Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 391372, 24 pages
doi:10.1155/2012/391372
Research Article
A New Positive Definite Expanded
Mixed Finite Element Method for Parabolic
Integrodifferential Equations
Yang Liu,1 Hong Li,1 Jinfeng Wang,2 and Wei Gao1
1
2
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
School of Statistics and Mathematics, Inner Mongolia Finance and Economics College,
Hohhot 010070, China
Correspondence should be addressed to Hong Li, smslh@imu.edu.cn
Received 4 February 2012; Revised 3 April 2012; Accepted 19 April 2012
Academic Editor: Jinyun Yuan
Copyright q 2012 Yang Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A new positive definite expanded mixed finite element method is proposed for parabolic partial
integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed
element system is symmetric positive definite and both the gradient equation and the flux equation
are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete
scheme are proved and error estimates are derived for both semidiscrete and fully discrete
schemes. Finally, some numerical results are provided to confirm our theoretical analysis.
1. Introduction
In this paper, we consider the following initial-boundary value problem of parabolic partial
integrodifferential equations:
ut − ∇ ·
ax, t∇u bx, t
t
∇uds
fx, t,
x, t ∈ Ω × J,
0
ux, t 0,
x, t ∈ ∂Ω × J,
ux, 0 u0 x,
1.1
x ∈ Ω,
where Ω is a bounded convex polygonal domain in Rd , d 1, 2, 3 with a smooth boundary
∂Ω, J 0, T is the time interval with 0 < T < ∞. The coefficients a ax, t, b bx, t are
2
Journal of Applied Mathematics
two functions, which satisfy the property that there exist some positive constants amin , amax ,
bmin , and bmax such that 0 < amin ≤ ax, t ≤ amax and 0 < bmin ≤ bx, t ≤ bmax .
Parabolic integrodifferential equations are a class of very important evolution equations which describe many physical phenomena such as heat conduction in material with
memory, compression of viscoelastic media, and nuclear reactor dynamics. In recent years,
a lot of researchers have studied the numerical methods for parabolic integrodifferential
equations, such as finite element methods 1–5, mixed finite element methods 6–9, and
finite volume element method 10 and so forth.
In 1994, Chen 11, 12 proposed a new mixed method, which is called a expanded
mixed finite element method and proved some mathematical theories for second-order
linear elliptic equation. Compared to standard mixed element methods, the expanded mixed
method is expanded in the sense that three variables are explicitly approximated, namely, the
scalar unknown, its gradient, and its flux the tensor coefficient times the gradient. In 1997,
Arbogast et al. 13 derived and exploited a connection between the expanded mixed method
and a certain cell-centered finite difference method. And Chen proved some mathematical
theories for second-order quasilinear elliptic equation 14 and fourth-order elliptic problems
15. With the development of the expanded mixed finite element method, the method was
applied to many evolution equations. In 16, some error estimates of the expanded mixed
element for a kind of parabolic equation were given. Woodward and Dawson 17 studied
the expanded mixed finite element method for nonlinear parabolic equation. Wu and Chen
et al. 18–22 studied the two-grid methods for expanded mixed finite-element solution of
semilinear reaction-diffusion equations. Song and Yuan 23 proposed the expanded upwindmixed multistep method for the miscible displacement problem in three dimensions. Guo and
Chen 24 developed and analysed an expanded characteristic-mixed finite element method
for a convection-dominated transport problem. In 2010, Chen and Wang 25 proposed an
H 1 -Galerkin expanded mixed method for a nonlinear parabolic equation in porous medium
flow, and Liu and Li 26 studied the H 1 -Galerkin expanded mixed method for pseudohyperbolic equation. Liu 27, studied the H 1 -Galerkin expanded mixed method for RLWBurgers equation and proved semidiscrete and fully discrete optimal error estimates. Jiang
and Li 28 studied an expanded mixed semidiscrete scheme for the problem of purely
longitudinal motion of a homogeneous bar. In 29, 30, the expanded mixed covolume
method was studied for the linear integrodifferential equation of parabolic type and elliptic
problems, respectively. In 31, a posteriori error estimator for expanded mixed hybrid
methods was studied and analysed.
In 2001, Yang 32 proposed a new mixed finite element method called the splitting
positive definite mixed finite element procedure to treat the pressure equation of parabolic
type in a nonlinear parabolic system describing a model for compressible flow displacement
in a porous medium. Compared to standard mixed methods whose numerical solutions have
been quite difficult because of losing positive definite properties, the proposed one does not
lead to some saddle point problems. From then on, the method was applied to the hyperbolic
equations 33 and pseudo-hyperbolic equations 34.
In this paper, our purpose is to propose and analyse a new expanded mixed method
based on the positive definite system 32–34 for parabolic integrodifferential equations.
Compared to expanded mixed methods, the proposed mixed element system is symmetric
positive definite and avoids some saddle point problems. What is more, both the gradient
equation and the flux equation are separated from its scalar unknown equation. The existence
and uniqueness for semidiscrete scheme are proved and error estimates are derived for both
semidiscrete and fully discrete schemes.
Journal of Applied Mathematics
3
The layout of the paper is as follows. In Section 2, the positive definite expanded
mixed weak formulation and semidiscrete mixed scheme are formulated, and the proof of
the existence and uniqueness of the discrete solutions is given. Error estimates are derived
for both semidiscrete and fully discrete schemes for problems, respectively, in Sections 3
and 4. In Section 5, some numerical results are provided to illustrate the efficiency of our
method. Finally in Section 6, we will give some concluding remarks about the positive
definite expanded mixed finite element method.
Throughout this paper, C will denote a generic positive constant which does not
depend on the spatial mesh parameters hu , hσ and time-discretization parameter δ and may
be different at their occurrences. Usual definitions, notations, and norms of the Sobolev spaces
as in 35–37 are used. We denote the natural inner product in L2 Ω or L2 Ωd by ·, ·
with norm · L2 Ω or · L2 Ω and introduce the function space W Hdiv; Ω {ω ∈
L2 Ωd ; ∇ · ω ∈ L2 Ω}.
2. A New Expanded Mixed Variational Formulation
Introducing the auxiliary variables:
λ ∇u,
σ ax, t∇u bx, t
t
∇u ds aλ b
0
t
λ ds,
2.1
0
Then we obtain the equivalent system of parabolic partial integrodifferential equations for
the problem 1.1:
a ut − ∇ · σ fx, t,
x, t ∈ Ω × J,
b λ − ∇u 0, x, t ∈ Ω × J,
t
c σ − aλ − b λ ds 0, x, t ∈ Ω × J,
2.2
0
with the initial values λx, 0 ∇u0 x, σx, 0 a∇u0 x, and ux, 0 u0 x.
Then, the following expanded mixed weak formulation of 2.2 can be given by:
a ut , v − ∇ · σ, v fx, t, v ,
b λ, w u, ∇ · w 0,
t
c σ, z − aλ, z −
b
∀v ∈ L2 Ω,
∀w ∈ W,
λ ds, z
0,
2.3
∀z ∈ W.
0
From 2.3b we derive
λt , w ut , ∇ · w 0.
2.4
4
Journal of Applied Mathematics
Taking v ∇ · w in 2.3a for w ∈ W and then substituting it into 2.4, we derive a new
equivalent expanded mixed weak formulation of the system 2.3:
a λt , w ∇ · σ, ∇ · w − fx, t, ∇ · w ,
t
b σ, z − aλ, z − b λ ds, z 0,
∀w ∈ W,
∀z ∈ W,
2.5
0
c ut , v − ∇ · σ, v fx, t, v ,
∀v ∈ L2 Ω.
Let Thu and Thσ be two families of quasi-regular partitions of the domain Ω, which
may be the same one or not, such that the elements in the partitions have the diameters
bounded by hu and hσ , respectively. Let Xhu ⊂ L2 Ω and Vhσ ⊂ W be finite element spaces
defined on the partitions Thu and Thσ .
Now the semidiscrete positive definite expanded mixed finite element method for
2.5 consists in determining uh , λh , σh ∈ Xhu × Vhσ × Vhσ such that
a λht , wh ∇ · σh , ∇ · wh − fx, t, ∇ · wh ,
t
b σh , zh − aλh , zh − b λh ds, zh 0,
0
c uht , vh − ∇ · σh , vh fx, t, vh ,
∀wh ∈ Vhσ ,
∀zh ∈ Vhσ ,
2.6
∀vh ∈ Xhu ,
with given an initial approximation u0h , λ0h , σh0 ∈ Xhu × Vhσ × Vhσ .
Remark 2.1. Compared to expanded mixed weak formulation 2.3, the new expanded mixed
element system 2.6 is symmetric positive definite, that is to say the gradient function and the
flux function system 2.6a,b is symmetric positive definite. And both the gradient equation
and the flux equation are separated from its scalar unknown equation 2.6c.
Theorem 2.2. There exists a unique discrete solution to the system 2.6.
N2
1
Proof. Let {ψi x}N
i1 and {ϕj x}j1 be bases of Xhu and Vhσ , respectively. Then, we have the
following expressions:
uh N1
i1
ui tψi x,
λh N2
j1
λj tϕj x,
σh N2
j1
σj tϕj x.
2.7
Journal of Applied Mathematics
5
Substituting these expressions into 2.6 and choosing vh ψm , wh zh ϕl , then the
problems 2.6 can be written in vector matrix form as: find {Ut, Λt, Σt} such that, for
all t ∈ 0, T a AΛ t BΣt Ft,
b AΣt − CΛt − H
t
2.8
Λsds 0,
0
c DU t − EΣt Gt,
where
A
C
aϕj , ϕl
E
ϕj , ϕl
N2 ×N2
N2 ×N2
D
,
∇ · ϕj , ψm
N1 ×N2
B
,
ψi , ψm
−ft, ∇ · ϕl
∇ · ϕj , ∇ · ϕl
N1 ×N1
N2 ×N2
H
,
,
bϕj , ϕl
N2 ×N2
Ut u1 t, u2 t, . . . , uN1 tT ,
,
Λt λ1 t, λ2 t, . . . , λN2 tT ,
Ft T
1×N2
,
2.9
Σt σ1 t, σ2 t, . . . , σN2 tT ,
Gt ,
ft, ψm
T
1×N1
.
It is easy to see that both A and D are symmetric positive definite. From 2.8, the problems
can be written as follows:
−1
a Λ t −A BA
−1
CΛt H
t
Λsds
A−1 Ft,
0
−1
−1
b Σt A CΛt A H
t
Λsds,
2.10
0
c U t D−1 EΣt D−1 Gt.
Thus, by the theory of differential equations 38, 39, 2.10 has a unique solution, and
equivalently 2.6 has a unique solution.
Remark 2.3. It is easy to see that the coefficient matrixes A, B, C, and H of system 2.8 are
symmetric positive definite. In view of this, the new expanded mixed element system 2.6 is
symmetric positive definite.
6
Journal of Applied Mathematics
3. Semidiscrete Error Estimates
Let Xhu and Vhσ be finite dimensional subspaces of L2 Ω and W, respectively, with the inverse property see 36 and the following approximation properties see 40–44: for 0 ≤
p ≤ ∞ and r, r ∗ , k positive integers
inf w − wh Lp Ω ≤ Chr1
σ wWr1,p Ω ,
wh ∈Vhσ
d
∀w ∈ Hdiv; Ω ∩ W r1,p Ω ,
d
∀w ∈ Hdiv; Ω ∩ W r1,p Ω ,
∗
inf ∇ · w − wh Lp Ω ≤ Chrσ ∇ · wW r ∗ ,p Ω ,
wh ∈Vhσ
inf v − vh Lp Ω ≤ Chk1
u vW k1,p Ω ,
vh ∈Xhu
3.1
∀v ∈ L2 Ω ∩ W k1,p Ω,
where r ∗ r 1 for the Brezzi-Douglas-Fortin-Marini spaces 43 and the Raviart-Thomas
spaces 42 and r ∗ r for the Brezzi-Douglas-Marini spaces 40, 43.
For our subsequent error analysis, we introduce two operators. It is well known that,
in any one of the classical mixed finite element spaces, there exists an operator Rh from
Hdiv; Ω onto Vhσ , see 40–44, such that, for 1 ≤ p ≤ ∞,
∇ · σ − Rh σ, φh 0,
∀φh ∈ ∇ · Vhσ φh ∇ · wh , wh ∈ Vhσ ;
σ − Rh σLp Ω ≤ Chr1
σ σWr1,p Ω ;
3.2
∗
∇ · σ − Rh σLp Ω ≤ Chrσ ∇ · σW r ∗ ,p Ω .
We also define the L2 -project operator Ph from L2 Ω onto Xhu such that
v − Ph v, vh 0,
∀v ∈ L2 Ω, vh ∈ Xhu ;
v − Ph vLp Ω ≤ Chk1
u vH k1,p Ω ,
∀v ∈ H k1 Ω.
3.3
Using the definitions of the operators Rh and Ph , we can easily obtain the following lemma.
Lemma 3.1. Assume that the solution of system 2.5 has the regular properties that ut ∈
L2 H k1 Ω, λt , λtt , σt ∈ L2 Hr1 Ω, then one has the following estimates:
λ − Rh λt Lp Ω ≤ Chr1
σ λt Wr1,p Ω ,
λ − Rh λtt Lp Ω ≤ Chr1
σ λtt Wr1,p Ω ,
σ − Rh σt Lp Ω ≤ Chr1
σ σt Wr1,p Ω ,
u − Ph ut Lp Ω ≤ Chk1
u ut H k1,p Ω .
3.4
Journal of Applied Mathematics
7
Let
u − uh u − Ph u Ph u − uh η ς,
3.5
λ − λh λ − Rh λ Rh λ − λh ρ ξ,
σ − σh σ − Rh σ Rh σ − σh γ θ.
Subtracting 2.6 from 2.5 and using projections 3.2 and 3.3, one obtains
ξt , wh ∇ · θ, ∇ · wh − ρt , wh ,
t
θ, zh − aξ, zh −
ξ ds, zh
b
∀wh ∈ Vhσ ,
− γ, zh aρ, zh
0
ςt , vh − ∇ · σ − σh , vh 0,
t
b ρ ds, zh ,
0
∀zh ∈ Vhσ ,
∀vh ∈ Xhu .
3.6
Theorem 3.2. Assume that the approximate properties 3.1 hold, and the solution of system 2.5
has regular properties that ut , utt ∈ L2 H k1 Ω, λt , λtt , σt , σtt ∈ L2 Hr1 Ω. Then one has the
error estimates
λ − λh L∞ L2 Ω σ − σh L∞ L2 Ω ≤ Chr1
σ ,
∗
∇ · σ − σh L∞ L2 Ω ≤ Chrσ ,
∗
.
u − uh L∞ L2 Ω ≤ C hrσ hk1
u
3.7
Proof . Choose wh θ in 3.6a and zh −ξt in 3.6b, and add the two equations to obtain
1 d
1/2 2
a ξ 2 ∇ · θ2L2 Ω
L Ω
2 dt
1
at ξ, ξ γ, ξt − aρ, ξt − ρt , θ −
2
t
t
b ξ ds, ξt − b ρ ds, ξt
0
d
1
d γ, ξ − γt , ξ −
aρ, ξ aρt at ρ, ξ
at ξ, ξ 2
dt
dt
t
t
d
b ξ ds, ξ bt ξ ds, ξ bξ, ξ
− ρt , θ −
dt
0
0
t
t
d
−
b ρ ds, ξ bt ρ ds, ξ bρ, ξ .
dt
0
0
0
3.8
8
Journal of Applied Mathematics
Integrate with respect to time from 0 to t and apply the Cauchy-Schwarz’s inequality and the
Young’s inequality to obtain
1/2 2
a ξ 2
L Ω
2
t
0
∇ · θ2L2 Ω ds
2
≤ a1/2 ξ0 2
L Ω
C
2
2
ρL2 Ω γ L2 Ω εξ2L2 Ω
3.9
t
0
2
ρ 2 ρs 2 2 γs 2 2 ξ2 2 θ2 2
ds.
L
L
Ω
Ω
L Ω
L Ω
L Ω
Choose zh θ in 3.6b to get
θ2L2 Ω
2
≤ C a1/2 ξ 2
L Ω
2
γ 2
2
ρ 2
L Ω
L Ω
t
0
ξ2L2 Ω
2
ρL2 Ω ds .
3.10
Combining 3.9 and 3.10, we obtain
ξ2L2 Ω θ2L2 Ω 2
t
0
2
≤ a1/2 ξ0 2
L Ω
C
t
0
∇ · θ2L2 Ω ds
2
2
ρL2 Ω γ L2 Ω εξ2L2 Ω
3.11
2
ρ 2 ρs 2 2 γs 2 2 ξ2 2 θ2 2
L Ω
L Ω ds.
L Ω
L Ω
L Ω
Using the fact that ξ0 0 and Gronwall’s lemma, we obtain
ξ2L2 Ω
θ2L2 Ω
2
t
0
∇ · θ2L2 Ω ds
2
2
≤ ρL2 Ω γ L2 Ω C
t
0
2
ρ 2 ρs 2 2 γs 2 2
ds.
L Ω
L Ω
L Ω
3.12
Differentiating 3.6b and taking zh ξt , we obtain
θt , ξt aξt 2L2 Ω at ξ, ξt − γt , ξt at ρ aρt , ξt
t
t
bt
ξ ds, ξt
0
bξ, ξt ρ ds, ξt
bt
bρ, ξt .
3.13
0
Choosing wh θt in 3.6a, we obtain
ξt , θt d
1 d
ρt , θ ρtt , θ .
∇ · θ2L2 Ω −
2 dt
dt
3.14
Journal of Applied Mathematics
9
Add 3.13 and 3.14 to obtain
1 d
d
ρt , θ ρtt , θ − at ξ, ξt γt , ξt − at ρ aρt , ξt
∇ · θ2L2 Ω −
2 dt
dt
t
t
− bt ξds, ξt − bξ, ξt − bt ρds, ξt − bρ, ξt .
aξt 2L2 Ω 0
0
3.15
Integrate 3.15 with respect to time from 0 to t to obtain
t
0
ξs 2L2 Ω ds
C
t
0
∇ ·
θ2L2 Ω
t
2
2
2
2
≤ ρL2 Ω γ L2 Ω C ξL2 Ω ξL2 Ω ds
0
2
ρ 2 ρs 2 2 ρss 2 2 γs 2 2
ds.
L Ω
L Ω
L Ω
L Ω
3.16
Substitute 3.12 into 3.16 to have
t
0
ξs 2L2 Ω ds ∇ · θ2L2 Ω
2
2
≤ ρL2 Ω γ L2 Ω C
t
0
2
ρ 2 γ 2 2
L Ω
L Ω
3.17
2
2
2
ρs L2 Ω ρss L2 Ω γs L2 Ω ds.
Choosing vh ς in 3.6b and applying Cauchy-Schwarz’s inequality, we obtain
ςL2 Ω ≤ ∇ · σ − σh L2 Ω .
3.18
Using Lemma 3.1, 3.17, and Gronwall’s lemma, we get
∗
.
ςL∞ L2 Ω ≤ C hrσ hk1
u
3.19
Using 3.12, 3.17, 3.19, 3.2, 3.3, and Lemma 3.1, we apply the triangle inequality to
complete the proof.
4. Fully Discrete Error Estimates
In this section, we get the error estimates of fully discrete schemes. For the backward Euler
procedure, let 0 t0 < t1 < t2 < · · · < tM T be a given partition of the time interval 0, T with
step length δ T/M, for some positive integer M. For a smooth function φ on 0, T , define
10
Journal of Applied Mathematics
φn φtn and ∂t φn φn − φn−1 /δ. For approximating the integrals, we use the composite
left rectangle rule
δ
n−1
φj ≈
tn
4.1
φsds.
0
j0
Note that φ ∈ C1 0, T , the quadrature error satisfies
tn
tn
n−1 j
δ φ −
φs sds.
φsds ≤ Cδ
j0
0
0
4.2
Equation 2.5 has the following equivalent formulation:
a ∂t λn , w ∇ · σ n , ∇ · w −f n , ∇ · w Rn1 , w , ∀w ∈ W,
⎛
⎞
n−1
b σ n , z − an λn , z − ⎝bn δ λj , z⎠ − Rn3 , z , ∀z ∈ W,
4.3
j0
c ∂t un , v − ∇ · σ n , v f n , v Rn2 , v ,
∀v ∈ L2 Ω,
where
Rn1
Rn3
∂2 λ
∂t λ − λt O δ 2
∂t
n
δ
n−1
j0
λ −
j
tn
0
Rn2
,
∂2 u
∂t u − ut O δ 2
∂t
n
,
∂λ
λsds O δ
.
∂t
4.4
Now we can formulate a fully discrete scheme based on 4.3.
Fully discrete scheme: find unh , λnh , σhn ∈ Xhu × Vhσ × Vhσ , n 1, 2, . . . , M − 1 such that
a ∂t λnh , wh ∇ · σhn , ∇ · wh −f n , ∇ · wh , ∀wh ∈ Vhσ ,
⎞
⎛
n−1
n n n j
b σh , zh − a λh , zh − ⎝bn δ λh , zh ⎠ 0, ∀zh ∈ Vhσ ,
j0
c ∂t unh , vh − ∇ · σhn , vh f n , vh ,
∀vh ∈ Xhu ,
with given an initial approximation u0h , λ0h , σh0 ∈ Xhu × Vhσ × Vhσ .
4.5
Journal of Applied Mathematics
11
For fully discrete error estimates, we now split the errors
un − unh un − Ph un Ph un − unh ηn ςn ,
4.6
λn − λnh λn − Rh λn Rh λn − λnh ρn ξn ,
σ n − σhn σ n − Rh σ n Rh σ n − σhn γ n θn .
From 4.3 to 4.5, we then obtain
a ∂t ξn , wh ∇ · θn , ∇ · wh − ∂t ρn , wh Rn1 , wh ,
⎞
⎛
n−1
b θn , zh − an ξn , zh − ⎝bn δ ξj , zh ⎠
∀wh ∈ Vhσ ,
j0
⎛
− γ n , zh an ρn , zh ⎝bn δ
n−1
4.7
⎞
ρj , zh ⎠ − Rn3 , zh ,
∀zh ∈ Vhσ ,
j0
c ∂t ςn , vh − ∇ · θn γ n , vh Rn2 , vh ,
∀vh ∈ Xhu .
Lemma 4.1. Assume that the solution of system 2.5 has regular properties that ut ∈ L2 H k1
Ω,λt , σt ∈ L2 Hr1 Ω. Then one has the estimates
max ∂t λ − Rh λn L2 Ω max ∂t σ − Rh σn L2 Ω ≤ Chr1
σ ,
0≤n≤M
0≤n≤M
max ∂t u − Ph un L2 Ω ≤ Chk1
u .
4.8
0≤n≤M
Theorem 4.2. Assume that ∂2 u/∂t2 , ∂u/∂t ∈ L2 H k1 Ω,∂λ/∂t, ∂2 λ/∂t2 , ∂σ/∂t, ∂2 σ/∂t2 ∈
L2 Hr1 Ω, u ∈ L∞ H k1 Ω, and λ, σ ∈ L∞ Hr1 Ω, then there exists a constant C such that
max λ − λh n L2 Ω max σ − σh n L2 Ω ≤ C hr1
σ δ ,
0≤n≤M
0≤n≤M
∗
max ∇ · σ − σh n L2 Ω ≤ C hrσ δ ,
0≤n≤M
4.9
r∗
max u − uh n L2 Ω ≤ C hk1
u hσ δ .
0≤n≤M
Proof . Set wh θn in 4.7a and zh −∂t ξn in 4.7b and add the two equations to obtain
∇ · θn 2L2 Ω an ∂t ξn , ξn − ∂t ρn , θn Rn1 , θn γ n , ∂t ξn − an ρn , ∂t ξn
⎛
⎞ ⎛
⎞
n−1
n−1
− ⎝bn δ ξj , ∂t ξn ⎠ − ⎝bn δ ρj , ∂t ξn ⎠ Rn3 , ∂t ξn .
j0
j0
4.10
12
Journal of Applied Mathematics
Note that
an ξn , ξn − an−1 ξn−1 , ξn−1
δ
an ξn , ξn − an ξn , ξn−1 an ξn , ξn−1 − an−1 ξn−1 , ξn−1
δ
an ∂t ξn , ξn an ∂t ξn , ξn−1 ∂t an ξn−1 , ξn−1
∂t an ξn 2L2 Ω 2an ∂t ξn , ξn −
2
n 1/2 n
ξ − ξn−1 2
a L Ω
δ
4.11
∂t an ξn−1 , ξn−1 .
So, we get
an ∂t ξn , ξn 1
∂t an ξn 2L2 Ω 2
2
n 1/2 n
ξ − ξn−1 2
a L Ω
2δ
−
1 n n−1 n−1 ∂t a ξ , ξ
.
2
4.12
Note that
ξn , γ n − ξn−1 , γ n−1
− ∂t γ n , ξn−1 ,
γ , ∂t ξ δ
n n n n−1 n−1 n−1 ξ ,a ρ − ξ ,a ρ
n n
− ∂t an ρn , ξn−1 ,
a ρ , ∂t ξn δ
n n n−1 n−1 ξ , R3 − ξ , R3
n
− ∂t Rn3 , ξn−1 .
R3 , ∂t ξn δ
⎛
⎞
j n
j n−1
n−1
bn δ n−1
− bn−1 δ n−2
j0 ξ , ξ
j0 ξ , ξ
⎝bn δ ξj , ∂t ξn ⎠ δ
j0
⎞
⎛
n−1
−⎝∂t bn δ ξj , ξn−1 ⎠ − bn−1 ξn−1 , ξn−1 ,
n
n
j0
⎛
⎝b n δ
n−1
j0
⎞
ρj , ∂t ξn ⎠ bn δ
n−1
j0
⎛
−⎝∂t bn δ
j n−1
ρj , ξn − bn−1 δ n−2
j0 ρ , ξ
n−1
j0
δ
⎞
ρj , ξn−1 ⎠ − bn−1 ρn−1 , ξn−1 .
4.13
Journal of Applied Mathematics
13
Substitute 4.12-4.13 into 4.10 to get
1
∇ · θn 2L2 Ω ∂t an ξn 2L2 Ω 2
2
n 1/2 n
ξ − ξn−1 2
a L Ω
2δ
n n n−1 n−1 ξ ,γ − ξ ,γ
1 n n−1 n−1 ∂t a ξ , ξ
− ∂t ρ , θ 2
δ
ξn , an ρn − ξn−1 , an−1 ρn−1
∂t an ρn , ξn−1
− ∂t γ n , ξn−1 −
δ
n
n−1 j
j
j
j
bn δ n−1
− bn−1 δ n−2
j0 ξ ρ , ξ
j0 ξ ρ , ξ
−
δ
⎛
⎞
n−1 ξj ρj , ξn−1 ⎠ bn−1 ξn−1 ρn−1 , ξn−1
⎝∂t bn δ
n
n
Rn1 , θn
4.14
j0
ξn , Rn3 − ξn−1 , Rn−1
3
− ∂t Rn3 , ξn−1 .
δ
Summing from 1 to n, we find that
n 1/2 n 2
a ξ 2
L Ω
0 1/2 0 2
≤ a
ξ 2
2δ
n 2
∇ · θj 2
L Ω
j1
L Ω
n 2
n 1/2 n
ξ − ξn−1 2
a L Ω
j1
2
2
− ξ0 , γ 0 ξ0 , a0 ρ0 εξn 2L2 Ω Cρn L2 Ω CRn3 L2 Ω
n−1 2
2
Cγ n L2 Ω Cδ ξj 2
j1
n j 2
Cδ
∂t ρ 2
j1
L Ω
L Ω
2
∂t γ j 2
Cδ
L Ω
n 2
j
θ 2
j1
L Ω
2
∂t aj ρj 2
L Ω
2
ρj 2
L Ω
2
j
R1 2
L Ω
j 2
∂t R3 2
L Ω
.
4.15
Choose zh θn in 3.6b to get
2
θn 2L2 Ω ≤ C an 1/2 ξn 2
L Ω
2
2
γ n L2 Ω ρn L2 Ω
Cδ
n−1 2
j
ξ 2
j1
L Ω
2
ρj 2
L Ω
.
4.16
14
Journal of Applied Mathematics
Substitute 4.16 into 4.15 and note that ξ0 0 to get
ξn 2L2 Ω θn 2L2 Ω 2δ
n 2
∇ · θj 2
L Ω
j1
2
2
2
≤ Cρn L2 Ω Cγ n L2 Ω CRn3 L2 Ω
Cδ
n−1 2
j
ξ 2
Cδ
L Ω
j1
2
∂t γ j 2
L Ω
2
ρj 2
L Ω
2
j
R1 2
L Ω
4.17
L Ω
j1
n j 2
Cδ
∂t ρ 2
j1
n 2
j
θ 2
L Ω
2
∂t aj ρj 2
L Ω
j 2
∂t R3 2
L Ω
.
Using Gronwall’s lemma, we obtain
ξn 2L2 Ω 1 − Cδθn 2L2 Ω 2δ
n 2
∇ · θj 2
L Ω
j1
2
2
2
≤ Cρn L2 Ω Cγ n L2 Ω CRn3 L2 Ω
Cδ
n j 2
∂t ρ 2
L Ω
j1
4.18
2
∂t γ j 2
2
∂t aj ρj 2
L Ω
L Ω
2
ρj 2
L Ω
2
j
R1 2
L Ω
j 2
∂t R3 2
L Ω
.
Note that
n j 2
a δ R1 2
j1
n 2 2
∂ λ
≤ Cδ
2
∂t 2
j1
L
3
L Ω
L2 Ω
≤
2 2
2 ∂ λ Cδ 2 ,
∂t ∞ 2
L L Ω
n 2
b δ ∂t ρj 2
n j 2
∂ρ ≤ Cδ ∂t 2
j1
≤C
n 2
c δ ∂t γ j 2
n j 2
∂γ ≤ Cδ ∂t 2
j1
T
· nh2r2
≤C
≤ Ch2r2
,
σ
σ
M
j1
j1
L Ω
L Ω
n j 2
d δ ∂t R3 2
j1
L Ω
L Ω
L Ω
T
· nh2r2
≤ Ch2r2
,
σ
σ
M
4.19
tj 2
j−1
n tj−1 λ − λ ds Cδ δ
2
j1 L Ω
2
2 ∂λ .
≤ Cδ ∂t L∞ L2 Ω
Journal of Applied Mathematics
15
Therefore, substituting the above estimates into 4.18 and choosing δ0 in such a way that for
0 < δ ≤ δ0 , 1 − Cδ > 0, we obtain
θn 2L2 Ω ξn 2L2 Ω δ
n 2
∇ · θj 2
L Ω
j1
2
.
≤ C h2r2
δ
σ
4.20
By 4.7b, we obtain
n an ρn − an−1 ρn−1
an ξn − an−1 ξn−1
, zh − ∂t γ , zh , zh
∂t θ , zh δ
δ
n−1 n−2 j
j
j
bn δ n−1
δ j0 ξ ρj , zh
j0 ξ ρ , zh − b
− ∂t Rn3 , zh
δ
n an ρn − an−1 ρn−1
an ξn − an−1 ξn−1
, zh − ∂t γ , zh , zh
δ
δ
⎛
⎞
n−2 n
n−1
n−1
n
j
j
, zh ⎝∂t b δ
ξ ρ , zh ⎠ − ∂t Rn3 , zh .
b ξ ρ
n
4.21
j0
Set zh ∂t ξn in 4.21 to obtain
n
an ξn − an−1 ξn−1
, ∂t ξn
δ
an ρn − an−1 ρn−1
n
, ∂t ξ
− ∂t γ , ∂t ξ ∂t θ , ∂t ξ δ
⎞
⎛
n−2 n
n−1
n−1
n
n
j
j
n
, ∂t ξ ⎝∂t b δ
ξ ρ , ∂t ξ ⎠ − ∂t Rn3 , ∂t ξn
b ξ ρ
n
n
n
j0
2
an 1/2 ∂t ξn 2
L Ω
∂t an ξn−1 , ∂t ξn an ∂t ρn , ∂t ξn
4.22
∂t an ρn−1 , ∂t ξn − ∂t γ n , ∂t ξn bn ξn−1 ρn−1 , ∂t ξn
⎛
⎝∂t bn δ
n−2 ⎞
ξj ρj , ∂t ξn ⎠ − ∂t Rn3 , ∂t ξn .
j0
Set wh ∂t θn in 4.7a to obtain
n
∇ · θ − θn−1 2 2
1
L Ω
− ∂t ρn , ∂t θn Rn1 , ∂t θn .
∂t ξn , ∂t θn ∂t ∇ · θn 2L2 Ω 2
2δ
4.23
16
Journal of Applied Mathematics
Substitute 4.22 into 4.23 to get
n
∇ · θ − θn−1 2 2
1
L Ω
n 2
∂t ∇ · θ L2 Ω L Ω
2
2δ
− ∂t ρn , ∂t θn − ∂t an ξn−1 , ∂t ξn − an ∂t ρn , ∂t ξn
n 1/2 n 2
a ∂t ξ 2
− ∂t an ρn−1 , ∂t ξn ∂t γ n , ∂t ξn − bn ξn−1 ρn−1 , ∂t ξn
Rn1 , ∂t θ
n
⎛
− ⎝∂t bn δ
n−2 4.24
⎞
ξj ρj , ∂t ξn ⎠ ∂t Rn3 , ∂t ξn .
j0
Take zh ∂t θn in 4.21 to obtain
an ξn − an−1 ξn−1
n
− ∂t γ n , ∂t θn
, ∂t θ
δ
an ρn − an−1 ρn−1
n
bn ξn−1 ρn−1 , ∂t θn
, ∂t θ
δ
⎛
⎞
n−2 ξj ρj , ∂t θn ⎠ − ∂t Rn3 , ∂t θn
⎝∂t bn δ
∂t θn 2L2 Ω
j0
4.25
an ∂t ξn , ∂t θn ∂t an ξn−1 , ∂t θn − ∂t γ n , ∂t θn an ∂t ρn , ∂t θn
∂t an ρn−1 , ∂t θn bn ξn−1 ρn−1 , ∂t θn
⎛
⎝∂t bn δ
n−2 ⎞
ξj ρj , ∂t θn ⎠ − ∂t Rn3 , ∂t θn .
j0
Add 4.24 and 4.25 to get
n
∇ · θ − θn−1 2 2
1
L Ω
2
∂t ∇ · θn L2 Ω L Ω
2
2δ
− ∂t ρn , ∂t θn − ∂t an ξn−1 , ∂t ξn − an ∂t ρn , ∂t ξn − ∂t an ρn−1 , ∂t ξn
n 1/2 n 2
a ∂t ξ 2
∂t θn 2L2 Ω
∂t γ n , ∂t ξn an ∂t ξn , ∂t θn ∂t an ξn−1 , ∂t θn − ∂t γ n , ∂t θn
an ∂t ρn , ∂t θn ∂t an ρn−1 , ∂t θn Rn1 , ∂t θn − ∂t Rn3 , ∂t θn ∂t ξn
⎛
⎞
n−2 ξj ρj , ∂t θn ∂t ξn ⎠.
bn ξn−1 ρn−1 , ∂t θn ∂t ξn ⎝∂t bn δ
j0
4.26
Journal of Applied Mathematics
17
Apply Cauchy-Schwarz’s inequality and Young’s inequality to obtain
1
amin ∂t ξn 2L2 Ω ∂t θn 2L2 Ω ∂t ∇ · θn 2L2 Ω
2
≤ ε ∂t ξn 2L2 Ω ∂t θn 2L2 Ω
2
C ξn−1 2
L Ω
Cδ2
2
2
2
∂t ρn L2 Ω ∂t γ n L2 Ω ρn−1 2
n−2 2
j
ξ 2
L Ω
L Ω
j0
2
2
Rn1 L2 Ω ∂t Rn3 L2 Ω
2
ρj 2
L Ω
.
4.27
Using 4.19 and 4.20 and summing from 1 to n, we obtain
∇ · θn 2L2 Ω δ
n δ2 .
∂t ξn 2L2 Ω ∂t θn 2L2 Ω ≤ C h2r2
σ
4.28
j1
Choosing vh ςn in 4.7c and applying Cauchy-Schwarz’s inequality, Young’s inequality,
and 4.28, we have
2
ςn 2L2 Ω − ςn−1 2
L Ω
2δ ∇ · σ − σh n , ςn 2δ Rn2 , ςn
4.29
2
2
≤ Cδ ∇ · σ − σh n L2 Ω Rn2 L2 Ω ςn 2L2 Ω .
Summing from 1 to n and using the Gronwall lemma, we obtain
2
ςn 2L2 Ω ≤ ς0 2
L Ω
Cδ
n 2
∇ · σ − σh j 2
L Ω
j1
2
j
R2 2
.
4.30
.
4.31
L Ω
Note that
n j 2
δ R2 2
j1
L Ω
n 2 2
∂ u
≤ Cδ
2
∂t 2
j1
L
3
L2 Ω
≤
2
∂2 u Cδ2 2 ∂t L∞ L2 Ω
Substituting 4.31 into 4.30 and using 3.2, 4.20, and the triangle inequality, we get
2
h2r∗
ςn 2L2 Ω ≤ C h2k2
u
σ δ .
4.32
Combining 3.2, 3.3, 4.20, 4.28, 4.32, and Lemma 4.1, we apply the triangle inequality
to complete the proof.
18
Journal of Applied Mathematics
Table 1: The errors and convergence order.
h, δ
√
2 1
,
8 8
√
2 1
,
16 16
√
2 1
,
32 32
u − uh L∞ L2 Ω
h, δ
√
2
,
8
√
2
,
16
√
2
,
32
1
8
1
16
1
32
λ − λh L∞ L2 Ω
Order
1.4527e − 002
Order
1.3532e − 001
6.5250e − 003
1.15
6.9202e − 002
0.97
3.0841e − 003
1.08
3.5363e − 002
0.97
σ − σh L∞ L2 Ω
Order
σ − σh L∞ Hdiv;Ω
Order
3.3512e − 001
3.8892e − 001
1.6734e − 001
1.00
1.7291e − 001
1.17
8.3746e − 002
1.00
8.4431e − 002
1.03
5. Numerical Example
In this section, we analyse some numerical results to illustrate the efficiency of the proposed
method. We consider the following 2D parabolic partial integrodifferential equations with
initial-boundary value condition:
ut − ∇ ·
ax, t∇u bx, t
t
∇u ds
fx, t,
x, t ∈ Ω × J,
0
ux, t 0,
x, t ∈ ∂Ω × J,
ux, 0 sinπx1 sinπx2 ,
5.1
x ∈ Ω,
where Ω 0, 1 × 0, 1, J 0, 1, ax, t 1 x12 2x22 , bx, t 1 2x12 x22 , x x1 , x2 , and
fx, t is chosen so that the exact solution for the scalar unknown function is
ux, t e−t sinπx1 sinπx2 .
5.2
The corresponding exact gradient is
λx, t λ1 , λ2 πe−t cosπx1 sinπx2 , πe−t sinπx1 cosπx2 ,
5.3
Journal of Applied Mathematics
19
The exact solution u at t = 1
0.4
0.3
0.2
1
0.1
0
1
0.8
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
Figure 1: The exact solution u.
The numerical solution uh at t = 1
0.4
0.3
0.2
1
0.1
0
1
0.8
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
Figure 2: The numerical solution uh .
and its exact flux is
σx, t σ1 , σ2 π 1 2 − e−t x12 1 e−t x22 cosπx1 sinπx2 ,
π 1 2 − e−t x12 1 e−t x22 sinπx1 cosπx2 .
5.4
Dividing the domain Ω into the triangulations of mesh size hu hσ h uniformly,
considering the piecewise constant space Xhu with index k 0 for the scalar unknown
function u and the lowest-order Raviart-Thomas triangular space Vhσ 42, 45 for the gradient
λ and the flux σ and using the backward Euler procedure with uniform time step length
λ − λh L∞ L2 Ω ,
δ 1/M, we obtain some convergence results √
for u −√uh L∞√L2 Ω , √
σ − σh L∞ L2 Ω and σ − σh L∞ Hdiv;Ω with h 2δ 2/8, 2/16, 2/32 in Table 1.
√
√
With time t 1, h 2δ 2/32, the exact solution u, λ, σ is shown in Figures 1, 3,
and 5, respectively, and the corresponding numerical solution uh , λh , σh is shown in Figures
2, 4, and 6, respectively.
20
Journal of Applied Mathematics
The exact solution λ = (λ1 , λ2 ) at t = 1
The exact solution λ = (λ1 , λ2 ) at t = 1
2
2
1
1
0
0
1
−1
−2
1
1
−1
−2
1
0.5
0.5
0.5
0.5
0
0
0
a
0
b
Figure 3: The exact gradient λ1 a and λ2 b.
The numerical solution λh = (λ1h , λ2h ) at t = 1
The numerical solution λh = (λ1h , λ2h ) at t = 1
2
2
1
1
0
0
1
−1
−2
1
0.5
0.5
1
−1
−2
1
0.5
0.5
0
a
0
0
0
b
Figure 4: The numerical gradient λ1h a and λ2h b.
We can see from Table 1 that the convergence rate is order 1 which confirms the
theoretical results of Theorem 4.2 for the above chosen spaces Xhu and Vhσ . The numerical
results in Table 1 and Figures 1–6 show that new positive definite expanded mixed scheme is
efficient.
Journal of Applied Mathematics
21
The exact solution σ = (σ 1 , σ 2 ) at t = 1
The exact solution σ = (σ 1 , σ 2 ) at t = 1
5
5
0
0
1
−5
−10
1
1
−5
−10
1
0.5
0.5
0.5
0.5
0
0
0
a
0
b
Figure 5: The exact flux σ1 a and σ2 b.
The numerical solution σ h = (σ 1h , σ 2h ) at t = 1
The numerical solution σ h = (σ 1h , σ 2h ) at t = 1
5
5
0
0
1
−5
−10
1
0.5
1
−5
−10
1
0.5
0.5
0.5
0
a
0
0
0
b
Figure 6: The numerical flux σ1h a and σ2h b.
6. Concluding Remarks
In the paper, a new expanded mixed finite element method based on a positive definite
system is proposed for parabolic partial integrodifferential equation. Compared to expanded
mixed method and standard mixed methods, the new expanded mixed element system
is symmetric positive definite and both the gradient equation and the flux equation are
separated from its scalar unknown equation. The existence and uniqueness for semidiscrete
22
Journal of Applied Mathematics
scheme are proved and error estimates are derived for both semidiscrete and fully discrete
schemes. Finally, some numerical results are provided to confirm our theoretical analysis.
In the near further, we will study the others evolution equations such as hyperbolic wave
equation, and miscible displacement of compressible flow in porous media.
Acknowledgments
The authors would like to thank the Editor and the referees for their valuable suggestions
and comments. This work is supported by the National Natural Science Fund of China no.
11061021, the Scientific Research Projection of Higher Schools of Inner Mongolia no.
NJ10006, no. NJ10016, no. NJZZ12011, the Program of Higher-level talents of Inner Mongolia
University SPH-IMU, no. Z200901004, no. 125119, and the YSF of Inner Mongolia University no. ND0702.
References
1 T. Zhang and C. J. Li, “Superconvergence of finite element approximations to parabolic and hyperbolic integro-differential equations,” Northeastern Mathematical Journal, vol. 17, no. 3, pp. 279–288, 2001.
2 A. K. Pani and T. E. Peterson, “Finite element methods with numerical quadrature for parabolic integrodifferential equations,” SIAM Journal on Numerical Analysis, vol. 33, no. 3, pp. 1084–1105, 1996.
3 A. K. Pani and R. K. Sinha, “Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data,” Calcolo, vol. 37, no. 4, pp. 181–
205, 2000.
4 Y. P. Lin, V. Thomée, and L. B. Wahlbin, “Ritz-Volterra projections to finite-element spaces and
applications to integrodifferential and related equations,” SIAM Journal on Numerical Analysis, vol.
28, no. 4, pp. 1047–1070, 1991.
5 T. Zhang, Finite Element Methods for Partial Differentio-Integral Equations, Chinese Science Press, Beijing,
China, 2009.
6 R. K. Sinha, R. E. Ewing, and R. D. Lazarov, “Mixed finite element approximations of parabolic
integro-differential equations with nonsmooth initial data,” SIAM Journal on Numerical Analysis, vol.
47, no. 5, pp. 3269–3292, 2009.
7 Z. Jiang, “L∞ L2 and L∞ L∞ error estimates for mixed methods for integro-differential equations of
parabolic type,” Mathematical Modelling and Numerical Analysis, vol. 33, no. 3, pp. 531–546, 1999.
8 H. Guo and H. X. Rui, “Least-squares Galerkin procedures for parabolic integro-differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 749–762, 2004.
9 D. Y. Shi and H. H. Wang, “An H 1 -Galerkin nonconforming mixed finite element method for integrodifferential equation of parabolic type,” Journal of Mathematical Research and Exposition, vol. 29, no. 5,
pp. 871–881, 2009.
10 H. R. Li and Q. Li, “Finite volume element methods for nonlinear parabolic integro-differential problems,” Journal of the Korean Society for Industrial and Applied Mathematics, vol. 7, no. 2, pp. 35–49, 2003.
11 Z. X. Chen, “Expanded mixed finite element methods for linear second order elliptic problems I,”
IMA preprint series # 1219, Institute for Mathematics and Its Applications, University of Minnesota,
Minneapolis, Minn, USA, 1994.
12 Z. X. Chen, “Expanded mixed finite element methods for linear second-order elliptic problems. I,”
RAIRO Modélisation Mathématique et Analyse Numérique, vol. 32, no. 4, pp. 479–499, 1998.
13 T. Arbogast, M. F. Wheeler, and I. Yotov, “Mixed finite elements for elliptic problems with tensor
coefficients as cell-centered finite differences,” SIAM Journal on Numerical Analysis, vol. 34, no. 2, pp.
828–852, 1997.
14 Z. X. Chen, “Expanded mixed finite element methods for quasilinear second order elliptic problems.
II,” RAIRO Modélisation Mathématique et Analyse Numérique, vol. 32, no. 4, pp. 501–520, 1998.
15 Z. X. Chen, “Analysis of expanded mixed methods for fourth-order elliptic problems,” Numerical
Methods for Partial Differential Equations, vol. 13, no. 5, pp. 483–503, 1997.
Journal of Applied Mathematics
23
16 Y. Liu, H. Li, and Z. C. Wen, “Expanded mixed finite element method for a second-order linear
parabolic differential equation,” Numerical Mathematics. A Journal of Chinese Universities, vol. 30, no. 3,
pp. 234–249, 2008.
17 C. S. Woodward and C. N. Dawson, “Analysis of expanded mixed finite element methods for a
nonlinear parabolic equation modeling flow into variably saturated porous media,” SIAM Journal
on Numerical Analysis, vol. 37, no. 3, pp. 701–724, 2000.
18 L. Wu and M. B. Allen, “A two-grid method for mixed finite-element solution of reaction-diffusion
equations,” Numerical Methods for Partial Differential Equations, vol. 15, no. 3, pp. 317–332, 1999.
19 Y. P. Chen, Y. Q. Huang, and D. H. Yu, “A two-grid method for expanded mixed finite-element
solution of semilinear reaction-diffusion equations,” International Journal for Numerical Methods in
Engineering, vol. 57, no. 2, pp. 193–209, 2003.
20 Y. P. Chen, H. W. Liu, and S. Liu, “Analysis of two-grid methods for reaction-diffusion equations by
expanded mixed finite element methods,” International Journal for Numerical Methods in Engineering,
vol. 69, no. 2, pp. 408–422, 2007.
21 Y. P. Chen and L. Li, “Lp error estimates of two-grid schemes of expanded mixed finite element
methods,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 197–205, 2009.
22 Y. P. Chen, P. Luan, and Z. L. Lu, “Analysis of two-grid methods for nonlinear parabolic equations by
expanded mixed finite element methods,” Advances in Applied Mathematics and Mechanics, vol. 1, no.
6, pp. 830–844, 2009.
23 H. L. Song and Y. R. Yuan, “The expanded upwind-mixed multi-step method for the miscible
displacement problem in three dimensions,” Applied Mathematics and Computation, vol. 195, no. 1,
pp. 100–109, 2008.
24 L. Guo and H. Z. Chen, “An expanded characteristic-mixed finite element method for a convectiondominated transport problem,” Journal of Computational Mathematics, vol. 23, no. 5, pp. 479–490, 2005.
25 H. Z. Chen and H. Wang, “An optimal-order error estimate on an H 1 -Galerkin mixed method for
a nonlinear parabolic equation in porous medium flow,” Numerical Methods for Partial Differential
Equations, vol. 26, no. 1, pp. 188–205, 2010.
26 Y. Liu and H. Li, “A new mixed finite element method for pseudo-hyperbolic equation,” Mathematica
Applicata, vol. 23, no. 1, pp. 150–157, 2010.
27 Y. Liu, Analysis and numerical simulation of nonstandard mixed element methods [Ph.D. thesis], Inner
Mongolia University, Hohhot, China, 2011.
28 Z. W. Jiang and A. Q. Li, “Expanded mixed finite element methods for the problem of purely longitudinal motion of a homogeneous bar,” Journal of Computational and Applied Mathematics, vol. 235, no. 8,
pp. 2157–2169, 2011.
29 A. L. Zhu, Z. W. Jiang, and Q. Xu, “Expanded mixed covolume method for a linear integro-differential
equation of parabolic type,” Numerical Mathematics. A Journal of Chinese Universities, vol. 31, no. 3, pp.
193–205, 2009.
30 H. X. Rui and T. C. Lu, “An expanded mixed covolume method for elliptic problems,” Numerical
Methods for Partial Differential Equations, vol. 21, no. 1, pp. 8–23, 2005.
31 D. Kim and E. J. Park, “A posteriori error estimator for expanded mixed hybrid methods,” Numerical
Methods for Partial Differential Equations, vol. 23, no. 2, pp. 330–349, 2007.
32 D. P. Yang, “A splitting positive definite mixed element method for miscible displacement of
compressible flow in porous media,” Numerical Methods for Partial Differential Equations, vol. 17, no. 3,
pp. 229–249, 2001.
33 J. S. Zhang and D. P. Yang, “A splitting positive definite mixed element method for second-order
hyperbolic equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 622–636,
2009.
34 Y. Liu, H. Li, J. F. Wang, and S. He, “Splitting positive definite mixed element methods for pseudohyperbolic equations,” Numerical Methods for Partial Differential Equations, vol. 28, no. 2, pp. 670–688,
2012.
35 Z. D. Luo, Mixed Finite Element Methods and Applications, Chinese Science Press, Beijing, China, 2006.
36 P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, New York, NY, USA, 1978.
37 R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.
38 H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3, NorthHolland, Amsterdam, The Netherlands, 1986.
39 D. Zwillinger, Handbook of Differential Equations, Academic Press, Boston, Mass, USA, 3rd edition,
1997.
24
Journal of Applied Mathematics
40 F. Brezzi, J. Douglas, Jr., and L. D. Marini, “Two families of mixed finite elements for second order
elliptic problems,” Numerische Mathematik, vol. 47, no. 2, pp. 217–235, 1985.
41 F. Brezzi, J. Douglas, Jr., M. Fortin, and L. D. Marini, “Efficient rectangular mixed finite elements in
two and three space variables,” RAIRO Modélisation Mathématique et Analyse Numérique, vol. 21, no. 4,
pp. 581–604, 1987.
42 P. A. Raviart and J. M. Thomas, “A mixed finite element method for 2nd order elliptic problems,” in
Mathematical Aspects of Finite Element Methods, vol. 606 of Lecture Notes in Mathematics, pp. 292–315,
Springer, Berlin, Germany, 1977.
43 Z. Chen, Finite Element Methods and Their Applications, Scientific Computation, Springer, Berlin,
Germany, 2005.
44 M. F. Wheeler, “A priori L2 -error estimates for Galerkin approximations to parabolic partial
differential equations,” SIAM Journal on Numerical Analysis, vol. 10, no. 4, pp. 723–749, 1973.
45 L. Chen and Y. P. Chen, “Two-grid method for nonlinear reaction-diffusion equations by mixed finite
element methods,” Journal of Scientific Computing, vol. 49, no. 3, pp. 383–401, 2011.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014