Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 683205, 11 pages
http://dx.doi.org/10.1155/2013/683205
Research Article
A Novel Characteristic Expanded Mixed Method for
Reaction-Convection-Diffusion Problems
Yang Liu, Hong Li, Wei Gao, Siriguleng He, and Zhichao Fang
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Correspondence should be addressed to Yang Liu; mathliuyang@yahoo.cn and Hong Li; smslh@imu.edu.cn
Received 3 November 2012; Revised 24 February 2013; Accepted 10 March 2013
Academic Editor: Alexander Timokha
Copyright © 2013 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 novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems.
The diffusion term ∇ ⋅ (𝑎(x, 𝑡)∇𝑢) is discretized by the novel expanded mixed method, whose gradient belongs to the square
integrable space instead of the classical H(div; Ω) space and the hyperbolic part 𝑑(x)(𝜕𝑢/𝜕𝑡) + c(x, 𝑡) ⋅ ∇𝑢 is handled by the
characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are
introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error
estimates in 𝐿2 - and 𝐻1 -norms for the scalar unknown 𝑢 and a priori error estimates in (𝐿2 )2 -norm for its gradient 𝜆 and its flux 𝜎
(the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.
and the bounded vector c(x, 𝑡) = (𝑐1 (x, 𝑡), 𝑐2 (x, 𝑡)) satisfies
1. Introduction
In this paper, we consider the following reaction-convectiondiffusion problems:
𝑑 (x)
𝜕𝑢
+ c (x, 𝑡) ⋅ ∇𝑢 − ∇ ⋅ (𝑎 (x, 𝑡) ∇𝑢) + 𝑅 (x, 𝑡) 𝑢
𝜕𝑡
= 𝑓 (x, 𝑡) ,
(x, 𝑡) ∈ Ω × 𝐽,
𝑢 (x, 𝑡) = 0,
(1)
(x, 𝑡) ∈ 𝜕Ω × 𝐽,
𝑢 (x, 0) = 𝑢0 (x) ,
x ∈ Ω,
where Ω is a bounded convex polygonal domain in R2 with
𝐿𝑖𝑝𝑠𝑐ℎ𝑖𝑡𝑧 continuous boundary 𝜕Ω and 𝐽 = (0, 𝑇] is the
time interval with 0 < 𝑇 < ∞. The initial value 𝑢0 (x)
and the source term 𝑓(x, 𝑡) are given functions. Throughout
this paper, we assume that the accumulation, diffusion, and
reaction coefficients, 𝑑 = 𝑑(x), 𝑎 = 𝑎(x, 𝑡), and 𝑅 = 𝑅(x, 𝑡),
satisfy
H1 : 0 < 𝑑0 ≤ 𝑑 (x) ≤ 𝑑1 < +∞,
H2 : 0 < 𝑎0 ≤ 𝑎 (x, 𝑡) ≤ 𝑎1 < +∞,
H3 : 0 ≤ 𝑅0 ≤ 𝑅 (x, 𝑡) ≤ 𝑅1 < +∞,
󵄨
󵄨󵄨
󵄨󵄨𝑎𝑡 (x, 𝑡)󵄨󵄨󵄨 ≤ 𝑎1 ,
󵄨
󵄨󵄨
󵄨󵄨𝑅𝑡 (x, 𝑡)󵄨󵄨󵄨 ≤ 𝑅1 ,
(2)
H4 : 0 < 𝑏0 ≤
2
( ∑ 𝑐𝑖2
𝑖=1
1/2
(x, 𝑡))
≤ 𝑏1 < +∞.
(3)
Reaction-convection-diffusion equations are a class of
important evolution partial differential equations and have
a lot of applications in many physical problems, such as the
infiltration of liquid, the proliferation of gas, the conduction
of heat, and the spread of impurities in semiconductor
materials. In recent years, a lot of numerical methods,
such as mixed finite element methods [1–3], Pod methods
[4, 5], characteristic-mixed covolume methods [6], spacetime discontinuous Galerkin methods [7, 8], least-squares
finite element methods [9, 10], nonconforming finite element
method [11], conforming rectangular element method [12],
and two-grid expanded mixed methods [13–16], have been
studied for reaction-convection-diffusion equations.
In 1994, Chen [17, 18] proposed an expanded mixed
finite element method for second-order linear elliptic equation. Compared to standard mixed element methods, the
expanded mixed method can approximate three variables
simultaneously, namely, the scalar unknown, its gradient, and
its flux (the tensor coefficient times the negative gradient).
From then on, the expanded mixed method was applied
2
Journal of Applied Mathematics
to many evolution equations [2, 19, 20], and some new
numerical methods based on the Chen’s expanded mixed
method were proposed, for example, expanded mixed covolume method [21], expanded mixed hybrid methods [22],
positive definite expanded mixed method [23], and so on.
From the above literature on the study of expanded mixed
method, we can find that all papers were studied based on
Chen’s expanded mixed method [17, 18].
In 2011, we developed a new expanded mixed finite
element method [24] based on the mixed weak formulations
[25–27]. In this paper, we develop and analyze a novel
characteristic expanded mixed finite element method, which
combines the novel expanded mixed method [24] applied
to approximating the diffusion term and the characteristic
method that handled the hyperbolic part, for reactionconvection-diffusion equations. The gradient for our method
belongs to the square integrable space instead of the classical
H(div; Ω) space of Chen’s expanded mixed method. We
derive a priori error estimates based on backward Euler
method. Moreover, we prove the optimal a priori error
estimates in 𝐿2 - and 𝐻1 -norms for the scalar unknown 𝑢 and
a priori error estimates in (𝐿2 )2 -norm for its gradient 𝜆 and
its flux 𝜎 (the coefficients times the negative gradient).
Throughout this paper, 𝐶 will denote a generic positive
constant which does not depend on the spatial mesh parameter ℎ and time discretization parameter Δ𝑡. At the same time,
we denote the natural inner product in 𝐿2 (Ω) or (𝐿2 (Ω))2 by
(⋅, ⋅) with the corresponding norm ‖ ⋅ ‖. The other notations
and definitions of Sobolev spaces as in [28] are used.
2. Novel Characteristic Expanded
Mixed System
Introducing the two auxiliary variables 𝜆 = ∇𝑢, 𝜎 =
−(𝑎(x, 𝑡)∇𝑢) = −(𝑎(x, 𝑡)𝜆), we obtain the following first-order
system for (1):
𝑑
𝜕𝑢
+ c ⋅ ∇𝑢 + ∇ ⋅ 𝜎 + 𝑅𝑢 = 𝑓 (x, 𝑡) ,
𝜕𝑡
𝜆 − ∇𝑢 = 0,
(4)
Using the similar method to the one in [29], the expanded
mixed weak formulation for problem (1) is to find {𝑢, 𝜆, 𝜎} :
[0, 𝑇] 󳨃→ 𝑊 × V × X such that
(𝜓
𝜕𝑢
, V) + (∇ ⋅ 𝜎, V) + (𝑅𝑢, V) = (𝑓, V) ,
𝜕𝜏
(𝜆, w) + (𝑢, ∇ ⋅ w) = 0,
(𝜎, z) + (𝑎𝜆, z) = 0,
𝑑 (x) 𝜕
c (x)
𝜕
=
⋅ ∇,
+
𝜕𝜏 (x) 𝜓 (x, 𝑡) 𝜕𝑡 𝜓 (x, 𝑡)
(𝜓
where 𝜓(x, 𝑡) = (𝑑2 (x) + |c(x, 𝑡)|2 )1/2 .
Then the first-order system (4) can be rewritten as
𝜓
𝜎 + 𝑎𝜆 = 0.
(7c)
(𝜆, w) − (∇𝑢, w) = 0,
(𝜎, z) + (𝑎𝜆, z) = 0,
∀V ∈ 𝐻01 ,
2
∀w ∈ (𝐿2 (Ω)) ,
2
∀z ∈ (𝐿2 (Ω)) .
(8a)
(8b)
(8c)
For approximating the solution at time 𝑡𝑛 = 𝑛Δ𝑡, the
characteristic derivative will be approximated by
x̂𝑛 = x −
Δ𝑡
𝑐 (x, 𝑡𝑛 ) ,
𝑑 (x)
(9)
and then we have the following approximation:
𝜓 (x, 𝑡𝑛 )
𝑢 (x, 𝑡𝑛 ) − 𝑢 (̂x𝑛 , 𝑡𝑛−1 )
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨󵄨 ≈ 𝜓 (x, 𝑡𝑛 )
𝜕𝜏 󵄨󵄨𝑡𝑛
√(x − x̂𝑛 )2 + (Δ𝑡)2
(10)
𝑢 (x, 𝑡𝑛 ) − 𝑢 (̂x𝑛 , 𝑡𝑛−1 )
.
Δ𝑡
Let (𝑉ℎ , Wℎ ) ⊂ (𝐻01 , (𝐿2 (Ω))2 ) be defined by the following
finite element pair 𝑃1 − 𝑃02 [25, 26]:
𝑉ℎ = {Vℎ ∈ 𝐶0 (Ω) ∩ 𝐻01 | Vℎ ∈ 𝑃1 (𝐾) , ∀𝐾 ∈ Kℎ } ,
Wℎ
𝜕𝑢
+ ∇ ⋅ 𝜎 + 𝑅𝑢 = 𝑓 (x, 𝑡) ,
𝜕𝜏
𝜆 − ∇𝑢 = 0,
(7b)
∀z ∈ X,
𝜕𝑢
, V) − (𝜎, ∇V) + (𝑅𝑢, V) = (𝑓, V) ,
𝜕𝜏
= 𝑑 (x)
(5)
∀w ∈ V,
(7a)
where 𝑊 = 𝐿2 (Ω) or 𝑊 = {𝑤 ∈ 𝐿2 (Ω)|𝑤|𝜕Ω = 0}, V =
H(div; Ω) = {v ∈ (𝐿2 (Ω))2 | ∇⋅v ∈ 𝐿2 (Ω)}, and X = (𝐿2 (Ω))2 .
In this paper, we propose and discuss a novel characteristic expanded mixed method. The new characteristic
expanded mixed weak formulation is to find {𝑢, 𝜆, 𝜎} :
[0, 𝑇] 󳨃→ 𝐻01 × (𝐿2 (Ω))2 × (𝐿2 (Ω))2 such that
𝜎 + 𝑎𝜆 = 0.
Let the characteristic direction corresponding to the hyperbolic part 𝑑𝜕𝑢/𝜕𝑡 + c ⋅ ∇𝑢 be denoted by 𝜏 = 𝜏(x), which is
subject to
∀V ∈ 𝑊,
2
= {wℎ = (𝑤1ℎ , 𝑤2ℎ ) ∈ (𝐿2 (Ω)) | 𝑤1ℎ , 𝑤2ℎ ∈ 𝑃0 (𝐾) ,
(6)
∀𝐾 ∈ Kℎ } .
(11)
Journal of Applied Mathematics
3
Now the novel characteristic expanded mixed finite element procedure for (8a), (8b), and (8c) is to find {𝑢ℎ𝑛 , 𝜆𝑛ℎ , 𝜎ℎ𝑛 } :
[0, 𝑇] 󳨃→ 𝑉ℎ × Wℎ × Wℎ satisfying
(𝑑
̂ 𝑛−1
𝑢ℎ𝑛 − 𝑢
ℎ
, Vℎ ) − (𝜎ℎ𝑛 , ∇Vℎ ) + (𝑅𝑢ℎ𝑛 , Vℎ )
Δ𝑡
𝑛
= (𝑓 , Vℎ ) ,
Lemma 5 (see [24]). There is a constant 𝐶 independent of ℎ
such that
󵄩󵄩 󵄩󵄩
2
󵄩󵄩𝜂󵄩󵄩 ≤ 𝐶ℎ ‖𝑢‖𝐻2 ,
󵄩󵄩 󵄩󵄩
2󵄩 󵄩
󵄩󵄩𝜂𝑡 󵄩󵄩 ≤ 𝐶ℎ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐻2 ,
(12a)
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜂󵄩󵄩1 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜆‖(𝐻1 (Ω))2 ) .
∀Vℎ ∈ 𝑉ℎ ,
(𝜆𝑛ℎ , wℎ ) − (∇𝑢ℎ𝑛 , wℎ ) = 0,
∀wℎ ∈ Wℎ ,
(12b)
(𝜎ℎ𝑛 , zℎ ) + (𝑎𝑛 𝜆𝑛ℎ , zℎ ) = 0,
∀zℎ ∈ Wℎ ,
(12c)
̂ 𝑛−1
where 𝑢ℎ𝑛 = 𝑢ℎ (𝑡𝑛 ), 𝑢
ℎ
(Δ𝑡/𝑑(x))𝑐(x, 𝑡𝑛 ), 𝑡𝑛−1 ).
=
𝑢ℎ (̂x𝑛 , 𝑡𝑛−1 )
=
𝑢ℎ (x −
Remark 1. From [25, 26], we find that (𝑉ℎ , Wℎ ) satisfies the
so-called discrete Ladyzhenskaya-Babuska-Brezzi condition.
Remark 2. Compared to the scheme (7a), (7b), and (7c) based
on Chen’s expanded mixed element method, the gradient in
the scheme (8a), (8b), and (8c) belongs to the simple square
integrable space instead of the classical H(div; Ω) space. For
H(div; Ω) ⊂ (𝐿2 (Ω))2 , we easily find that our method reduces
the regularity requirement on the gradient solution 𝜆 = ∇𝑢.
Remark 3. Based on finite element space (𝑉ℎ , Wℎ ) in (11), the
number of total degrees of freedom for our scheme ((12a),
(12b), and (12c)) is less than that for the scheme in [29]. By
the same discussion as Remark 1 in [30], we can obtain the
detailed analysis for degrees of freedom.
3. Some Lemmas and Error Estimates
3.1. Novel Expanded Mixed Projection and Lemmas. We
first introduce the novel expanded mixed elliptic projection
[24] associated with our equations to derive a priori error
estimates for the proposed method.
̃ ,𝜎
Let (̃
𝑢ℎ , 𝜆
ℎ ̃ ℎ ) : [0, 𝑇] → 𝑉ℎ × Wℎ × Wℎ be given by the
following mixed relations:
̃ ℎ , ∇Vℎ ) = 0,
(𝜎 − 𝜎
∀Vℎ ∈ 𝑉ℎ ,
(13a)
̃ , w ) − (∇ (𝑢 − 𝑢
̃ ℎ ) , wℎ ) = 0,
(𝜆 − 𝜆
ℎ
ℎ
∀wℎ ∈ Wℎ ,
(13b)
̃ ) , z ) = 0,
̃ ℎ , zℎ ) + (𝑎 (𝜆 − 𝜆
(𝜎 − 𝜎
ℎ
ℎ
∀zℎ ∈ Wℎ .
(13c)
In the following discussion, we will give some important
lemmas based on the novel expanded mixed projection.
Lemma 4 (see [24]). There is a constant 𝐶 independent of ℎ
such that
‖𝛿‖ ≤ 𝐶ℎ (‖𝜆‖(𝐻1 (Ω))2 + ‖𝑢‖𝐻2 ) ,
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜌󵄩󵄩 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜎‖(𝐻1 (Ω))2 + ‖𝜆‖(𝐻1 (Ω))2 ) ,
󵄩󵄩 󵄩󵄩
󵄩󵄩∇𝜂󵄩󵄩 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜆‖(𝐻1 (Ω))2 ) ,
̃ , and 𝜌 = 𝜎 − 𝜎
̃ℎ.
̃ℎ , 𝛿 = 𝜆 − 𝜆
where 𝜂 = 𝑢 − 𝑢
ℎ
(14)
(15)
(16)
(17)
Lemma 6 (see [6]). For each 𝑛 one has
(𝑑̂V, ̂V) − (𝑑V, V) ≤ 𝐶Δ𝑡 (𝑑V, V) ,
∀V ∈ 𝐿2 (Ω) ,
(18)
where ̂V = V(x − c(x, 𝑡𝑛 )Δ𝑡/𝑑(x)).
Remark 7. For proving Lemmas 4 and 5, we introduce two
linear operators [25, 26] Πℎ : (𝐿2 (Ω))2 → Wℎ and 𝑃ℎ :
𝐻01 (Ω) → 𝑉ℎ and consider the following auxiliary elliptic
problem:
−∇ ⋅ (𝑎∇𝜒) = 𝜂,
𝜒 = 0,
in Ω,
(19)
on 𝜕Ω.
The detailed proofs for Lemmas 4 and 5 are shown in [24].
3.2. A Priori Error Estimates. In the following discussion, we
will derive a priori error estimates based on fully discrete
backward Euler method. Let 0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑀 = 𝑇
be a given partition of the time interval [0, 𝑇] with step length
Δ𝑡 = 𝑇/𝑀 and nodes 𝑡𝑛 = 𝑛Δ𝑡, for some positive integer 𝑀.
For a smooth function 𝜙 on [0, 𝑇], define 𝜙𝑛 = 𝜙(𝑡𝑛 ).
In order to derive a priori error estimates, we now write
̃ 𝑛ℎ ) + (̃
𝑢𝑛ℎ − 𝑢ℎ𝑛 ) = 𝜂𝑛 + 𝜍𝑛 ,
𝑢 (𝑡𝑛 ) − 𝑢ℎ𝑛 = (𝑢 (𝑡𝑛 ) − 𝑢
̃ 𝑛 ) + (𝜆
̃ 𝑛 − 𝜆𝑛 ) = 𝛿𝑛 + 𝜃𝑛 ,
𝜆 (𝑡𝑛 ) − 𝜆𝑛ℎ = (𝜆 (𝑡𝑛 ) − 𝜆
ℎ
ℎ
ℎ
(20)
̃ 𝑛ℎ ) + (̃
𝜎𝑛ℎ − 𝜎ℎ𝑛 ) = 𝜌𝑛 + 𝜉𝑛 .
𝜎 (𝑡𝑛 ) − 𝜎ℎ𝑛 = (𝜎 (𝑡𝑛 ) − 𝜎
Combining (8a), (8b), and (8c), (12a), (12b), and (12c) and
(13a), (13b), and (13c) at 𝑡 = 𝑡𝑛 , we get the error equations
(𝑑
𝜍𝑛 − ̂𝜍𝑛−1
, Vℎ ) − (𝜉𝑛 , ∇Vℎ ) + (𝑅𝜍𝑛 , Vℎ )
Δ𝑡
= (𝜓𝑛
̂ 𝑛−1
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
, Vℎ )
𝜕𝜏
Δ𝑡
− (𝑑
𝜂𝑛 − 𝜂̂𝑛−1
, Vℎ ) − (𝑅𝜂𝑛 , Vℎ ) ,
Δ𝑡
∀Vℎ ∈ 𝑉ℎ ,
(21a)
(𝜃𝑛 , wℎ ) − (∇𝜍𝑛 , wℎ ) = 0,
∀wℎ ∈ Wℎ ,
(21b)
(𝜉𝑛 , zℎ ) + (𝑎𝑛 𝜃𝑛 , zℎ ) = 0,
∀zℎ ∈ Wℎ .
(21c)
Based on the error equations (21a), (21b), and (21c), we obtain
the following theorem for the fully discrete error estimates.
4
Journal of Applied Mathematics
̃ ℎ (0); then there exist some
Theorem 8. Assume that 𝑢ℎ0 = 𝑢
positive constants independent of ℎ = 𝑂(Δ𝑡) such that
󵄩
󵄨󵄩
󵄩󵄨
󵄩󵄩
󵄩󵄩𝑢 (𝑡𝐽 ) − 𝑢ℎ𝐽 󵄩󵄩󵄩 ≤ 𝐶 (𝑎0 , 𝑑1 ) 󵄨󵄨󵄨󵄩󵄩󵄩𝑂 (ℎ2 , Δ𝑡)󵄩󵄩󵄩󵄨󵄨󵄨
󵄩
󵄨󵄩
󵄩󵄨
󵄩
󵄩󵄩 1/2 𝑛 󵄩󵄩2 󵄩󵄩 1/2 𝑛−1 󵄩󵄩2
󵄩󵄩𝑑 𝜍 󵄩󵄩 − 󵄩󵄩𝑑 ̂𝜍 󵄩󵄩
󵄩
󵄩 󵄩
󵄩 + 𝑎 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩(𝑅𝑛 )1/2 𝜍𝑛 󵄩󵄩󵄩󵄩2
0󵄩 󵄩
󵄩󵄩
󵄩󵄩
2Δ𝑡
󵄩󵄩 1/2 𝑛 󵄩󵄩2 󵄩󵄩 1/2 𝑛−1 󵄩󵄩2 󵄩󵄩 1/2 𝑛 𝑛−1 󵄩󵄩2
󵄩󵄩𝑑 𝜍 󵄩󵄩 − 󵄩󵄩𝑑 ̂𝜍 󵄩󵄩 + 󵄩󵄩𝑑 (𝜍 − ̂𝜍 )󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
≤󵄩
2Δ𝑡
󵄩󵄩
1/2 󵄩
1/2 󵄩
󵄩2 󵄩󵄩
󵄩2
+ 󵄩󵄩󵄩(𝑎𝑛 ) 𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩
󵄩 󵄩
󵄩
󵄩
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ℎ2 ‖𝑢‖𝐿∞ (𝐻2 ) ,
󵄩󵄩
󵄩
󵄨󵄩
󵄩󵄨
󵄩󵄩𝑢 (𝑡𝐽 ) − 𝑢ℎ𝐽 󵄩󵄩󵄩 ≤ 𝐶 (𝑎0 , 𝑑1 ) 󵄨󵄨󵄨󵄩󵄩󵄩𝑂 (ℎ1 , Δ𝑡)󵄩󵄩󵄩󵄨󵄨󵄨
󵄩
󵄩1
󵄨󵄩
󵄩󵄨
+ ℎ [𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ‖𝑢‖𝐿∞ (𝐻2 )
+ 𝐶‖𝜆‖𝐿∞ ((𝐻1 )2 ) ] ,
󵄩󵄩
󵄩
󵄨󵄩
󵄩󵄨
󵄩󵄩𝜆 (𝑡𝐽 ) − 𝜆𝐽ℎ 󵄩󵄩󵄩 ≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) 󵄨󵄨󵄨󵄩󵄩󵄩𝑂 (ℎ1 , Δ𝑡)󵄩󵄩󵄩󵄨󵄨󵄨
󵄩
󵄩
󵄨󵄩
󵄩󵄨
Noting that −𝑑̂𝜍𝑛−1 𝜍𝑛 /Δ𝑡 = (||𝑑1/2 (𝜍𝑛 −̂𝜍𝑛−1 )||2 −(||𝑑1/2̂𝜍𝑛−1 ||2 +
||𝑑1/2 𝜍𝑛 ||2 ))/2Δ𝑡, (24) may be written as
= (𝜓𝑛
(22)
+ ℎ [𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) ‖𝑢‖𝐿∞ (𝐻2 )
󵄩󵄩
󵄩
󵄨󵄩
󵄩󵄨
󵄩󵄩𝜎 (𝑡𝐽 ) − 𝜎ℎ𝐽 󵄩󵄩󵄩 ≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) 󵄨󵄨󵄨󵄩󵄩󵄩𝑂 (ℎ1 , Δ𝑡)󵄩󵄩󵄩󵄨󵄨󵄨
󵄩
󵄩
󵄨󵄩
󵄩󵄨
(𝜓𝑛
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) ‖𝑢‖𝐿∞ (𝐻2 ) ] ,
where |‖𝑂(ℎ2−𝑗 , Δ𝑡)‖| = ℎ2−𝑗 (||𝑢||𝐿∞ (𝐻2 ) + ||𝑢𝑡 ||𝐿2 (𝐻2 ) ) +
Δ𝑡||𝑢𝑡𝑡 ||𝐿2 (𝐿2 ) , 𝑗 = 0, 1.
Proof. Set Vℎ = 𝜍𝑛 in (21a), wℎ = 𝜉𝑛 in (21b), and zℎ = 𝜃𝑛 in
(21c) to obtain
𝜍𝑛 − ̂𝜍𝑛−1 𝑛
, 𝜍 ) − (𝜉𝑛 , ∇𝜍𝑛 ) + (𝑅𝑛 𝜍𝑛 , 𝜍𝑛 )
Δ𝑡
= (𝜓𝑛
̂ 𝑛−1 𝑛
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,𝜍 )
𝜕𝜏
Δ𝑡
− (𝑑
(23a)
𝜂𝑛 − 𝜂̂𝑛−1 𝑛
, 𝜍 ) − (𝑅𝑛 𝜂𝑛 , 𝜍𝑛 ) ,
Δ𝑡
(∇𝜍𝑛 , 𝜉𝑛 ) − (𝜃𝑛 , 𝜉𝑛 ) = 0,
(23b)
󵄩󵄩
1/2 󵄩
󵄩2
(𝜉𝑛 , 𝜃𝑛 ) + 󵄩󵄩󵄩(𝑎𝑛 ) 𝜃𝑛 󵄩󵄩󵄩 = 0.
󵄩
󵄩
(23c)
̂ 𝑛−1 𝑛
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,𝜍 )
𝜕𝜏
Δ𝑡
− (𝑑
𝑛
𝑛−1
𝜂 − 𝜂̂
Δ𝑡
, 𝜍𝑛 ) − (𝑅𝑛 𝜂𝑛 , 𝜍𝑛 ) .
= (𝜓𝑛
̂ 𝑛−1 𝑛
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,𝜍 )
𝜕𝜏
Δ𝑡
− (𝑑
𝜂𝑛 − 𝜂𝑛−1 + 𝜂𝑛−1 − 𝜂̂𝑛−1 𝑛
, 𝜍 ) − (𝑅𝑛 𝜂𝑛 , 𝜍𝑛 )
Δ𝑡
󵄩󵄩
󵄩 󵄩
󵄩
̂ 𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜂𝑛 − 𝜂𝑛−1 󵄩󵄩󵄩
𝑢𝑛 − 𝑢
󵄩 𝜕𝑢𝑛
󵄩󵄩 + 󵄩󵄩𝑑
󵄩󵄩
≤ (󵄩󵄩󵄩󵄩𝜓𝑛
−𝑑
Δ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩
Δ𝑡 󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏
󵄩󵄩 𝑛−1 ̂𝑛−1 󵄩󵄩
󵄩 𝜂 − 𝜂 󵄩󵄩 󵄩󵄩 𝑛 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩 + 󵄩󵄩𝑅 𝜂 󵄩󵄩) 󵄩󵄩𝜍 󵄩󵄩
+ 󵄩󵄩󵄩󵄩𝑑
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
󵄩2
𝑡𝑛 󵄩
1 𝑡𝑛 󵄩󵄩󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩󵄩2
󵄩󵄩 𝜕2 𝑢 󵄩󵄩
󵄩 󵄩2
≤ 𝐶 (𝑑1 ) [Δ𝑡 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 +
∫ 󵄩󵄩 󵄩󵄩 𝑑𝑠 + 󵄩󵄩󵄩𝜍𝑛 󵄩󵄩󵄩 ]
󵄩
󵄩
𝜕𝜏
Δ𝑡
𝜕𝜏
󵄩
𝑡𝑛−1 󵄩
𝑡
󵄩
󵄩
𝑛−1
󵄩
󵄩
󵄩󵄩 𝑛−1 ̂𝑛−1 󵄩󵄩
󵄩2 󵄩 𝜂 − 𝜂 󵄩󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩
󵄩
+ 󵄩󵄩󵄩𝑅𝑛 𝜂𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑑
󵄩󵄩󵄩 󵄩󵄩𝜍 󵄩󵄩 .
Δ𝑡
󵄩󵄩
󵄩
(26)
󵄩󵄩 𝑛−1 ̂𝑛−1 󵄩󵄩
ℎ + Δ𝑡 󵄩󵄩 𝑛−1 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩 𝜂 − 𝜂 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩𝑑
󵄩󵄩 󵄩󵄩𝜍 󵄩󵄩 ≤ 𝐶 (
) 󵄩󵄩󵄩𝜂 󵄩󵄩󵄩 󵄩󵄩𝜃 󵄩󵄩
󵄩󵄩
󵄩
Δ𝑡
Δ𝑡
󵄩󵄩
󵄩
𝑎
󵄩 󵄩2
≤ 𝐶 (𝑎0 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) + 0 ‖𝜃‖2 .
2
󵄩󵄩
𝜍𝑛 − ̂𝜍𝑛−1 𝑛
1/2 󵄩
1/2 󵄩
󵄩2 󵄩󵄩
󵄩2
, 𝜍 ) + 󵄩󵄩󵄩(𝑎𝑛 ) 𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩
󵄩 󵄩
󵄩
󵄩
Δ𝑡
= (𝜓𝑛
𝜂𝑛 − 𝜂̂𝑛−1 𝑛
̂ 𝑛−1 𝑛
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
, 𝜍 ) − (𝑑
, 𝜍 ) − (𝑅𝑛 𝜂𝑛 , 𝜍𝑛 )
𝜕𝜏
Δ𝑡
Δ𝑡
Using the assumption ℎ = 𝑂(Δ𝑡), we obtain
Adding the above three equations, we obtain
(𝑑
𝜂𝑛 − 𝜂̂𝑛−1 𝑛
, 𝜍 ) − (𝑅𝑛 𝜂𝑛 , 𝜍𝑛 ) .
Δ𝑡
Using the same analysis as that in [31], the right-hand side of
(25) can be rewritten as
+ ℎ [𝐶 (‖𝜆‖𝐿∞ ((𝐻1 )2 ) + ‖𝜎‖𝐿∞ ((𝐻1 )2 ) )
(𝑑
̂ 𝑛−1 𝑛
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,𝜍 )
𝜕𝜏
Δ𝑡
− (𝑑
+ 𝐶‖𝜆‖𝐿∞ ((𝐻1 )2 ) ] ,
(25)
(24)
(27)
By Lemma 6, we have
󵄩󵄩 1/2 𝑛−1 󵄩󵄩2 󵄩󵄩 1/2 𝑛−1 󵄩󵄩2
󵄩2
󵄩
󵄩󵄩𝑑 𝜍 󵄩󵄩 − 󵄩󵄩𝑑 ̂𝜍 󵄩󵄩 ≥ −𝐶Δ𝑡󵄩󵄩󵄩𝑑1/2 𝜍𝑛−1 󵄩󵄩󵄩 .
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
(28)
Journal of Applied Mathematics
5
Choose wℎ = 𝜉𝑛 in (32), Vℎ = (𝜍𝑛 − 𝜍𝑛−1 )/Δ𝑡 in (21a), and
zℎ = (𝜃𝑛 − 𝜃𝑛−1 )/Δ𝑡 in (21c) to obtain
Using (28), we obtain
󵄩󵄩 1/2 𝑛 󵄩󵄩2 󵄩󵄩 1/2 𝑛−1 󵄩󵄩2
󵄩󵄩𝑑 𝜍 󵄩󵄩 − 󵄩󵄩𝑑 ̂𝜍 󵄩󵄩
󵄩
󵄩 󵄩
󵄩 + 𝑎 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩(𝑅𝑛 )1/2 𝜍𝑛 󵄩󵄩󵄩󵄩2
0󵄩 󵄩
󵄩󵄩
󵄩󵄩
2Δ𝑡
󵄩󵄩 1/2 𝑛 󵄩󵄩2
󵄩2
󵄩
󵄩󵄩𝑑 𝜍 󵄩󵄩 − (1 + 𝐶Δ𝑡) 󵄩󵄩󵄩𝑑1/2 𝜍𝑛−1 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≥
2Δ𝑡
1/2 󵄩
󵄩2
󵄩 󵄩2 󵄩󵄩
+ 𝑎0 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 1/2 𝑛 󵄩󵄩2 󵄩󵄩 1/2 𝑛−1 󵄩󵄩2
󵄩2
󵄩
󵄩󵄩𝑑 𝜍 󵄩󵄩 − 󵄩󵄩𝑑 𝜍 󵄩󵄩 − 𝐶Δ𝑡󵄩󵄩󵄩𝑑1/2 𝜍𝑛−1 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
=
2Δ𝑡
1/2 󵄩
󵄩2
󵄩 󵄩2 󵄩󵄩
+ 𝑎0 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩 .
󵄩
󵄩
(𝑑
∇𝜍𝑛 − ∇𝜍𝑛−1
𝜍𝑛 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
,
) − (𝜉𝑛 ,
)
Δ𝑡
Δ𝑡
Δ𝑡
= (𝜓𝑛
Multiplying by 2Δ𝑡, summing (29) from 𝑛 = 1 to 𝐽, and using
(25)–(29), the resulting inequality becomes
(
𝐽
󵄩 󵄩2
1/2 󵄩
󵄩2
󵄩 󵄩2 󵄩󵄩
𝑑0 󵄩󵄩󵄩󵄩𝜍𝐽 󵄩󵄩󵄩󵄩 + Δ𝑡 ∑ (𝑎0 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩 )
󵄩
󵄩
𝑛=1
𝐽
󵄩
󵄩2
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) + 𝐶 (𝑑1 ) Δ𝑡 ∑ 󵄩󵄩󵄩󵄩𝜍𝑛−1 󵄩󵄩󵄩󵄩
󵄩 󵄩2
= 𝑑1 󵄩󵄩󵄩󵄩𝜍0 󵄩󵄩󵄩󵄩 + 𝐶 (𝑎0 , 𝑑1 )
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
× [(Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠]
𝑡0 󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 𝜕𝜏 󵄩󵄩
(33b)
𝜃𝑛 − 𝜃𝑛−1
𝜃𝑛 − 𝜃𝑛−1
) + (𝑎𝑛 𝜃𝑛 ,
) = 0.
Δ𝑡
Δ𝑡
(33c)
Adding the three equations and using Cauchy-Schwarz
inequality and Young inequality, we obtain
(𝑑
𝜍𝑛 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜃𝑛 − 𝜃𝑛−1
,
) + (𝑎𝑛 𝜃𝑛 ,
)
Δ𝑡
Δ𝑡
Δ𝑡
+ (𝑅𝑛 𝜍𝑛 ,
𝐽
󵄩
󵄩2
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) + 𝐶 (𝑑1 ) Δ𝑡 ∑ 󵄩󵄩󵄩󵄩𝜍𝑛−1 󵄩󵄩󵄩󵄩 .
= (𝜓𝑛
𝑛=1
(30)
𝐽
󵄩 󵄩2
1/2 󵄩
󵄩2
󵄩 󵄩2 󵄩󵄩
𝑑0 󵄩󵄩󵄩󵄩𝜍𝐽 󵄩󵄩󵄩󵄩 + Δ𝑡 ∑ (𝑎0 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩 )
󵄩
󵄩
𝑛=1
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
(31)
≤ 𝐶 (𝑎0 , 𝑑1 ) [(Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠]
󵄩
󵄩
𝜕𝜏
𝜕𝜏
󵄩
𝑡0 󵄩
𝑡
󵄩
󵄩
0
󵄩
󵄩
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) .
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
̂ 𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,
)
𝜕𝜏
Δ𝑡
Δ𝑡
− (𝑑
Using Gronwall lemma, we have
𝜂𝑛 − 𝜂̂𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜍𝑛 − 𝜍𝑛−1
,
) − (𝑅𝑛 𝜂𝑛 ,
)
Δ𝑡
Δ𝑡
Δ𝑡
󵄩󵄩
󵄩 󵄩
󵄩
̂ 𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜂𝑛 − 𝜂𝑛−1 󵄩󵄩󵄩
𝑢𝑛 − 𝑢
󵄩 𝜕𝑢𝑛
󵄩󵄩 + 󵄩󵄩𝑑
󵄩󵄩
≤ ( 󵄩󵄩󵄩󵄩𝜓𝑛
−𝑑
Δ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩
Δ𝑡 󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏
󵄩󵄩 𝑛−1 ̂𝑛−1 󵄩󵄩
󵄩
󵄩
󵄩 𝜂 − 𝜂 󵄩󵄩 󵄩󵄩 𝑛 𝑛 󵄩󵄩 󵄩󵄩󵄩 𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩
󵄩󵄩 + 󵄩󵄩𝑅 𝜂 󵄩󵄩) 󵄩󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝑑
󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩 2 󵄩󵄩2
𝐶 (𝑑1 ) 𝑡𝑛 󵄩󵄩󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩󵄩2
󵄩󵄩 𝜕 𝑢 󵄩󵄩
󵄩󵄩 2 󵄩󵄩 𝑑𝑠 +
∫ 󵄩󵄩 󵄩󵄩 𝑑𝑠
󵄩 𝜕𝜏 󵄩󵄩
Δ𝑡
𝑡𝑛−1 󵄩
𝑡𝑛−1 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩
󵄩
𝑡𝑛
≤ 𝐶 (𝑑1 ) Δ𝑡 ∫
From (21b), we get
𝜃𝑛 − 𝜃𝑛−1
∇𝜍𝑛 − ∇𝜍𝑛−1
, wℎ ) − (
, wℎ ) = 0.
Δ𝑡
Δ𝑡
𝜂𝑛 − 𝜂̂𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜍𝑛 − 𝜍𝑛−1
,
) − (𝑅𝑛 𝜂𝑛 ,
),
Δ𝑡
Δ𝑡
Δ𝑡
(33a)
𝜃𝑛 − 𝜃𝑛−1 𝑛
∇𝜍𝑛 − ∇𝜍𝑛−1 𝑛
,𝜉 ) − (
, 𝜉 ) = 0,
Δ𝑡
Δ𝑡
(𝜉𝑛 ,
𝑛=1
(
̂ 𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜕𝑢𝑛
𝑢𝑛 − 𝑢
−𝑑
,
)
𝜕𝜏
Δ𝑡
Δ𝑡
− (𝑑
󵄩 󵄩2
≤ 𝑑1 󵄩󵄩󵄩󵄩𝜍0 󵄩󵄩󵄩󵄩 + 𝐶 (𝑎0 , 𝑑1 )
𝐽
𝑡𝑛 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
1 𝑡𝑛 󵄩󵄩󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩󵄩2
× [Δ𝑡 ∑ (Δ𝑡 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 +
∫ 󵄩 󵄩 𝑑𝑠)]
Δ𝑡 𝑡𝑛−1 󵄩󵄩󵄩 𝜕𝜏 󵄩󵄩󵄩
𝑡𝑛−1 󵄩
󵄩 𝜕𝜏 󵄩󵄩
𝑛=1
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
+ (𝑅𝑛 𝜍𝑛 ,
(29)
(32)
󵄩 󵄩2
+ 𝐶 (𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) +
󵄩
󵄩2
1 󵄩󵄩󵄩 𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 .
2 󵄩󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
(34)
6
Journal of Applied Mathematics
Summing (36) from 𝑛 = 1 to 𝐽, the resulting inequality
becomes
The left-hand side of (34) can be written as
(𝑑
𝜍𝑛 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜃𝑛 − 𝜃𝑛−1
,
) + (𝑎𝑛 𝜃𝑛 ,
)
Δ𝑡
Δ𝑡
Δ𝑡
+ (𝑅𝑛 𝜍𝑛 ,
𝑛
𝜍 −𝜍
Δ𝑡
𝑛−1
󵄩2
𝐽 󵄩
󵄩󵄩
𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩
󵄩󵄩
Δ𝑡 ∑ 󵄩󵄩󵄩󵄩𝑑1/2
Δ𝑡 󵄩󵄩󵄩
󵄩
𝑛 = 1󵄩
)
󵄩2
󵄩󵄩
𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
󵄩
󵄩󵄩 + (𝑑
,
)
= 󵄩󵄩󵄩󵄩𝑑1/2
Δ𝑡 󵄩󵄩󵄩
Δ𝑡
Δ𝑡
󵄩󵄩
󵄩󵄩 𝑛 1/2 𝑛
󵄩2
󵄩󵄩(𝑎 ) (𝜃 − 𝜃𝑛−1 )󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
+
2Δ𝑡
󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
󵄩󵄩(𝑎 ) 𝜃 󵄩󵄩 − 󵄩󵄩(𝑎 ) 𝜃 󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩
󵄩󵄩
+
2Δ𝑡
󵄩󵄩 𝑛 1/2 𝑛 𝑛−1 󵄩󵄩2
󵄩󵄩(𝑅 ) (𝜍 − 𝜍 )󵄩󵄩
󵄩
󵄩󵄩
𝑎𝑛 − 𝑎𝑛−1 𝑛−1 𝑛−1
−(
𝜃 ,𝜃 ) + 󵄩
2Δ𝑡
2Δ𝑡
𝐽
󵄩󵄩
󵄩󵄩2 󵄩󵄩
󵄩󵄩2
1/2
1/2
+ ∑ (󵄩󵄩󵄩(𝑎𝑛 ) (𝜃𝑛 − 𝜃𝑛−1 )󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) (𝜍𝑛 − 𝜍𝑛−1 )󵄩󵄩󵄩 )
󵄩
󵄩 󵄩
󵄩
𝑛=1
󵄩󵄩
1/2 󵄩
1/2 󵄩
󵄩2 󵄩󵄩
󵄩2
+ 󵄩󵄩󵄩󵄩(𝑎𝐽 ) 𝜃𝐽 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(𝑅𝐽 ) 𝜍𝐽 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
󵄩󵄩
1/2 󵄩
󵄩2
≤ 󵄩󵄩󵄩󵄩(𝑎0 ) 𝜃0 󵄩󵄩󵄩󵄩 + 𝐶 (𝑑1 ) ((Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠
󵄩
󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
+ ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠)
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
𝐽−1
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶 (𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) + 𝐶 (𝑎1 ) Δ𝑡 ∑ 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩
󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
󵄩󵄩(𝑅 ) 𝜍 󵄩󵄩 − 󵄩󵄩(𝑅 ) 𝜍 󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩
󵄩󵄩
+
2Δ𝑡
−(
𝑛=0
𝐽−1
󵄩 󵄩2
+ 𝐶 (𝑅1 ) Δ𝑡 ∑ 󵄩󵄩󵄩𝜍𝑛 󵄩󵄩󵄩
𝑅𝑛 − 𝑅𝑛−1 𝑛−1 𝑛−1
𝜍 ,𝜍 ).
2Δ𝑡
𝑛=0
𝐽
(35)
− Δ𝑡 ∑ (𝑑
𝑛=1
Substitute (35) into (34) and multiply by 2Δ𝑡 to get
(37)
󵄩󵄩
󵄩2
󵄩2
𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑛 1/2 𝑛
󵄩
󵄩󵄩 + 󵄩󵄩(𝑎 ) (𝜃 − 𝜃𝑛−1 )󵄩󵄩󵄩󵄩
Δ𝑡󵄩󵄩󵄩󵄩𝑑1/2
󵄩
󵄩
󵄩
Δ𝑡 󵄩󵄩
󵄩󵄩
󵄩󵄩2
1/2
󵄩󵄩
+ 󵄩󵄩󵄩(𝑅𝑛 ) (𝜍𝑛 − 𝜍𝑛−1 )󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩2
󵄩󵄩
1/2
1/2 󵄩
󵄩2 󵄩󵄩
+ (󵄩󵄩󵄩(𝑎𝑛 ) 𝜃𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩(𝑎𝑛−1 ) 𝜃𝑛−1 󵄩󵄩󵄩󵄩 )
󵄩 󵄩
󵄩
󵄩
Using the same analysis as that in [29], we obtain
𝐽
Δ𝑡 ∑ (𝑑
𝑛=1
󵄩󵄩 2 󵄩
󵄩󵄩 𝜕 𝑢 󵄩󵄩
󵄩󵄩 2 󵄩󵄩 𝑑𝑠
󵄩 𝜕𝜏 󵄩󵄩
𝑡𝑛−1 󵄩
󵄩
󵄩
2
󵄩 𝜕𝜂 󵄩󵄩
𝑡𝑛 󵄩
󵄩 󵄩
󵄩 󵄩2
+ 𝐶 (𝑑1 ) ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠 + 𝐶 (𝑅1 ) Δ𝑡󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 )
󵄩
𝜕𝜏
𝑡𝑛−1 󵄩
󵄩 󵄩
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
,
)
Δ𝑡
Δ𝑡
𝐽
󵄩󵄩2
󵄩󵄩
1/2
1/2 󵄩
󵄩2 󵄩󵄩
+ (󵄩󵄩󵄩(𝑅𝑛 ) 𝜍𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩(𝑅𝑛−1 ) 𝜍𝑛−1 󵄩󵄩󵄩󵄩 )
󵄩
󵄩 󵄩
󵄩
≤ 𝐶 (𝑑1 ) (Δ𝑡)2 ∫
= Δ𝑡 ∑ (𝑑
𝑛=1
󵄩2
𝑡𝑛
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
− Δ𝑡 (𝑑
,
)
Δ𝑡
Δ𝑡
+ Δ𝑡 (
𝑛−1
2Δ𝑡
𝐽
𝑛=1
(36)
𝐽
= Δ𝑡 ∑ (𝑑
𝑛=1
𝐽
𝑛=1
𝑛−1
𝜃
𝑛−1
,𝜃
)
𝐽
∫𝑡 𝑛 𝑅𝑡 𝑑𝑠
− Δ𝑡 ∑ (𝑑
𝑛=1
𝑛−1
2Δ𝑡
𝜍𝑛−1 , 𝜍𝑛−1 ) .
𝐽
= Δ𝑡 ∑ (𝑑
𝑛=1
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛−1
,
)
Δ𝑡
Δ𝑡
𝜍𝑛−1 − 𝜍𝑛 − ̂𝜍𝑛−1 + ̂𝜍𝑛 𝜍𝑛
, )
Δ𝑡
Δ𝑡
+ Δ𝑡 ∑ (𝑑
𝑡
− Δ𝑡 (
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛
, )
Δ𝑡
Δ𝑡
− Δ𝑡 ∑ (𝑑
𝑡
∫𝑡 𝑛 𝑎𝑡 𝑑𝑠
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛 − 𝜍𝑛−1
,
).
Δ𝑡
Δ𝑡
𝜍𝑛 − ̂𝜍𝑛 𝜍𝑛
, )
Δ𝑡 Δ𝑡
𝜍𝑛−1 − ̂𝜍𝑛−1 𝜍𝑛−1
,
)
Δ𝑡
Δ𝑡
𝜍𝑛−1 − 𝜍𝑛 − ̂𝜍𝑛−1 + ̂𝜍𝑛 𝜍𝑛
𝜍𝐽 − ̂𝜍𝐽 𝐽
, ) + (𝑑
,𝜍 )
Δ𝑡
Δ𝑡
Δ𝑡
Journal of Applied Mathematics
7
𝐽 󵄩
𝐽
𝐽
󵄩󵄩 𝜍𝑛−1 − 𝜍𝑛 − ̂𝜍𝑛−1 + ̂𝜍𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝜍𝑛 󵄩󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩 + (𝑑 𝜍 − ̂𝜍 , 𝜍𝐽 )
≤ Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩
Δ𝑡
Δ𝑡
󵄩󵄩 󵄩 Δ𝑡 󵄩󵄩
󵄩
𝑛=1 󵄩
𝐽 󵄩
𝐽
󵄩󵄩 𝜍𝑛−1 − 𝜍𝑛 󵄩󵄩󵄩2
󵄩󵄩 + 𝐶 (𝜀) Δ𝑡 ∑ 󵄩󵄩󵄩𝜍𝑛 󵄩󵄩󵄩2 + 𝐶 (𝑑 ) 󵄩󵄩󵄩𝜍𝐽 󵄩󵄩󵄩2 .
≤ 𝜀Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
1 󵄩
󵄩 󵄩
󵄩 󵄩󵄩
󵄩 Δ𝑡 󵄩󵄩
𝑛 = 1󵄩
𝑛=1
(38)
Combining (14)–(16), (31), (40), (43), (42), and the triangle
inequality, we complete the proof.
Remark 9. Compared to the results in [29], we can obtain the
optimal a priori error estimates in 𝐻1 -norm for the scalar
unknown 𝑢 in Theorem 8.
4. Numerical Experiment
Substitute (31) and (38) into (37) to obtain
𝐽 󵄩
󵄩󵄩 𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩2
󵄩󵄩
(𝑑0 − 𝜀) Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩 Δ𝑡 󵄩󵄩
𝑛 = 1󵄩
In this section, in order to confirm our theoretical results
for the novel characteristic expanded mixed finite element
method, we consider the following test problem:
𝐽
󵄩󵄩
󵄩󵄩2 󵄩󵄩
󵄩󵄩2
1/2
1/2
+ ∑ (󵄩󵄩󵄩(𝑎𝑛 ) (𝜃𝑛 − 𝜃𝑛−1 )󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝑅𝑛 ) (𝜍𝑛 − 𝜍𝑛−1 )󵄩󵄩󵄩 )
󵄩
󵄩
󵄩
󵄩
𝑛=1
(x, 𝑡) ∈ 𝜕Ω × 𝐽,
(45)
𝑢 (x, 0) = 𝑥1 𝑥2 (𝑥1 − 1) (𝑥2 − 1) (2𝑥2 − 1) ,
x ∈ Ω, (46)
and initial condition
𝑛=0
(39)
Using Gronwall lemma, we have
𝐽 󵄩
󵄩󵄩 𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩2 󵄩 𝐽 󵄩2 󵄩󵄩 𝐽 1/2 𝐽 󵄩󵄩2
󵄩󵄩 + 󵄩󵄩𝜃 󵄩󵄩 + 󵄩󵄩(𝑅 ) 𝜍 󵄩󵄩
Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩
Δ𝑡
󵄩
󵄩
󵄩
󵄩󵄩
𝑛 =1󵄩
where Ω = [0, 1] × [0, 1], x = (𝑥1 , 𝑥2 ), 𝐽 = (0, 1], 𝑎(x, 𝑡) =
1 + 2𝑥12 + 𝑥22 , c(x, 𝑡) = (1 + 𝑥12 + 𝑥22 + 𝑡2 , 1 + 𝑥12 + 𝑥22 + 𝑡2 ),
𝑅(x, 𝑡) = 1 + 𝑥12 + 2𝑥22 , and 𝑑(x) = 1 and 𝑓(x, 𝑡) is chosen so
that the exact solution for the scalar unknown function is
𝑢 (x, 𝑡) = 𝑒−𝑡 𝑥1 𝑥2 (𝑥1 − 1) (𝑥2 − 1) (2𝑥2 − 1) .
󵄩2
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ((Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠)
𝑡0 󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 𝜕𝜏 󵄩󵄩
(40)
𝑛
Taking zℎ = 𝜉 in (21a), (21b), and (21c), we get
(47)
The corresponding exact gradient function is
𝑥2 (2𝑥1 − 1) (𝑥2 − 1) (2𝑥2 − 1)
],
𝑥1 (𝑥1 − 1) (6𝑥22 − 6𝑥2 + 1)
𝜆 = ∇𝑢 = 𝑒−𝑡 [
(48)
and its exact flux function is
𝜎 = − 𝑎∇𝑢
(41)
Substitute (40) into (41) to get
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
󵄩 𝜕𝜂 󵄩󵄩2
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩𝜉 󵄩󵄩 ≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ((Δ𝑡)2 ∫ 󵄩󵄩󵄩 2 󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 𝑑𝑠)
󵄩 󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩
𝑡0 󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩
(42)
Taking wℎ = ∇𝜍𝑛 in (21b), we have
󵄩󵄩 𝑛 󵄩󵄩2 󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩∇𝜍 󵄩󵄩 ≤ 󵄩󵄩𝜃 󵄩󵄩
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ((Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠)
𝑡0 󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) .
(x, 𝑡) ∈ Ω × 𝐽,
𝑢 (x, 𝑡) = 0,
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) + 𝐶 (𝑎1 ) Δ𝑡 ∑ 󵄩󵄩󵄩𝜃𝑛 󵄩󵄩󵄩 .
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) .
(44)
with boundary condition
𝐽−1
󵄩󵄩 𝑛 󵄩󵄩2
󵄩 𝑛 󵄩2
󵄩󵄩𝜉 󵄩󵄩 ≤ 𝐶󵄩󵄩󵄩𝜃 󵄩󵄩󵄩 .
𝜕𝑢
+ c (x, 𝑡) ⋅ ∇𝑢 − ∇ ⋅ (𝑎 (x, 𝑡) ∇𝑢) + 𝑅 (x, 𝑡) 𝑢
𝜕𝑡
= 𝑓 (x, 𝑡) ,
󵄩󵄩
1/2 󵄩
1/2 󵄩
󵄩2 󵄩󵄩
󵄩2
+ 󵄩󵄩󵄩󵄩(𝑎𝐽 ) 𝜃𝐽 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩(𝑅𝐽 ) 𝜍𝐽 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
󵄩 2 󵄩󵄩2
𝑡𝐽 󵄩
𝑡𝐽 󵄩
󵄩󵄩 𝜕𝜂 󵄩󵄩󵄩2
󵄩𝜕 𝑢󵄩
≤ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 ) ((Δ𝑡)2 ∫ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑑𝑠)
𝑡0 󵄩
𝑡0 󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 𝜕𝜏 󵄩󵄩
󵄩 󵄩2
+ 𝐶 (𝑎0 , 𝑑1 , 𝑅1 , 𝑅1 ) 󵄩󵄩󵄩𝜂󵄩󵄩󵄩𝐿∞ (𝐿2 ) .
𝑑 (x)
(43)
𝑥2 (2𝑥1 − 1) (𝑥2 − 1) (2𝑥2 − 1)
].
𝑥1 (𝑥1 − 1) (6𝑥22 − 6𝑥2 + 1)
(49)
= − 𝑒−𝑡 (1 + 2𝑥12 + 𝑥22 ) [
We divide the domain Ω = [0, 1] × [0, 1] into the triangulations of spatial mesh parameter ℎ uniformly and use the
backward Euler procedure with uniform time discretization
parameter Δ𝑡. We consider the piecewise linear space 𝑉ℎ with
index 𝑘 = 1 and the corresponding piecewise constant space
Wℎ with index 𝑘 = 0.
In Table 1, we get the optimal a priori error estimate in
𝐿2 -norm for the scalar unknown 𝑢 with ℎ = 2√2Δ𝑡 =
√2/16, √2/32, √2/64. At the same time, we also obtain the
optimal a priori error estimate in 𝐻1 -norm for the scalar
unknown 𝑢.
In Table 2, we obtain some convergence results in
(𝐿2 (Ω))2 -norm for the gradient 𝜆 with ℎ = 2√2Δ𝑡 = √2/16,
√2/32, √2/64 in Table 2. The similar results are obtained for
the flux 𝜎 in Table 2. From the data obtained in Table 2, we can
8
Journal of Applied Mathematics
Table 1: The errors and convergence rate for 𝑢.
(ℎ, Δ𝑡)
(√2/8, 1/16)
(√2/16, 1/32)
‖𝑢 − 𝑢ℎ ‖𝐿∞ (𝐿2 (Ω))
1.3622𝑒 − 003
Rate
‖𝑢 − 𝑢ℎ ‖𝐿∞ (𝐻1 (Ω))
2.6286𝑒 − 002
Rate
4.1958𝑒 − 004
1.6989
1.3583𝑒 − 002
0.9525
(√2/32, 1/64)
1.3172𝑒 − 004
1.6715
6.8843𝑒 − 003
0.9804
Table 2: The errors and convergence rate for 𝜆 and 𝜎.
(ℎ, Δ𝑡)
(√2/8, 1/16)
(√2/16, 1/32)
‖𝜆 − 𝜆 ℎ ‖𝐿∞ ((𝐿2 (Ω))2 )
2.6251𝑒 − 002
Rate
‖𝜎 − 𝜎ℎ ‖𝐿∞ ((𝐿2 (Ω))2 )
5.4313𝑒 − 002
Rate
1.3576𝑒 − 002
0.9513
2.9081𝑒 − 002
0.9013
(√2/32, 1/64)
6.8830𝑒 − 003
0.9800
1.5018𝑒 − 002
0.9534
The exact solution 𝑢 at 𝑡 = 1
The numerical solution 𝑢ℎ at 𝑡 = 1
0.01
0.01
0.005
0.005
0
0
−0.005
−0.01
1 0.9 0.8 0.7 0.6
0.5 0.4 0.3 0.2 0.1
0
1
0.5
0
Figure 1: The surface for the exact solution 𝑢.
find that the numerical results confirm the theoretical results
of Theorem 8.
Figure 1 shows the surface for the exact solution 𝑢 at 𝑡 =
1, and Figure 2 describes the corresponding surface for the
numerical solution 𝑢ℎ with ℎ = 2√2Δ𝑡 = √2/16 at 𝑡 = 1. In
Figures 1 and 2, we can find easily that the exact solution 𝑢 is
approximated very well by the numerical solution 𝑢ℎ .
Figures 3 and 4 show the surface of the exact gradient
function 𝜆 and the surface of the numerical gradient function
𝜆 ℎ with ℎ = 2√2Δ𝑡 = √2/16 at 𝑡, respectively. A similar
comparison between the surface of the exact flux function 𝜎
and the surface of the numerical flux function 𝜎ℎ in Figures 5
and 6 also is made.
From the convergence results for ‖𝑢 − 𝑢ℎ ‖𝐿∞ (𝐿2 (Ω)) , ‖𝑢 −
𝑢ℎ ‖𝐿∞ (𝐻1 (Ω)) , ‖𝜆 − 𝜆 ℎ ‖𝐿∞ ((𝐿2 (Ω))2 ) , and ‖𝜎 − 𝜎ℎ ‖𝐿∞ ((𝐿2 (Ω))2 ) in
Tables 1 and 2 and Figures 1 and 2, we find that the numerical
results confirm our theoretical analysis.
5. Concluding Remarks
In this paper, we propose and study a novel characteristic
expanded mixed finite element method, which combines the
novel expanded mixed method [24] applied to approximating
the diffusion term and the characteristic method that handled the hyperbolic part, for reaction-convection-diffusion
equations. Compared to Chen’s expanded mixed method, the
−0.005
1
−0.01
1
0.5
0.9 0.8 0.7 0.6 0.5
0.4 0.3 0.2 0.1 0
0
Figure 2: The surface for the numerical solution 𝑢ℎ .
gradient for our method belongs to the square integrable
space instead of the classical H(div; Ω) space. We derive a priori error estimates based on backward Euler method. Moreover, we prove the optimal a priori error estimates in 𝐿2 - and
𝐻1 -norms for the scalar unknown 𝑢 and a priori error estimates in (𝐿2 )2 -norm for its gradient 𝜆 and its flux 𝜎. Finally,
we choose a test problem to confirm our theoretical results. In
the near future, the proposed characteristic expanded mixed
scheme will be applied to other linear/nonlinear evolution
equations, such as nonlinear reaction-diffusion equations,
linear/nonlinear convection-dominated Sobolev equations,
and time-dependent convection-diffusion optimal control
problems.
Acknowledgments
The authors thank the anonymous referees and editors
for their helpful comments and suggestions, which greatly
improve the paper. This work is supported by the National
Natural Science Fund (11061021), Natural Science Fund
of Inner Mongolia Autonomous Region (2012MS0108,
2012MS0106, and 2011BS0102), Scientific Research Projection
of Higher Schools of Inner Mongolia (NJZZ12011, NJ10006,
NJ10016, and NJZY13199), Key Project of Chinese Ministry
of Education (12024), and the Program of Higher-Level
Journal of Applied Mathematics
9
The exact solution 𝜆 2 at 𝑡 = 1
The exact solution 𝜆 1 at 𝑡 = 1
0.04
0.1
0.02
0.05
0
0
1
−0.02
0.8
−0.04
1
1
−0.05
0.8
−0.1
1
0.6
0.6
0.4
0.5
0.4
0.5
0.2
0.2
0 0
0
(a)
0
(b)
Figure 3: The surface for the exact solution 𝜆 = (𝜆 1 , 𝜆 2 ).
The numerical solution 𝜆 1ℎ at 𝑡 = 1
The numerical solution 𝜆 2ℎ at 𝑡 = 1
0.04
0.1
0.02
0.05
0
0
1
−0.02
0.8
−0.04
1
0.6
1
−0.05
0.8
−0.1
1
0.6
0.4
0.4
0.5
0.2
0.5
0.2
0 0
(a)
0
(b)
Figure 4: The surface for the numerical solution 𝜆 ℎ = (𝜆 1ℎ , 𝜆 2ℎ ).
0
10
Journal of Applied Mathematics
The exact solution 𝜎1 at 𝑡 = 1
The exact solution 𝜎2 at 𝑡 = 1
0.2
0.3
0.1
0.2
0
0.1
1
−0.1
0.8
−0.2
1
1
0
0.8
−0.1
1
0.6
0.6
0.4
0.5
0.4
0.5
0.2
0
0.2
0
0
(a)
0
(b)
Figure 5: The surface for the exact solution 𝜎 = (𝜎1 , 𝜎2 ).
The numerical solution 𝜎1ℎ at 𝑡 = 1
The numerical solution 𝜎2ℎ at 𝑡 = 1
0.15
0.3
0.1
0.2
0.05
0.1
0
1
−0.05
0.8
−0.1
1
0.6
1
0
0.8
−0.1
1
0.6
0.4
0.5
0.2
0.4
0.5
0.2
0 0
(a)
0 0
(b)
Figure 6: The surface for the numerical solution 𝜎ℎ = (𝜎1ℎ , 𝜎2ℎ ).
Journal of Applied Mathematics
Talents of Inner Mongolia University (125119, Z200901004,
and 30105-125132).
References
[1] C. Johnson and V. Thomée, “Error estimates for some mixed
finite element methods for parabolic type problems,” RAIRO
Analyse Numérique, vol. 15, no. 1, pp. 41–78, 1981.
[2] 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.
[3] H. Z. Chen and H. Wang, “An optimal-order error estimate
on an 𝐻1 -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.
[4] Z. D. Luo, Z. H. Xie, Y. Q. Shang, and J. Chen, “A reduced finite
volume element formulation and numerical simulations based
on POD for parabolic problems,” Journal of Computational and
Applied Mathematics, vol. 235, no. 8, pp. 2098–2111, 2011.
[5] P. Sun, Z. Luo, and Y. Zhou, “Some reduced finite difference
schemes based on a proper orthogonal decomposition technique for parabolic equations,” Applied Numerical Mathematics,
vol. 60, no. 1-2, pp. 154–164, 2010.
[6] Z. X. Chen, S. H. Chou, and D. Y. Kwak, “Characteristicmixed covolume methods for advection-dominated diffusion
problems,” Numerical Linear Algebra with Applications, vol. 13,
no. 9, pp. 677–697, 2006.
[7] S. He, H. Li, Y. Liu, Z. C. Fang et al., “A splitting mixed spacetime discontinuous Galerkin method for parabolic problems,”
Procedia Engineering, vol. 31, pp. 1050–1059, 2012.
[8] Y. Liu, H. Li, and S. He, “Mixed time discontinuous spacetime finite element method for convection diffusion equations,”
Applied Mathematics and Mechanics, vol. 29, no. 12, pp. 1579–
1586, 2008.
[9] D. P. Yang, “Least-squares mixed finite element methods for
nonlinear parabolic problems,” Journal of Computational Mathematics, vol. 20, no. 2, pp. 153–164, 2002.
[10] H. X. Rui, S. D. Kim, and S. Kim, “Split least-squares finite
element methods for linear and nonlinear parabolic problems,”
Journal of Computational and Applied Mathematics, vol. 223, no.
2, pp. 938–952, 2009.
[11] D. Shi and B. Y. Zhang, “Nonconforming finite element method
for nonlinear parabolic equations,” Journal of Systems Science &
Complexity, vol. 23, no. 2, pp. 395–402, 2010.
[12] J. C. Li, “Optimal uniform convergence analysis for a twodimensional parabolic problem with two small parameters,”
International Journal of Numerical Analysis and Modeling, vol.
2, no. 1, pp. 107–126, 2005.
[13] Y. Chen, Y. Huang, and D. 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.
[14] Y. 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.
[15] Y. Chen, P. Luan, and Z. 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.
11
[16] W. Liu, H. X. Rui, and H. Guo, “A two-grid method with
expanded mixed element for nonlinear reaction-diffusion equations,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 3, pp.
495–502, 2011.
[17] Z. X. Chen, Expanded Mixed Finite Element Methods For Linear
Second-Order Elliptic Problems I, vol. 1219 of IMA preprint series,
Institute for Mathematics and its Applications, University of
Minnesota, Minneapolis, Minn, USA, 1994.
[18] Z. 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.
[19] 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.
[20] Z. 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.
[21] H. Rui and T. Lu, “An expanded mixed covolume method for
elliptic problems,” Numerical Methods for Partial Differential
Equations, vol. 21, no. 1, pp. 8–23, 2005.
[22] 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.
[23] Y. Liu, H. Li, J. Wang, and W. Gao, “A new positive definite
expanded mixed finite element method for parabolic integrodifferential equations,” Journal of Applied Mathematics, vol. 2012,
Article ID 391372, 24 pages, 2012.
[24] Y. Liu, H. Li et al., “A new expanded mixed finite element
method for elliptic equations,” http://nmg.zsksedu.com/3267/
2012011690642.htm.
[25] S. C. Chen and H. R. Chen, “New mixed element schemes for a
second-order elliptic problem,” Mathematica Numerica Sinica,
vol. 32, no. 2, pp. 213–218, 2010.
[26] F. Shi, J. P. Yu, and K. T. Li, “A new stabilized mixed finiteelement method for Poisson equation based on two local Gauss
integrations for linear element pair,” International Journal of
Computer Mathematics, vol. 88, no. 11, pp. 2293–2305, 2011.
[27] D. Y. Shi and Y. D. Zhang, “High accuracy analysis of a
new nonconforming mixed finite element scheme for Sobolev
equations,” Applied Mathematics and Computation, vol. 218, no.
7, pp. 3176–3186, 2011.
[28] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,
North-Holland, Amsterdam, The Netherlands, 1978.
[29] L. Guo and H. Z. Chen, “An expanded characteristic-mixed
finite element method for a convection-dominated transport
problem,” Journal of Computational Mathematics, vol. 23, no. 5,
pp. 479–490, 2005.
[30] L. Li, P. Sun, and Z. D. Luo, “A new mixed finite element
formulation and error estimates for parabolic problems,” Acta
Mathematica Scientia, vol. 32A, no. 6, pp. 1158–1165, 2012.
[31] J. Douglas, Jr. and T. F. Russell, “Numerical methods for
convection-dominated diffusion problems based on combining
the method of characteristics with finite element or finite
difference procedures,” SIAM Journal on Numerical Analysis,
vol. 19, no. 5, pp. 871–885, 1982.
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