Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 808514, 26 pages
doi:10.1155/2012/808514
Research Article
Adaptive Finite Element Method for Optimal
Control Problem Governed by Linear
Quasiparabolic Integrodifferential Equations
Wanfang Shen1, 2 and Hua Su2
1
2
School of Mathematics, Shandong University, Jinan 250100, China
School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics,
Jinan 250014, China
Correspondence should be addressed to Wanfang Shen, wanfangshen@gmail.com
Received 21 June 2012; Accepted 10 July 2012
Academic Editor: Xinguang Zhang
Copyright q 2012 W. Shen and H. Su. 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.
The mathematical formulation for a quadratic optimal control problem governed by a linear
quasiparabolic integrodifferential
equation is studied. The control constrains are given in an
integral sense: Uad {u ∈ X; ΩU u 0, t ∈ 0, T }. Then the a posteriori error estimates in
L∞ 0, T ; H 1 Ω-norm and L2 0, T ; L2 Ω-norm for both the state and the control approximation
are given.
1. Introduction
Integrodifferential equations of quasiparabolic and their control of this nature appear in
applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and viscoelasticity. All these models express a conservation of a certain
quantity in any moment for any subdomain and the historical accumulation feature in the
physical models. This in many applications is the most desirable feature of the approximation
method when it comes to numerical solution of the corresponding initial boundary value
problem. The existence and uniqueness of the solution of the quasiparabolic Integrodifferential equations has been studied in 1. Finite element methods for quasiparabolic Integrodifferential equations problems with a smooth kernel have been discussed in Cui 2. Although
there is so much work for the finite element approximation of this problem, to our knowledge,
there has been a lack of a posteriori error estimates for finite element approximation of any
quasiparabolic Integrodifferential optimal control problem.
2
Abstract and Applied Analysis
The finite element approximation of optimal control problems has been an important
topic in engineering design works. There have been extensive theoretical and numerical
studies for various optimal control problems, see, for instance, 3–11, although it is impossible to give even a very brief review here. And research on finite element approximation
of parabolic optimal control problems can be found in, for example, 12, 13.
Among many finite element methods, the adaptive finite element method based on a
posteriori error estimates has become a central theme in scientific and engineering computations for its high efficiency. In order to obtain a numerical solution of acceptable accuracy, it is
essential for the adaptive finite element method to use a posteriori error estimate indicators
to guide the mesh refinement procedure. We only need refine the area where the error
indicators are larger, so that a higher density of nodes are distributed over the area where
the solution is difficult to approximate. In this sense, adaptive finite element approximation
relies very much on the error indicators used, which are often based on a posteriori error
estimates of the solutions.
The purpose of this paper is to derive the a posteriori error estimates for the semidiscrete finite element approximation of a quadratic optimal control problem governed by
a linear quasiparabolic Integrodifferential equation, which paves a way to derive the a
posteriori error estimates for the full discrete finite element approximation for this control
problem and thus to develop its adaptive finite element schemes. We extend the existing
techniques and results in 14–16 to the optimal control problem governed by the Integrodifferential equation of quasiparabolic type.
The outline of the paper is as follows. In Section 2, we first briefly introduce the optimal
control problem and give the optimality conditions, then construct the finite element approximation schemes for the optimal control problem. In Section 3, we give the a posteriori error
bounds in L∞ 0, T ; H 1 Ω-norm for the control problem. And the a posteriori error bounds
in L2 0, T ; L2 Ω-norm for the control problem are derived in Section 4.
2. Optimal Control Problem and Its Finite Element Approximation
Let Ω and ΩU be bounded convex polygon domains in Rd with Lipschitz boundary ∂Ω and
∂ΩU . In this paper, we adopt the standard notation W m,q Ω for Sobolev spaces on Ω with
m,q
norm ·m,q,Ω , and seminorm |·|m,q,Ω . We set W0 Ω {w ∈ W m,q Ω : w|∂Ω 0}. We denote
W m,2 ΩW0m,2 Ω by H m ΩH0m Ω, with norm · m,Ω , and seminorm | · |m,Ω .
We denote by Ls 0, T ; W m,q Ω the Banach space of all Ls integrable functions from
T
0, T into W m,q Ω with norm vLs 0,T ;W m,q Ω 0 vsW m,q Ω dt1/s for s ∈ 1, ∞ and the
standard modification for s ∞. Similarly, one can define the spaces H 1 0, T ; W m,q Ω and
Ck 0, T ; W m,q Ω. The details can be found in 17. In addition, c or C denotes a general
positive constant independent of the mesh size h.
In the following, we will give semi-discrete finite element approximation schemes for
the optimal control problem governed by a linear quasiparabolic Integrodifferential equation.
2.1. Model Problem and Its Weak Formulation
We will take the state space W L2 0, T ; V with V H01 Ω and the control space X L2 0, T ; U with U L2 ΩU . Let the observation space Y L2 0, T ; H with H L2 Ω and
Uad ⊆ X a convex subset.
Abstract and Applied Analysis
3
We are interested in the following optimal control problem:
1
min J u, yu u∈Uad ⊂X
2
T
0
y − zd 2 dt 0,Ω
T
0
u20,ΩU dt
,
2.1
subject to
yt − div A∇yt D∇y t
Ct, τ∇yx, τdτ
f Bu,
in Ω × 0, T ,
0
y 0,
on ∂Ω × 0, T ,
y|t0 y0 ,
2.2
in Ω,
where u is control, y is state, zd is the observation, Uad is a closed convex subset, fx, t ∈
L2 0, T ; L2 Ω, and zd and y0 ∈ H 1 Ω are some suitable functions to be specified later. B is
a linear bounded operator from L2 ΩU to L2 Ω independent of t. And
n×n
n×n
,
D Dx di,j · n×n ∈ C∞ Ω
,
A Ax ai,j · n×n ∈ C∞ Ω
2.3
such that there is a constant c > 0 satisfying that for any vector X ∈ Rn as follows:
X t AX ≥ cX2Rn ,
X t DX ≥ cX2Rn ,
C Cx, t, τ ci,j x, t, τn×n ∈ C∞ 0, T ; L2 Ω
Let
f1 , f2 Ω
f1 f2 ,
n×n
2.4
.
∀ f1 , f2 ∈ H × H,
u, vU ΩU
uv,
az, ω A∇z, ∇ω,
∀u, v ∈ U × U,
2.5
dz, ω D∇z, ∇ω,
ct, τ; z, ω Ct, τ∇z, ∇ω,
∀z, w ∈ V × V.
In the case that f1 ∈ V and f2 ∈ V ∗ , the dual pair f1 , f2 is understood as f1 , f2 V ×V ∗ .
Assume that there are constants c and C, such that for all t and τ in 0, T as follows:
a az, z cz21,Ω ,
b |az, w| Cz1,Ω w1,Ω ,
c |ct, τ; z, w| Cz1,Ω w1,Ω .
for any z and w in V .
|dz, w| Cz1,Ω w1,Ω ,
2.6
4
Abstract and Applied Analysis
Then the weak form of the state equation reads as
yt , w a yt , w d y, w t
c t, τ; yτ, w dτ f Bu, w
0
∀w ∈ V, t ∈ 0, T ,
2.7
y |t0 y0 .
It is well known see, e.g., 1 that the above weak formulation has at least one solution
in y ∈ W0, T {w ∈ L∞ 0, T ; H 1 Ω, wt ∈ L2 0, T ; H 1 Ω}.
Therefore, the weak form of the control problem 2.1 and 2.2 reads as OCP
min J u, yu ,
yt , w a yt , w d y, w t
u∈Uad
c t, τ; yτ , w dτ f Bu, w
∀w ∈ V, t ∈ 0, T ,
0
y |t0 y0 .
2.8
In the following, we first give the existence and uniqueness of the solution of the system 2.8.
Theorem 2.1. Assume that the condition 2.6 (a)–(c) holds. There exists the unique solution u, y
for the minimization problem 2.8 such that u ∈ L2 0, T ; L2 ΩU , y ∈ L∞ 0, T ; H 1 Ω, and
yt ∈ L2 0, T ; H 1 Ω.
Proof. Let {un , yn }∞
n1 be a minimization sequence for the system 2.8, then the sequence
is
bounded
in L2 0, T ; L2 ΩU . Thus there is a subsequence of {un }∞
{un }∞
n1
n1 still denote
n ∞
by {u }n1 such that un converges to u∗ weakly in L2 0, T ; L2 ΩU . For the subsequence
{un }∞
n1 , we have
ytn , w a ytn , w d yn , w t
c t, τ; yn τ, wt dτ f Bun , w
0
2.9
∀w ∈ V, t ∈ 0, T .
By setting w yn and integrating from 0 to t in 2.9, we give
n 2
y t 1,Ω
t
0
C y0 n 2
y dτ
1,Ω
1,Ω
C
t
0
2
f un 20,ΩU dt −1,Ω
t τ
0
0
ys2 ds dτ .
1,Ω
2.10
Applying Gronwall’s inequality to 2.10 yields
n 2
y ∞
L
T
2
n 2
2
0
n 2
0,T ;H 1 Ω y L2 0,T ;H 1 Ω C y 1,Ω 0 f −1,Ω u 0,ΩU .
2.11
Abstract and Applied Analysis
5
2
2
n ∞
∞
So {un }∞
n1 is a bounded set in L 0, T ; L ΩU and {y }n1 is a bounded set in L 0, T ;
1
H Ω. Thus
un −→ u
weakly in L2 0, T ; L2 ΩU ,
yn −→ y
weakly in L∞ 0, T ; H 1 Ω ,
2.12
yn T −→ yT weakly in H 1 Ω.
Let W {w; w ∈ L∞ 0, T ; H 1 Ω, wt ∈ L2 0, T ; H 1 Ω}.
By integrating time from 0 to T in 2.9 and taking limit as n → ∞, we obtain
yT , wT a yT , wT −
T t
0
T
0
y, wt a y, wt d y, w
c t, τ; yτ, wt dτ dt
2.13
0
T
y0 , w0 a y0 , w0 f Bu, w ,
∀w ∈ W.
0
Then,
yt , w a yt , w d y, w t
c t, τ; yτ, w dτ f Bu, w ,
∀w ∈ V, t ∈ 0, T .
0
2.14
Furthermore, we have
T
yt , yt a yt , yt d y, yt 0
t
T
c t, τ; yτ, yt t dτ dt f Bu, yt .
0
2.15
0
Then, we get
T
0
2
yt C
1,Ω
T
t
2
2
2
2
f u0,ΩU y1,Ω y1,Ω dτ .
−1,Ω
0
2.16
0
This means yt ∈ L2 0, T ; H 1 Ω. So u, y is one solution of 2.8.
T
T
Since 0 y − zd 20,Ω is a convex function on space L2 0, T ; L2 Ω and α/2 0 u20,ΩU
is a strictly convex function on U, hence Ju, yu is a strictly convex function on U, so the
minimization problem 2.8 has one unique solution.
6
Abstract and Applied Analysis
2.2. Optimality Conditions and Their Finite Element Approximation
By the theory of optimal control problem see 18, we can similarly deduce the following
optimality conditions of the problem 2.8.
Theorem 2.2. A pair y, u ∈ L2 0, T ; H01 Ω × L2 0, T ; L2 ΩU is the solution of the optimal
control problem 2.8, if and only if there exists a costate p ∈ L2 0, T ; H01 Ω such that the triple
y, p, u satisfies the following optimality conditions:
yt , w a yt , w d y, w t
c t, τ; yτ, wt dτ f Bu, w
0
∀w ∈ V, t ∈ 0, T ,
2.17
y |t0 y0 ;
− q, pt − a q, pt d q, p T
c τ, t; qt, pτ dτ y − zd , q
∀q ∈ V, t ∈ 0, T ,
t
p |tT 0;
T
u B∗ p, v − u
0
U
dt 0,
2.18
2.19
∀v ∈ Uad ,
where B∗ is the adjoint operator of B.
Let us consider the semi-discrete finite element approximation of the control problem
2.8. Here, we only consider triangular and conforming elements.
Let Ωh be a polygonal approximation to Ω with boundary ∂Ωh . Let T h be a partitioning
h
of Ωh into disjoint regular n-simplices τ, so that Ω τ∈T h τ. Each element has at most one
face on ∂Ωh , and τ and τ have either only one common vertex or a whole edge or face if τ
and τ ∈ T h . We further require that Pi ∈ ∂Ωh ⇒ Pi ∈ ∂Ω where Pi i 1, . . . , J is the vertex
set associated with the triangulation Th . As usual, h denotes the diameter of the triangulation
T h . For simplicity, we assume that Ω is a convex polygon so that Ω Ωh .
h
Associated with T h is a finite-dimensional subspace Sh of CΩ , such that χ|τ are
polynomials of order m m ≥ 1 for all χ ∈ Sh and τ ∈ T h . Let V h {vh ∈ Sh : vh Pi 0
i 1, . . . , J}, W h L2 0, T ; V h . Note that we do not impose a continuity requirement. It is
easy to see that V h ⊂ V, W h ⊂ W.
h
Let TUh be a partitioning of ΩhU into disjoint regular n-simplices τU , so that ΩU τU ∈T h τ U . τ U and τ U have either only one common vertex or a whole edge or face if τ U and
U
τ U ∈ TUh . We further require that Pi ∈ ∂ΩhU ⇒ Pi ∈ ∂ΩU , where Pi i 1, . . . , J is the vertex
set associated with the triangulation TUh . For simplicity, we again assume that ΩU is a convex
polygon so that ΩU ΩhU .
Associated with TUh is another finite-dimensional subspace Uh of L2 ΩhU , such that
χ|τU are polynomials of order m m 0 for all χ ∈ Uh and τU ∈ TUh . Here there is no
requirement for the continuity. Let X h L2 0, T ; Uh . It is easy to see that X h ⊂ X. Let hτ hτU denote the maximum diameter of the element ττU in T h TUh .
Abstract and Applied Analysis
7
Due to the limited regularity of the optimal control u in general, there will be no advantage in considering higher-order finite element spaces than the piecewise constant space
for the control. We therefore only consider the piecewise constant finite element space for
the approximation of the control, though higher-order finite element spaces will be used to
approximate the state and the co-state. Let P0 Ω denote all the 0-order polynomial over Ω.
h
is a
Therefore, we always take X h {u ∈ X : ux, t|x∈τU ∈ P0 τU , for all t ∈ 0, T }. Uad
h
h
closed convex set in X . For ease of exposition, in this paper, we assume that Uad ⊂ Uad ∩ X h .
Then a possible semi-discrete finite element approximation of OCP is as follows
OCPh :
min J uh , yh
h
uh ∈Uad
1
2
T
0
yh − zd 2 0,Ω
T
0
uh 20,ΩU
2.20
,
such that
∂yh
, wh
∂t
t
∂yh
, wh d yh , wh c t, τ; yh τ, wh t dτ f Buh , wh ,
a
∂t
0
∀wh ∈ V h , t ∈ 0, T ,
yh t0 yh0 ,
2.21
where yh ∈ W h and yh0 ∈ V h is the approximation of y0 .
In the same way of proving Theorem 2.1, we can easily prove that the problem 2.20h
.
2.21 has a unique solution yh , uh ∈ W h × Uad
h
is a solution of 2.20It is well known see 18 that a pair yh , uh ∈ W h × Uad
h
2.21, if and only if there exists a co-state ph ∈ W such that the triple yh , ph , uh satisfies the
following optimality conditions:
∂yh
, wh
∂t
t
∂yh
, wh d yh , wh c t, τ; yh τ, wh t dτ f Buh , wh ,
a
∂t
0
∀wh ∈ V h ,
yh |t0 yh0 ,
2.22
T
∂ph
∂ph
− qh ,
c τ, t; qh , ph τ dτ yh − zd , qh ,
− a qh ,
d qh , ph ∂t
∂t
t
ph |tT 0,
T
0
uh B∗ ph , vh − uh
U
dt 0,
∀qh ∈ V h ,
2.23
h
∀vh ∈ Uad
.
2.24
8
Abstract and Applied Analysis
The optimality conditions in 2.22–2.24 are the semi-discrete approximation to the
problem 2.17–2.19.
Introduce the local averaging operator πh given by
τ
w
τU
1
πh w |τU : U
Then, we have
valent to
ΩU
w T
0
ΩU
,
∀τU ∈ TUh .
2.25
πh w for any w ∈ L2 0, T ; L2 ΩU , t ∈ 0, T and 2.24 is equi-
uh πh B∗ ph , vh − uh U dt 0,
h
∀vh ∈ Uad
.
2.26
In the following, we derive the a posteriori error estimates for semi-discrete finite element
approximation 2.22–2.24, allowing different meshes to be used for the state and the
control.
The following lemmas are important in deriving the a posteriori error estimates of
residual type.
Lemma 2.3 see 19. Let πh be the standard Lagrange interpolation operator. For m 0 or 1,
q > n/2 and v ∈ W 2,q Ω as
|v − πh v|m,q,Ω Ch2−m |v|2,q,Ω .
2.27
Lemma 2.4 see 20. Let πh be the average interpolation operator defined in 2.25. For m 0 or
1, 1 q ∞ and for all v ∈ W 1,q Ωh as
|v − πh v|m,q,τ τ
Ch1−m
|v|1,q,τ .
τ
τ
/∅
2.28
Lemma 2.5 see 21. For all v ∈ W 1,q Ω, 1 q < ∞ as
−1/q
1−1/q
|v|1,q,τ .
v0,q,∂τ C hτ v0,q,τ hτ
2.29
3. A Posteriori Error Estimates in L∞ 0, T ; H 1 Ω-Norm
In this paper, the control constraints are given in an integral sense as follows:
Uad v ∈ X;
ΩU
v 0, t ∈ 0, T .
3.1
The following lemma is the first step to derive the a posteriori error estimates of residual type.
Abstract and Applied Analysis
9
Lemma 3.1. Let y, p, u and yh , ph , uh be the solutions of 2.17–2.19 and 2.22–2.24. Then,
we have
2
u − uh 2L2 0,T ;L2 Ω Cη12 Cph − puh L2 0,T ;L2 Ω ,
U
3.2
where
η12
T 0
τU
∗
∗
B ph − Ph B ph
2
3.3
dt,
τU
Ph is the L2 -projection from L2 Ω to Uh , and puh is defined by the following system:
t
∂
∂
yuh , ω a
yuh , ω d yuh , ω c t, τ; yuh τ, ωt dτ
∂t
∂t
0
f Buh , ω , ∀ ω ∈ V,
3.4
yuh 0 y0h x, x ∈ Ω,
T
∂
∂
− q, puh − a q, puh d q, puh c τ, t; qt, puh τ dτ
∂t
∂t
t
yuh − zd , q , ∀q ∈ V.
3.5
Proof. From 2.19, we have
u, u − uh U − B∗ p, u − uh U .
3.6
Then, by 2.24 and 3.6, we have
u − uh 2L2 0,T ;L2 Ω U
T
0
−
u, u − uh U − uh , u − uh U dt T
B∗ ph uh , u − vh
0
T
T
h
vh ∈Uad
T
0
U
dt −
∗
B∗ ph uh , vh − u
0
U
U
B∗ ph uh , vh − uh
dt T
U
dt
B∗ puh − B∗ p, u − uh
0
U
dt
dt
B∗ ph − puh , u − uh U dt 0
− B∗ p uh , u − uh U dt
0
B ph − B puh , u − uh
0
inf
∗
T
T
T
0
B∗ puh − p , u − uh U dt
I1 I 2 I 3 .
3.7
Next, we will estimate I1 , I2 , and I3 , respectively.
10
Abstract and Applied Analysis
1 We first estimate I1 . Let Ph be the L2 -projection from L2 Ω to Uh .
We have
ΩU
Ph v − vφ 0,
∀φ ∈ X h , v ∈ Uad , t ∈ 0, T .
3.8
h
. So that we can take vh Ph u in I1 .
Since v ∈ Uad , so ΩU Ph v 0, then Ph v ∈ Uad
For given t ∈ 0, T , let
B ∗ ph
ΩU
∗
uh Ph −B ph max 0,
ΩU
3.9
.
1
We have uh ∈ X h . We will show that uh is the solution of the variational inequality in 2.24
assuming ph is known.
Since ΩU Ph −B∗ ph max{0, ΩU B∗ ph / ΩU 1}−−B∗ ph max{0, ΩU B∗ ph / ΩU 1} 0,
we have
ΩU
uh −
Thus,
ΩU
ΩU
B ph uh B∗ ph , vh − uh
ΩU
U
∗
ΩU
Ph −B ph max 0,
− −B ph max 0,
ΩU
ΩU
max 0,
ΩU
B ∗ ph
ΩU
1
B ∗ ph
ΩU
∗
If
1
ΩU
B ∗ ph
ΩU
ΩU
1
0,
ΩU
ΩU
ΩU
max 0,
1
B∗ ph < 0,
B∗ ph 0.
3.10
B ∗ ph
ΩU
vh − uh 1
3.11
vh − uh .
B∗ ph < 0, then
∗
uh B ph , vh − uh
− Ω U B ∗ ph ,
ΩU
If
max 0,
ΩU
B ∗ ph
ΩU
h
h
uh 0, we have uh ∈ Uad
. Note that for all vh ∈ Uad
, t ∈ 0, T , we have
∗
U
ΩU
0 · vh − uh 0.
3.12
B∗ ph 0, since
ΩU
uh ∗
ΩU
−B ph max 0,
ΩU
B ∗ ph
ΩU
1
0.
3.13
Abstract and Applied Analysis
11
we have
∗
uh B ph , vh − uh
U
ΩU
B ∗ ph ΩU
1
ΩU
vh − uh B ∗ ph ·
vh 0.
1
ΩU
ΩU
ΩU
3.14
From 3.11–3.14, we obtain
uh B∗ ph , vh − uh
U
0,
3.15
h
∀vh ∈ Uad
.
So uh Ph −B∗ ph max{0, ΩU B∗ ph / ΩU 1} is the solution of the variational inequality in
2.24 assuming ph is known.
Then,
I1 T
B ∗ p h uh , P h u − u
0
U
dt
T B ∗ ph
ΩU
∗
∗
Ph −B ph max 0, B ph Ph u − u dt.
1
0
τU τU
ΩU
Since
τU
3.16
Ph u − u 0, we have
I1 T 0
0
∗
∗
∗
∗
−Ph B ph B ph Ph u − u dt
−Ph B ph B ph Ph u − uh − u − uh dt
τU
τU
Cδ
τU
τU
T T 0
τU
2
−Ph B∗ ph B∗ ph
τU
Cη12 δu − uh 2L2
0,T ;L2 ΩU dt δu − uh 2L2
3.17
0,T ;L2 ΩU .
2 Consider
I2 T
0
2
B∗ ph − puh , u − uh U dt Cph − puh L2 0,T ;L2 Ω δu − uh 2L2
.
0,T ;L2 ΩU 3.18
3 By 3.4 and 2.17, we have for t ∈ 0, T ∂
∂
y − yuh , ω a
y − yuh , ω d y − yuh , w
∂t
∂t
t
c t, τ; y − yuh τ, ωt dτ Bu − uh , ω, ∀ω ∈ V,
0
3.19
12
Abstract and Applied Analysis
and from 3.5 and 2.18, we have
∂
∂
− q,
p − puh − a q,
p − puh d q, p − puh ∂t
∂t
T
c τ, t; qt, p − puh τ dτ y − yuh , q , ∀q ∈ V.
3.20
t
Then, from 3.19, 3.20, and integrating by part we have
I3 T
∗
B puh − p , u − uh
0
T 0
U
dt T
0
puh − p, Bu − uh U dt
∂
a
y − yuh , puh − p
∂t
t
d y − yuh , puh − p c t, τ; y − yuh τ, puh − p t dτ dt
∂
y − yuh , puh − p
∂t
0
T ∂
∂
− y − yuh ,
puh − p − a y − yuh ,
puh − p
∂t
∂t
0
T
d y − yuh , puh − p c τ, t; y − yuh t, puh − p τ dτ dt
t
T
− y − yuh , y − yuh dt 0.
0
3.21
Following from 3.17–3.21, let δ be small enough as
2
u − uh 2L2 0,T ;L2 Ω Cη12 Cph − puh L2 0,T ;L2 Ω .
U
3.22
This completes the proof.
Lemma 3.2. Let y, p, u and yh , ph , uh be the solutions of 2.17–2.19, and 2.22–2.24 respectively. Then, there hold the a posteriori error estimates as
∂
2
yh − yuh 2 ∞
−
yu
y
h ∂t h
L 0,T ;H 1 Ω
L2 0,T ;H 1 Ω
6
∂
2
2
C
ph − puh ph − puh L∞ 0,T ;H 1 Ω ηi2 ,
2
∂t
L 0,T ;H 1 Ω
i2
3.23
Abstract and Applied Analysis
13
where
η22
T 0
h2τ
τ
τ
∂ph
∗ ∂ph
− div A ∇
div D∗ ∇ph
∂t
∂t
T
⎫
⎬
2
div C∗ τ, t∇ph τ dτ yh − zd
t
dτ
dt,
⎭
2
T
∗
∗
∗ ∂ph
− A∇
· n D ∇ph · n hl
C τ, t∇ph τ · ndτ dl dt,
∂t
0 τ
t
∂τ
⎧
T ⎨ ∂yh
∂
η42 yh − div A∇
h2τ
− div D∇yh
∂t
∂t
0⎩ τ
τ
2 ⎫
t
⎬
− div Ct, τ∇yh τ dτ − f − Buh
dt,
⎭
0
T η32
η52
T 0
hl
∂τ
τ
2
η62 y0h − y0 ,
∂yh
A∇
∂t
· n A∇yh · n t
Ct, τ∇yh τ · ndτ
2
dl dt,
0
1,Ω
3.24
where l is a face of an element τ, hl is the maximum diameter of l, and ∇ph · n and ∇yh · n are the
normal derivative jumps over the interior face l defined by
∇ph · n l ∇ph h|τl1 − ∇ph |τl2 · n,
3.25
∇yh · n l ∇yh |τl1 − ∇yh |τl2 · n,
where n is the unit normal vector on l τl1 ∩ τl2 outwards τl1 . For later convenience, one can define
∇ph · nl 0 and ∇yh · nl 0 when l ⊂ ∂Ω.
Proof. Let
∂
∂
ph − puh − a v,
ph − puh d v, ph − puh Ruh , v − v,
∂t
∂t
T
c τ, t; vt, ph − puh τ dτ,
3.26
t
and πh the average interpolation operator defined as in 2.25 and e ph − puh . Then, it
follows from 2.23 and 3.5 that
∂
∂
− qh ,
ph − puh − a qh ,
ph − puh d qh , ph − puh ∂t
∂t
3.27
T
h
c τ, t; qh t, ph − puh τ dτ yh − yuh , qh , ∀qh ∈ V .
t
14
Abstract and Applied Analysis
We have
Ruh , v
∂
∂
− v − πh v,
ph − puh − a v − πh v,
ph − puh d v − πh v, ph − puh ∂t
∂t
T
∂
c τ, t; v − πh vt, ph − puh τ dτ − πh v,
ph − puh ∂t
t
T
∂
− a πh v,
c τ, t; πh vt, ph − puh τ dτ
ph − puh d πh v, ph − puh ∂t
t
T
∂ph
∂ph
c τ, t; v − πh vt, ph τ dτ
− v − πh v,
− a v − πh v,
d v − πh v, ph ∂t
∂t
t
− yuh − zd , v − πh v yh − yuh , πh v
T
∗
∗
∂ph
∗ ∂ph
−
div A ∇
− div D ∇ph −
div C τ, t∇ph τ dτ − yh zd
∂t
∂t
t
τ
τ
T
∗
∗
∗ ∂ph
× v − πh v − A∇
· n D ∇ph · n C τ, t∇ph τ · ndτ
∂t
t
∂τ
τ
× v − πh v yh − yuh , v
∂ph
2
∗ ∂ph
div A ∇
− div D∗ ∇ph
hτ −
∂t
∂t
τ
τ
2
T
∗
−
div C τ, t∇ph τ dτ − yh zd
t
2 ⎫1/2
T
⎬
∂p
h
hl − A∗ ∇
C∗ τ, t∇ph τ · ndτ
· n D∗ ∇ph · n ⎭
∂t
t
∂τ
τ
× v1,Ω yh − yuh , v .
3.28
Taking v ph − puh in 3.28 and from 2.6, we have
−
1 d
ph − puh 2 − 1 d a ph − puh , ph − puh cph − puh 2
0,Ω
1,Ω
2 dt
2 dt
⎧
T
⎨ ∗
∂ph
2
∗ ∂ph
div A ∇
h −
div C∗ τ, t∇ph τ dτ
− div D ∇ph −
⎩ τ τ τ
∂t
∂t
t
2
−yh zd
τ
⎫
2 ⎬1/2
∂p
h
· n D∗ ∇ph · n
hl − A∗ ∇
⎭
∂t
∂τ
Abstract and Applied Analysis
15
×ph − puh 1,Ω yh − yuh , ph − puh T
−
c τ, t; ph − puh t, ph − puh τ dτ.
t
3.29
Integrating time from t to T in 3.29 and by Schwartz inequality, Lemmas 2.4 and 2.5, we
have
T
1
ph − puh 2 dτ
ph − puh 2 cph − puh 2 c
0,Ω
1,Ω
1,Ω
2
t
T ∂ph
2
∗ ∂ph
− div A ∇
div D∗ ∇ph
hτ ×
∂t
∂t
t τ
τ
T
div C∗ s, τ∇ph s ds yh − zd
2
dτ
τ
T t
δ
τ
T
t
2
T
∗
∗
∗ ∂ph
C s, τ∇ph s · nds dτ
− A∇
· n D ∇ph · n hl
∂t
τ
∂τ
ph − puh 2 dτ C
1,Ω
T
t
yh − yuh 2 dτ C
0,Ω
T T
t
τ
ph − puh s2 ds dτ.
1,Ω
3.30
Letting δ be small enough, we have
T
ph − puh 2 dτ
1,Ω
t
T ∂ph
∂ph
− div A∗ ∇
C
h2τ
div D∗ ∇ph
∂t
∂t
t τ
τ
T
div C s, τ∇ph s ds yh − zd
∗
2
dτ
τ
T ∂ph
hl
C
− A∗ ∇
· n D∗ ∇ph · n
∂t
t τ
∂τ
T
∗
3.31
2
C s, τ∇ph s · nds dτ
τ
C
T
t
yh − yuh 2 dτ C
0,Ω
T T
t
τ
ph − puh s2 ds dτ.
1,Ω
Then, from Gronwall inequality and 3.28–3.31 we have
2
ph − puh 2 2
Cη22 Cη32 Cyh − yuh L2 0,T ;L2 Ω .
L 0,T ;H 1 Ω
3.32
16
Abstract and Applied Analysis
Similarly,
2
ph − puh 2 ∞
C η22 η32 yh − yuh L2 0,T ;L2 Ω
L 0,T ;H 1 Ω
C
T T
0
t
ph − puh τ2 dτ dt
1,Ω
3.33
2
Cη22 Cη32 Cyh − yuh L2 0,T ;L2 Ω .
In the same way of getting 3.32,by setting v ∂/∂tph − puh in 3.28, we have
∂
2
2
Cη22 Cη32 Cyh − yuh L2 0,T ;L2 Ω .
∂t ph − puh 2
1
L 0,T ;H Ω
3.34
Similarly analysis for yh − yuh L∞ 0,T ;H 1 Ω , we let
Quh , v ∂
∂
yh − yuh , v a
yh − yuh , v d yh − yuh , v
∂t
∂t
t
c t, τ; yh − yuh τ, vt dτ.
3.35
0
From 2.22 and 3.4, we obtain
∂
∂
ωh ,
yh − yuh a
yh − yuh , ωh d yh − yuh , ωh
∂t
∂t
t
c t, τ; yh − yuh τ, ωh t dτ 0, ∀ωh ∈ V h .
3.36
0
We have
Quh , v
∂
∂
yh − yuh , v − πh v a
yh − yuh , v − πh v d yh − yuh , v − πh v
∂t
∂t
t
c t, τ; yh − yuh τ, v − πh vt dτ
0
∂yh
∂yh
, v − πh v a
, v − πh v d yh , v − πh v
∂t
∂t
t
c t, τ; yh τ, v − πh vt dτ − f Buh , v − πh v
0
τ
τ
t
∂yh
∂yh
− div A∇
− div D∇yh − div Ct, τ∇yh dτ − f − Buh v − πh v
∂t
∂t
0
Abstract and Applied Analysis
17
t
∂yh
Ct, τ∇yh · ndτ v − πh v
A∇
· n D∇yh · n ∂t
0
∂τ
τ
⎧
2
t
⎨ ∂yh
∂
2
y − div A∇
− div D∇yh − div Ct, τ∇yh dτ − f − Buh
h
⎩ τ τ τ ∂t h
∂t
0
τ
hl
∂τ
∂yh
A∇
∂t
· n D∇yh · n t
Ct, τ∇yh · ndτ
0
2 ⎫1/2
⎬
⎭
v1,Ω .
3.37
By setting v yh − yuh and Swartz inequality, we have
1 d
yh − yuh 2 1 d a yh − yuh , yh − yuh cyh − yuh 2
0,Ω
1,Ω
2 dt
2 dt
2
t
∂yh
∂yh
2
− div A∇
− div D∇yh − div Ct, τ∇yh dτ − f − Buh
hτ
∂t
∂t
0
τ
τ
2
t
∂yh
· n D∇yh · n hl A∇
Ct, τ∇yh · ndτ
∂t
0
∂τ
τ
t
2
δ yh − yuh 1,Ω − c t, τ; yh − yuh τ, yh − yuh t dτ.
0
3.38
Integrating time from 0 to t in 3.38, we obtain
yh − yuh 2 c
1,Ω
C
t
0 τ
h2τ
τ
t
0
yh − yuh 2 dτ
1,Ω
∂yh
∂yh
− div A∇
− div D∇yh
∂t
∂t
τ
2
− div Cτ, s∇yh ds − f − Buh dτ
3.39
0
t τ
2 ∂yh
hl
A∇
Cτ, s∇yh · nds dt
· n D∇yh · n ∂t
0 τ
0
∂τ
t τ
t
2
2
h
yh − yuh 2 ds dτ C
yh − yuh 1,Ω dτ C
−
y
δ
.
y
0
0
1,Ω
0
0
0
Since δ is small enough, then from 3.39 and Gronwall inequality, we have
t
0
yh − yuh 2 dt
1,Ω
1,Ω
18
Abstract and Applied Analysis
C
t 0 τ
h2τ
τ
∂yh
∂yh
− div A∇
− div D∇yh
∂t
∂t
−
τ
div Cτ, s∇yh ds − f − Buh
2
dτ
0
C
t τ
2
2
∂yh
A∇
hl
Cτ, s∇yh · nds dt Cy0 − y0h .
· n D∇yh · n 1,Ω
∂t
0 τ
0
∂τ
3.40
Then,
yh − yuh 2 2
C η42 η52 η62 ,
L 0,T ;H 1 Ω
3.41
2
2
2
yh − yuh 2 ∞
η
η
C
η
1
5
6 .
4
L 0,T ;H Ω
In the same way of getting 3.34, we can similarly obtain
∂
2
C η42 η52 η62 .
∂t yh − yuh 2
L 0,T ;H 1 Ω
3.42
Then the desired results 3.23 follow from 3.32–3.34 and 3.41-3.42.
From Lemmas 3.1 and 3.2, we have the following results.
Theorem 3.3. Let y, p, u and yh , ph , uh be the solutions of 2.17–2.19 and 2.22–2.24
respectively. Then, there hold the a posteriori error estimates as
∂
2
2
2
y − yh u − uh L2 0,T ;L2 Ω y − yh L∞ 0,T ;H 1 Ω 2
U
∂t
L 0,T ;H 1 Ω
3.43
6
2
∂
2
2
C ηi ,
p − ph L∞ 0,T ;H 1 Ω ∂t p − ph 2
L 0,T ;H 1 Ω
i1
where η12 is defined in Lemma 3.1.
Proof. First, from 3.27 and 3.36, and 2, we have the following stability results:
∂
2
y − yuh 2 ∞
Cu − uh 2L2 0,T ;L2 Ω ,
y
−
yu
h ∂t
L 0,T ;H 1 Ω
U 2
1
L 0,T ;H Ω
2
∂
2
p − puh 2 ∞
Cy − yuh L2 0,T ;L2 Ω
p − puh L 0,T ;H 1 Ω
∂t
L2 0,T ;H 1 Ω
Cu − uh 2L2
0,T ;L2 ΩU .
3.44
Abstract and Applied Analysis
19
Then, the desired results 3.43 follows from triangle inequality, 3.44 and Lemmas 3.1 and
3.2.
This completes the proof.
4. A Posteriori Error Estimates in L2 0, T ; L2 Ω-Norm
In the following, we will derive the a posteriori error estimates in L2 0, T ; L2 Ω-norm.
For given F ∈ L2 0, T ; L2 Ω, we have
t
∂φ
∂φ
− div A∇
− div D∇φ − div Ct, τ∇φτ dτ F,
∂t
∂t
0
x, t ∈ Ω × 0, T ,
φ
0, t ∈ 0, T ,
∂Ω
φx, 0 0, x ∈ Ω,
4.1
and its dual equation
T
∗
∂ψ
∗ ∂ψ
div A ∇
− div D ∇ψ −
div C∗ τ, t∇ψτ dτ F,
−
∂t
∂t
t
ψ
0, t ∈ 0, T ,
∂Ω
ψx, T 0, x ∈ Ω.
x, t ∈ Ω × 0, T ,
4.2
From 1, 2, we have the following stability results.
Lemma 4.1. Assume that Ω is a convex domain. Let φ and ψ be the solution of 4.1 and 4.2,
respectively. Then,
φ ∞
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
∇φ 2
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
2 CFL2 0,T ;L2 Ω ,
D φ 2
L 0,T ;L2 Ω
∂ φ
CFL2 0,T ;L2 Ω ,
∂t 2
L 0,T ;L2 Ω
4.3
ψ ∞
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
∇ψ 2
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
2 CFL2 0,T ;L2 Ω ,
D ψ 2
L 0,T ;L2 Ω
∂ ψ
CFL2 0,T ;L2 Ω ,
∂t 2
L 0,T ;L2 Ω
where D2 φ ∂2 φ/∂xi ∂xj , 1 i, j n, and D2 ψ is defined similarly.
20
Abstract and Applied Analysis
Using Lemmas 3.1 and 4.1, we have the following upperbounds.
Lemma 4.2. Let y, p, u and yh , ph , uh be the solutions of 2.17–2.19 and 2.22–2.24,
respectively. Then, there hold the a posteriori error estimates as
5
2
2
2
yh − yuh 2 2
ξi η6 ,
ph − puh L2 0,T ;L2 Ω C
L 0,T ;L2 Ω
4.4
i2
where η62 is defined in Lemma 3.2, and
ξ22
T ∂ph
∂ph
− div A∗ ∇
div D∗ ∇ph
h4τ
∂t
∂t
0
τ
τ
T
div C∗ τ, t∇ph τ dτ yh − zd
t
2 ⎫
⎬
dt,
⎭
2
T T
∗
∗
3
∗ ∂ph
− A∇
· n D ∇ph · n hl
C τ, t∇ph τ · ndτ dl dt,
∂t
0 τ
t
∂τ
T ∂yh
∂yh
2
4
− div A∇
ξ4 hτ
− div D∇yh
∂t
∂t
0
τ
τ
2 ⎫
t
⎬
dt,
− div Ct, τ∇yh τ dτ − f − Buh
⎭
0
ξ32
ξ52
4.5
2
T t
∂yh
3
A∇
· n D∇yh · n hl
Ct, τ∇yh τ · ndτ dl dt.
∂t
0 τ
0
∂τ
Proof. We first estimate ph − puh 2L2 0,T ;L2 Ω .
Let φ be the solution of 4.1 with F ph − puh , and φI πh φ the interpolation of φ
in Lemma 2.3.
From 4.1, 3.27, and by integrating by parts we obtain
ph − puh 2 2
L 0,T ;L2 Ω
T
Ft, ph − puh t dt
0
T
−
0
∂
∂
ph − puh , φ − a φ,
ph − puh ∂t
∂t
d φ, ph − puh t
0
c t, τ; φτ, ph − puh t dτ dt
Abstract and Applied Analysis
21
T ∂
∂
−
ph − puh , φ − φI − a φ − φI ,
ph − puh ∂t
∂t
0
t
d φ − φI , ph − puh c t, τ; φ − φI τ, ph − puh t dτ
0
∂
∂
ph − puh , φI − a φI ,
ph − puh ∂t
∂t
t
d φI , ph − puh c t, τ; φI τ, ph − puh t dτ dt
−
0
T
−
0
∂ph
, φ − φI
∂t
d φ − φI , ph ∂ph
− a φ − φI ,
∂t
t
c t, τ; φ − φI τ, ph t dτ 0
∂puh , φ − φI
∂t
t
∂puh − d φ − φI , puh − c t, τ; φ − φI τ, puh t dτ
a φ − φI ,
∂t
0
∂
∂
−
ph − puh , φI − a φI ,
ph − puh ∂t
∂t
t
d φI , ph − puh c t, τ; φI τ, ph − puh t dτ dt
0
T
−
0
∂ph
, φ − φI
∂t
d φ − φI , ph ∂ph
− a φ − φI ,
∂t
T
c τ, t; φ − φI t, ph τ dτ
t
∂puh a φ − φI ,
∂t
T
− d φ − φI , puh −
c τ, t; φ − φI t, puh τ dτ
∂puh , φ − φI
∂t
t
∂
∂
−
ph − puh , φI − a φI ,
ph − puh ∂t
∂t
T
d φI , ph − puh c τ, t; φI t, ph − puh τ dτ dt
t
T T
∂ph
∂ph
−
, φ − φI − a φ − φI ,
c τ, t; φ − φI t, ph τ dτ dt
d φ − φI , ph ∂t
∂t
0
t
T
T
yuh − zd , φ − φI dt yh − yuh , φI dt
−
0
T 0
τ
τ
0
∂ph
∂ph
−
div A∗ ∇
∂t
∂t
− div D∗ ∇ph
22
Abstract and Applied Analysis
T
∗
div C τ, t∇ph τ dτ yh − zd φ − φI dt
t
T T
∗
∗
∗ ∂ph
− A∇
C τ, t∇ph τ · ndτ
· n D ∇ph · n ∂t
0 τ
t
∂τ
× φ − φI dl dt T
yh − yuh , φ dt
0
J1 J2 J3 .
4.6
It follows from Lemmas 2.3, 2.5, and 4.1 that
T ∂ph
∂ph
J1 Cδ
− div A∗ ∇
div D∗ ∇ph
h4τ
∂t
∂t
0
τ
τ
T
∗
div C τ, t∇ph τ dτ yh − zd
t
δ
T
0
2 ⎫
⎬
dt
⎭
2
φ dt Cδξ2 δph − puh 2 2
,
2
2,Ω
L 0,T ;L2 Ω
2
T T
∗
∗
3
∗ ∂ph
− A∇
hl
J2 Cδ
C τ, t∇ph τ · ndτ dl dt
· n D ∇ph · n ∂t
0 τ
t
∂τ
δ
T
0
2
φ dt
0,Ω
2
Cδξ32 δph − puh L2 0,T ;L2 Ω .
4.7
By Schwartz inequality, we have
2
2
J3 Cδyh − yuh L2 0,T ;L2 Ω δph − puh L2 0,T ;L2 Ω .
4.8
Letting δ be small enough, it follows from 4.6–4.8 that
3
2
ph − puh 2 2
ξi2 Cyh − yuh L2 0,T ;L2 Ω .
C
2
L 0,T ;L Ω
4.9
i2
Next, we estimate yh − yuh 2L2 0,T ;L2 Ω . Similarly let ψ be the solution of 4.2 with F yh − yuh , and ψI πh ψ the interpolation of ψ in Lemma 2.3.
Abstract and Applied Analysis
23
Then, it follows from 3.36 and integrating by parts that
yh − yuh 2 2
L 0,T ;L2 Ω
T
Ft, yh − yuh t dt
0
T ∂
a
yh − yuh , ψ d yh − yuh , ψ
∂t
T
c τ, t; yh − yuh t, ψτ dτ dt
0
∂
yh − yuh , ψ
∂t
t
a y0h − y0 , ψ0 y0h − y0 , ψ0
T 0
∂
yh − yuh , ψ − ψI
∂t
∂
a
yh − yuh , ψ − ψI
∂t
d yh − yuh , ψ − ψI T
c τ, t; yh − yuh t, ψ − ψI τ dτ dt
t
T 0
∂
yh − yuh , ψI
∂t
d yh − yuh , ψI ∂
a
yh − yuh , ψI
∂t
T
c τ, t; yh − yuh t, ψI τ dτ dt
t
a y0h − y0 , ψ0 y0h − y0 , ψ0
T ∂yh
∂yh
, ψ − ψI a
, ψ − ψI d yh , ψ − ψI
∂t
∂t
t
c t, τ; yh τ, ψ − ψI t dτ dt
0
−
T
0
0
f Buh , ψ − ψI dt a y0h − y0 , ψ0 y0h − y0 , ψ0
T 0
τ
τ
∂yh
∂yh
− div A∇
− div D∇yh
∂t
∂t
t
− div Cτ, t∇yh τ dτ − f − Buh ψ − ψI dt
0
T t
∂yh
· n D∇yh · n −
Ct, τ∇yh τ · n dτ ψ − ψI dl dt
A∇
∂t
0 τ
0
∂τ
a y0h − y0 , ψ0 y0h − y0 , ψ0 D1 D2 D3 D4 .
4.10
24
Abstract and Applied Analysis
Similarly, it follows from Lemmas 2.3, 2.5 and 4.1 that
D1 Cδ
T 0
h4τ
τ
τ
∂yh
∂yh
− div A∇
− div D∇yh
∂t
∂t
−
t
div Cτ, t∇yh τ dτ − f − Buh
2 ⎫
⎬
0
δ
T
0
D2 Cδ
2
ψ dt Cξ2 δyh − yuh 2 2
,
4
2,Ω
L 0,T ;L2 Ω
T 0
δ
T
0
dt
⎭
τ
h3l
A∇
∂τ
∂yh
∂t
· n D∇yh · n −
t
2
4.11
Ct, τ∇yh τ · n dl dt
0
2
ψ dt Cξ2 δyh − yuh 2 2
,
5
2,Ω
L 0,T ;L2 Ω
2
D3 D4 Cη62 δyh − yuh L2 0,T ;L2 Ω
Letting δ be small enough, then from 4.10–4.11, we have
yh − yuh 2 2
L
2
2
2
ξ
η
C
ξ
5
6 .
4
4.12
0,T ;L2 Ω
The desired results 4.4 follows from 4.9–4.12.This completes the proof.
Using Lemmas 3.1 and 4.2, we have the following upper bounds.
Theorem 4.3. Let y, p, u and yh , ph , uh be the solutions of 2.17–2.19 and 2.22–2.24,
respectively. Then, there hold the a posteriori error estimates as
u −
uh 2L2 0,T ;L2 Ω U 5
2
2
2
2
2
y − yh L2 0,T ;L2 Ω p − ph L2 0,T ;L2 Ω C η1 ξi η6 .
i2
4.13
Proof. By triangle inequality, 3.44 and 3.38, Lemmas 3.1 and 4.2, we can easily prove 4.13
in the same way of getting 3.43.
This completes the proof.
5. Conclusion
In this paper, we study the semi-discrete adaptive finite element method for optimal control
problem governed by a linear quasiparabolic Integrodifferential equation. We extend the
existing methods in studying adaptive finite element approximation of optimal control
governed by a parabolic Integrodifferential equation to the control governed by a quasiparabolic Integrodifferential equation. After presenting the weak form and the existence and
Abstract and Applied Analysis
25
uniqueness of the solution for the optimal control problem, the a posteriori error estimates
for semi-discrete finite element approximations in L∞ 0, T ; H 1 Ω-norm and L2 0, T ; L2 Ωnorm are derived. The work will pave a way to derive the a posteriori error estimates of full
discrete finite element approximations of this optimal control problem
Acknowledgments
This research was supported by Science and Technology Development Planning project of
Shandong Province no. 2011GGH20118, Shandong Province Natural Science Foundation
no. ZR2009AQ004, and National Natural Science Foundation of China no. 11071141.
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