Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 951692, 10 pages
http://dx.doi.org/10.1155/2013/951692
Research Article
A New Characteristic Nonconforming Mixed Finite Element
Scheme for Convection-Dominated Diffusion Problem
Dongyang Shi,1 Qili Tang,1,2 and Yadong Zhang3
1
Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China
3
School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China
2
Correspondence should be addressed to Qili Tang; tql132@163.com
Received 14 December 2012; Accepted 23 March 2013
Academic Editor: Junjie Wei
Copyright © 2013 Dongyang Shi 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 characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion
problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable 𝑢 and
the auxiliary variable 𝜎 with respect to the space are obtained by employing some typical characters of the interpolation operator
instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At
last, we give some numerical results to confirm the theoretical analysis.
1. Introduction
Consider the following convection-dominated diffusion problem:
𝑢 (𝑥, 𝑦, 𝑡) = 0,
(𝑥, 𝑦, 𝑡) ∈ Ω × (0, 𝑇) ,
(𝑥, 𝑦, 𝑡) ∈ 𝜕Ω × (0, 𝑇) ,
𝑢 (𝑥, 𝑦, 0) = 𝑢0 (𝑥, 𝑦) ,
(A2) a(𝑥, 𝑦) = (𝑎1 (𝑥, 𝑦), 𝑎2 (𝑥, 𝑦)) represents Darcy velocity of mixed fluid, and 𝑓 a source term;
(A3) 𝑏(𝑥, 𝑦) is sufficiently smooth and there exist constants
𝑏1 and 𝑏2 , such that
𝑢𝑡 + a (𝑥, 𝑦) ⋅ ∇𝑢 − ∇ ⋅ (𝑏 (𝑥, 𝑦) ∇𝑢)
= 𝑓 (𝑥, 𝑦, 𝑡) ,
(A1) 𝑢 denotes, for example, the concentration or saturation of soluble substances;
(1)
(𝑥, 𝑦) ∈ Ω,
where Ω is a bounded polygonal domain in R2 with Lipschitz
continuous boundary 𝜕Ω, 𝐽 = (0, 𝑇], 0 < 𝑇 < +∞. ∇ and
∇⋅ denote the gradient and the divergence operators, respectively.
Model (1) has been widely used to describe the conduction of heat in fluid, the diffusion of soluble minerals or pollutants in ground water, the incompressible miscible displacement in porous media, and so on. The parameters appearing
in (1) satisfy the following assumptions [1, 2]:
0 < 𝑏1 ≤ 𝑏 (𝑥, 𝑦) ≤ 𝑏2 < +∞,
∀ (𝑥, 𝑦) ∈ Ω.
(2)
It is well known that convection dominated-diffusion
problem (1) often presents serious numerical difficulties.
The standard numerical methods, such as finite difference
method (FDM), FEM and MFEM, usually produce numerical
diffusion along sharp fronts. In order to overcome this fatal
defect, Douglas et al. [3] combined the method of characteristics with FE procedures so as to reduce the truncation
error, and it allows us to use large time steps without lose of
accuracy. Moreover, there have appeared many effective discretization schemes concentrating on the hyperbolic nature
of the equation, for example, characteristic FD streamline diffusion method [4, 5], Eulerian-Lagrangian method [6, 7],
characteristic-finite volume element method [2, 8, 9],
2
characteristics-mixed covolume method [10, 11], the modified method of characteristic-Galerkin FE procedure [12],
characteristic nonconforming FEM [13–15], characteristic
MFEM [16–19] and expanded characteristic MFEM [1, 20],
and so forth.
As for the characteristic MFEM or expanded characteristic MFEM, the convergence rates of 𝑢 and 𝜎 in existing
literature were suboptimal [11, 18, 21, 22] and the convergence
analysis was valid only to the case of the lowest order MFE
approximation [10, 17]. So far, to our best knowledge
there are few studies on the optimal order error estimates
except for [23], in which a family of characteristic MFEM
with arbitrary degree of Raviart-Thomas-Nédélec space in
[24, 25] for transient convection diffusion equations was
studied.
Recently, based on the low regularity requirement of the
flux variable in practical problems, a new mixed variational
form for second elliptic problem was proposed in [26]. It has
two typical advantages: the flux space belongs to the square
integrable space instead of the traditional 𝐻( div ; Ω), which
makes the choices of MFE spaces sufficiently simple and easy;
the LBB condition is automatically satisfied when the gradient
of approximation space for the original variable is included
in approximation space for the flux variable. Motivated by
this idea, this paper will construct a characteristic nonconforming MFE scheme for (1) with a new mixed variational
formulation. Similar to the expanded characteristic MFEM,
the coefficient 𝑏 of (1) in this proposed scheme does not
need to be inverted; therefore, it is also suitable for the case
when 𝑏 is small. By employing some distinct characters of
the interpolation operators on the element instead of the
mixed or expanded mixed elliptic projection used in [1, 17, 20]
which is an indispensable tool in the traditional characteristic
MFEM analysis, the 𝑂(ℎ2 ) order error estimate in 𝐿2 -norm
for original variable 𝑢, which is one order higher than [1, 20]
and half order higher than [18], is derived, and the optimal
error estimates with order 𝑂(ℎ) for auxiliary variable 𝜎 in 𝐿2 norm and for 𝑢 in broken 𝐻1 -norm are obtained, respectively.
It seems that the result for 𝑢 in broken 𝐻1 -norm has never
been seen in the existing literature by making full use of the
high-accuracy estimates of the lowest order Raviart-Thomas
element proved by the technique of integral identities in [27]
and the special properties of nonconforming 𝐸𝑄1rot element
(see Lemma 1 below).
The paper is organized as follows. Section 2 is devoted
to the introduction of the nonconforming FE approximation
spaces and their corresponding interpolation operators. In
Section 3, we will give the construction of the new characteristic nonconforming MFE scheme and two important lemmas, and the existence and uniqueness of the discrete scheme
solution will be proved. In Section 4, the convergence analysis
and optimal error estimates for both the original variable
𝑢 and the flux variable 𝜎 are obtained. In Section 5, some
numerical results are provided to illustrate the effectiveness
of our proposed method.
Throughout this paper, 𝐶 denotes a generic positive constant independent of the mesh parameters ℎ and Δ𝑡 with
respect to domain Ω and time 𝑡.
Journal of Applied Mathematics
2. Construction of Nonconforming MFEs
As in [28], we frequently employ the space 𝐿2 (Ω) of square
integrable functions with scalar product and norm
(𝑢, V) = (𝑢, V)𝐿2 (Ω) = (∫ 𝑢V𝑑𝑥𝑑𝑦)
Ω
2
‖V‖ = ‖V‖𝐿2 (Ω) = (∫ V 𝑑𝑥𝑑𝑦)
1/2
1/2
Ω
,
(3)
.
We also employ the Sobolev space 𝐻𝑚 (Ω), 𝑚 ≥ 1, of functions V such that 𝐷𝛽 V ∈ 𝐿2 (Ω) for all |𝛽| ≤ 𝑚, equipped with
the norm and seminorm
1/2
‖V‖𝑚,Ω = ‖V‖𝐻𝑚 (Ω)
󵄩
󵄩2
= ( ∑ 󵄩󵄩󵄩󵄩𝐷𝛽 V󵄩󵄩󵄩󵄩 )
|𝛽|≤𝑚
󵄩
󵄩2
|V|𝑚,Ω = |V|𝐻𝑚 (Ω) = ( ∑ 󵄩󵄩󵄩󵄩𝐷𝛽 V󵄩󵄩󵄩󵄩 )
|𝛽|=𝑚
,
(4)
1/2
.
The space 𝐻01 (Ω) denotes the closure of the set of infinitely
differentiable functions with compact supports in Ω. For any
Sobolev space 𝑌, 𝐿𝑝 (0, 𝑇; 𝑌) is the space of measurable 𝑌𝑇
𝑝
valued functions Φ of 𝑡 ∈ (0, 𝑇), such that ∫0 ‖Φ(⋅, 𝑡)‖𝑌 𝑑𝑡 <
∞ if 1 ≤ 𝑝 < ∞, or such that ess sup0<𝑡<𝑇 ‖Φ(⋅, 𝑡)‖𝑌 < ∞ if
𝑝 = ∞.
We now introduce the nonconforming MFE space described in [29] for and summarize it as follows.
Let Ω ⊂ R2 be a polygon domain with edges parallel to
the coordinate axes on 𝑥𝑦 plane, and let 𝑇ℎ be a rectangular
subdivision of Ω satisfying the regular condition [30]. For a
given element 𝑒 ∈ 𝑇ℎ , denote the barycenter of element 𝑒 by
(𝑥𝑒 , 𝑦𝑒 ), denote the length of edges parallel to 𝑥-axis and 𝑦axis by 2ℎ𝑥𝑒 and 2ℎ𝑦𝑒 , respectively, ℎ𝑒 = max𝑒∈𝑇ℎ {ℎ𝑥𝑒 , ℎ𝑦𝑒 }, ℎ =
max𝑒∈𝑇ℎ ℎ𝑒 .
Let 𝑒̂ = [−1, 1] × [−1, 1] be the reference element on 𝑥̂𝑦̂
plane and four vertices 𝑑̂1 = (−1, −1), 𝑑̂2 = (1, −1), 𝑑̂3 =
(1, 1), and 𝑑̂ = (−1, 1), the four edges ̂𝑙 = 𝑑̂ 𝑑̂ , ̂𝑙 = 𝑑̂ 𝑑̂ ,
4
1
1 2
2
2 3
̂𝑙 = 𝑑̂ 𝑑̂ , and ̂𝑙 = 𝑑̂ 𝑑̂ . Then there exists an affine mapping
3
3 4
4
4 1
𝐹𝑒 : 𝑒̂ → 𝑒 as
̂
𝑥 = 𝑥𝑒 + ℎ𝑥𝑒 𝑥,
(5)
̂
𝑦 = 𝑦𝑒 + ℎ𝑦𝑒 𝑦.
̂𝑖 , ∑
̂ 𝑖 ), (𝑖 = 1, 2, 3) by
Define the FE spaces (̂
𝑒, 𝑃
̂ 1 = {̂V , ̂V , ̂V , ̂V , ̂V } ,
∑
1 2 3 4 5
̂ 𝑦,
̂ 𝜙 (𝑥)
̂ , 𝜙 (𝑦)}
̂ ,
𝑃̂1 = span {1, 𝑥,
̂ 2 = {𝑝̂ , 𝑝̂ } ,
∑
1 2
̂ ,
𝑃̂2 = span {1, 𝑥}
̂ 3 = {̂
𝑞1 , 𝑞̂2 } ,
∑
̂ ,
𝑃̂3 = span {1, 𝑦}
(6)
where ̂V𝑖 = (1/|̂𝑙𝑖 |) ∫̂𝑙 ̂V𝑑̂𝑠, (𝑖 = 1, 2, 3, 4), ̂V5 = (1/|̂
𝑒|)
𝑖
2
̂
̂ 𝑠, 𝑞̂𝑖 =
̂ 𝜙(𝑡) = (1/2)(3𝑡 − 1), 𝑝̂𝑖 = (1/|𝑙2𝑖 |) ∫̂𝑙 𝑝𝑑̂
∫𝑒̂ ̂V𝑑𝑥̂ 𝑑𝑦,
2𝑖
̂
(1/|𝑙 |) ∫ 𝑞̂𝑑̂𝑠, (𝑖 = 1, 2).
2𝑖−1
̂𝑙2𝑖−1
Journal of Applied Mathematics
3
The interpolation operators on 𝑒̂ are defined as follows:
Then (1) can be put in the following system:
̂ 1 : ̂V ∈ 𝐻1 (̂
̂ 1 ̂V ∈ 𝑃̂1 ,
Π
𝑒) 󳨀→ Π
̂ 1 ̂V − ̂V) 𝑑̂𝑠 = 0,
∫ (Π
𝜓 (𝑥, 𝑦)
(𝑖 = 1, 2, 3, 4) ,
̂𝑙𝑖
∀ (𝑥, 𝑦, 𝑡) ∈ Ω × (0, 𝑇] ,
𝑢 (𝑥, 𝑦, 𝑡) = 0,
̂ 1 ̂V − ̂V) 𝑑𝑥̂ 𝑑𝑦̂ = 0,
∫ (Π
𝑒̂
̂ 2 𝑝̂ − 𝑝)
̂ 𝑑̂𝑠 = 0,
∫ (Π
̂𝑙2𝑖
(7)
(𝑖 = 1, 2) ,
̂ 3 : 𝑞̂ ∈ 𝐻1 (̂
̂ 3 𝑞̂ ∈ 𝑃̂3 ,
Π
𝑒) 󳨀→ Π
̂𝑙2𝑖−1
̂ 3 𝑞̂ − 𝑞̂) 𝑑̂𝑠 = 0,
(Π
(𝑖 = 1, 2) .
𝐹
(8)
Vℎ = {wℎ = (𝑤ℎ1 , 𝑤ℎ2 ) :
̂2 ∘ 𝐹𝑒−1 ) ,
𝑤1 ∘ 𝐹𝑒−1 , 𝑤
wℎ |𝑒 = (̂
Let Δ𝑡 > 0, 𝑁 = 𝑇/Δ𝑡 ∈ Z, 𝑡𝑛 = 𝑛Δ𝑡, and 𝜙𝑛 = 𝜙(𝑥, 𝑦, 𝑡𝑛 ).
When solving 𝑢ℎ𝑛+1 , we would like to make the scheme as
implicit as possible by using of the characteristic vector 𝜏.
Denote 𝑋 = (𝑥, 𝑦) ∈ Ω and
∀V ∈ 𝐻1 (Ω) ,
𝑢 (𝑋, 𝑡𝑛 ) − 𝑢 (𝑋, 𝑡𝑛−1 )
Δ𝑡
=
This leads to the following characteristic nonconforming
MFE scheme. Find (𝑢ℎ , 𝜎ℎ ) : {𝑡0 , 𝑡1 , . . ., 𝑡𝑁} → 𝑀ℎ ×Vℎ , such
that
𝑢ℎ𝑛 − 𝑢𝑛−1
ℎ
, Vℎ ) − (𝜎ℎ𝑛 , ∇Vℎ )ℎ = (𝑓𝑛 , Vℎ ) ,
Δ𝑡
∀Vℎ ∈ 𝑀ℎ ,
(15a)
(𝜎ℎ𝑛 , wℎ ) + (𝑏∇𝑢ℎ𝑛 , wℎ )ℎ = 0,
𝑢ℎ0 = 𝜋ℎ1 𝑢0 (𝑥, 𝑦) ,
2
∀w = (𝑤1 , 𝑤2 ) ∈ (𝐻1 (Ω)) .
(14)
𝑢𝑛 − 𝑢𝑛−1
.
Δ𝑡
(9)
̂2𝑤
̂3𝑤
̂1 ) ∘ 𝐹𝑒−1 , (Π
̂2 ) ∘ 𝐹𝑒−1 ) ,
𝜋𝑒2 w = ((Π
∀wℎ ∈ Vℎ ,
𝜎ℎ0 = 𝜋ℎ2 (𝑏∇𝑢0 (𝑥, 𝑦)) ,
(15b)
∀ (𝑥, 𝑦) ∈ Ω,
(15c)
where 𝑢𝑛ℎ = 𝑢ℎ (𝑋, 𝑡𝑛 ), (𝑢, V)ℎ = ∑𝑒∈𝑇ℎ ∫𝑒 𝑢V𝑑𝑥𝑑𝑦. Generally
speaking, 𝑢𝑛−1
ℎ (𝑛 = 2, . . . , 𝑁) are not node values and should
be derived by interpolation formulas on 𝑢ℎ𝑛−1 .
3. New Characteristic Nonconforming MFE
Scheme and Two Lemmas
1/2
Let 𝜓(𝑥, 𝑦) = (1 + |a(𝑥, 𝑦)|2 ) and 𝜏 = 𝜏(𝑥, 𝑦) be the characteristic direction associated with 𝑢𝑡 + a(𝑥, 𝑦) ⋅ ∇𝑢, such that
a (𝑥, 𝑦)
𝜕
𝜕
1
⋅ ∇.
=
+
𝜕𝜏 𝜓 (𝑥, 𝑦) 𝜕𝑡 𝜓 (𝑥, 𝑦)
𝑢 (𝑋, 𝑡𝑛 ) − 𝑢 (𝑋, 𝑡𝑛−1 )
𝜕𝑢 󵄨󵄨󵄨󵄨
󵄨󵄨 ≈ 𝜓 (𝑥, 𝑦)
𝜕𝜏 󵄨󵄨𝑡𝑛
2
√ (𝑋 − 𝑋) + (Δ𝑡)2
=
(
𝜋ℎ2 |𝑒 = 𝜋𝑒2 ,
(13)
similar to [1, 3], and then we have the following approximation:
𝜓 (𝑥, 𝑦)
respectively, where [𝜑] represents the jump value of 𝜑 across
the boundary 𝐹, and [𝜑] = 𝜑 if 𝐹 ⊂ 𝜕Ω.
Similarly, the interpolation operators 𝜋ℎ1 and 𝜋ℎ2 are
defined as
󵄨
𝜋ℎ1 󵄨󵄨󵄨󵄨𝑒 = 𝜋𝑒1 ,
𝜋ℎ1 : 𝐻1 (Ω) 󳨀→ 𝑀ℎ ,
2
2
∀w ∈ (𝐿2 (Ω)) .
(12)
̂2 ) ∈ 𝑃̂2 × 𝑃̂3 } ,
w
̂ = (̂
𝑤1 , 𝑤
𝜋ℎ2 : (𝐻1 (Ω)) 󳨀→ Vℎ ,
∀V ∈ 𝐻01 (Ω) ,
𝑋 = 𝑋 − a (𝑥, 𝑦) Δ𝑡,
∫ [Vℎ ] 𝑑𝑠 = 0, 𝐹 ⊂ 𝜕𝑒} ,
̂ 1 ̂V) ∘ 𝐹−1 ,
𝜋𝑒1 V = (Π
𝑒
𝜕𝑢
, V) − (𝜎, ∇V) = (𝑓 (𝑥, 𝑦, 𝑡) , V)
𝜕𝜏
(𝜎, w) + (𝑏 (𝑥, 𝑦) ∇𝑢, w) = 0,
𝑀ℎ = {Vℎ : Vℎ |𝑒 = ̂V ∘ 𝐹𝑒−1 , ̂V ∈ 𝑃̂1 ,
(𝑥, 𝑦) ∈ Ω.
By introducing 𝜎 = −𝑏(𝑥, 𝑦)∇𝑢 and using Green’s
formula, we obtain the new characteristic mixed form of (11).
2
Find (𝑢, 𝜎) : (0, 𝑇] → 𝐻01 (Ω) × (𝐿2 (Ω)) , such that
(𝜓 (𝑥, 𝑦)
Then the associated nonconforming 𝐸𝑄1rot element space 𝑀ℎ
[29] and lowest order Raviart-Thomas element space Vℎ [25,
27] are defined as
(11)
(𝑥, 𝑦, 𝑡) ∈ 𝜕Ω × (0, 𝑇] ,
𝑢 (𝑥, 𝑦, 0) = 𝑢0 (𝑥, 𝑦) ,
̂ 2 : 𝑝̂ ∈ 𝐻1 (̂
̂ 2 𝑝̂ ∈ 𝑃̂2 ,
Π
𝑒) 󳨀→ Π
∫
𝜕𝑢
− ∇ ⋅ (𝑏 (𝑥, 𝑦) ∇𝑢) = 𝑓 (𝑥, 𝑦, 𝑡) ,
𝜕𝜏
(10)
Remark 1. In [1], the expanded characteristic MFE scheme
was presented by introducing two new auxiliary variables
which avoided the inversion of the coefficient 𝑏 when 𝑏 is
small. The new mixed schemes (15a), (15b), and (15c) not only
keep the advantage of expanded characteristic MFE scheme,
but also donot need to solve three variables.
4
Journal of Applied Mathematics
Now, we prove the existence and uniqueness of the solution of (15a), (15b), and (15c).
Theorem 1. Under assumption (A3), there exists a unique
solution (𝑢ℎ , 𝜎ℎ ) ∈ 𝑀ℎ × Vℎ to the schemes (15a), (15b), and
(15c).
Proof. The linear system generated by (15a), (15b), and (15c)
is square, so the existence of the solution is implied by its uniqueness. From (15a), (15b), and (15c), we have
𝑢𝑛−1
𝑢𝑛
( ℎ , Vℎ ) − (𝜎ℎ𝑛 , ∇Vℎ )ℎ = ( ℎ , Vℎ ) + (𝑓𝑛 , Vℎ ) , ∀Vℎ ∈ 𝑀ℎ ,
Δ𝑡
Δ𝑡
(𝜎ℎ𝑛 , wℎ ) + (𝑏∇𝑢ℎ𝑛 , wℎ )ℎ = 0,
𝑢ℎ𝑛 ,
Taking 𝑡 = 𝑡𝑛 in (12) yields
(𝜓
ℎ
∀Vℎ ∈ 𝑀ℎ ,
(23a)
(𝜎𝑛 , wℎ ) + (𝑏∇𝑢𝑛 , wℎ )ℎ = 0,
(
(16)
𝜌𝑛 − 𝜌𝑛−1
𝜕𝑢𝑛 𝑢𝑛 − 𝑢𝑛−1
−
, Vℎ ) − (
, Vℎ )
𝜕𝜏
Δ𝑡
Δ𝑡
+ (𝜂𝑛 , ∇Vℎ )ℎ + ∑ ∫ 𝜎𝑛 Vℎ ⋅ n 𝑑𝑠,
𝑒∈𝑇ℎ 𝜕𝑒
(17)
(∇ (𝑢 −
𝜋ℎ1 𝑢) , ∇Vℎ )ℎ
= 0,
(∇ (𝑢 −
𝜋ℎ1 𝑢) , wℎ )ℎ
= 0,
(18)
󵄩 󵄩
(p − 𝜋ℎ2 p, wℎ ) ≤ 𝐶ℎ2 |p|2,Ω 󵄩󵄩󵄩wℎ 󵄩󵄩󵄩 ,
(19)
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 ∑ ∫ pV ⋅ n 𝑑𝑠󵄨󵄨󵄨 ≤ 𝐶ℎ2 |p| 󵄩󵄩󵄩V 󵄩󵄩󵄩 ,
󵄨󵄨
󵄨󵄨
ℎ
2,Ω 󵄩 ℎ 󵄩1,ℎ
󵄨󵄨𝑒∈𝑇ℎ 𝜕𝑒
󵄨󵄨
󵄨
󵄨
(20)
(𝜉𝑛 , wℎ ) + (𝑏∇𝑒𝑛 , wℎ )ℎ = − (𝜂𝑛 , wℎ ) − (𝑏∇𝜌𝑛 , wℎ )ℎ ,
∀wℎ ∈ Vℎ .
(24b)
We are now in a position to prove the optimal order error
estimates.
Theorem 2. Let (𝑢, 𝜎) and (𝑢ℎ𝑛 , 𝜎ℎ𝑛 ) be the solutions of (12),
(15a), (15b), and (15c), respectively, (𝜕2 𝑢/𝜕𝜏2 ) ∈ 𝐿2 (0, 𝑇;
𝐿2 (Ω)), 𝑢𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻2 (Ω)), 𝑢 ∈ 𝐿∞ (0, 𝑇; 𝐻2 (Ω)), 𝜎 ∈
𝐿∞ (0, 𝑇; 𝐻2 (Ω)) and assume that Δ𝑡 = 𝑂(ℎ2 ). Then under
assumption (A3), we have
󵄩
󵄩
max 󵄩󵄩(𝑢ℎ − 𝑢) (𝑡𝑛 )󵄩󵄩󵄩1,ℎ ≤ 𝐶 (Δ𝑡 + ℎ) ,
(25)
0≤𝑛≤𝑁󵄩
where ‖ ⋅ ‖1,ℎ = (∑𝑒∈𝑇ℎ | ⋅ |1,𝑒 ) is a norm on 𝑀ℎ , and n
denotes the outward unit normal vector on 𝜕𝑒.
where ‖𝜑‖−1 = sup𝜙∈𝐻1 (Ω) ((𝜑, 𝜙)/‖𝜙‖1,Ω ).
󵄩
󵄩
max 󵄩󵄩(𝑢ℎ − 𝑢) (𝑡𝑛 )󵄩󵄩󵄩 ≤ 𝐶 (Δ𝑡 + ℎ2 ) ,
(26)
󵄩
󵄩
max 󵄩󵄩(𝜎ℎ − 𝜎) (𝑡𝑛 )󵄩󵄩󵄩 ≤ 𝐶 (Δ𝑡 + ℎ) .
(27)
0≤𝑛≤𝑁 󵄩
1/2
Lemma 2 (see [1, 3]). Let 𝜑 ∈ 𝐿2 (Ω), and 𝜑 = 𝜑(𝑋 − 𝑔(𝑋)Δ𝑡),
where function 𝑔 and its gradient ∇𝑔 are bounded, then
󵄩
󵄩 󵄩
󵄩󵄩
(21)
󵄩󵄩𝜑 − 𝜑󵄩󵄩󵄩−1 ≤ 𝐶 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 Δ𝑡,
0≤𝑛≤𝑁 󵄩
Proof. Taking Vℎ = 𝑒𝑛 in (24a) and wℎ = ∇𝑒𝑛 in (24b), and
adding them, we have
(
𝑒𝑛 − 𝑒𝑛−1 𝑛
, 𝑒 ) + (𝑏∇𝑒𝑛 , ∇𝑒𝑛 )ℎ
Δ𝑡
= (𝜓
𝜌𝑛 − 𝜌𝑛−1 𝑛
𝜕𝑢𝑛 𝑢𝑛 − 𝑢𝑛−1 𝑛
−
,𝑒 ) − (
,𝑒 )
𝜕𝜏
Δ𝑡
Δ𝑡
4. Convergence Analysis and Optimal Order
Error Estimates
−(
In this section, we aim to analyze the convergence analysis
and error estimates of characteristic nonconforming MFEM.
In order to do this, let
+ ∑ ∫ 𝜎𝑛 𝑒𝑛 ⋅ n 𝑑𝑠 − (𝑏∇𝜌𝑛 , ∇𝑒𝑛 )ℎ
𝑢ℎ − 𝑢 = 𝑢ℎ −
𝜋ℎ1 𝑢
+
𝜋ℎ1 𝑢
− 𝑢 = 𝑒 + 𝜌,
𝜎ℎ − 𝜎 = 𝜎ℎ − 𝜋ℎ2 𝜎 + 𝜋ℎ2 𝜎 − 𝜎 = 𝜉 + 𝜂.
(22)
∀Vℎ ∈ 𝑀ℎ ,
(24a)
To get error estimates, we state the following two important lemmas.
Lemma 1 (see [27, 29, 31]). Assume that 𝑢 ∈ 𝐻1 (Ω), p ∈
2
(𝐻2 (Ω)) , for all Vℎ ∈ 𝑀ℎ , wℎ ∈ Vℎ , and then there hold
(23b)
𝑒𝑛 − 𝑒𝑛−1
, Vℎ ) − (𝜉𝑛 , ∇Vℎ )ℎ
Δ𝑡
= (𝜓
Let 𝑢ℎ𝑛 and 𝑓 be zero, and thus 𝑢𝑛ℎ is zero too; taking Vℎ =
wℎ = (1/𝑏)𝜎ℎ𝑛 in (16) and adding them together, we have
Thus assumption (A3) implies that 𝑢ℎ𝑛 = 𝜎ℎ𝑛 = 0. The proof is
complete.
∀wℎ ∈ Vℎ .
From (23a), (23b), (15a), (15b), and (15c) we get
∀wℎ ∈ Vℎ .
1 󵄩󵄩 𝑛 󵄩󵄩2
1 𝑛 𝑛
󵄩𝑢 󵄩 + ( 𝜎ℎ , 𝜎ℎ ) = 0.
Δ𝑡 󵄩 ℎ 󵄩
𝑏
𝜕𝑢𝑛
, V ) − (𝜎𝑛 , ∇Vℎ )ℎ + ∑ ∫ 𝜎𝑛 Vℎ ⋅ n𝑑𝑠 = (𝑓𝑛 , Vℎ ) ,
𝜕𝜏 ℎ
𝑒∈𝑇 𝜕𝑒
𝜌𝑛−1 − 𝜌𝑛−1 𝑛
,𝑒 )
Δ𝑡
𝑒∈𝑇ℎ 𝜕𝑒
5
= ∑(Err)𝑖 .
𝑖=1
(28)
Journal of Applied Mathematics
5
On the one hand, we consider the right hand of (28).
Using the method similar to [3], we have
󵄩󵄩
󵄩2
𝜀
󵄩 𝜕𝑢𝑛 𝑢𝑛 − 𝑢𝑛−1 󵄩󵄩󵄩
󵄩󵄩 + 1 󵄩󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩󵄩2
−
(Err)1 ≤ 𝐶󵄩󵄩󵄩󵄩𝜓
󵄩
Δ𝑡 󵄩󵄩
2
󵄩󵄩 𝜕𝜏
󵄩󵄩 2 󵄩󵄩2
𝜀 󵄩 󵄩2
󵄩𝜕 𝑢󵄩
+ 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩 .
≤ 𝐶Δ𝑡󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (𝑡𝑛−1 ,𝑡𝑛 ;𝐿2 (Ω)) 2
Combining (29)–(34) with (28) gives
1 󵄩󵄩 𝑛 󵄩󵄩2 󵄩󵄩 𝑛−1 󵄩󵄩2
󵄩 󵄩2
(󵄩𝑒 󵄩 − 󵄩󵄩𝑒 󵄩󵄩󵄩 ) + 𝑏1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ
2Δ𝑡 󵄩 󵄩 󵄩
𝑛
󵄩󵄩 2 󵄩󵄩2
ℎ4 𝑡 󵄩󵄩 󵄩󵄩2
󵄩𝜕 𝑢󵄩
≤ 𝐶 (Δ𝑡󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
+
∫ 󵄩󵄩𝑢𝑡 󵄩󵄩2,Ω 𝑑𝑠
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (𝑡𝑛−1 , 𝑡𝑛 ; 𝐿2 (Ω)) Δ𝑡 𝑡𝑛−1
(29)
󵄩2
󵄩
󵄩 󵄩2
󵄩 󵄩2
+ℎ (󵄩󵄩󵄩󵄩𝑢𝑛−1 󵄩󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩𝜎𝑛 󵄩󵄩󵄩2,Ω ) )
(Err)2 can be estimated as
𝑛
2
𝑡
1
󵄨󵄨
󵄨
󵄨󵄨(Err)2 󵄨󵄨󵄨 ≤ (∫ (∫ 𝜌𝑡 𝑑𝑠) 𝑑𝑥 𝑑𝑦)
Δ𝑡 Ω 𝑡𝑛−1
𝑛
𝑡
1
≤
(∫ ∫ 𝜌𝑡2 𝑑𝑠 𝑑𝑥 𝑑𝑦)
√Δ𝑡 Ω 𝑡𝑛−1
≤
𝑡
1/2
1/2
󵄩2 𝑏 󵄩 󵄩
󵄩
󵄩 󵄩2
+ 𝜀1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩 + 𝐶󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩 + 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ .
2
󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩𝑒 󵄩󵄩
󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩𝑒 󵄩󵄩
Taking 1 − 2Δ𝑡𝜀1 > 0, multiplying (35) by 2Δ𝑡, summing over
from 𝑖 = 1 to 𝑖 = 𝑛, and noticing that 𝑒0 = 0, we obtain
(30)
𝑛
𝜀 󵄩 󵄩2
𝐶
󵄩 󵄩2
∫ 󵄩󵄩𝜌 󵄩󵄩 𝑑𝑠 + 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩
Δ𝑡 𝑡𝑛−1 󵄩 𝑡 󵄩
2
𝑛
󵄩 𝑖 󵄩2
󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩𝑒 󵄩󵄩 + Δ𝑡∑󵄩󵄩󵄩󵄩𝑒 󵄩󵄩󵄩󵄩1,ℎ
𝑖=1
󵄩󵄩 2 󵄩󵄩2
𝑡𝑛
󵄩𝜕 𝑢󵄩
󵄩 󵄩2
≤ 𝐶 ((Δ𝑡)2 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
+ ℎ4 ∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩2,Ω 𝑑𝑠
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
0
𝑛
𝜀 󵄩 󵄩2
𝐶ℎ4 𝑡 󵄩󵄩 󵄩󵄩2
≤
∫ 󵄩𝑢 󵄩 𝑑𝑠 + 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩 .
Δ𝑡 𝑡𝑛−1 󵄩 𝑡 󵄩2,Ω
2
𝑛
𝑛−1
󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
+Δ𝑡ℎ4 ∑ (󵄩󵄩󵄩󵄩𝑢𝑖 󵄩󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩󵄩𝜎𝑖 󵄩󵄩󵄩󵄩2,Ω )) + 𝐶 ∑ 󵄩󵄩󵄩󵄩𝑒𝑖 󵄩󵄩󵄩󵄩 .
By Lemma 2, we obtain
1 󵄩 𝑛−1
󵄨
󵄨󵄨
𝑛−1 󵄩 󵄩 𝑛 󵄩
󵄨󵄨(Err)3 󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝜌 − 𝜌 󵄩󵄩󵄩󵄩−1 󵄩󵄩󵄩𝑒 󵄩󵄩󵄩1,ℎ
Δ𝑡
󵄩2 𝑏 󵄩 󵄩2
󵄩
≤ 𝐶󵄩󵄩󵄩󵄩𝜌𝑛−1 󵄩󵄩󵄩󵄩 + 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ
6
≤
󵄩
󵄩2
𝐶ℎ4 󵄩󵄩󵄩󵄩𝑢𝑛−1 󵄩󵄩󵄩󵄩2,Ω
𝑖=1
(31)
It follows from discrete Gronwall’s lemma that
𝑛
󵄩 𝑖 󵄩2
󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩𝑒 󵄩󵄩 + Δ𝑡∑󵄩󵄩󵄩󵄩𝑒 󵄩󵄩󵄩󵄩1,ℎ
𝑏 󵄩 󵄩2
+ 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ .
6
𝑏 󵄩 𝑛 󵄩2
󵄨
󵄨󵄨
4 󵄩 𝑛 󵄩2
󵄨󵄨(Err)4 󵄨󵄨󵄨 ≤ 𝐶ℎ 󵄩󵄩󵄩𝜎 󵄩󵄩󵄩2,Ω + 1 󵄩󵄩󵄩𝑒 󵄩󵄩󵄩1,ℎ .
6
𝑖=1
(32)
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
≤ 𝐶 ((Δ𝑡)2 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
≤
+‖𝜎‖2𝐿∞ (0,𝑡𝑛 ;
(33)
𝑏 󵄩 󵄩2
+ 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ .
6
On the other hand, the left hand of (28) can be bounded by
(
𝑒𝑛 − 𝑒𝑛−1 𝑛
, 𝑒 ) + (𝑏∇𝑒𝑛 , ∇𝑒𝑛 )ℎ
Δ𝑡
1
󵄩 󵄩2
≥
((𝑒𝑛 , 𝑒𝑛 ) − (𝑒𝑛−1 , 𝑒𝑛−1 )) + 𝑏1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ
2Δ𝑡
≥
(34)
1 󵄩󵄩 𝑛 󵄩󵄩2
󵄩
󵄩2
󵄩 󵄩2
(󵄩𝑒 󵄩 − (1 + 𝐶Δ𝑡) 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩 ) + 𝑏1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ ,
2Δ𝑡 󵄩 󵄩
2
2
(37)
󵄩 󵄩2
+ ℎ (󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω)) + ‖𝑢‖2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω))
4
Let 𝑏 = (1/|𝑒|) ∫𝑒 𝑏(𝑥, 𝑦)𝑑𝑥 𝑑𝑦. By Lemma 1, we have
󵄨󵄨
󵄨 󵄨
𝑛
𝑛 󵄨
󵄨󵄨(Err)5 󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨−((𝑏 − 𝑏) ∇𝜌 , ∇𝑒 )ℎ 󵄨󵄨󵄨󵄨
󵄩 󵄩 󵄩 󵄩
≤ 𝐶ℎ|𝑏|𝑊1,∞ (Ω) 󵄩󵄩󵄩𝜌𝑛 󵄩󵄩󵄩1,ℎ 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ
𝑖=1
(36)
It follows from Lemma 1 that
󵄩 󵄩2
𝐶ℎ4 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩2,Ω
(35)
4
where the inequality ‖𝑒𝑛−1 ‖ ≤ (1 + 𝐶Δ𝑡)‖𝑒𝑛−1 ‖ proved in [3]
is used in the last step.
2
(𝐻2 (Ω)) )
)).
From (37) we get the optimal order error estimate of ‖𝑒𝑛 ‖
rather than ‖𝑒𝑛 ‖1,ℎ . So we start to reestimate ‖𝑒𝑛 ‖1,ℎ in the
following manner and derive the estimation of ‖𝜉𝑛 ‖ simultaneously.
Firstly, choosing Vℎ = ((𝑒𝑛 − 𝑒𝑛−1 )/Δ𝑡) in (24a) and wℎ =
∇((𝑒𝑛 − 𝑒𝑛−1 )/Δ𝑡) in (24b), and adding them, we have
(
𝑒𝑛 − 𝑒𝑛−1
𝑒𝑛 − 𝑒𝑛−1 𝑒𝑛 − 𝑒𝑛−1
,
) + (𝑏∇𝑒𝑛 , ∇
)
Δ𝑡
Δ𝑡
Δ𝑡
ℎ
= (𝜓
𝜕𝑢𝑛 𝑢𝑛 − 𝑢𝑛−1 𝑒𝑛 − 𝑒𝑛−1
−
,
)
𝜕𝜏
Δ𝑡
Δ𝑡
−(
𝜌𝑛 − 𝜌𝑛−1 𝑒𝑛 − 𝑒𝑛−1
,
)
Δ𝑡
Δ𝑡
6
Journal of Applied Mathematics
−(
𝜌𝑛−1 − 𝜌𝑛−1 𝑒𝑛 − 𝑒𝑛−1
,
)
Δ𝑡
Δ𝑡
+ ∑ ∫ 𝜎𝑛
𝑒∈𝑇ℎ 𝜕𝑒
𝑒𝑛 − 𝑒𝑛−1
𝑒𝑛 − 𝑒𝑛−1
⋅ n 𝑑𝑠 − (𝑏∇𝜌𝑛 , ∇
)
Δ𝑡
Δ𝑡
ℎ
5
= ∑(Err)󸀠𝑖 .
𝑖=1
Multiplying (41) by 2Δ𝑡 and summing over in time from 𝑖 = 1
to 𝑖 = 𝑛 yield
󵄩󵄩 𝑛 𝑛−1 󵄩󵄩2
󵄩 𝑒 − 𝑒 󵄩󵄩
󵄩󵄩 + 𝑏1 󵄩󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩󵄩21,ℎ
Δ𝑡󵄩󵄩󵄩󵄩
󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
󵄩 󵄩2
≤ 𝐶 [(Δ𝑡)2 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
+ ℎ4 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω))
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
𝑛
󵄩 󵄩2
󵄩 󵄩2
+ℎ4 ∑ (󵄩󵄩󵄩󵄩𝑢𝑖 󵄩󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩󵄩𝜎𝑖 󵄩󵄩󵄩󵄩2,Ω )]
(38)
The left hand can be estimated as
𝑖=1
𝑒𝑛 − 𝑒𝑛−1 𝑒𝑛 − 𝑒𝑛−1
𝑒𝑛 − 𝑒𝑛−1
(
,
) + (𝑏∇𝑒𝑛 , ∇
)
Δ𝑡
Δ𝑡
Δ𝑡
ℎ
󵄩󵄩 𝑛 𝑛−1 󵄩󵄩2
1
󵄩 𝑒 − 𝑒 󵄩󵄩
󵄩󵄩 +
≥ 󵄩󵄩󵄩󵄩
[(𝑏∇𝑒𝑛 , ∇𝑒𝑛 ) − (𝑏∇𝑒𝑛−1 , ∇𝑒𝑛−1 )]
2Δ𝑡
󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
+(
𝑒𝑛−1 − 𝑒𝑛−1 𝑒𝑛 − 𝑒𝑛−1
,
),
Δ𝑡
Δ𝑡
(Err)󸀠𝑖 ,
(𝑖 = 1, 2, 3, 4, 5) can be bounded by
󵄩
󵄩2
󵄩󵄩󵄩 𝜕2 𝑢 󵄩󵄩󵄩2
1 󵄩󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
󵄨
󵄨󵄨
󵄩󵄩
󵄩󵄩 ,
󵄨󵄨(Err)󸀠1 󵄨󵄨󵄨 ≤ 𝐶Δ𝑡󵄩󵄩󵄩 2 󵄩󵄩󵄩
+
󵄩󵄩 𝜕𝜏 󵄩󵄩 2 𝑛−1 𝑛 2
󵄩
󵄩
󵄨
󵄨
󵄩
󵄩𝐿 (𝑡 ,𝑡 ;𝐿 (Ω)) 4 󵄩󵄩 Δ𝑡 󵄩󵄩
𝑛
󵄨󵄨
󵄨 𝐶ℎ4 𝑡 󵄩󵄩 󵄩󵄩2
󵄨󵄨(Err)󸀠2 󵄨󵄨󵄨 ≤
∫ 󵄩𝑢 󵄩 𝑑𝑠 +
󵄨
󵄨
Δ𝑡 𝑡𝑛−1 󵄩 𝑡 󵄩2,Ω
∑(
𝑖=1
󵄩
󵄩2
𝐶ℎ4 󵄩󵄩 𝑛 󵄩󵄩2
𝜀 󵄩󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
󵄨
󵄨󵄨
󵄩
󵄩󵄩 .
󵄨󵄨(Err)󸀠5 󵄨󵄨󵄨 ≤
𝑢
+
Δ𝑡
󵄩
󵄩
󵄩
󵄨
󵄨
Δ𝑡 󵄩 󵄩2,Ω 3 󵄩󵄩󵄩 Δ𝑡 󵄩󵄩󵄩1,ℎ
(44)
𝑒𝑖−1 − 𝑒𝑖−1 𝑖 𝑖−1
,𝑒 − 𝑒 )
Δ𝑡
𝑖−1
𝑛−1 𝑒
𝑒𝑛−1 − 𝑒𝑛−1 𝑛
=(
,𝑒 ) + ∑ (
Δ𝑡
𝑖=1
󵄩
󵄩2
𝜀 󵄩󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
󵄨 𝐶ℎ4 󵄩󵄩 𝑛 󵄩󵄩2
󵄨󵄨
󵄩
󵄩󵄩 ,
󵄨󵄨(Err)󸀠4 󵄨󵄨󵄨 ≤
𝜎
+
Δ𝑡
󵄩
󵄩
󵄩
󵄨
󵄨
Δ𝑡 󵄩 󵄩2,Ω 3 󵄩󵄩󵄩 Δ𝑡 󵄩󵄩󵄩1,ℎ
(43)
and by inverse inequality, we have
󵄩 𝑖 𝑖−1 󵄩󵄩2
𝑛 󵄩
𝑛 󵄩
󵄩󵄩 𝑒𝑖 − 𝑒𝑖−1 󵄩󵄩󵄩2
󵄩󵄩 𝑒 − 𝑒 󵄩󵄩
2
󵄩
󵄩󵄩 .
󵄩
󵄩
(Δ𝑡) ∑󵄩󵄩
󵄩󵄩 ≤ 𝐶Δ𝑡∑󵄩󵄩󵄩
󵄩󵄩
󵄩 Δ𝑡 󵄩󵄩1,ℎ
󵄩 Δ𝑡 󵄩󵄩
𝑖=1 󵄩
𝑖=1 󵄩
At the same time, using Lemma 2, we obtain
𝑛
󵄩
󵄩2
1 󵄩󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 ,
4 󵄩󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
󵄩
󵄩2
𝜀 󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
󵄨󵄨
󵄨 𝐶ℎ4 󵄩󵄩 𝑛−1 󵄩󵄩2
󵄩󵄩𝑢 󵄩󵄩 + Δ𝑡󵄩󵄩󵄩
󵄩
󵄨󵄨(Err)󸀠3 󵄨󵄨󵄨 ≤
󵄨
󵄨
󵄩2,Ω 3 󵄩󵄩 Δ𝑡 󵄩󵄩󵄩 ,
Δ𝑡 󵄩
󵄩
󵄩1,ℎ
− 𝑒𝑖 − (𝑒𝑖−1 − 𝑒𝑖 )
Δ𝑡
, 𝑒𝑖 )
𝑛−1
󵄩
󵄩󵄩 󵄩
󵄩
󵄩󵄩 󵄩
≤ 𝐶 󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ + ∑ 󵄩󵄩󵄩󵄩𝑒𝑖 − 𝑒𝑖−1 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑒𝑖 󵄩󵄩󵄩󵄩1,ℎ
𝑖=1
𝑛−1󵄩
󵄩󵄩 𝑒𝑖 − 𝑒𝑖−1 󵄩󵄩󵄩2
󵄩2 𝑏 󵄩 󵄩2
󵄩
󵄩󵄩
≤ 𝐶󵄩󵄩󵄩󵄩𝑒𝑛−1 󵄩󵄩󵄩󵄩 + 1 󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ + Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
2
󵄩 Δ𝑡 󵄩󵄩
𝑖=1 󵄩
(40)
From (38)–(40), we get
󵄩
󵄩2
1 󵄩󵄩󵄩 𝑒𝑛 − 𝑒𝑛−1 󵄩󵄩󵄩
1
󵄩󵄩
󵄩󵄩 +
[(𝑏∇𝑒𝑛 , ∇𝑒𝑛 )ℎ − (𝑏∇𝑒𝑛−1 , ∇𝑒𝑛−1 )ℎ ]
󵄩
󵄩
2 󵄩󵄩 Δ𝑡 󵄩󵄩
2Δ𝑡
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
≤ 𝐶 [Δ𝑡󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (𝑡𝑛−1 ,𝑡𝑛 ;𝐿2 (Ω))
𝑛
+
Secondly, we take Δ𝑡 → 0 and Δ𝑡 must approach zero in such
a way that Δ𝑡 and ℎ satisfy
Δ𝑡 = 𝑂 (ℎ2 ) ,
(39)
and
𝑛 󵄩
𝑛
𝑖−1
𝑖−1
󵄩󵄩 𝑒𝑖 − 𝑒𝑖−1 󵄩󵄩󵄩2
󵄩󵄩 + ∑ ( 𝑒 − 𝑒 , 𝑒𝑖 − 𝑒𝑖−1 ) .
+ 𝜀(Δ𝑡)2 ∑󵄩󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩 Δ𝑡 󵄩󵄩1,ℎ 𝑖=1
𝑖=1 󵄩
(42)
𝑡
ℎ4
󵄩2
󵄩
󵄩 󵄩2
󵄩 󵄩2
(∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩2,Ω 𝑑𝑠 + 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩󵄩𝑢𝑛−1 󵄩󵄩󵄩󵄩2,Ω
Δ𝑡 𝑡𝑛−1
󵄩󵄩 𝑛 𝑛−1 󵄩󵄩2
󵄩 𝑒 − 𝑒 󵄩󵄩
󵄩 󵄩2
󵄩󵄩
+󵄩󵄩󵄩𝜎𝑛 󵄩󵄩󵄩2,Ω )] + 𝜀Δ𝑡󵄩󵄩󵄩󵄩
󵄩󵄩 Δ𝑡 󵄩󵄩󵄩1,ℎ
𝑛−1
󵄩 󵄩2
+ 𝐶Δ𝑡 ∑ 󵄩󵄩󵄩󵄩𝑒𝑖 󵄩󵄩󵄩󵄩1,ℎ .
𝑖=1
(45)
From (42)–(45), taking suitable small 𝜀 such that 1 − 𝜀𝐶 > 0,
we have
󵄩󵄩 𝑛 𝑛−1 󵄩󵄩2
󵄩 𝑒 − 𝑒 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩 + 󵄩󵄩𝑒 󵄩󵄩1,ℎ
Δ𝑡󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩 Δ𝑡 󵄩󵄩
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
󵄩 󵄩2
≤ 𝐶 [(Δ𝑡)2 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
+ ℎ4 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω))
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
𝑛
󵄩 󵄩2
󵄩 󵄩2
+ℎ4 ∑ (󵄩󵄩󵄩󵄩𝑢𝑖 󵄩󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩󵄩𝜎𝑖 󵄩󵄩󵄩󵄩2,Ω )]
𝑖=1
𝑒𝑛−1 − 𝑒𝑛−1 𝑒𝑛 − 𝑒𝑛−1
+(
,
).
Δ𝑡
Δ𝑡
(41)
𝑛−1󵄩
𝑛−1
󵄩󵄩 𝑒𝑖 − 𝑒𝑖−1 󵄩󵄩󵄩2
󵄩󵄩 𝑛−1 󵄩󵄩2
󵄩󵄩 + 𝐶Δ𝑡 ∑ 󵄩󵄩󵄩𝑒𝑖 󵄩󵄩󵄩2 .
󵄩
󵄩
+ 󵄩󵄩𝑒 󵄩󵄩 + 𝐶Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩1,ℎ
󵄩 Δ𝑡 󵄩󵄩
𝑖=1 󵄩
𝑖=1
(46)
Journal of Applied Mathematics
7
Table 1: Numerical results of ‖𝑢 − 𝑢ℎ ‖1,ℎ .
Finally, applying discrete Gronwall’s lemma yields
󵄩󵄩 2 󵄩󵄩2
󵄩󵄩 𝑛 󵄩󵄩2
󵄩 󵄩2
2󵄩 𝜕 𝑢 󵄩
+ ℎ4 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω))
󵄩󵄩𝑒 󵄩󵄩1,ℎ ≤ 𝐶 [(Δ𝑡) 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
+ ℎ2 (‖𝑢‖2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω)) + ‖𝜎‖2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω)) ) ] .
(47)
In order to derive (27), set wℎ = 𝜉𝑛 in (24b) and employ
Lemma 1 and assumption (A3) to give
𝑡 = 0.2
0.75277
0.42984
0.21758
𝑡 = 0.5
0.55291
0.29234
0.14466
𝛼
/
0.81
0.99
𝛼
/
0.92
1.02
𝑡 = 0.3
0.75017
0.41849
0.21412
𝑡 = 0.8
0.42211
0.23117
0.10807
𝛼
/
0.84
0.97
𝛼
/
0.87
1.10
𝑡 = 0.4
0.66433
0.35474
0.17552
𝑡 = 0.9
0.40937
0.21120
0.09343
𝛼
/
0.91
1.02
𝛼
/
0.96
1.18
Table 2: Numerical results of ‖𝑢 − 𝑢ℎ ‖.
󵄩󵄩 𝑛 󵄩󵄩2
𝑛 𝑛
𝑛 𝑛
𝑛 𝑛
󵄩󵄩𝜉 󵄩󵄩 = −(𝑏∇𝑒 , 𝜉 )ℎ − (𝜂 , 𝜉 ) − (𝑏∇𝜌 , 𝜉 )ℎ
󵄩 󵄩
󵄩 󵄩2
≤ 𝐶 (󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ + ℎ4 󵄩󵄩󵄩𝜎𝑛 󵄩󵄩󵄩2,Ω )
1 󵄩 󵄩2
− ((𝑏 − 𝑏) ∇𝜌𝑛 , 𝜉𝑛 )ℎ + 󵄩󵄩󵄩𝜉𝑛 󵄩󵄩󵄩
4
󵄩 󵄩2
󵄩 󵄩
󵄩 󵄩
≤ 𝐶 (󵄩󵄩󵄩𝑒𝑛 󵄩󵄩󵄩1,ℎ + ℎ4 (󵄩󵄩󵄩𝜎𝑛 󵄩󵄩󵄩2,Ω + 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩2,Ω )) +
𝑚×𝑛
8×8
16 × 16
32 × 32
𝑚×𝑛
8×8
16 × 16
32 × 32
1 󵄩󵄩 𝑛 󵄩󵄩2
󵄩𝜉 󵄩 .
2󵄩 󵄩
(48)
𝑚×𝑛
8×8
16 × 16
32 × 32
𝑚×𝑛
8×8
16 × 16
32 × 32
𝑡 = 0.4
0.0298190
0.0073087
0.0020769
𝑡 = 0.8
0.0198730
0.0044472
0.0011894
𝛼
/
2.03
1.82
𝛼
/
2.16
1.90
𝑡 = 0.5
0.0276370
0.0062445
0.0017926
𝑡 = 0.9
0.0175900
0.0041982
0.0010738
𝛼
/
2.15
1.80
𝛼
/
2.07
1.97
𝑡 = 0.7
0.0223240
0.0048038
0.0013309
𝑡 = 1.0
0.0154090
0.0039150
0.0009466
𝛼
/
2.22
1.85
𝛼
/
1.98
2.05
Combining (47) with (48) yields
󵄩󵄩 2 󵄩󵄩2
󵄩󵄩 𝑛 󵄩󵄩2
󵄩 󵄩2
2󵄩 𝜕 𝑢 󵄩
+ ℎ4 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω))
󵄩󵄩𝜉 󵄩󵄩 ≤ 𝐶 [(Δ𝑡) 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
+ℎ2 (|𝑢|2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω)) + ‖𝜎‖2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω)) ) ] .
(49)
By using of interpolation theory and the triangle inequality,
(37), (47), and (49) lead to (25), (26), and (27), respectively,
which are the desired results.
Remark 2. From (37), we have
𝑛
𝑛 󵄩
𝑖󵄩
󵄩 󵄩2
󵄩
󵄩2
Δ𝑡∑󵄩󵄩󵄩󵄩𝑒𝑖 󵄩󵄩󵄩󵄩1,ℎ = Δ𝑡∑󵄩󵄩󵄩(𝜋ℎ1 𝑢 − 𝑢ℎ ) 󵄩󵄩󵄩
󵄩
󵄩1,ℎ
𝑖=1
𝑖=1
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
≤ 𝐶 ((Δ𝑡)2 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝜏 󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐿2 (Ω))
successful optimal order error estimations. If we want to
get higher order accuracy, similar to Lemma 1, the nonconforming finite elements for approximating 𝑢 should also
possess a very special property, that is, the consistency error
estimates with 𝑂(ℎ2 ) order, and satisfy (18). For the famous
nonconforming Wilson element [32] whose shape function is
span{1, 𝑥, 𝑦, 𝑥2 , 𝑦2 }, by a counter-example, it has been proven
in [32] that its consistency error estimate is of 𝑂(ℎ) order and
cannot be improved any more. For the rotated bilinear 𝑄1
element [33] whose shape function is span{1, 𝑥, 𝑦, 𝑥2 − 𝑦2 },
although its consistency error with 𝑂(ℎ2 ) order and (∇(𝑢 −
𝜋ℎ1 𝑢), ∇Vℎ )ℎ = 0 on square meshes is satisfied, the second term
of (18) is not valid. Thus when they are applied to (1) on new
characteristic mixed finite element scheme, up to now, the
optimal order error estimates of (25), (26), and (27) cannot
be obtained directly.
5. Numerical Example
(50)
󵄩 󵄩2
+ ℎ4 (󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (0,𝑡𝑛 ;𝐻2 (Ω)) + ‖𝑢‖2𝐿∞ (0,𝑡𝑛 ;𝐻2 (Ω))
+‖𝜎‖2𝐿∞ (0,𝑡𝑛 ;(𝐻2 (Ω))2 ) ) ) .
This byproduct can be regarded as the superclose result
between 𝜋ℎ1 𝑢 and 𝑢ℎ in mean broken 𝐻1 -norm. It seems that
both (25) and (50) have never been seen in the existing studies. At the same time, by employing the new characteristic
nonconforming MFE scheme, we can also obtain the same
error estimate of (27) as traditional characteristic MFEM [10].
Remark 3. From the analysis of Theorem 2 in this paper,
we may see that Lemma 1 is the key result leading to the
In order to verify our theoretical analysis in previous sections,
we consider the convection-dominated diffusion problem (1)
as follows:
𝑢𝑡 + 𝑢𝑥 + 𝑢𝑦 − 10−4 (𝑢𝑥𝑥 + 𝑢𝑦𝑦 )
= 𝑓 (𝑥, 𝑦, 𝑡) ,
𝑢 (𝑥, 𝑦, 𝑡) = 0,
(𝑥, 𝑦, 𝑡) ∈ Ω × (0, 𝑇) ,
(𝑥, 𝑦, 𝑡) ∈ 𝜕Ω × (0, 𝑇) ,
𝑢 (𝑥, 𝑦, 0) = 𝑢0 (𝑥, 𝑦) ,
(51)
(𝑥, 𝑦) ∈ Ω
with Ω = [0, 1] × [0, 1], a(𝑥, 𝑦) = (1, 1), and 𝑏(𝑥, 𝑦) = 10−4 .
The right hand term 𝑓(𝑥, 𝑦, 𝑡) is taken such that 𝑢 =
𝑒−𝑡 sin(𝜋𝑥) sin(2𝜋𝑦), 𝜎 = −10−4 𝑒−𝑡 (𝜋 cos(𝜋𝑥) sin(2𝜋𝑦),
2𝜋 sin(𝜋𝑥) cos(2𝜋𝑦)) are the exact solutions.
8
Journal of Applied Mathematics
Table 3: Numerical results of ‖𝜎 − 𝜎ℎ ‖.
𝑚×𝑛
𝑡 = 0.1
𝛼
𝑡 = 0.4
𝛼
𝑡 = 0.5
𝛼
8×8
16 × 16
4.9528𝑒 − 005
2.3945𝑒 − 005
/
1.05
4.2661𝑒 − 005
1.8843𝑒 − 005
/
1.18
3.8292𝑒 − 005
1.6806𝑒 − 005
/
1.19
32 × 32
1.1749𝑒 − 005
1.03
9.0029𝑒 − 006
1.07
8.0521𝑒 − 006
1.06
𝑡 = 0.7
𝛼
𝑡 = 0.8
𝛼
𝑡 = 0.9
𝛼
8×8
16 × 16
3.0714𝑒 − 005
1.3326𝑒 − 005
/
1.20
2.7735𝑒 − 005
1.224𝑒 − 005
/
1.18
2.524𝑒 − 005
1.1443𝑒 − 005
/
1.14
32 × 32
6.455𝑒 − 006
1.05
5.8353𝑒 − 006
1.07
5.3751𝑒 − 006
1.09
𝑚×𝑛
𝑡 = 0.4
0
−2
−2
−4
log(error)
−4
log(error)
𝑡 = 0.8
0
−6
−8
−6
−8
−10
−10
−12
−12
−3.5
−3
−2.5
−14
−3.5
−2
−3
log(ℎ)
Figure 3: Errors at 𝑡 = 0.8.
Figure 1: Errors at 𝑡 = 0.4.
𝑡 = 0.5
0
𝑡 = 0.9
0
−2
−2
−4
log(error)
−4
log(error)
−2
‖𝑢 − 𝑢ℎ ‖1,ℎ
‖𝜎 − 𝜎ℎ ‖
ℎ
ℎ2
‖𝑢 − 𝑢ℎ ‖
‖𝑢 − 𝑢ℎ ‖1,ℎ
‖𝜎 − 𝜎ℎ ‖
ℎ
ℎ2
‖𝑢 − 𝑢ℎ ‖
−2.5
log(ℎ)
−6
−8
−6
−8
−10
−10
−12
−12
−3.5
−3
−2.5
−2
−14
−3.5
−3
ℎ
ℎ2
‖𝑢 − 𝑢ℎ ‖
‖𝑢 − 𝑢ℎ ‖1,ℎ
‖𝜎 − 𝜎ℎ ‖
Figure 2: Errors at 𝑡 = 0.5.
−2.5
log(ℎ)
log(ℎ)
ℎ
ℎ2
‖𝑢 − 𝑢ℎ ‖
‖𝑢 − 𝑢ℎ ‖1,ℎ
‖𝜎 − 𝜎ℎ ‖
Figure 4: Errors at 𝑡 = 0.9.
−2
Journal of Applied Mathematics
We first divide the domain Ω into 𝑚 and 𝑛 equal intervals
along 𝑥-axis and 𝑦-axis and the numerical results at different
times are listed in Tables 1, 2, and 3 and pictured in Figures
1, 2, 3, and 4, respectively. (𝑢ℎ , pℎ ) denotes the characteristic
nonconforming MFE solution of the problem (15a), (15b), and
(15c). Δ𝑡 represents the time step and the experiment is done
with Δ𝑡 = ℎ2 . 𝛼 stands for the convergence order.
It can be seen from the above Tables 1, 2, and 3 that
‖𝑢 − 𝑢ℎ ‖1,ℎ and ‖𝜎−𝜎ℎ ‖ are convergent at optimal rate of 𝑂(ℎ)
and ‖𝑢 − 𝑢ℎ ‖ is convergent at optimal rate of 𝑂(ℎ2 ), respectively, which coincide with our theoretical investigation in
Section 4.
Acknowledgments
The research was supported by the National Natural Science
Foundation of China (Grant nos. 10971203, 11101384, and
11271340) and the Specialized Research Fund for the Doctoral
Program of Higher Education (Grant no. 20094101110006).
The author would like to thank the referees for their helpful
suggestions.
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