, η
1
)+


a(c)R[[η

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
]]) +


have
p − p
h

2
1
,
h
≤ C( c − c
h

2
0
+ h
2l−2
)
.
(7:19)
We easily deduce,using (7.19)
|α
1/2
(c)(u − u
h
)|
2
0
≤ Cp − p
h

2

h
c ·∇
h
c ≥
(
φd
m
+ min
(
d
l
, d
t
)
|u|
)
|∇
h
c|
2
.
(7:21)
If, in addition, d
m
(x) ≥ d
m,*
>0uniformly in the domain Ω, then D(u) is uniformly
positive definite in Ω:
D
(

*
and d
t
(x) ≤ d
t
*
.
Then
D(u) − D(v)
(
L
2
(

))
d×d ≤ k
D
u − v
(
L
2
(

))
d
.
(7:23)
where
k
D

(T
h
)). We further assume that p, ∇p, cand∇c are essentially bounded. If the constant
M for the cut-off operator is sufficiently large, then there exists a constant C indepen-
dent of h and Δt such that


φ(c − c
h
)
L
∞
(0,T;L
2
())
+(
N

i=1
t(|
D
1/2
(
u
i−1
)∇
h
(c
i
− c

h
− c
n
h

t
, w)+B(c
n+1
h
, w; u
n
M
)=L(w; u
n
M
, c
n+1
h
)
.
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 12 of 17
It can be written as
(φ
τ
n+1
2
− τ
n

, c
n+1
h
)
.
Subtracting the DG scheme equation from the weak formulation, we have for any w
Î D
k-1
(T
h
)
(φc
t
, w) −

φ
τ
n+1
2
− τ
n
2
t
, w

+

φ
τ
n+1

, c
n+1
) − L(w; u
n
M
, c
n+1
h
).
that is
(φc
t
, w)+

φ
τ
n+
1
1
− τ
n
1
t
, w

+ B(τ
n+1
1
, w; u
n

)+B(c
n+1
, w; u
n
M
) − B(c
n+1
, w; u
n
)
+L(w;
u
n
, c
n+1
) − L(w; u
n
M
, c
n+1
h
).
Choosing w = τ
1
n+1
, we obtain

φ
τ
n+


− (φc
t
, τ
n+1
1
)
+

φ
τ
n+1
2
− τ
n
2
t
, τ
n+1
1

+ B(τ
n+1
2
, τ
n+1
1
; u
n
M

n+1
h
).
(7:25)
Let us first consider the left side of the error equation (7.25). The first term can be
bounded as

φ
τ
n+1
1
− τ
n
1
t
, τ
n+1
1

≥
φ
2t
((τ
n+1
1
, τ
n+1
1
) − (τ
n

)+(u
n
M
·∇
h
τ
n+1
1
, τ
n+1
1
)
−


q
−
(τ
n+1
1
)
2
dx +


h
C
11
[[τ
n+1

2
0
+ C

φτ
n+1
1

2
0
.
:
(u
n
M
·∇
h
τ
n+1
1
, τ
n+1
1
) ≤ ε|D
1/2
(u
n
)∇
h
τ

n+1
1
, τ
n+1
1
; u
n
M
)
≥
φ
2t
(τ
n+1
1

2
0
−τ
n
1

2
0
)+|D
1/2
(u
n
)∇
h

; u
n
, c
n+1
) − L(τ
n+1
1
; u
n
M
, c
n+1
h
) ≤ C

φτ
n+1
1

2
0
.
(7:29)
Using Taylor series expansion, we have

φ
c
n+1
− c
n

1

2
0
.
(7:30)
The fourth term in the right side of the error equation (7.25) is
B(τ
n+1
2
, τ
n+1
1
; u
n
M
)=(D(u
n
M
)∇
h
τ
n+1
2
, ∇
h
τ
n+1
1
)+(u

1
}}[[τ
n+1
2
]]d
s
−


h
{{D(u
n
M
)∇
h
τ
n+1
2
}}[[τ
n+1
1
]]ds
+


h
C
11
[[τ
n+1

|
2
0
+ C∇
h
(c
n+1
−
˜
c
n+1
)
2
0
,
≤ ε|D
1/2
(u
n
)∇
h
τ
n+1
1
|
2
0
+ Ch
2k−2
,

0
+ h
2k
)
.
(7:32)
Terms T
4
and T
5
can be estimated using inverse inequalities,
|T
4
|≤εh

K∈
h
|D
1/2
(u
n
)∇
h
τ
n+1
1
|
0,∂K
+
C

5
|≤ε[[τ
n+1
1
]]
2
0,
h
+
C
h

K∈
h
τ
n+1
2

2
0,∂K
,
≤ ε[[τ
n+1
1
]]
2
0,
h
+ Ch
2k−2

0,
h
+ Ch
2k−2
.
(7:35)
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
Luo et al. Boundary Value Problems 2011, 2011:48
http://www.boundaryvalueproblems.com/content/2011/1/48
Page 14 of 17
B(c
n+1
, τ
n+1
1
; u
n
M
) − B(c
n+1
, τ
n+1
1
; u
n
)=((D(u
n

n
M
) − D(u
n
))∇
h
c
n+1
}}[[τ
n+1
1
]]ds
=:
3

i
=1
S
i
.
Noting that
|u
n
− u
n
M
| = |u
n
− u
n

,
≤ ε|D
1/2
(u
n
)∇
h
τ
n+1
1
|
2
0
+ Cu
n
− u
n
M

2
0
,
≤ ε|
D
1/2
(u
n
)∇
h
τ

n
− u
n
h

2
0
.
(7:37)
Term S
3
can be bounded using the penalty term and continuity of dispersion-diffu-
sion tensor
|
S
3
|≤ε


h
C
11
[[τ
n+1
1
]]
2
ds + D(u
n
) − D(u

1

2
0
−

φτ
n
1

2
0
)+
1
2
(|
D
1/2
(u
n
)∇
h
τ
n+1
1
|
2
0
+ [[τ
n+1

())
+ t∂
t
τ
2

2
L
2
(t
k
,t
k+1
;L
2
())
)
+C(h
2k−2
+ u
n
− u
n
h

2
0
).
Suppose that m is an integer, 0 ≤ m ≤ N - 1. Multiplying by 2Δt , summing from n =
0ton = m, we obtain

h
)
≤ Ct(
m

n=1
τ
n
1

2
0
+
m

n=0
e
u

2
0
)+C(t
2
D
2
t
c
2
L
2

2
(0, T; H
l
(T
h
)), u Î (L
2
(0, T; H
l-1
(T
h
)))
d
and c Î L
2
(0,
T; H
k
(T
h
)). We further assume that p, ∇p, cand∇c are essentially bounded. If the con-
stant M for the cut-off operator is sufficient ly large, then there exists a constant C inde-
pendent of h and Δt such that
max
0≤
t
≤
T
(u − u
h

h

2
L∞(0,T;L
2
)
+ h
2l−2
)
.
(7:40)
Substituting (7.24) into the above inequality, we obtain (7.39). □
Acknowledgments
The work was supported by National Natural Science Foundation of China (Grant
Nos.11101069, Grant Nos.11126105)and the Youth Research Foundation of Sichuan
University (no. 2009SCU11113).
Author details
1
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan
610054, PR China
2
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
Authors’ contributions
YL participated in the design and theoretical analysis of the study, drafted the manuscript. MF conceived the study,
and participated in its design and coordination. YX participated in the design and the revision of the study. All
authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 8 May 2011 Accepted: 25 November 2011 Published: 25 November 2011
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on local projections. Appl Math Mech. 29, 171–183 (2008). doi:10.1007/s10483-008-0205-z
18. Sun, S, Rivière, B, Wheeler, MF: A combined mixed element and discontinuous Galerkin method for miscible
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http://www.boundaryvalueproblems.com/content/2011/1/48
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doi:10.1186/1687-2770-2011-48
Cite this article as: Luo et al.: A stabilized mixed discontinuous Galerkin method for the incompressible miscible
displacement problem. Boundary Value Problems 2011 2011:48.
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