In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the costate are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a secondorder convergence in
L
^{2}
norm and superconvergence between the numerical solution and elliptic projection of the exact solution in
H
^{1}
norm or the gradient of the exact solution and recovery gradient in
L
^{2}
norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.
AMS Mathematics Subject Classification : 49J20, 65M60.
1. Introduction
Optimal control problems have been extensively used in many aspects of the modern life such as social, economic, scientific and engineering numerical simulation. Finite element approximation seems to be the most widely used method in computing optimal control problems. A systematic introduction of finite element method for PDEs and optimal control problems can be found in
[14
,
17
,
18
,
23
,
26]
.
For a constrained optimal control problem, the control has lower regularity than the state and the costate. So most researchers considered using piecewise linear functions to approximate the state and the costate and using piecewise constant functions to approximate the control. They constructed a projection gradient algorithm in which the control is firstorder convergent (see e.g.,
[16
,
20]
). Recently, Borzì considered a secondorder discretization and multigrid solution of constrained nonlinear elliptic control problems in
[4]
, Hinze introduced a variational discretization concept for optimal control problems and derived a secondorder convergence for the control in
[11
,
12]
.
There has been extensive research on the superconvergence of finite element methods for optimal control problems in the literature, most of which focused upon elliptic control problems. Superconvergence properties of linear and semilinear elliptic control problems are established in
[24]
and
[6]
, respectively. Superconvergence of finite element approximation for bilinear elliptic optimal control problems is studied in
[32]
. Recently, superconvergence of fully discrete finite element and variational discretization approximation for linear and semilinear parabolic control problems are derived in
[27
,
28
,
29]
.
The literature on a posteriori error estimation of finite element method is huge. Some internationally known works can be found in
[1
,
2
,
3
,
5]
. Concerning finite element methods of elliptic optimal control problems, a posteriori error estimates of residual type are investigated in
[10
,
21
,
33
,
34]
, a posteriori error estimates of recovery type are derived in
[16
,
20
,
31]
. Some error estimates and superconvergence results have been established in
[6
,
7
,
36]
, and some adaptive finite element methods can be found in
[3
,
13
,
15
,
35]
. For parabolic optimal control problems, residual type a posteriori error estimates of finite element methods are investigated in
[22
,
30]
and an adaptive spacetime finite element method is investigated in
[25]
.
The purpose of this paper is to consider the superconvergence and recovery type a posteriori error estimates of variational discretization for elliptic optimal control problems with pointwise control constraints.
We are interested in the following quadratic optimal control problem:
where Ω is a bounded domain in ℝ
^{2}
with a Lipschitz boundary ∂Ω, the coefficient
such that (
A
(
x
)
ξ
) ·
ξ
≥
c

ξ

^{2}
, ∀
ξ
∈ ℝ
^{2}
,
B
is a linear continuous operator,
y_{d}
,
u_{d}
,
f
∈
L
^{2}
(Ω), and
K
is defined by
where
a
and
b
are constants.
In this paper, we adopt the standard notation
W^{m,q}
(Ω) for Sobolev spaces on Ω with norm
and seminorm
We set
and denote
W
^{m,2}
(Ω) by
H^{m}
(Ω). In addition,
c
or
C
denotes a generic positive constant.
The outline of this paper is as follows. In Section 2, we introduce a variational discretization approximation for the model problem. In Section 3, we derive the convergence. In Section 4, we obtain the superconvergence and a posteriori error estimates. We present some numerical examples to demonstrate our theoretical results in the last section.
2. variational discretization approximation for the model problem
We now consider a variational discretization approximation for the model problem (1). For ease of exposition, we set
and
It follows from the assumption on A that
Then the model problem (1) can be restated as:
It is well known (see e.g.,
[17]
) that the control problem (2) has a unique solution (
y, u
) ∈
W
×
K
, and a pair (
y, u
) ∈
W
×
K
is the solution of (2) if and only if there is a costate
p
∈
W
such that the triplet (
y, p, u
) ∈
W
×
W
×
K
satisfies the following optimality conditions:
where
B
^{*}
is the adjoint operator of
B
.
We introduce the following pointwise projection operator:
It is clear that Π
_{[a,b]}
(·) is Lipschitz continuous with constant 1. As in
[8]
, it is easy to prove the following lemma:
Lemma 2.1.
Let
(
y, p, u
)
be the solution of (3)(5). Then
Remark 2.1.
We should point out that (5) and (6) are equivalent. This theory can be used to another situation, for example,
K
is characterized by a bound on the integral on
u
over Ω, namely,
∫
_{Ω}
u
(
x
)
dx
≥ 0, we have similar results.
Let
τ^{h}
be a regular triangulation of Ω, such that
. Let
h
= max
_{τ∈τh}
{
h_{τ}
}, where
h_{τ}
denotes the diameter of the element
τ
. Associated with
τ^{h}
is a finite dimensional subspace
S_{h}
of
, such that
χ

_{τ}
are polynomials of morder(
m
= 1) for all
χ
∈
S_{h}
and
_{τ}
∈
τ^{h}
. Let
W_{h}
= {
v_{h}
∈
S_{h}
:
v_{h}

_{∂Ω}
= 0}. It is easy to see that
W_{h}
⊂
W
. Then a possible variational discretization approximation scheme of (2) is as follows:
It is well known (see e.g.,
[21]
) that the control problem (7) has a unique solution (
y_{h}, u_{h}
) ∈
W_{h}
×
K
, and that if the pair (
y_{h}
,
u_{h}
) ∈
W_{h}
×
K
is the solution of (7), if and only if there is a costate
p_{h}
∈
W_{h}
such that the triplet (
y_{h}, p_{h}, u_{h}
) ∈
W_{h}
×
W_{h}
×
K
satisfies the following optimality conditions:
Similar to Lemma 2.1, it is easy to show the following lemma:
Lemma 2.2.
Suppose
(
y_{h}, p_{h}, u_{h}
)
be the solution of (8)(10). Then , we have
Remark 2.2.
According to (11), we can replace
u_{h}
by
in our program. Thus the control need not be discretized directly.
3. Convergence analysis
We first introduce the following intermediate variables. For any
u_{h}
∈
K
, let (
y
(
u_{h}
),
p
(
u_{h}
)) ∈
W
×
W
satisfies the following system:
The following lemmas are very important in deriving the convergence.
Lemma 3.1
(
[9]
).
Let π_{h}
be the standard Lagrange interpolation operator. For m
= 0
or
1,
and
∀
v
∈
W
^{2,q}
(Ω),
we have
Lemma 3.2.
Let
(
y_{h}
,
p_{h}
,
u_{h}
)
and
(
y
(
u_{h}
),
p
(
u_{h}
))
be the solutions of (8)(10) and (12)(13), respectively. Assume that p
(
u_{h}
),
y
(
u_{h}
) ∈
H
^{2}
(Ω).
Then there exists a constant C independent of h such that
Proof
. From (9), (13)(14) and Young’s inequality, we have
Let
δ
be small enough, we obtain
Similarly, we can prove that
Then (15) follows from (16)(18).
We introduce the following auxiliary problems:
From the regularity estimates (see e.g.,
[9]
), we obtain
Lemma 3.3.
Let
(
y_{h}
,
p_{h}
,
u_{h}
)
be the solution of (8)(10). Suppose that y
(
u_{h}
),
p
(
u_{h}
) ∈
H
^{2}
(Ω).
Then there exists a constant C independent of h such that
Proof
. Let
F
_{1}
=
y
(
u_{h}
) −
y_{h}
in (19) and
ξ_{h}
=
π_{h}ξ
. From (8), (12) and (15), we have
Note that
Thus,
Similarly, let
F
_{2}
=
p
(
u_{h}
) −
p_{h}
in (20) and
ζ_{h}
=
π_{h}ζ
. From (9), (13) and (15), we obtain
From (24) and (25), we get (21).
Lemma 3.4.
Let
(
y, p, u
)
and
(
y_{h}, p_{h}, u_{h}
)
be the solutions of (3)(5) and (8)(10), respectively. Assume that all the conditions in Lemma 3.3 are valid. Then there exists a constant C independent of h such that
Proof
. By selecting
v
=
u_{h}
and
v
=
u
in (5) and (10), respectively. From (3)(4) and (12)(13), we have
From (21) and (27), we derive (26).
Now we combine lemmas 3.23.4 to come up with the following main result.
Theorem 3.5.
Let
(
y, p, u
)
and
(
y_{h}, p_{h}, u_{h}
)
be the solutions of (3)(5) and (8)(10), respectively. Assume that all the conditions in lemmas 3.23.4 are valid. Then we have
4. Superconvergence and a posteriori error estimates
We now derive the superconvergence and a posteriori error estimates for the variational discretization approximation. To begin with, let us introduce the elliptic projection operator
P_{h}
:
W
→
W_{h}
, which satisfies: for any
ϕ
∈
W
It has the following approximation properties:
Theorem 4.1.
Let
(
y, p, u
)
and
(
y_{h}, p_{h}, u_{h}
)
be the solutions of (3)(5) and (8)(10), respectively. Assume that all the conditions in lemmas 3.5 are valid. Then we have
Proof
. From (3)(8), we have the following error equation:
By using the definition of
P_{h}
and choosing
w_{h}
=
P_{h}y
−
y_{h}
, we have
Let us note that
B
is linear continuous operator. From (26), (29) and Holder inequality, we get
Similarly, we can derive
Thus, (30) follows from (33) and (34).
In the second, let us recall recovery operators
R_{h}
and
G_{h}
, respectively. Let
R_{h}v
be a continuous piecewise linear function (without zero boundary constraint). Similar to the
Z

Z
patch recovery in
[37
,
38]
, the value of
R_{h}v
on the nodes are defined by leastsquares argument on an element patches surrounding the nodes. The gradient recovery operator
G_{h}v
= (
R_{h}
v
_{x1}
, (
R_{h}
v
_{x2}
), where
R_{h}
is the recovery operator defined above for the recovery of the control. The details can be found in
[16]
.
Theorem 4.2.
Let
(
y, p, u
)
and
(
y_{h}, p_{h}, u_{h}
)
be the solutions of (3)(5) and (8)(10), respectively. Suppose that all the conditions in Theorem 4.1 are valid and y, p
∈
H
^{3}
(Ω).
Then
Proof
. Let
y_{I}
be the piecewise linear Lagrange interpolation of
y
. According to Theorem 2.1.1 in
[19]
, we have
From the standard interpolation error estimate technique (see, e.g.,
[9]
) that
By using (36)(37), we get
Therefore,
From Theorem 4.1 and (39), we derive
Similarly, we can prove that
Then (36) follows from (40)(41).
By using the superconvergence results above, we obtain the following a posteriori error estimates of variational discretization approximation for the elliptic optimal control problems.
Theorem 4.3.
Assume that all the conditions in Theorem 4.1 and Theorem 4.2 are valid. Then
Proof
. From Theorem 4.1 and Theorem 4.2, it is easy to obtain the above results.
5. Numerical experiments
In this section, we present some numerical examples which is solved numerically with codes developed based on AFEPack. The package provide a freely available tool of finite element approximation for PDEs and the details can be found in
[15]
.
We solve the following optimal control problems:
where
and the domain Ω is the unit square [0, 1] × [0, 1] and
B
=
I
is the identity operator and
E
is the 2 × 2 identity matrix.
Example 1.
The data are as follows:
The numerical results are listed in
Table 1
. In
Figure 1
, we show the profiles of the exact solution
u
alongside the solution error.
Numerical results, Example 1.
Numerical results, Example 1.
The exact solution u (top) and the error u_{h} − u (bottom), Example 1.
The results in
Table 1
indicate that ∥
u
−
u_{h}
∥ =
O
(
h
^{2}
), ∥
y
−
y_{h}
∥ =
O
(
h
^{2}
) and ∥
p
−
p_{h}
∥ =
O
(
h
^{2}
). It is consistent with our theoretical result obtained in the Theorem 3.5.
Example 2.
The data are as follows:
The numerical results based on adaptive mesh and uniform mesh are presented in
Table 2
. In
Figure 2
, we show the profiles of the exact solution
u
alongside the solution error. From
Table 2
, it is clear that the adaptive mesh generated via the error indicators in Theorem 4.3 are able to save substantial computational work, in comparison with the uniform mesh. Our numerical results confirm our theoretical results.
Numerical results, Example 2.
Numerical results, Example 2.
The exact solution u (top) and the error u_{h} − u (bottom), Example 2.
BIO
Yuchun Hua received M.Sc. form Yunnan University. His research interests include differential equations and dynamical system.
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China.
email:huayuchun306@sina.com
Yuelong Tang received M.Sc. and Ph.D from Xiangtan University. His research interests include numerical optimization and numerical methods for partial differential equations.
Department of Mathematics and Computational Science, Hunan University of Science and Engineering, Yongzhou 425100, Hunan, China.
email:tangyuelonga@163.com
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