Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2013, Article ID 787891, 14 pages
http://dx.doi.org/10.1155/2013/787891
Research Article
A Coupling Method of New EMFE and FE for Fourth-Order
Partial Differential Equation of Parabolic Type
Yang Liu,1 Hong Li,1 Zhichao Fang,1 Siriguleng He,1 and Jinfeng Wang2
1
2
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
Correspondence should be addressed to Yang Liu; mathliuyang@yahoo.cn and Hong Li; smslh@imu.edu.cn
Received 29 November 2012; Accepted 11 March 2013
Academic Editor: B. G. Konopelchenko
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.
We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE)
and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic
equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the
2
other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable (𝐿2 (Ω))
space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution 𝜆 = ∇𝑢.
For a priori error estimates based on both semidiscrete and fully discrete schemes, we introduce a new expanded mixed projection
and some important lemmas. We derive the optimal a priori error estimates in 𝐿2 and 𝐻1 -norm for both the scalar unknown 𝑢 and
2
the diffusion term 𝛾 and a priori error estimates in (𝐿2 ) -norm for its gradient 𝜆 and its flux 𝜎 (the coefficients times the negative
gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.
1. Introduction
In recent years, many researchers have studied some numerical methods for fourth-order elliptic equations [1–6], fourthorder parabolic equations [5–9], fourth-order wave equations
[10, 11], and so on. Chen [1] proposed and analyzed an
expanded mixed finite element method for fourth-order
elliptic problems. In [2], Chen et al. studied an anisotropic
nonconforming element for fourth-order elliptic singular
perturbation problem. In [3–6], some mixed finite element (MFE) methods were studied for fourth-order linear/nonlinear elliptic equations. In [7], the FE method was
studied for nonlinear Cahn-Hilliard equation. The optimalorder error estimates were obtained in 𝐿2 -norm by means of
an FE biharmonic projection approximation. In [12], the MFE
methods were studied for solving a fourth-order nonlinear
reaction diffusion equation. In [8], an 𝐻1 -Galerkin MFE
method was studied for solving the fourth-order parabolic
partial differential equations. Optimal error estimates were
derived for both semidiscrete and fully discrete schemes
for problems in one space dimension, and error estimates
were derived for semidiscrete scheme for several space
dimensions, and the stability for fully discrete scheme was
proved by the iteration method. In [9], the FE method was
studied for fourth-order nonlinear parabolic equation. He
et al. [10] studied and analyzed the explicit/implicit MFE
methods for a class of fourth-order wave equations. Shi and
Peng [13] studied the finite element methods for fourth-order
eigenvalue problems on anisotropic meshes. Liu et al. [11]
studied a 𝐶1 -conforming FE method for nonlinear fourthorder wave equation. In this paper, we consider the following
fourth-order partial differential equations of parabolic type:
𝑢𝑡 +∇ ⋅ (𝑏 (x, 𝑡) ∇ (∇ ⋅ (𝑎 (𝑡) ∇𝑢))) = 𝑓 (x, 𝑡) ,
𝑢 (x, 𝑡) = Δ𝑢 (x, 𝑡) = 0,
𝑢 (x, 0) = 𝑢0 (x) ,
(x, 𝑡) ∈ Ω × 𝐽,
(x, 𝑡) ∈ 𝜕Ω × 𝐽,
x ∈ Ω,
(1)
2
Advances in Mathematical Physics
where Ω ⊂ 𝑅2 is a bounded convex polygonal domain with
the Lipschitz continuous boundary 𝜕Ω and 𝐽 = (0, 𝑇] is
the time interval with 0 < 𝑇 < ∞. The initial value 𝑢0 (x)
and 𝑓(x, 𝑡) are given functions; coefficients 𝑎 = 𝑎(𝑡) and
𝑏 = 𝑏(x, 𝑡) satisfy
󵄨
󵄨󵄨
󵄨󵄨𝑎𝑡 (𝑡)󵄨󵄨󵄨 ≤ 𝑐0 ,
󵄨
󵄨
A2 : 0 < 𝑏0 ≤ 𝑏 (x, 𝑡) ≤ 𝑏1 < +∞, 󵄨󵄨󵄨𝑏𝑡 (x, 𝑡)󵄨󵄨󵄨 ≤ 𝑐0 ,
A1 : 0 < 𝑎0 ≤ 𝑎 (𝑡) ≤ 𝑎1 < +∞,
(2)
for some positive constants 𝑎0 , 𝑎1 , 𝑏0 , 𝑏1 , and 𝑐0 .
Chen proposed the EMFE method for second-order
linear/quasilinear elliptic equation [14–16] and fourth-order
linear elliptic equation [1]. Then the method was applied
to other partial differential equations [17, 18]. Compared to
standard MFE methods, the EMFE method is expanded in
the sense that three variables are explicitly approximated,
namely, the scalar unknown, its gradient, and its flux. In
recent years, many numerical method based on the expanded
mixed scheme, such as two-grid EMFE methods [19–22], 𝐻1 Galerkin EMFE methods [23–25], expanded mixed covolume
method [26], expanded mixed hybrid method [27], and
positive definite EMFE method [28], have been proposed and
discussed.
In [29], we developed and analyzed a new EMFE method
for second-order elliptic problems based on the mixed weak
formulation [30–32]. The new EMFE method [29], whose
gradient belongs to the square integrable (𝐿2 (Ω))2 space
instead of the classical H(div; Ω) space, is different from
Chen’s EMFE method [14–16].
In this paper, we present a new coupling method of new
EMFE scheme [29] and FE scheme for fourth-order partial
differential equation of parabolic type. We first introduce an
auxiliary variable 𝛾 = −∇ ⋅ (𝑎(𝑡)∇𝑢) to reduce the fourthorder parabolic equation to a coupled system of second-order
equations, then solve a second-order equation by FE method.
For the other one, we introduce the two auxiliary variables,
𝜆 = ∇𝑢 and 𝜎 = −𝑎(𝑡)∇𝑢 = −𝑎𝜆, to split the equation
into a first-order system of equations, then approximate that
by a new EMFE method. Because the new EMFE method’s
gradient belongs to the simple square integrable (𝐿2 (Ω))2
space, the regularity requirement on the gradient solution
𝜆 = ∇𝑢 is reduced. We apply the new coupling method of
new EMFE and FE to derive a priori error estimates based on
both semidiscrete and fully discrete schemes; that is to say,
we obtain the optimal a priori error estimates in 𝐿2 and 𝐻1 norm for both the scalar unknown 𝑢 and the diffusion term
𝛾 and a priori error estimates in (𝐿2 )2 -norm for its gradient 𝜆
and its flux 𝜎. Finally, we provide some numerical results to
verify our theoretical analysis.
Throughout this paper, 𝐶 will denote a generic positive
constant which does not depend on the spatial mesh parameter ℎ or the time step 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
for the Sobolev spaces as in [33, 34] are used.
2. New Expanded Mixed Scheme
We first introduce an auxiliary variable 𝛾 = −∇ ⋅ (𝑎(𝑡)∇𝑢)
to obtain the following coupled system of second-order
equations:
𝑢𝑡 − ∇ ⋅ (𝑏 (x, 𝑡) ∇𝛾) = 𝑓 (x, 𝑡) ,
(3a)
𝛾 + ∇ ⋅ (𝑎 (𝑡) ∇𝑢) = 0,
(3b)
𝑢 (x, 𝑡) = 𝛾 (x, 𝑡) = 0.
(3c)
For (3b), we introduce the two auxiliary variables, 𝜆 = ∇𝑢
and 𝜎 = −𝑎(𝑡)∇𝑢 = −𝑎𝜆, to obtain the following first-order
system of equations:
𝑢𝑡 − ∇ ⋅ (𝑏∇𝛾) = 𝑓 (x, 𝑡) ,
(4a)
𝛾 − ∇ ⋅ 𝜎 = 0,
(4b)
𝜎 + 𝑎𝜆 = 0,
(4c)
𝜆 − ∇𝑢 = 0,
(4d)
𝑢 (x, 𝑡) = 𝛾 (x, 𝑡) = 0.
(4e)
Using Green’s formula, the new mixed weak formulation of
(4a)–(4e) is to determine {𝑢, 𝛾, 𝜆, 𝜎} : [0, 𝑇] 󳨃→ 𝐻01 (Ω) ×
𝐻01 (Ω) × (𝐿2 (Ω))2 × (𝐿2 (Ω))2 such that
∀V ∈ 𝐻01 (Ω) ,
(𝑢𝑡 , V) + (𝑏∇𝛾, ∇V) = (𝑓, V) ,
(𝛾, 𝑔) + (𝜎, ∇𝑔) = 0,
∀𝑔 ∈ 𝐻01 (Ω) ,
2
(𝜎, z) + (𝑎𝜆, z) = 0,
∀z ∈ (𝐿2 (Ω)) ,
(𝜆, w) − (∇𝑢, w) = 0,
∀w ∈ (𝐿2 (Ω)) .
2
(5a)
(5b)
(5c)
(5d)
Then, the semidiscrete coupled EMFE scheme-FE scheme for
(5a)–(5d) is to find {𝑢ℎ , 𝛾ℎ , 𝜆ℎ , 𝜎ℎ } : [0, 𝑇] 󳨃→ 𝑉ℎ × 𝑉ℎ × Wℎ ×
Wℎ such that
(𝑢ℎ𝑡 , Vℎ ) + (𝑏∇𝛾ℎ , ∇Vℎ ) = (𝑓, Vℎ ) ,
∀Vℎ ∈ 𝑉ℎ ,
(6a)
(𝛾ℎ , 𝑔ℎ ) + (𝜎ℎ , ∇𝑔ℎ ) = 0,
∀𝑔ℎ ∈ 𝑉ℎ ,
(6b)
(𝜎ℎ , zℎ ) + (𝑎𝜆ℎ , zℎ ) = 0,
∀zℎ ∈ Wℎ ,
(6c)
(𝜆ℎ , wℎ ) − (∇𝑢ℎ , wℎ ) = 0,
∀wℎ ∈ Wℎ .
(6d)
By the theory of differential equations [35], we can prove
the existence and uniqueness of solution for semidiscrete
scheme (6a)–(6d).
Remark 1. From the new coupling weak formulation (5a)–
(5d), we can find that the gradient belongs to the weaker
square integrable (𝐿2 (Ω))2 space taking the place of the
classical H(div; Ω) space. It is easy to see that our method
reduces the regularity requirement on the gradient solution
𝜆 = ∇𝑢.
Advances in Mathematical Physics
3
Remark 2. In the semidiscrete coupling scheme (6a)–(6d),
the MFE space (𝑉ℎ , Wℎ ) based on the FE pair 𝑃1 − 𝑃02 is
chosen as follows:
𝑉ℎ = {Vℎ ∈ 𝐶0 (Ω) ∩ 𝐻01 (Ω) | Vℎ ∈ 𝑃1 (𝐾) , ∀𝐾 ∈ Kℎ } ,
2
𝑤1ℎ , 𝑤2ℎ ∈ 𝑃0 (𝐾) , ∀𝐾 ∈ Kℎ } .
(7)
From [30, 31], we know that MFE space (𝑉ℎ , Wℎ ) satisfies the
discrete LBB condition.
Remark 3. From (6a)–(6d), we know that (6a) is an FE
scheme for 𝑢ℎ , 𝛾ℎ , Vℎ ∈ 𝑉ℎ and that (6b)–(6d) is a new EMFE
scheme for (𝛾ℎ , 𝜎ℎ , 𝜆ℎ ), (𝑔ℎ , zℎ , wℎ ) ∈ 𝑉ℎ × Wℎ × Wℎ . So
the scheme (6a)–(6d) is called as the coupling system of new
EMFE scheme and FE scheme.
Remark 4. The finite element spaces 𝑉ℎ and Wℎ for the new
EMFE method are chosen as finite dimensional subspaces
of 𝐻01 (Ω) and (𝐿2 (Ω))2 , respectively. However, 𝑉ℎ and Wℎ
for 𝐻1 -Galerkin MFE method in two space variables in [8]
are finite dimensional subspaces of 𝐻01 (Ω) and H(div; Ω),
respectively. Compared to the complex H(div; Ω) space used
in the 𝐻1 -Galerkin MFE method, we use the weaker (𝐿2 (Ω))2
space for our EMFE method.
For deriving some a priori error estimates of our method,
we first introduce the projection operator Pℎ and the new
expanded mixed elliptic projection operator Rℎ associated
with the coupled equations.
Lemma 5. One now defines a linear projection operator Pℎ :
𝐻01 (Ω) → 𝑉ℎ as
∀Vℎ ∈ 𝑉ℎ ,
and then
󵄩󵄩
󵄩
󵄩
󵄩
2󵄩 󵄩
󵄩󵄩𝛾 − Pℎ 𝛾󵄩󵄩󵄩𝐿2 (Ω) + ℎ󵄩󵄩󵄩𝛾 − Pℎ 𝛾󵄩󵄩󵄩𝐻1 ≤ 𝐶ℎ 󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐻2 ,
󵄩
󵄩󵄩
󵄩 󵄩
2 󵄩 󵄩
󵄩󵄩𝛾𝑡 − Pℎ 𝛾𝑡 󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶ℎ (󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐻2 + 󵄩󵄩󵄩𝛾𝑡 󵄩󵄩󵄩𝐻2 ) ,
󵄩
󵄩󵄩
󵄩 󵄩
󵄩 󵄩
2 󵄩 󵄩
󵄩󵄩𝛾𝑡𝑡 − Pℎ 𝛾𝑡𝑡 󵄩󵄩󵄩𝐿2 (Ω) ≤ 𝐶ℎ (󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐻2 + 󵄩󵄩󵄩𝛾𝑡 󵄩󵄩󵄩𝐻2 + 󵄩󵄩󵄩𝛾𝑡𝑡 󵄩󵄩󵄩𝐻2 ) .
(8)
(9)
Let (Rℎ 𝑢, Rℎ 𝜆, Rℎ 𝜎) : [0, 𝑇] → 𝑉ℎ × Wℎ × Wℎ be given
by the following new expanded mixed relations [29]
∀𝑔ℎ ∈ 𝑉ℎ ,
Lemma 7. There is a constant 𝐶 > 0 independent of the spatial
mesh parameter ℎ such that
󵄩
󵄩󵄩
2
󵄩󵄩𝑢 − Rℎ 𝑢󵄩󵄩󵄩 ≤ 𝐶ℎ ‖𝑢‖𝐻2 ,
󵄩
󵄩󵄩
2󵄩 󵄩
󵄩󵄩𝑢𝑡 − Rℎ 𝑢𝑡 󵄩󵄩󵄩 ≤ 𝐶ℎ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐻2 ,
󵄩
󵄩
󵄩󵄩
2󵄩
󵄩󵄩𝑢𝑡𝑡 − Rℎ 𝑢𝑡𝑡 󵄩󵄩󵄩 ≤ 𝐶ℎ 󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩𝐻2 ,
󵄩󵄩
󵄩
󵄩󵄩𝑢 − Rℎ 𝑢󵄩󵄩󵄩1 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜆‖(𝐻1 )2 ) .
∀zℎ ∈ Wℎ ,
(10b)
(𝜆 − Rℎ 𝜆, wℎ ) − (∇ (𝑢 − Rℎ 𝑢) , wℎ ) = 0,
∀wℎ ∈ Wℎ .
(10c)
(12)
In [29], we can obtain the detailed proof for Lemmas 6-7.
For a priori error estimates, we write the errors as
𝛾 − 𝛾ℎ = 𝛾 − Pℎ 𝛾 + Pℎ 𝛾 − 𝛾ℎ = 𝜓 + 𝜛;
𝜆 − 𝜆ℎ = 𝜆 − Rℎ 𝜆 + Rℎ 𝜆 − 𝜆ℎ = 𝛿 + 𝜃;
(13)
𝜎 − 𝜎ℎ = 𝜎 − Rℎ 𝜎 + Rℎ 𝜎 − 𝜎ℎ = 𝜌 + 𝜉.
Using (5a), (5b), (5c), (5d), (6a), (6b), (6c), (6d), (8), (10a),
(10b), and (10c) we can obtain the error equations
(𝜍𝑡 , Vℎ ) + (𝑏∇𝜛, ∇Vℎ ) = − (𝜂𝑡 , Vℎ ) ,
∀Vℎ ∈ 𝑉ℎ ,
∀𝑔ℎ ∈ 𝑉ℎ ,
(𝜛, 𝑔ℎ ) + (𝜉, ∇𝑔ℎ ) = − (𝜓, 𝑔ℎ ) ,
(14a)
(14b)
(𝜉, zℎ ) + (𝑎𝜃, zℎ ) = 0,
∀zℎ ∈ Wℎ ,
(14c)
(𝜃, wℎ ) − (∇𝜍, wℎ ) = 0,
∀wℎ ∈ Wℎ .
(14d)
We will prove the error estimates for semidiscrete scheme.
Theorem 8. Suppose that 𝛾, 𝛾𝑡 , 𝑢𝑡 ∈ 𝐿2 (𝐻2 (Ω)), 𝜆, 𝜎 ∈
𝐿∞ ((𝐻1 (Ω))2 ), and 𝑢, 𝛾 ∈ 𝐿∞ (𝐻2 (Ω)); then there exists a
constant 𝐶 > 0 independent of the spatial mesh parameter ℎ
such that
(10a)
(𝜎 − Rℎ 𝜎, zℎ ) + (𝑎 (𝜆 − Rℎ 𝜆) , zℎ ) = 0,
(11)
󵄩
󵄩󵄩
󵄩󵄩∇ (𝑢 − Rℎ 𝑢)󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜆‖(𝐻1 )2 ) .
𝑢 − 𝑢ℎ = 𝑢 − Rℎ 𝑢 + Rℎ 𝑢 − 𝑢ℎ = 𝜂 + 𝜍;
3. Semidiscrete Error Estimates
(𝜎 − Rℎ 𝜎, ∇𝑔ℎ ) = 0,
Lemma 6. There is a constant 𝐶 > 0 independent of the spatial
mesh parameter ℎ such that
󵄩
󵄩󵄩
󵄩󵄩𝜆 − Rℎ 𝜆󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝜆‖(𝐻1 )2 + ‖𝑢‖𝐻2 ) ,
󵄩
󵄩󵄩
󵄩󵄩𝜎 − Rℎ 𝜎󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝑢‖𝐻2 + ‖𝜎‖(𝐻1 )2 + ‖𝜆‖(𝐻1 )2 ) ,
Wℎ = {wℎ = (𝑤1ℎ , 𝑤2ℎ ) ∈ (𝐿2 (Ω)) |
(𝑏∇ (𝛾 − Pℎ 𝛾) , ∇Vℎ ) = 0,
Then the following two important lemmas based on the
new expanded mixed projection (10a)–(10c) hold.
󵄩
󵄩 󵄩
󵄩󵄩
2
󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ [‖𝑢‖𝐿∞ (𝐻2 ) + 󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 )
𝑡
+(∫ |||◼|||2 𝑑𝑠)
0
1/2
],
4
Advances in Mathematical Physics
󵄩
󵄩󵄩
󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩1 ≤ 𝐶ℎ [‖𝜆‖𝐿∞ ((𝐻1 )2 ) + ‖𝑢‖𝐿∞ (𝐻2 )
Using (18), (19), Cauchy-Schwarz inequality and Young
inequality, we have
𝑡
1/2
𝑡
󵄩 󵄩
+󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 ) + (∫ |||◼|||2 𝑑𝑠)
0
𝑎0 ‖∇𝜍‖2 + 2𝑏0 ∫ ‖∇𝜛‖2 𝑑𝑠
],
0
𝑡
󵄩
󵄩 󵄩
󵄩󵄩
󵄩󵄩𝜆 − 𝜆ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ [‖𝜆‖𝐿∞ ((𝐻1 )2 ) + 󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 )
𝑡
󵄩 󵄩
󵄩 󵄩
≤ 2𝐶1 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ‖∇𝜍‖ + 2𝐶2 ∫ 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 ‖∇𝜛‖ 𝑑𝑠
0
1/2
𝑡
+(∫ |||◼|||2 𝑑𝑠)
],
0
𝑡
󵄩 󵄩2
+ 𝐶 ∫ (𝐶12 ‖∇𝜍‖2 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠
󵄩
󵄩󵄩
󵄩󵄩𝜎 − 𝜎ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ [‖𝑢‖𝐿∞ (𝐻2 ) + ‖𝜎‖𝐿∞ ((𝐻1 )2 ) + ‖𝜆‖𝐿∞ ((𝐻1 )2 )
1/2
𝑡
󵄩 󵄩
+󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 ) + (∫ |||◼|||2 𝑑𝑠)
0
󵄩
󵄩2
(𝜍𝑡 , 𝜛) + 󵄩󵄩󵄩󵄩𝑏1/2 ∇𝜛󵄩󵄩󵄩󵄩 = − (𝜂𝑡 , 𝜛) ,
(16a)
− (𝜛, 𝜍𝑡 ) − (𝜉, ∇𝜍𝑡 ) = (𝜓, 𝜍𝑡 ) ,
(16b)
(𝜉, ∇𝜍𝑡 ) + (𝑎𝜃, ∇𝜍𝑡 ) = 0,
(16c)
1 𝑑 󵄩󵄩 1/2 󵄩󵄩2 1
󵄩󵄩𝑎 ∇𝜍󵄩󵄩 = (𝑎𝑡 ∇𝜍, ∇𝜍) .
󵄩
2 𝑑𝑡 󵄩
2
(16d)
Adding the above four equations and using Cauchy-Schwarz
inequality and Young’s inequality, we have
0
𝑡
󵄩 󵄩2
+ 𝐶 ∫ (𝐶12 ‖∇𝜍‖2 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠.
(20)
Using Gronwall’s lemma, we get
𝑡
‖∇𝜍‖2 + ∫ ‖∇𝜛‖2 𝑑𝑠
0
𝑡
󵄩 󵄩2
󵄩 󵄩2 󵄩 󵄩 2 󵄩 󵄩2
≤ 𝐶 (󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠) .
0
𝑡
‖𝜍‖2 + ∫ ‖𝜛‖2 𝑑𝑠
0
𝑡
󵄩2
1
(𝑎 ∇𝜍, ∇𝜍)
2 𝑡
𝑑
1
= − (𝜂𝑡 , 𝜛) +
(𝜓, 𝜍) − (𝜓𝑡 , 𝜍) + (𝑎𝑡 ∇𝜍, ∇𝜍)
𝑑𝑡
2
(21)
Substitute (19) into (21) to have
󵄩 󵄩2
󵄩 󵄩2 󵄩 󵄩 2 󵄩 󵄩2
≤ 𝐶 (󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠) .
0
1 𝑑 󵄩󵄩 1/2 󵄩 󵄩󵄩 1/2 󵄩
󵄩󵄩𝑎 ∇𝜍󵄩󵄩 + 󵄩󵄩𝑏 ∇𝜛󵄩󵄩
󵄩 󵄩
󵄩
2 𝑑𝑡 󵄩
≤
4𝐶2 𝑡 󵄩 󵄩2
󵄩2 𝑎
2
󵄩󵄩𝜓󵄩󵄩󵄩 + 0 ‖∇𝜍‖ + 2 ∫ 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 𝑑𝑠
𝑎0
2
𝑏0 0
0
Proof. Choose Vℎ = 𝜛, 𝑔ℎ = 𝜍𝑡 , zℎ = ∇𝜍𝑡 , and wℎ = ∇𝜍𝑡 in
(14a)–(14d) to obtain
= − (𝜂𝑡 , 𝜛) + (𝜓, 𝜍𝑡 ) +
≤
+ 𝑏0 ∫ ‖∇𝜛‖2 𝑑𝑠
where |||◼|||2 ≜ ‖𝑢𝑡 ‖2𝐻2 + ‖𝛾‖2𝐻2 + ‖𝛾𝑡 ‖2𝐻2 .
󵄩2
0
8𝐶12 󵄩󵄩
𝑡
],
(15)
− (𝑎𝜃, ∇𝜍𝑡 ) +
𝑡
󵄩 󵄩
󵄩 󵄩2
󵄩 󵄩
≤ 2 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ‖𝜍‖ + 2 ∫ 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 ‖𝜛‖ 𝑑𝑠 + 𝐶 ∫ (‖𝜍‖21 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠
0
0
(22)
Taking wℎ = 𝜃 in (14d) and using (22), we get
(17)
𝑑
󵄩 󵄩2
󵄩 󵄩
(𝜓, 𝜍) + 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 ‖𝜛‖ + 𝐶 (‖𝜍‖21 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) .
𝑑𝑡
𝑡
󵄩 󵄩2
󵄩 󵄩2 󵄩 󵄩 2 󵄩 󵄩2
‖𝜃‖2 ≤ 𝐶 (󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠) .
0
(23)
Choose zℎ = 𝜉 in (14c) and use (22) to obtain
𝑡
󵄩 󵄩2
󵄩 󵄩2 󵄩 󵄩 2 󵄩 󵄩2
‖𝜉‖2 ≤ 𝐶 (󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠) .
0
Integrate with respect to time from 0 to 𝑡 to obtain
(24)
𝑡
𝑎0 ‖∇𝜍‖2 + 2𝑏0 ∫ ‖∇𝜛‖2 𝑑𝑠
Combining Lemmas 5–7, (21)–(24) and using the triangle
inequality, we get the error estimates for Theorem 8.
0
𝑡
𝑡
󵄩 󵄩
󵄩 󵄩2
≤ 2 (𝜓, 𝜍) + 2 ∫ 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 ‖𝜛‖ 𝑑𝑠 + 𝐶 ∫ (‖𝜍‖21 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 ) 𝑑𝑠.
0
0
(18)
Noting that 𝜍, 𝜛 ∈
𝐻01 (Ω),
we use Poincaré inequality to have
‖𝜍‖1 ≤ 𝐶1 ‖∇𝜍‖ ,
‖𝜛‖ ≤ 𝐶2 ‖∇𝜛‖ .
(19)
Remark 9. From Theorem 8, we can see that
󵄩
󵄩
󵄩󵄩
󵄩
2
󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩𝐿∞ (𝐿2 ) + ℎ󵄩󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩𝐿∞ (𝐻1 ) = 𝑂 (ℎ ) ;
(25)
that is to say, a priori error estimates in 𝐿2 and 𝐻1 -norms for
the scalar unknown 𝑢 are optimal.
Advances in Mathematical Physics
5
Theorem 10. Suppose that 𝜕𝑗 𝑢/𝜕𝑡𝑗 , 𝜕𝑗 𝛾/𝜕𝑡𝑗 ∈ 𝐿2 (𝐻2 (Ω)) and
𝛾, 𝛾𝑡 , 𝑢𝑡 ∈ 𝐿∞ (𝐻2 (Ω)); then there exists a constant 𝐶 > 0
independent of the spatial mesh parameter ℎ such that
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝛾 − 𝛾ℎ 󵄩󵄩󵄩 + ℎ󵄩󵄩󵄩𝛾 − 𝛾ℎ 󵄩󵄩󵄩1
𝑡
󵄩 󵄩2
𝑏0 ‖∇𝜛‖2 + 2𝑎0 ∫ 󵄩󵄩󵄩∇𝜍𝑡 󵄩󵄩󵄩 𝑑𝑠
0
𝑡
󵄩 󵄩
≤ −2 (𝜂𝑡 , 𝜛) + 2𝐶2 ∫ 󵄩󵄩󵄩𝜂𝑡𝑡 󵄩󵄩󵄩 ‖∇𝜛‖ 𝑑𝑠 + 2 (𝜓𝑡 , 𝜍)
0
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
≤ 𝐶ℎ2 [󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿∞ (𝐻2 ) + 󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 ) + 󵄩󵄩󵄩𝛾𝑡 󵄩󵄩󵄩𝐿∞ (𝐻2 )
[
𝑡
1/2
󵄩󵄩 𝑗 󵄩󵄩2 󵄩󵄩 𝑗 󵄩󵄩2
󵄩󵄩 𝜕 𝛾 󵄩󵄩
󵄩󵄩 𝜕 𝑢 󵄩󵄩
󵄩
󵄩
󵄩
󵄩
+ (∫ ( ∑󵄩󵄩 𝑗 󵄩󵄩 + 󵄩󵄩 𝑗 󵄩󵄩 ) 𝑑𝑠) ] .
0
󵄩 𝜕𝑡 󵄩󵄩𝐻2 󵄩󵄩 𝜕𝑡 󵄩󵄩𝐻2
𝑗=0󵄩
]
(26)
𝑡
Integrate with respect to time from 0 to 𝑡 and use (19),
Cauchy-Schwarz inequality, and Young inequality to get
2
Proof. Differentiating (14b), (14c), and (14d) with respect to
time 𝑡, respectively, we get
(𝜍𝑡 , Vℎ ) + (𝑏∇𝜛, ∇Vℎ ) = − (𝜂𝑡 , Vℎ ) ,
(27a)
(𝜛𝑡 , 𝑔ℎ ) + (𝜉𝑡 , ∇𝑔ℎ ) = − (𝜓𝑡 , 𝑔ℎ ) ,
(27b)
(𝜉𝑡 , zℎ ) + (𝑎𝑡 𝜃, zℎ ) + (𝑎𝜃𝑡 , zℎ ) = 0,
(27c)
(𝜃𝑡 , wℎ ) − (∇𝜍𝑡 , wℎ ) = 0.
(27d)
Choose Vℎ = 𝜛𝑡 , 𝑔ℎ = 𝜍𝑡 , zℎ = ∇𝜍𝑡 , and wℎ = ∇𝜍𝑡 in (27a)–
(27d) to get
1 𝑑 󵄩󵄩 1/2 󵄩󵄩2
1
󵄩󵄩𝑏 ∇𝜛󵄩󵄩 = − (𝜂𝑡 , 𝜛𝑡 ) + (𝑏𝑡 ∇𝜛, ∇𝜛) ,
(𝜍𝑡 , 𝜛𝑡 ) +
󵄩
󵄩
2 𝑑𝑡
2
(28a)
− (𝜛𝑡 , 𝜍𝑡 ) − (𝜉𝑡 , ∇𝜍𝑡 ) = (𝜓𝑡 , 𝜍𝑡 ) ,
(28b)
(𝜉𝑡 , ∇𝜍𝑡 ) + (𝑎𝑡 𝜃, ∇𝜍𝑡 ) + (𝑎𝜃𝑡 , ∇𝜍𝑡 ) = 0,
(28c)
󵄩2
󵄩
− (𝑎𝜃𝑡 , ∇𝜍𝑡 ) + 󵄩󵄩󵄩󵄩𝑎1/2 ∇𝜍𝑡 󵄩󵄩󵄩󵄩 = 0.
(28d)
𝑡
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶 ∫ (󵄩󵄩󵄩𝜓𝑡𝑡 󵄩󵄩󵄩 + ‖𝜍‖2 + ‖𝜃‖2 ) 𝑑𝑠 + 𝑎0 ∫ 󵄩󵄩󵄩∇𝜍𝑡 󵄩󵄩󵄩 𝑑𝑠
0
0
≤
𝑡
8𝐶22 󵄩󵄩 󵄩󵄩2 𝑏0
󵄩 󵄩2
󵄩 󵄩2
2
2
󵄩󵄩𝜂𝑡 󵄩󵄩 + ‖∇𝜛‖ + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 + ‖𝜍‖ + 𝑎0 ∫ 󵄩󵄩󵄩∇𝜍𝑡 󵄩󵄩󵄩 𝑑𝑠
𝑏0
2
0
𝑡
󵄩 󵄩2
󵄩 󵄩2
+ 𝐶 ∫ (󵄩󵄩󵄩𝜓𝑡𝑡 󵄩󵄩󵄩 + ‖𝜍‖2 + ‖𝜃‖2 + 󵄩󵄩󵄩𝜂𝑡𝑡 󵄩󵄩󵄩 + ‖∇𝜛‖2 ) 𝑑𝑠.
0
(30)
Substituting (22) and (23) into (30) and using Gronwall
lemma, we get
𝑡
󵄩 󵄩2
‖∇𝜛‖2 + ∫ 󵄩󵄩󵄩∇𝜍𝑡 󵄩󵄩󵄩 𝑑𝑠
0
󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2
≤ 𝐶󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩
𝑡
󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2
+ 𝐶 ∫ (󵄩󵄩󵄩𝜂󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜂𝑡𝑡 󵄩󵄩󵄩
0
(31)
󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2
+󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡𝑡 󵄩󵄩󵄩 ) 𝑑𝑠.
Use (19) to obtain
󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2
‖𝜛‖2 ≤ 𝐶󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩
𝑡
󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2
+ 𝐶 ∫ (󵄩󵄩󵄩𝜂󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜂𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜂𝑡𝑡 󵄩󵄩󵄩
0
(32)
󵄩 󵄩 2 󵄩 󵄩2 󵄩 󵄩2
+ 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡𝑡 󵄩󵄩󵄩 ) 𝑑𝑠.
Combining Lemmas 5–7, (31), and (32) and using the triangle
inequality, we obtain the error estimates for Theorem 10.
Remark 11. From Theorem 10, we can see that
󵄩󵄩
󵄩
󵄩
󵄩
2
󵄩󵄩𝛾 − 𝛾ℎ 󵄩󵄩󵄩𝐿∞ (𝐿2 ) + ℎ󵄩󵄩󵄩𝛾 − 𝛾ℎ 󵄩󵄩󵄩𝐿∞ (𝐻1 ) = 𝑂 (ℎ ) ;
Adding the four equations for (28a)–(28d), we have
that is to say, a priori error estimates in 𝐿2 and 𝐻1 -norm for
the diffusion term 𝛾 are optimal.
1 𝑑 󵄩󵄩 1/2 󵄩󵄩2 󵄩󵄩 1/2 󵄩󵄩2
󵄩󵄩𝑏 ∇𝜛󵄩󵄩 + 󵄩󵄩𝑎 ∇𝜍𝑡 󵄩󵄩
󵄩
󵄩 󵄩
2 𝑑𝑡 󵄩
= − (𝜂𝑡 , 𝜛𝑡 ) + (𝜓𝑡 , 𝜍𝑡 ) − (𝑎𝑡 𝜃, ∇𝜍𝑡 ) +
=−
1
(𝑏 ∇𝜛, ∇𝜛)
2 𝑡
𝑑
𝑑
(𝜂 , 𝜛) + (𝜂𝑡𝑡 , 𝜛) +
(𝜓 , 𝜍)
𝑑𝑡 𝑡
𝑑𝑡 𝑡
− (𝜓𝑡𝑡 , 𝜍) − (𝑎𝑡 𝜃, ∇𝜍𝑡 ) +
(33)
1
(𝑏 ∇𝜛, ∇𝜛) .
2 𝑡
(29)
4. Fully Discrete Error Estimates
In the following analysis, we will derive a priori error
estimates based on fully discrete backward Euler scheme. 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 ).
6
Advances in Mathematical Physics
For (5a), (5b), (5c), and (5d), we have the following
equivalent formulation:
(
𝑢𝑛 − 𝑢𝑛−1
, V)+(𝑏𝑛 ∇𝛾𝑛 , ∇V) = (𝑓𝑛 +𝑅1𝑛 , V) ,
Δ𝑡
∀V ∈ 𝐻01 (Ω) ,
(34a)
(𝛾𝑛 , 𝑔) + (𝜎𝑛 , ∇𝑔) = 0,
Proof. Using (8), (10a), (10b), (10c), (34a), (34b), (34c), (34d),
(36a), (36b), (36c), and (36d) at 𝑡 = 𝑡𝑛 , we obtain the error
equations
∀𝑔 ∈ 𝐻01 (Ω) ,
(34b)
2
(𝜎𝑛 , z) + (𝑎𝑛 𝜆𝑛 , z) = 0,
∀z ∈ (𝐿2 (Ω)) ,
(𝜆𝑛 , w) − (∇𝑢𝑛 , w) = 0,
∀w ∈ (𝐿2 (Ω)) ,
(34c)
2
(34d)
(
𝜍𝑛 − 𝜍𝑛−1
, Vℎ ) + (𝑏𝑛 ∇𝜛𝑛 , ∇Vℎ )
Δ𝑡
𝜂𝑛 − 𝜂𝑛−1
= −(
, Vℎ ) + (𝑅1𝑛 , Vℎ ) ,
Δ𝑡
(𝜛𝑛 , 𝑔ℎ ) + (𝜉𝑛 , ∇𝑔ℎ ) = − (𝜓𝑛 , 𝑔ℎ ) ,
𝑛
(𝜉 , zℎ ) + (𝑎 𝜃 , zℎ ) = 0,
where
𝑅1𝑛 =
𝑛
𝑢 −𝑢
Δ𝑡
𝑛−1
− 𝑢𝑡 (𝑡𝑛 ) =
𝑡𝑛
1
∫ (𝑡 − 𝑠) 𝑢𝑡𝑡 𝑑𝑠.
Δ𝑡 𝑡𝑛−1 𝑛−1
(35)
Now a fully discrete procedure is to find (𝑢ℎ𝑛 , 𝛾ℎ𝑛 , 𝜆𝑛ℎ , 𝜎𝑛ℎ ) ∈ 𝑉ℎ ×
𝑉ℎ × Wℎ × Wℎ (𝑛 = 0, 1, . . . , 𝑀) such that
(
𝑛 𝑛
𝑢ℎ𝑛 − 𝑢ℎ𝑛−1
, Vℎ ) + (𝑏𝑛 ∇𝛾ℎ𝑛 , ∇Vℎ ) = (𝑓𝑛 , Vℎ ) ,
Δ𝑡
∀Vℎ ∈ 𝑉ℎ ,
(36a)
(𝛾ℎ𝑛 , 𝑔ℎ )
+
(𝜎𝑛ℎ , ∇𝑔ℎ )
= 0,
∀𝑔ℎ ∈ 𝑉ℎ ,
(36b)
(𝜎𝑛ℎ , zℎ ) + (𝑎𝑛 𝜆𝑛ℎ , zℎ ) = 0,
∀zℎ ∈ Wℎ ,
(36c)
(𝜆𝑛ℎ , wℎ ) − (∇𝑢ℎ𝑛 , wℎ ) = 0,
∀wℎ ∈ Wℎ .
𝛾𝑛 − 𝛾ℎ𝑛 = 𝛾𝑛 − Pℎ 𝛾𝑛 + Pℎ 𝛾𝑛 − 𝛾ℎ𝑛 = 𝜓𝑛 + 𝜛𝑛 ;
𝑢𝑛 − 𝑢ℎ𝑛 = 𝑢𝑛 − Rℎ 𝑢𝑛 + Rℎ 𝑢𝑛 − 𝑢ℎ𝑛 = 𝜂𝑛 + 𝜍𝑛 ;
𝜆 −
𝜆𝑛ℎ
𝑛
𝑛
𝑛
= 𝜆 − Rℎ 𝜆 + Rℎ 𝜆 −
𝜆𝑛ℎ
𝑛
𝑛
=𝛿 +𝜃 ;
𝜎𝑛 − 𝜎𝑛ℎ = 𝜎𝑛 − Rℎ 𝜎𝑛 + Rℎ 𝜎𝑛 − 𝜎𝑛ℎ = 𝜌𝑛 + 𝜉𝑛 .
We will prove the theorem for the fully discrete error
estimates.
Theorem 12. Suppose that 𝛾, 𝛾𝑡 , 𝑢𝑡 ∈ 𝐿2 (𝐻2 (Ω)), 𝜆, 𝜎 ∈
𝐿∞ ((𝐻1 (Ω))2 ), 𝛾 ∈ 𝐿∞ (𝐻2 (Ω)), and 𝑢𝑡𝑡 ∈ 𝐿2 (𝐿2 (Ω)); then
there exists a constant 𝐶 > 0 independent of the spatial mesh
parameter ℎ and the time step Δ𝑡 such that
󵄩
󵄩󵄩 𝐽
󵄩󵄩𝑢 − 𝑢ℎ𝐽 󵄩󵄩󵄩 ≤ 𝐶ℎ2 (‖𝑢‖𝐿∞ (𝐻2 ) + |||⧫|||) + 𝐶Δ𝑡󵄩󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐿2 ) ,
󵄩
󵄩
󵄩󵄩 𝐽
𝐽󵄩
󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝑢‖𝐿∞ (𝐻2 ) + ‖𝜆‖𝐿∞ ((𝐻1 )2 ) + |||⧫|||)
󵄩1
󵄩
󵄩 󵄩
+ 𝐶Δ𝑡󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩𝐿2 (𝐿2 ) ,
󵄩
󵄩󵄩 𝐽
󵄩󵄩𝜆 − 𝜆𝐽ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝜆‖𝐿∞ ((𝐻1 )2 ) + |||⧫|||) + 𝐶Δ𝑡󵄩󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐿2 ) ,
󵄩
󵄩
󵄩󵄩 𝐽
𝐽󵄩
󵄩󵄩𝜎 − 𝜎ℎ 󵄩󵄩󵄩 ≤ 𝐶ℎ (‖𝜎‖𝐿∞ ((𝐻1 )2 ) + |||⧫|||) + 𝐶Δ𝑡󵄩󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐿2 ) ,
󵄩
󵄩
(38)
where |||⧫||| ≜ ‖𝑢𝑡 ‖𝐿2 (𝐻2 ) + ‖𝛾‖𝐿∞ (𝐻2 ) + ‖𝛾‖𝐿2 (𝐻2 ) + ‖𝛾𝑡 ‖𝐿2 (𝐻2 ) .
∀𝑔ℎ ∈ 𝑉ℎ ,
∀zℎ ∈ Wℎ ,
∀wℎ ∈ Wℎ .
(39b)
(39c)
(39d)
󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩2
󵄩󵄩(𝑏 ) ∇𝜛 󵄩󵄩
󵄩󵄩
󵄩󵄩
= −(
𝜂𝑛 − 𝜂𝑛−1 𝑛
𝜍𝑛 − 𝜍𝑛−1 𝑛
,𝜛 ) − (
, 𝜛 ) + (𝑅1𝑛 , 𝜛𝑛 )
Δ𝑡
Δ𝑡
= (𝜉𝑛 ,
𝜂𝑛 − 𝜂𝑛−1 𝑛
∇𝜍𝑛 − ∇𝜍𝑛−1
)−(
, 𝜛 ) + (𝑅1𝑛 , 𝜛𝑛 )
Δ𝑡
Δ𝑡
+ (𝜓𝑛 ,
(37)
∀Vℎ ∈ 𝑉ℎ ,
Take Vℎ = 𝜛𝑛 in (39a), 𝑔ℎ = (𝜍𝑛 − 𝜍𝑛−1 )/Δ𝑡 in (39b), zℎ =
(∇𝜍𝑛 − ∇𝜍𝑛−1 )/Δ𝑡 in (39c), and wℎ = (∇𝜍𝑛 − ∇𝜍𝑛−1 )/Δ𝑡 in (39d)
to get
(36d)
For deriving the fully discrete error estimates, we now
decompose the errors as
𝑛
(𝜃𝑛 , wℎ ) − (∇𝜍𝑛 , wℎ ) = 0,
(39a)
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
= − (𝑎𝑛 𝜃𝑛 ,
𝜂𝑛 − 𝜂𝑛−1 𝑛
∇𝜍𝑛 − ∇𝜍𝑛−1
)−(
,𝜛 )
Δ𝑡
Δ𝑡
+ (𝑅1𝑛 , 𝜛𝑛 ) + (𝜓𝑛 ,
= − (𝑎𝑛
𝜂𝑛 − 𝜂𝑛−1 𝑛
∇𝜍𝑛 − ∇𝜍𝑛−1
, ∇𝜍𝑛 ) − (
,𝜛 )
Δ𝑡
Δ𝑡
+ (𝑅1𝑛 , 𝜛𝑛 ) + (𝜓𝑛 ,
=−
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
1 󵄩󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
(󵄩(𝑎 ) ∇𝜍 󵄩󵄩 − 󵄩󵄩󵄩(𝑎 ) ∇𝜍 󵄩󵄩󵄩 )
󵄩 󵄩
2Δ𝑡 󵄩󵄩
󵄩
−
1 󵄩󵄩 𝑛
󵄩2
󵄩󵄩𝑎 (∇𝜍𝑛 − ∇𝜍𝑛−1 )󵄩󵄩󵄩
󵄩
2Δ𝑡 󵄩
+
𝜂𝑛 − 𝜂𝑛−1 𝑛
1 𝑎𝑛 − 𝑎𝑛−1 𝑛−1 𝑛−1
(
∇𝜍 , ∇𝜍 ) − (
,𝜛 )
2
Δ𝑡
Δ𝑡
+ (𝑅1𝑛 , 𝜛𝑛 ) + (𝜓𝑛 ,
𝜍𝑛 − 𝜍𝑛−1
).
Δ𝑡
(40)
Advances in Mathematical Physics
7
By (40), we have
Use (42) and Gronwall lemma to get
𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩∇𝜍 󵄩󵄩 + Δ𝑡 ∑ 󵄩󵄩󵄩󵄩∇𝜛𝑛 󵄩󵄩󵄩󵄩2
󵄩 󵄩
󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩2
󵄩󵄩(𝑏 ) ∇𝜛 󵄩󵄩 + 1 󵄩󵄩󵄩󵄩𝑎𝑛 (∇𝜍𝑛 − ∇𝜍𝑛−1 )󵄩󵄩󵄩󵄩2
󵄩󵄩
󵄩󵄩
󵄩
2Δ𝑡 󵄩
𝑛=1
󵄩 󵄩2
≤ 𝐶󵄩󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩󵄩
1 󵄩󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
+
(󵄩(𝑎 ) ∇𝜍 󵄩󵄩 − 󵄩󵄩󵄩(𝑎 ) ∇𝜍 󵄩󵄩󵄩 )
󵄩 󵄩
2Δ𝑡 󵄩󵄩
󵄩
=
󵄩󵄩 𝑛
󵄩2 󵄩
󵄩2
󵄩󵄩 𝜂 − 𝜂𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩2
󵄩
󵄩
󵄩
󵄩󵄩 + 󵄩󵄩𝑅1 󵄩󵄩 ) .
+ 𝐶Δ𝑡 ∑ (󵄩󵄩
󵄩󵄩 + 󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩 Δ𝑡 󵄩󵄩 󵄩󵄩
󵄩
𝑛=1 󵄩
(43)
𝐽
𝜂𝑛 − 𝜂𝑛−1 𝑛
1 𝑎𝑛 − 𝑎𝑛−1 𝑛−1 𝑛−1
(
∇𝜍 , ∇𝜍 ) − (
,𝜛 )
2
Δ𝑡
Δ𝑡
Note that
𝜍𝑛 − 𝜍𝑛−1
+ (𝑅1𝑛 , 𝜛𝑛 ) + (𝜓𝑛 ,
)
Δ𝑡
=
𝑡
𝑛
󵄩 󵄩2
󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩𝑅1 󵄩󵄩 ≤ 𝐶Δ𝑡 ∫ 󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩 𝑑𝑠,
𝑡
𝑛−1
𝜂𝑛 − 𝜂𝑛−1 𝑛
1 1 𝑡𝑛
( ∫ 𝑎𝑡 (𝑠) 𝑑𝑠∇𝜍𝑛−1 , ∇𝜍𝑛−1 ) − (
,𝜛 )
2 Δ𝑡 𝑡𝑛−1
Δ𝑡
+ (𝑅1𝑛 , 𝜛𝑛 ) +
− (𝜍𝑛−1 ,
(𝜓𝑛 , 𝜍𝑛 ) − (𝜓𝑛−1 , 𝜍𝑛−1 )
Δ𝑡
𝑛
𝜓 −𝜓
Δ𝑡
𝑛−1
).
Multiplying by 2Δ𝑡, summing (41) from 𝑛 = 1 to 𝐽, and using
(19), the resulting equation becomes
𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩∇𝜍 󵄩󵄩 + Δ𝑡 ∑ 󵄩󵄩󵄩󵄩∇𝜛𝑛 󵄩󵄩󵄩󵄩2
󵄩 󵄩
𝑛=1
𝑡𝐽
󵄩 󵄩2
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐶 (󵄩󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 (𝑠)󵄩󵄩󵄩 ) 𝑑𝑠
𝑡
𝐽
󵄩 󵄩2
+ (Δ𝑡)2 ∫ 󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩 𝑑𝑠) .
𝑡
Combining (19) and (45), we obtain
𝑛=1
𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩𝜍 󵄩󵄩 + Δ𝑡 ∑ 󵄩󵄩󵄩󵄩𝜛𝑛 󵄩󵄩󵄩󵄩2
󵄩 󵄩
󵄩󵄩 𝑛
󵄩
𝐽
󵄩 𝜂 − 𝜂𝑛−1 󵄩󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩
󵄩󵄩 + 󵄩󵄩𝑅1 󵄩󵄩) 󵄩󵄩𝜛 󵄩󵄩
≤ 2Δ𝑡 ∑ (󵄩󵄩󵄩󵄩
󵄩
󵄩 Δ𝑡 󵄩󵄩
𝑛=1 󵄩
𝑛=1
𝑡𝐽
󵄩 󵄩2
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐶 (󵄩󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 (𝑠)󵄩󵄩󵄩 ) 𝑑𝑠
(46)
𝑡0
𝐽−1
󵄨
󵄨
󵄩 󵄩2
sup 󵄨󵄨󵄨𝑎𝑡 (𝑡)󵄨󵄨󵄨 Δ𝑡 ∑ 󵄩󵄩󵄩∇𝜍𝑛 󵄩󵄩󵄩
𝑡∈[𝑡𝑛−1 ,𝑡𝑛 ]
𝑛=1
𝑡
𝐽
󵄩 󵄩2
+ (Δ𝑡)2 ∫ 󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩 𝑑𝑠) .
𝑡
0
Taking wℎ = 𝜃 in (39d) and zℎ = 𝜉𝑛 in (39c) and using (45),
we have
󵄩󵄩󵄩𝜃𝐽 󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝜉𝐽 󵄩󵄩󵄩2
󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩
𝑛
󵄩
󵄩2
󵄩󵄩 𝑛−1 󵄩󵄩2 󵄩󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩
󵄩 󵄩󵄩 󵄩
󵄩
󵄩󵄩 ) + 󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩 󵄩󵄩󵄩𝜍𝐽 󵄩󵄩󵄩
󵄩
󵄩
+ Δ𝑡 ∑ (󵄩󵄩𝜍 󵄩󵄩 + 󵄩󵄩
󵄩
󵄩 󵄩󵄩 󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩
𝑛=1
𝐽
󵄩󵄩 𝑛
󵄩2
𝐽
𝐽
󵄩 𝜂 − 𝜂𝑛−1 󵄩󵄩󵄩 󵄩󵄩 𝑛 󵄩󵄩2
󵄩󵄩 + 󵄩󵄩𝑅1 󵄩󵄩 ) + 𝑏0 Δ𝑡 ∑ 󵄩󵄩󵄩󵄩∇𝜛𝑛 󵄩󵄩󵄩󵄩2
≤ 𝐶Δ𝑡 ∑ (󵄩󵄩󵄩󵄩
󵄩
󵄩 Δ𝑡 󵄩󵄩
𝑛=1 󵄩
𝑛=1
𝑡
𝐽
󵄩 󵄩2
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐶 (󵄩󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩󵄩 + ∫ (󵄩󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜓𝑡 (𝑠)󵄩󵄩󵄩 ) 𝑑𝑠
𝑡
0
(47)
𝑡
𝐽
󵄩 󵄩2
+(Δ𝑡)2 ∫ 󵄩󵄩󵄩𝑢𝑡𝑡 󵄩󵄩󵄩 𝑑𝑠) .
𝑡
󵄩
󵄩2
󵄩󵄩 𝑛−1 󵄩󵄩2 󵄩󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩
󵄩 󵄩2
󵄩󵄩 ) + 𝐶󵄩󵄩󵄩𝜓𝐽 󵄩󵄩󵄩
+ 𝐶Δ𝑡 ∑ (󵄩󵄩󵄩∇𝜍 󵄩󵄩󵄩 + 󵄩󵄩󵄩
󵄩
󵄩 󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩
𝑛=1
𝐽
𝑎0 󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩∇𝜍 󵄩󵄩 .
2󵄩 󵄩
(45)
𝑡0
0
𝐽
󵄩 󵄩2
󵄩
󵄩2
𝑎0 󵄩󵄩󵄩󵄩∇𝜍𝐽 󵄩󵄩󵄩󵄩 + 2𝑏0 Δ𝑡 ∑ 󵄩󵄩󵄩∇𝜛𝑛 󵄩󵄩󵄩
+
(44)
Substitute (44) into (43) to get
(41)
+
󵄩󵄩 𝑛
1 𝑡𝑛 󵄩󵄩
󵄩󵄩 𝜂 − 𝜂 󵄩󵄩󵄩
󵄩2
󵄩󵄩
󵄩󵄩 ≤
∫ 󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩 𝑑𝑠,
󵄩󵄩 Δ𝑡 󵄩󵄩
Δ𝑡
𝑡𝑛−1
󵄩
󵄩
2
󵄩󵄩 𝑛
󵄩
1 𝑡𝑛 󵄩󵄩
󵄩󵄩 𝜓 − 𝜓𝑛−1 󵄩󵄩󵄩
󵄩2
󵄩󵄩
󵄩󵄩 ≤
∫ 󵄩𝜓 (𝑠)󵄩󵄩 𝑑𝑠.
Δ𝑡
Δ𝑡 𝑡𝑛−1 󵄩 𝑡 󵄩
󵄩󵄩󵄩
󵄩󵄩󵄩
𝑛−1 󵄩2
0
Combining Lemmas 5–7, (45), (46), and (47) with the triangle
inequality, we accomplish the proof for Theorem 12.
(42)
In the following analysis, we will derive the optimal a
priori error estimates in 𝐿2 and 𝐻1 -norm for 𝛾.
8
Advances in Mathematical Physics
Table 1: 𝐿2 and 𝐻1 -norms and convergence order for 𝑢.
(ℎ, Δ𝑡)
‖𝑢 − 𝑢ℎ ‖𝐿∞ (𝐿2 (Ω))
‖𝑢 − 𝑢ℎ ‖𝐿∞ (𝐻1 (Ω))
Order
Order
(√2/8, 1/8)
2.8933𝑒 − 002
(√2/16, 1/16)
8.2158𝑒 − 003
1.8162
1.2986𝑒 − 001
0.8719
(√2/32, 1/32)
2.1012𝑒 − 003
1.9672
6.8466𝑒 − 002
0.9235
(√2/64, 1/64)
5.3511𝑒 − 004
1.9733
3.5245𝑒 − 002
0.9580
2.3766𝑒 − 001
Table 2: 𝐿2 and 𝐻1 -norms and convergence order for 𝛾.
(ℎ, Δ𝑡)
‖𝛾 − 𝛾ℎ ‖𝐿∞ (𝐿2 (Ω))
‖𝛾 − 𝛾ℎ ‖𝐿∞ (𝐻1 (Ω))
Order
Order
(√2/8, 1/8)
3.4238𝑒 − 001
(√2/16, 1/16)
9.2701𝑒 − 002
1.8849
2.5429𝑒 + 000
0.8381
(√2/32, 1/32)
2.3840𝑒 − 002
1.9592
1.3493𝑒 + 000
0.9143
(√2/64, 1/64)
5.8649𝑒 − 003
2.0232
6.9558𝑒 − 001
0.9559
4.5457𝑒 + 000
Theorem 13. Suppose that 𝛾, 𝛾𝑡 , 𝑢𝑡 ∈ 𝐿2 (𝐻2 (Ω)), 𝜕2 𝑢/𝜕𝑡2 ,
𝜕4 𝑢/𝜕𝑡4 ∈ 𝐿2 (𝐿2 (Ω)), 𝛾 ∈ 𝐿∞ (𝐻2 (Ω)), and 𝜕3 𝑢/𝜕𝑡3 ∈
𝐿∞ (𝐿2 (Ω)); then there exists a constant 𝐶 > 0 independent of
the spatial mesh parameter ℎ and time discretization parameter
Δ𝑡 such that
󵄩󵄩 𝐽
󵄩 󵄩
𝐽󵄩
2−𝑗 󵄩 󵄩
󵄩
󵄩󵄩󵄩𝛾 − 𝛾ℎ 󵄩󵄩󵄩𝑗 ≤ 𝐶ℎ (󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿∞ (𝐻2 ) + 󵄩󵄩󵄩𝛾󵄩󵄩󵄩𝐿2 (𝐻2 )
𝑢 at 𝑡 = 1
0.2
0.15
󵄩 󵄩
󵄩 󵄩
+󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝐿2 (𝐻2 ) + 󵄩󵄩󵄩𝛾𝑡 󵄩󵄩󵄩𝐿2 (𝐻2 ) )
0.1
1
0.05
0
1
0.8
0.6
0.8
0.6
0.4
0.4
0.2
02
0.2
0
0
Figure 1: Surface for exact solution 𝑢.
󵄩󵄩 3 󵄩󵄩
󵄩󵄩 2 󵄩󵄩
󵄩𝜕 𝑢󵄩
󵄩𝜕 𝑢󵄩
+ 󵄩󵄩󵄩󵄩 3 󵄩󵄩󵄩󵄩
+ Δ𝑡 (󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 ) 󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿∞ (𝐿2 )
󵄩󵄩 4 󵄩󵄩
󵄩𝜕 𝑢󵄩
+󵄩󵄩󵄩󵄩 4 󵄩󵄩󵄩󵄩
) , 𝑗 = 0, 1.
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 )
Proof. Use (39a)–(39d) to get
(
𝜂𝑛 − 𝜂𝑛−1
= −(
, Vℎ ) + (𝑅1𝑛 , Vℎ ) ,
Δ𝑡
𝑢ℎ at 𝑡 = 1, Δ𝑡 = 1/16
(
0.15
0.1
1
0.05
(
0.8
0.6
0.8
0.6
0.4
0.2
0.2
0
0
Figure 2: Surface for numerical solution 𝑢ℎ .
(
∀Vℎ ∈ 𝑉ℎ ,
(49c)
∀zℎ ∈ Wℎ ,
∇𝜍𝑛 − ∇𝜍𝑛−1
𝜃𝑛 − 𝜃𝑛−1
, wℎ ) − (
, wℎ )
Δ𝑡
Δ𝑡
= 0,
(49b)
∀𝑔ℎ ∈ 𝑉ℎ ,
𝑎𝑛 𝜃𝑛 − 𝑎𝑛−1 𝜃𝑛−1
𝜉𝑛 − 𝜉𝑛−1
, zℎ ) + (
, zℎ )
Δ𝑡
Δ𝑡
= 0,
0.4
(49a)
𝜉𝑛 − 𝜉𝑛−1
𝜛𝑛 − 𝜛𝑛−1
, 𝑔ℎ ) + (
, ∇𝑔ℎ )
Δ𝑡
Δ𝑡
𝜓𝑛 − 𝜓𝑛−1
= −(
, 𝑔ℎ ) ,
Δ𝑡
0.2
0
1
𝜍𝑛 − 𝜍𝑛−1
, Vℎ ) + (𝑏𝑛 ∇𝜛𝑛 , ∇Vℎ )
Δ𝑡
(48)
∀wℎ ∈ Wℎ .
(49d)
Advances in Mathematical Physics
9
𝜎1 (left) and 𝜎2 (right) at 𝑡 = 1
1
1
0.5
0.5
0
0
1
−0.5
1
−0.5
−1
1
−1
1
0.5
0.5
0.5
0.5
0 0
0
0
Figure 3: Surface for exact solution 𝜎 = (𝜎1 , 𝜎2 ).
𝜎1ℎ (left) and 𝜎2ℎ (right) at 𝑡 = 1, Δ𝑡 = 1/16
1
1
0.5
0.5
0
0
1
−0.5
−1
1
0.5
1
−0.5
−1
1
0.5
0.5
0.5
0
0
0
0
Figure 4: Surface for numerical solution 𝜎ℎ = (𝜎1ℎ , 𝜎2ℎ ).
Taking Vℎ = (𝜛𝑛 − 𝜛𝑛−1 )/Δ𝑡, 𝑔ℎ = (𝜍𝑛 − 𝜍𝑛−1 )/Δ𝑡, zℎ = (∇𝜍𝑛 −
∇𝜍𝑛−1 )/Δ𝑡, and wℎ = (∇𝜍𝑛 − ∇𝜍𝑛−1 )/Δ𝑡, respectively, in (49a)–
(49d), we get
1 󵄩󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
(󵄩(𝑏 ) ∇𝜛 󵄩󵄩 − 󵄩󵄩󵄩(𝑏 ) ∇𝜛 󵄩󵄩󵄩 )
󵄩 󵄩
2Δ𝑡 󵄩󵄩
󵄩
1 󵄩󵄩 𝑛
󵄩2
󵄩󵄩𝑏 (∇𝜛𝑛 − ∇𝜛𝑛−1 )󵄩󵄩󵄩
+
󵄩
2Δ𝑡 󵄩
= (𝑏𝑛 ∇𝜛𝑛 , ∇
+
𝜛𝑛 − 𝜛𝑛−1
)
Δ𝑡
1 𝑏𝑛 − 𝑏𝑛−1 𝑛−1
(
∇𝜛 , ∇𝜛𝑛−1 )
2
Δ𝑡
= −(
𝜍𝑛 − 𝜍𝑛−1 𝜛𝑛 − 𝜛𝑛−1
,
)
Δ𝑡
Δ𝑡
10
Advances in Mathematical Physics
𝜆1 (left) and 𝜆2 (right) at 𝑡 = 1
0.5
0.5
0
0
1
1
−0.5
1
0.5
−0.5
1
0.5
0.5
0.5
0
0
0
0
Figure 5: Surface for exact solution 𝜆 = (𝜆1 , 𝜆2 ).
Table 3: (𝐿2 )2 -norm and convergence order for 𝜆 and 𝜎.
(ℎ, Δ𝑡)
‖𝜎 − 𝜎ℎ ‖𝐿∞ ((𝐿2 (Ω))2 )
‖𝜆 − 𝜆ℎ ‖𝐿∞ ((𝐿2 (Ω))2 )
Order
Order
(√2/8, 1/8)
2.3958𝑒 − 001
(√2/16, 1/16)
1.3011𝑒 − 001
0.8808
1.2960𝑒 − 001
0.8641
(√2/32, 1/32)
6.8501𝑒 − 002
0.9255
6.8434𝑒 − 002
0.9213
(√2/64, 1/64)
3.5250𝑒 − 002
0.9585
3.5242𝑒 − 002
0.9574
−(
𝜂𝑛 − 𝜂𝑛−1 𝜛𝑛 − 𝜛𝑛−1
,
)
Δ𝑡
Δ𝑡
+ (𝑅1𝑛 ,
+ (𝑅1𝑛 ,
𝜛𝑛 − 𝜛𝑛−1
1 𝑏𝑛 − 𝑏𝑛−1 𝑛−1
)+ (
∇𝜛 , ∇𝜛𝑛−1 )
Δ𝑡
2
Δ𝑡
𝜓𝑛 − 𝜓𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜉𝑛 − 𝜉𝑛−1 ∇𝜍𝑛 − ∇𝜍𝑛−1
=(
,
)+(
,
)
Δ𝑡
Δ𝑡
Δ𝑡
Δ𝑡
+
+
(𝑅1𝑛 ,
+(
󵄩󵄩
𝑛
𝑛−1 󵄩
󵄩󵄩2
󵄩
1/2 ∇𝜍 − ∇𝜍
󵄩󵄩
= −󵄩󵄩󵄩󵄩(𝑎𝑛 )
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩
𝑎𝑛 − 𝑎𝑛−1 𝑛−1 ∇𝜍𝑛 − ∇𝜍𝑛−1
𝜃 ,
)
Δ𝑡
Δ𝑡
𝑛
+(
𝜓𝑛 − 𝜓𝑛−1 𝜍𝑛 − 𝜍𝑛−1
,
)
Δ𝑡
Δ𝑡
𝑛−1
𝑛−1
𝑎 −𝑎
Δ𝑡
𝑛
1 𝑏𝑛 − 𝑏𝑛−1 𝑛−1
(
∇𝜛 , ∇𝜛𝑛−1 )
2
Δ𝑡
−(
∇𝜛𝑛−1 , ∇𝜛𝑛−1 )
𝑎𝑛 (𝜃𝑛 − 𝜃𝑛−1 ) ∇𝜍𝑛 − ∇𝜍𝑛−1
= −(
,
)
Δ𝑡
Δ𝑡
−(
+
𝜂𝑛 − 𝜂𝑛−1 𝜛𝑛 − 𝜛𝑛−1
𝜛𝑛 − 𝜛𝑛−1
)−(
,
)
Δ𝑡
Δ𝑡
Δ𝑡
𝜂𝑛 − 𝜂𝑛−1 𝜛𝑛 − 𝜛𝑛−1
𝜛𝑛 − 𝜛𝑛−1
)−(
,
)
Δ𝑡
Δ𝑡
Δ𝑡
1 𝑏 −𝑏
(
2
Δ𝑡
𝑛
2.3589𝑒 − 001
𝜓 −𝜓
Δ𝑡
𝜃𝑛−1 ,
𝑛−1
,
𝑛
∇𝜍 − ∇𝜍
Δ𝑡
𝜍𝑛 − 𝜍𝑛−1
)
Δ𝑡
𝑛−1
)
+ (𝑅1𝑛 ,
−(
+
𝜛𝑛 − 𝜛𝑛−1
)
Δ𝑡
𝜂𝑛 − 𝜂𝑛−1 𝜛𝑛 − 𝜛𝑛−1
,
)
Δ𝑡
Δ𝑡
1 𝑏𝑛 − 𝑏𝑛−1 𝑛−1
(
∇𝜛 , ∇𝜛𝑛−1 ) .
2
Δ𝑡
(50)
Advances in Mathematical Physics
11
𝜆1ℎ (left) and 𝜆2ℎ (right) at 𝑡 = 1, Δ𝑡 = 1/16
0.5
0.5
0
0
1
1
−0.5
1
−0.5
1
0.5
0.5
0.5
0.5
0
0
0
0
Figure 6: Surface for numerical solution 𝜆ℎ = (𝜆1ℎ , 𝜆2ℎ ).
Multiply (51) by 2Δ𝑡, sum from 𝑛 = 1 to 𝐽, and use (19) to get
By using (50), we obtain
1 󵄩󵄩󵄩 𝑛 1/2 𝑛 󵄩󵄩󵄩2 󵄩󵄩󵄩 𝑛−1 1/2 𝑛−1 󵄩󵄩󵄩2
(󵄩(𝑏 ) ∇𝜛 󵄩󵄩 − 󵄩󵄩󵄩(𝑏 ) ∇𝜛 󵄩󵄩󵄩 )
󵄩 󵄩
2Δ𝑡 󵄩󵄩
󵄩
+
𝐽 󵄩
󵄩󵄩 ∇𝜍𝑛 − ∇𝜍𝑛−1 󵄩󵄩󵄩2
󵄩
󵄩2
󵄩󵄩
𝑏0 󵄩󵄩󵄩󵄩∇𝜛𝐽 󵄩󵄩󵄩󵄩 + 2𝑎0 Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩
󵄩󵄩
𝑛=1󵄩
1 󵄩󵄩 𝑛
󵄩2
󵄩󵄩𝑏 (∇𝜛𝑛 − ∇𝜛𝑛−1 )󵄩󵄩󵄩
󵄩
2Δ𝑡 󵄩
󵄩󵄩
󵄩󵄩 𝑛
󵄩
𝐽
󵄩 𝑎 − 𝑎𝑛−1 𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 ∇𝜍𝑛 − ∇𝜍𝑛−1 󵄩󵄩󵄩
󵄩󵄩
≤ 2Δ𝑡 ∑ (󵄩󵄩󵄩󵄩
𝜃 󵄩󵄩󵄩 󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩 󵄩󵄩
󵄩 Δ𝑡
󵄩
𝑛=1 󵄩
󵄩󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜍𝑛 − 𝜍𝑛−1 󵄩󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩)
+ 󵄩󵄩󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩 󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
󵄩 𝑅𝑛 − 𝑅𝑛−1 󵄩󵄩
𝐽 󵄩
󵄩 󵄩󵄩 󵄩
󵄩
1 󵄩
󵄩󵄩 󵄩󵄩󵄩𝜛𝑛−1 󵄩󵄩󵄩
+ 2 󵄩󵄩󵄩󵄩𝜛𝐽 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑅1𝐽 󵄩󵄩󵄩󵄩 + 2Δ𝑡 ∑ 󵄩󵄩󵄩󵄩 1
󵄩󵄩 󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
𝑛=1 󵄩
󵄩󵄩 𝐽
󵄩
󵄩 𝜂 − 𝜂𝐽−1 󵄩󵄩󵄩 󵄩󵄩 𝐽 󵄩󵄩
+ 2 󵄩󵄩󵄩󵄩
𝑐󵄩󵄩󵄩 󵄩󵄩󵄩𝜛 󵄩󵄩󵄩
󵄩󵄩 Δ𝑡
󵄩󵄩
󵄩
𝐽 󵄩
󵄩󵄩 𝜂𝑛 − 2𝜂𝑛−1 + 𝜂𝑛−2 󵄩󵄩󵄩 󵄩󵄩 𝑛−1 󵄩󵄩
󵄩󵄩 󵄩󵄩𝜛 󵄩󵄩
+ 2Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩 󵄩
󵄩
(Δ𝑡)2
󵄩
󵄩
𝑛=1 󵄩
󵄩󵄩
󵄩2
󵄩󵄩 𝑛 1/2 ∇𝜍𝑛 − ∇𝜍𝑛−1 󵄩󵄩󵄩
󵄩
󵄩󵄩
+ 󵄩󵄩(𝑎 )
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
= −(
𝑎𝑛 − 𝑎𝑛−1 𝑛−1 𝜍𝑛 − 𝜍𝑛−1
𝜃 ,∇
)
Δ𝑡
Δ𝑡
+(
+
(𝜛𝑛 , 𝑅1𝑛 ) − (𝜛𝑛−1 , 𝑅1𝑛−1 )
Δ𝑡
−(
−
𝑅1𝑛
𝑅1𝑛−1
−
Δ𝑡
, 𝜛𝑛−1 )
𝐽
((𝜂𝑛 − 𝜂𝑛−1 ) /Δ𝑡, 𝜛𝑛 ) − ((𝜂𝑛−1 − 𝜂𝑛−2 ) /Δ𝑡, 𝜛𝑛−1 )
Δ𝑡
+(
+
𝜓𝑛 − 𝜓𝑛−1 𝜍𝑛 − 𝜍𝑛−1
,
)
Δ𝑡
Δ𝑡
𝜂𝑛 − 2𝜂𝑛−1 + 𝜂𝑛−2
(Δ𝑡)2
󵄩
󵄩2
+ 𝐶Δ𝑡 ∑ 󵄩󵄩󵄩󵄩∇𝜛𝑛−1 󵄩󵄩󵄩󵄩
𝑛=1
󵄩
󵄩2
󵄩󵄩 𝑛−1 󵄩󵄩 󵄩󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
󵄩
≤ 𝐶Δ𝑡 ∑ (󵄩󵄩𝜃 󵄩󵄩 + 󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
𝑛=1
𝐽
, 𝜛𝑛−1 )
󵄩2 󵄩 𝑛
󵄩󵄩 𝑛
𝑛−1 󵄩2
󵄩 𝜂 − 2𝜂𝑛−1 + 𝜂𝑛−2 󵄩󵄩󵄩 󵄩󵄩󵄩 𝑅1 − 𝑅1 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 )
+ 󵄩󵄩󵄩󵄩
+
󵄩󵄩 󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩 󵄩󵄩
󵄩󵄩
(Δ𝑡)2
󵄩
1 𝑏𝑛 − 𝑏𝑛−1 𝑛−1
(
∇𝜛 , ∇𝜛𝑛−1 ) .
2
Δ𝑡
(51)
𝐽 󵄩
󵄩󵄩 ∇𝜍𝑛 − ∇𝜍𝑛−1 󵄩󵄩󵄩2
󵄩󵄩
+ 𝑎0 Δ𝑡 ∑ 󵄩󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩
𝑛=1󵄩
12
Advances in Mathematical Physics
󵄩
󵄩2
𝑏 󵄩
󵄩2
󵄩󵄩 𝐽 󵄩󵄩2 󵄩󵄩󵄩 𝜂𝐽 − 𝜂𝐽−1 󵄩󵄩󵄩
󵄩
󵄩󵄩 ) + 0 󵄩󵄩󵄩∇𝜛𝐽 󵄩󵄩󵄩
󵄩
󵄩
+ 𝐶 (󵄩󵄩𝑅1 󵄩󵄩 + 󵄩󵄩
󵄩
󵄩
󵄩
Δ𝑡
2
󵄩󵄩
󵄩󵄩
Use (55) and Poincaré inequality (19) to get
𝑡𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩𝜛 󵄩󵄩 ≤ 𝐶 (󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩2𝐿∞ 𝐿2 + ∫ (󵄩󵄩󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩𝜓𝑡 (𝑠)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
(
)
󵄩 󵄩
𝑡
𝐽
󵄩
󵄩2
+ 𝐶Δ𝑡 ∑ 󵄩󵄩󵄩󵄩∇𝜛𝑛−1 󵄩󵄩󵄩󵄩 .
0
𝑛=1
(52)
󵄩󵄩 3 󵄩󵄩2
󵄩󵄩 2 󵄩󵄩2
󵄩𝜕 𝑢󵄩
󵄩𝜕 𝑢󵄩
+ (Δ𝑡)2 [󵄩󵄩󵄩󵄩 3 󵄩󵄩󵄩󵄩
+ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿∞ (𝐿2 ) 󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 )
󵄩󵄩 4 󵄩󵄩2
󵄩𝜕 𝑢󵄩
+ 󵄩󵄩󵄩󵄩 4 󵄩󵄩󵄩󵄩
]) .
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 )
Use Gronwall lemma to obtain
󵄩
󵄩2
𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩 󵄩󵄩 𝜓𝑛 − 𝜓𝑛−1 󵄩󵄩󵄩
󵄩
󵄩󵄩∇𝜛 󵄩󵄩 ≤ 𝐶Δ𝑡 ∑ (󵄩󵄩󵄩𝜃𝑛−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩
󵄩
󵄩 󵄩󵄩
󵄩
Δ𝑡
󵄩
󵄩
𝑛=1
(56)
󵄩󵄩 𝑅𝑛 − 𝑅𝑛−1 󵄩󵄩2 󵄩󵄩 𝑛
𝑛−1
𝑛−2 󵄩
󵄩2
󵄩
1 󵄩
󵄩󵄩 + 󵄩󵄩󵄩 𝜂 − 2𝜂 + 𝜂 󵄩󵄩󵄩 )
+󵄩󵄩󵄩󵄩 1
󵄩󵄩
󵄩󵄩 󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩
(Δ𝑡)2
󵄩
󵄩2
󵄩󵄩 𝐽 󵄩󵄩2 󵄩󵄩󵄩 𝜂𝐽 − 𝜂𝐽−1 󵄩󵄩󵄩
󵄩󵄩 ) .
+ 𝐶 (󵄩󵄩󵄩𝑅1 󵄩󵄩󵄩 + 󵄩󵄩󵄩
󵄩󵄩 Δ𝑡 󵄩󵄩󵄩
(53)
Note that
Combining Lemmas 6-7, (55), (56) and using the triangle
inequality, we complete the proof.
5. Numerical Results
In this section, we would like to give some numerical
results for the coupling method of EMFE method and FE
method proposed and analyzed in this paper. We consider
the following initial-boundary value problem of fourth-order
parabolic system:
𝑢𝑡 + ∇ ⋅ (𝑏∇ (∇ ⋅ (𝑎 (𝑡) ∇𝑢))) = 𝑓 (x, 𝑡) ,
󵄩󵄩 𝑅𝑛 −
󵄩󵄩 1
󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩
󵄩󵄩 𝑛
󵄩󵄩 𝑢 − 2𝑢𝑛−1 + 𝑢𝑛−2 𝑢𝑡 (𝑡𝑛 ) − 𝑢𝑡 (𝑡𝑛−1 ) 󵄩󵄩󵄩
󵄩󵄩
−
󵄩󵄩 = 󵄩󵄩󵄩
󵄩󵄩
Δ𝑡
󵄩󵄩
󵄩󵄩
(Δ𝑡)2
󵄩
2
𝑅1𝑛−1 󵄩󵄩󵄩󵄩
󵄩
󵄩2
󵄩󵄩 𝑛
󵄩󵄩2
󵄩 𝑢 − 2𝑢𝑛−1 + 𝑢𝑛−2
󵄩󵄩
󵄩
= 󵄩󵄩󵄩󵄩
−
𝑢
(𝑡
)
𝑡𝑡
Δ
2
∗ 󵄩
󵄩󵄩
󵄩󵄩
(Δ𝑡)
󵄩
2
󵄩󵄩 3 󵄩󵄩
󵄩󵄩 4 󵄩󵄩
󵄩𝜕 𝑢󵄩
󵄩󵄩 𝜕 𝑢 󵄩󵄩
󵄩󵄩 4 󵄩󵄩 𝑑𝑠 + Δ𝑡󵄩󵄩󵄩 3 󵄩󵄩󵄩
]
󵄩󵄩 𝜕𝑡 󵄩󵄩 ∞ 2
󵄩 𝜕𝑡 󵄩󵄩
𝑡𝑛−2 󵄩
󵄩
󵄩
󵄩
󵄩𝐿 (𝐿 )
𝑡𝑛
󵄩󵄩 4 󵄩
󵄩󵄩 3 󵄩
󵄩󵄩 𝜕 𝑢 󵄩󵄩
󵄩𝜕 𝑢󵄩
󵄩󵄩 4 󵄩󵄩 𝑑𝑠 + (Δ𝑡)2 󵄩󵄩󵄩 3 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 𝜕𝑡 󵄩󵄩 ∞ 2 ,
𝜕𝑡
󵄩
󵄩
𝑡𝑛−2 󵄩
󵄩
󵄩
󵄩𝐿 (𝐿 )
≤ 𝐶(Δ𝑡)2 ∫
𝑡𝑛
𝑢 (x, 0) = sin (𝜋𝑥1 ) sin (𝜋𝑥2 ) ,
(x, 𝑡) ∈ 𝜕Ω × 𝐽,
x = (𝑥1 , 𝑥2 ) ∈ Ω,
(57)
≤ 𝐶[Δ𝑡 ∫
󵄩2
𝑢 (x, 𝑡) = Δ𝑢 (x, 𝑡) = 0,
(x, 𝑡) ∈ Ω × 𝐽,
󵄩2
where 𝑡Δ ∗ ∈ (𝑡𝑛−1 , 𝑡𝑛 ) .
where Ω = [0, 1] × [0, 1], 𝐽 = (0, 1], 𝑎(𝑡) = 1 + 𝑡2 , 𝑏 = 1,
and 𝑓(x, 𝑡) is chosen so that the exact solution for the scalar
unknown function is
𝑢 (x, 𝑡) = 𝑒−2𝑡 sin (𝜋𝑥1 ) sin (𝜋𝑥2 ) ,
(58)
its exact gradient is
𝜆 = ∇𝑢 = 𝑒−2𝑡 𝜋 [
(54)
cos (𝜋𝑥1 ) sin (𝜋𝑥2 )
sin (𝜋𝑥1 ) cos (𝜋𝑥2 )
],
(59)
its exact flux function is
cos (𝜋𝑥1 ) sin (𝜋𝑥2 )
],
𝜎 = −𝑎𝜆 = 𝑒−2𝑡 (1 + 𝑡2 ) 𝜋 [
sin (𝜋𝑥1 ) cos (𝜋𝑥2 )
Substitute (44), (47) and (54) into (53) to obtain
𝑡𝐽
󵄩󵄩 𝐽 󵄩󵄩2
󵄩󵄩∇𝜛 󵄩󵄩 ≤ 𝐶 (󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩2𝐿∞ 𝐿2 + ∫ (󵄩󵄩󵄩󵄩𝜂𝑡 (𝑠)󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩𝜓𝑡 (𝑠)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
( )
󵄩
󵄩
𝑡
0
󵄩󵄩 2 󵄩󵄩2
󵄩󵄩 3 󵄩󵄩2
󵄩𝜕 𝑢󵄩
󵄩𝜕 𝑢󵄩
+ 󵄩󵄩󵄩󵄩 2 󵄩󵄩󵄩󵄩
+ (Δ𝑡) [󵄩󵄩󵄩󵄩 3 󵄩󵄩󵄩󵄩
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿∞ (𝐿2 ) 󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 )
(60)
and the diffusion term is
𝛾 = ∇ ⋅ 𝜎 = 2𝜋2 𝑒−2𝑡 (1 + 𝑡2 ) sin (𝜋𝑥1 ) sin (𝜋𝑥2 ) .
2
󵄩󵄩 4 󵄩󵄩2
󵄩𝜕 𝑢󵄩
+󵄩󵄩󵄩󵄩 4 󵄩󵄩󵄩󵄩
]) .
󵄩󵄩 𝜕𝑡 󵄩󵄩𝐿2 (𝐿2 )
(55)
(61)
Dividing the domain Ω into the triangulations of mesh
size ℎ uniformly and choosing the piecewise linear space 𝑉ℎ
with index 𝑘 = 1 and ℎ = √2Δ𝑡 = √2/8, √2/16, √2/32,
√2/64, we obtain the optimal a priori error estimates in 𝐿2
and 𝐻1 -norm for the scalar unknown 𝑢 in Table 1. Similarly,
from Table 2, we can find that both ‖𝛾 − 𝛾ℎ ‖𝐿∞ (𝐻1 (Ω)) and
Advances in Mathematical Physics
13
(𝐿2 (Ω))2 space; therefore, the regularity requirement on the
gradient solution 𝜆 = ∇𝑢 is reduced. We obtain the optimal
priori error estimates in 𝐿2 and 𝐻1 -norm for both the
scalar unknown 𝑢 and the diffusion term 𝛾 and the priori
error estimates in (𝐿2 )2 -norm for its gradient 𝜆 and its flux
𝜎. Finally, a numerical example is provided to verify the
efficiency of our methods. At the same time, we apply the
new coupling method based on new EMFE method and
FE method to solving the fourth-order eigenvalue problems,
which will be shown in another article.
In the near future, we will study the new expanded mixed
finite element method for other partial differential equations,
such as fourth-order wave equations. Moreover, we will apply
some new techniques based on the combination of two-grid
methods [36, 37] and new (expanded) mixed methods [29–
32, 38] for solving fourth-order nonlinear elliptic equation
and fourth-order nonlinear parabolic equations.
𝛾 at 𝑡 = 1
6
4
1
2
0
1
0.8
0.6
0.8
0.4
0.6
0.4
0.2
0.2
0
0
Figure 7: Surface for exact solution 𝛾.
Acknowledgments
𝛾ℎ at 𝑡 = 1, Δ𝑡 = 1/16
6
4
1
2
The authors thank the referees and editor for their valuable suggestions and comments, which greatly improve the
paper. This work is supported by National Natural Science
Fund (11061021), Natural Science Fund of Inner Mongolia
Autonomous Region (2012MS0108, 2012MS0106), Scientific
Research Projection of Higher Schools of Inner Mongolia
(NJZZ12011, NJ10006, NJZY13199), and Program of Higherlevel talents of Inner Mongolia University (125119, 30105125132).
0.8
0
1
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0
Figure 8: Surface for numerical solution 𝛾ℎ .
‖𝛾 − 𝛾ℎ ‖𝐿∞ (𝐿2 (Ω)) for the diffusion term 𝛾 are optimal, too. At
the same time, we take the corresponding piecewise constant
space Wℎ with index 𝑘 = 0 for the gradient 𝜆 and the flux
𝜎 and obtain some convergence results in (𝐿2 )2 -norm with
ℎ = √2Δ𝑡 = √2/8, √2/16, √2/32, √2/64 in Table 3.
Taking 𝑡 = 1 with ℎ = √2Δ𝑡 = √2/32, we show the
surfaces of the exact solution 𝑢, 𝜎, 𝜆, and 𝛾 in Figures 1, 3, 5,
and 7, respectively. Figures 2, 4, 6, and 8 show the surfaces
of the numerical solution 𝑢ℎ , 𝜎ℎ , 𝜆ℎ and 𝛾ℎ at 𝑡 = 1 with
ℎ = √2Δ𝑡 = √2/16, respectively.
From the convergence results in Tables 1–3 and Figures 1–
8, we can find that the numerical results verify our theoretical
analysis.
6. Concluding Remarks
In this article, we study a new coupling method of new
EMFE scheme [29] and FE scheme for fourth-order partial
differential equation of parabolic type. The new EMFE
method’s gradient belongs to the simple square integrable
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Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Discrete 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
Volume 2014
Volume 2014