Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 217493, 21 pages
doi:10.1155/2011/217493
Research Article
Adaptive Mixed Finite Element Methods for
Parabolic Optimal Control Problems
Zuliang Lu1, 2
1
2
School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China
College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China
Correspondence should be addressed to Zuliang Lu, zulianglux@126.com
Received 12 May 2011; Accepted 30 June 2011
Academic Editor: Kue-Hong Chen
Copyright q 2011 Zuliang Lu. 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 will investigate the adaptive mixed finite element methods for parabolic optimal control problems. The state and the costate are approximated by the lowest-order Raviart-Thomas mixed
finite element spaces, and the control is approximated by piecewise constant elements. We derive
a posteriori error estimates of the mixed finite element solutions for optimal control problems. Such
a posteriori error estimates can be used to construct more efficient and reliable adaptive mixed
finite element method for the optimal control problems. Next we introduce an adaptive algorithm
to guide the mesh refinement. A numerical example is given to demonstrate our theoretical results.
1. Introduction
Optimal control problems are very important models in science and engineering numerical
simulation. Finite element method of optimal control problems plays an important role in
numerical methods for these problems. Let us mention two early papers devoted to linear
optimal control problems by Falk 1 and Geveci 2. Knowles was concerned with standard
finite element approximation of parabolic time optimal control problems in 3. In 4 Gunzburger and Hou investigated the finite element approximation of a class of constrained
nonlinear optimal control problems. For quadratic optimal control problem governed by
linear parabolic equation, Liu and Yan derived a posteriori error estimates for both the state
and the control approximation in 5. Systematic introductions of the finite element method
for optimal control problems can be found in 6–10.
Adaptive finite element approximation was the most important means of boosting the
accuracy and efficiency of finite element discretization. The literature in this aspect was
huge, see, for example, 11, 12. Adaptive finite element method was widely used in engineering numerical simulation. There has been extensive studies on adaptive finite element
2
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approximation for optimal control problems. In 13, the authors have introduced some basic
concept of adaptive finite element discretization for optimal control of partial differential
equations. A posteriori error estimators for distributed elliptic optimal control problems were
contained in Li et al. 14. Recently an adaptive finite element method for the estimation of
distributed parameter in elliptic equation was discussed by Feng et al. 15. Note that all the
above works aimed at standard finite element method.
In many control problems, the objective functional contains the gradient of the state
variables. Thus, the accuracy of the gradient is important in numerical discretization of the
coupled state equations. When the objective functional contains the gradient of the state
variable, mixed finite element methods should be used for discretization of the state equation
with which both the scalar variable and its flux variable can be approximated in the same
accuracy. In 16–20 we have done some primary works on a priori error estimates and
superconvergence for linear optimal control problems by mixed finite element methods. We
considered a posteriori error estimates of mixed finite element methods for quadratic and
general optimal control problems in 21–23.
In 24, the authors discussed the mixed finite element approximation for general
optimal control problems governed by parabolic equation. And then, they derived a posteriori error estimates of mixed finite element solution. In this paper, we study the adaptive
mixed finite element methods for the parabolic optimal control problems. We construct the
mixed finite element discretization for the original problems and derive a useful posteriori
error indicators. Furthermore, we provide an adaptive algorithm to guide the multimesh
refinement. Finally, a numerical experiment shows that this algorithm works very well with
the adaptive multimesh discretization.
The plan of this paper is as follows. In the next section, we construct the mixed finite
element discretization for the parabolic optimal control problems. Then, we derive a posteriori error estimates for the mixed finite element solutions in Section 3. Next, we introduce an
adaptive algorithm to guide the mesh refinement in Section 4. Finally, a numerical example
is given to demonstrate our theoretical results in Section 5.
2. Mixed Methods of Optimal Control Problems
In this section, we investigate the mixed finite element approximation for parabolic optimal
control problems. We adopt the standard notation W m,p Ω for Sobolev spaces on Ω with
p
p
p
α
a norm · m,p given by vm,p
|α|≤m D vLp Ω , a seminorm | · |m,p given by |v|m,p
p
m,p
α
{v ∈ W m,p Ω : v|∂Ω
0}. For p
2, we denote
|α| m D vLp Ω . We set W0 Ω
H m Ω W m,2 Ω, H0m Ω W0m,2 Ω, and · m · m,2 , · · 0,2 .
We denote by Ls 0, T ; W m,p Ω the Banach space of all Ls integrable functions from J
T
into W m,p Ω with norm vLs J;W m,p Ω 0 vsW m,p Ω dt1/s for s ∈ 1, ∞ and the standard
modification for s ∞. The details can be found in 25.
The parabolic optimal control problems that we are interested in are as follows:
T
min
u∈K⊂U
yt x, t
g1 p
g2 y
ju dt ,
0
div px, t
f
ux, t,
x ∈ Ω,
Mathematical Problems in Engineering
px, t
yx, t
3
−Ax∇yx, t,
x ∈ ∂Ω, t ∈ J,
0,
x ∈ Ω,
y0 x,
yx, 0
x ∈ Ω,
2.1
where the bounded open set Ω ⊂ R2 is a convex polygon with the boundary ∂Ω. Let K be
0, T , y0 x ∈ H01 Ω.
a closed convex set in U
L2 J; L2 Ω, f ∈ L2 J; L2 Ω, J
Furthermore, we assume that the coefficient matrix Ax
ai,j x2×2 ∈ L∞ Ω; R2×2 is
a symmetric 2 × 2-matrix and there is a constant c > 0 satisfying for any vector X ∈ R2 ,
Xt AX ≥ cX2R2 . j is positive, g1 , g2 , and j are locally Lipschitz on L2 Ω2 , W, U, and that
there is a c > 0 such that j u − j u, u − u ≥ cu − u0 , for all u, u ∈ U.
Now we will describe the mixed finite element discretization of parabolic optimal control problems 2.1. Let
V
2
v ∈ L2 Ω , div v ∈ L2 Ω ,
Hdiv; Ω
W
L2 Ω.
2.2
The Hilbert space V is equipped with the following norm:
div v20,Ω
v20,Ω
vHdiv;Ω
vdiv
1/2
.
2.3
We recast 2.1 as the following weak form: find p, y, u ∈ V × W × K such that
T
min
u∈K⊂U
g1 p
g2 y
yt , w
ju dt ,
2.4
∀v ∈ V,
2.5
0
A−1 p, v − y, div v
div p, w
yx, 0
f
y0 x,
0,
u, w ,
∀w ∈ W,
∀x ∈ Ω.
2.6
2.7
Similar to 26, the optimal control problems 2.4–2.7 have a unique solution p,
y, u, and a triplet p, y, u is the solution of 2.4–2.7 if and only if there is a costate q, z ∈
V × W such that p, y, q, z, u satisfies the following optimality conditions:
A−1 p, v − y, div v
yt , w
div p, w
f
0,
u, w ,
∀v ∈ V,
∀w ∈ W,
2.8
2.9
4
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yx, 0
∀x ∈ Ω,
y0 x,
2.10
A−1 q, v − z, div v
− g1 p, v ,
∀v ∈ V,
2.11
−zt , w
g2 y , w ,
∀w ∈ W,
2.12
div q, w
zx, T T
∀x ∈ Ω,
0,
z, u − u
j u
0
U
dt ≥ 0,
2.13
∀u ∈ K,
2.14
where ·, ·U is the inner product of U. In the rest of the paper, we will simply write the
product as ·, · whenever no confusion should be caused.
Let Th be regular triangulation of Ω. They are assumed to satisfy the angle condition
which means that there is a positive constant C such that, for all τ ∈ Th , C−1 h2τ ≤ |τ| ≤ Ch2τ ,
where |τ| is the area of τ, hτ is the diameter of τ and h max hτ . In addition C or c denotes a
general positive constant independent of h.
Let Vh × Wh ⊂ V × W denote the Raviart-Thomas space 27 of the lowest order
associated with the triangulation Th of Ω. Pk denotes the space of polynomials of total degree
at most k. Let Vτ {v ∈ P02 τ x · P0 τ}, Wτ P0 τ. We define
Vh : {vh ∈ V : ∀τ ∈ Th , vh |τ ∈ Vτ},
Wh : {wh ∈ W : ∀τ ∈ Th , wh |τ ∈ Wτ},
Kh : {uh ∈ K : ∀τ ∈ Th , uh |τ
2.15
constant}.
The mixed finite element discretization of 2.4–2.7 is as follows: compute ph , yh , uh ∈ Vh × Wh × Kh such that
T
min
uh ∈Kh
g1 ph g2 yh
yht , wh
juh dt ,
0
A−1 ph , vh − yh , div vh
div ph , wh yh x, 0
f
y0h x,
0,
∀vh ∈ Vh ,
uh , w h ,
2.16
∀wh ∈ Wh ,
∀x ∈ Ω,
where y0h x ∈ Wh is an approximation of y0 . The optimal control problem 2.16 again has
a unique solution ph , yh , uh , and a triplet ph , yh , uh is the solution of 2.16 if and only
Mathematical Problems in Engineering
5
if there is a costate qh , zh ∈ Vh × Wh such that ph , yh , qh , zh , uh satisfies the following
optimality conditions:
A−1 ph , vh − yh , div vh
yht , wh
div ph , wh yh x, 0
div qh , wh ∀vh ∈ Vh ,
uh , w h ,
f
y0 x,
A−1 qh , vh − zh , div vh −zht , wh 0,
∀wh ∈ Wh ,
∀x ∈ Ω,
− g1 ph , vh ,
g2 yh , wh ,
∀vh ∈ Vh ,
2.17
∀wh ∈ Wh ,
0, ∀x ∈ Ω,
zh , uh − uh ≥ 0, ∀uh ∈ Kh .
zh x, T j uh We now consider the fully discrete approximation for the semidiscrete problem. Let
Δt > 0, N T/Δt ∈ Z, and ti iΔt, i ∈ Z. Also, let
ψi
ψ i x
ψx, ti ,
ψ i − ψ i−1
.
Δt
dt ψ i
2.18
For i
1, 2, . . . , N, construct the finite element spaces Vih ∈ V, Whi ∈ W similar as Vh .
Similarly, construct the finite element spaces Khi ∈ Kh with the mesh Tih . Let hiτ denote
the maximum diameter of the element τ i in Tih . Define mesh functions τ· and mesh size
functions hτ · such that τt|t∈ti−1 ,ti τ i , hτ t|t∈ti−1 ,ti hτi . For ease of exposition, we will
denote τt and hτ t by τ and hτ , respectively.
The following fully discrete approximation scheme is to find pih , yhi , uih ∈ Vih × Whi ×
i
Kh , i 1, 2, . . . , N, such that
min
uih ∈Khi
t
N
i
i 1
ti−1
g1 pih
g2 yhi
A−1 pih , vh − yhi , div vh
dt yhi , wh
div pih , wh
yh0 x, 0
0,
fx, ti y0h x,
,
2.19
∀vh ∈ Vih ,
2.20
j
uih
uih , wh ,
∀x ∈ Ω.
∀wh ∈ Whi ,
2.21
2.22
It follows that the optimal control problems 2.19–2.22 have a unique solution pih , yhi , uih ,
1, 2, . . . , N, is the
i
1, 2, . . . , N, and that a triplet pih , yhi , uih ∈ Vih × Whi × Khi , i
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Mathematical Problems in Engineering
, zi−1
∈ Vih × Whi such that
solution of 2.19–2.22 if and only if there is a costate qi−1
h
h
2
, zi−1
, uih ∈ Vih × Whi × Khi satisfies the following optimality conditions:
pih , yhi , qi−1
h
h
A−1 pih , vh − yhi , div vh
div pih , wh
dt yhi , wh
fx, ti yh0 x, 0
y0h x,
i−1
A−1 qi−1
h , vh − zh , div vh
div qi−1
h , wh
− dt zih , wh
zN
h x, T 2.23
uih , wh ,
∀wh ∈ Whi ,
∀x ∈ Ω,
2.24
2.25
− g1 pih , vh ,
∀vh ∈ Vih ,
2.26
g2 yhi , wh ,
∀wh ∈ Whi ,
2.27
0,
i
zi−1
h , uh − uh ≥ 0,
j uih
For i
∀vh ∈ Vih ,
0,
∀x ∈ Ω,
2.28
∀uh ∈ Khi .
2.29
1, 2, . . . , N, let
Yh |ti−1 ,ti Zh |ti−1 ,ti Ph |ti−1 ,ti Qh |ti−1 ,ti Uh |ti−1 ,ti t − ti−1 yhi
,
Δt
t − ti−1 zih
ti − tzi−1
h
,
Δt
t − ti−1 pih
ti − tpi−1
h
,
Δt
t − ti−1 qih
ti − tqi−1
h
,
Δt
ti − tyhi−1
2.30
uih .
For any function w ∈ CJ; L2 Ω, let
wx,
t|t∈ti−1 ,ti wx, ti ,
wx, t|t∈ti−1 ,ti wx, ti−1 .
2.31
Then the optimality conditions 2.23–2.29 can be restated as.
A−1 Ph , vh − Yh , div vh
Yht , wh div Ph , wh
Yh x, 0
0,
∀vh ∈ Vih ,
f Uh , wh ,
y0h x,
∀x ∈ Ω,
∀wh ∈ Whi ,
2.32
2.33
2.34
Mathematical Problems in Engineering
7
− g1 Ph , vh ,
A−1 Qh , vh − Zh , div vh
−Zht , wh g2 Yh , wh ,
div Qh , wh
Zh x, T j Uh ∀vh ∈ Vih ,
2.35
∀wh ∈ Whi ,
2.36
∀x ∈ Ω,
0,
Zh , uh − Uh ≥ 0,
2.37
∀uh ∈ Khi .
2.38
In the rest of the paper, we will use some intermediate variables. For any control function Uh ∈ K, we first define the state solution pUh , yUh , qUh , zUh which satisfies
A−1 pUh , v − yUh , div v
yt Uh , w
div pUh , w
yUh x, 0
f
y0 x,
div qUh , w
zUh x, T Uh , w ,
∀x ∈ Ω,
2.40
2.41
g2 yUh , w ,
0,
2.39
∀w ∈ W,
− g1 pUh , v ,
A−1 qUh , v − zUh , div v
−zt Uh , w
∀v ∈ V,
0,
∀v ∈ V,
2.42
∀w ∈ W,
2.43
∀x ∈ Ω.
2.44
3. A Posteriori Error Estimates
In this section we study a posteriori error estimates of the mixed finite element approximation
for the parabolic optimal control problems. Fixed given u ∈ K, let M1 , M2 be the inverse
operators of the state equation 2.6, such that pu M1 u and yu M2 u are the solutions
of the state equations 2.6. Similarly, for given Uh ∈ Kh , Ph Uh M1h Uh , Yh Uh M2h Uh
are the solutions of the discrete state equation 2.33. Let
Fu
Fh Uh g1 M1 U
g1 M1h Uh g2 M2 U
g2 M2h Uh ju,
jUh .
3.1
8
Mathematical Problems in Engineering
It can be shown that
j u z, v ,
F u, v
j Uh zUh , v ,
F Uh , v
Fh Uh , v
j Uh 3.2
Zh , v .
It is clear that F and Fh are well defined and continuous on K and Khi . Also the functional Fh can be naturally extended on K. Then 2.4 and 2.19 can be represented as
min{Fu},
3.3
min {Fh Uh }.
3.4
u∈K
Uh ∈Khi
In many application, F· is uniform convex near the solution u. The convexity of F·
is closely related to the second order sufficient conditions of the optimal control problems,
which are assumed in many studies on numerical methods of the problem. For instance, in
many applications, u → g1 M1 U and u → g2 M2 U are convex. Then there is a constant
c > 0, independent of h, such that
T
F u − F Uh , u − Uh
0
U
≥ cu − Uh 2L2
J;L2 Ω
.
3.5
Theorem 3.1. Let u and Uh be the solutions of 3.3 and 3.4, respectively. Assume that Khi ⊂ K. In
addition, assume that Fh Uh |τ ∈ H s τ, for all τ ∈ Th , s 0, 1, and there is a vh ∈ Khi such
that
F Uh , vh − u ≤ C
hτ Fh Uh H s τ u − Uh sL2 τ .
h
3.6
τ∈Th
Then one has
u − Uh 2L2 J;L2 Ω ≤ Cη12
2
CzUh − Zh 2
L
J;L2 Ω
,
3.7
where
η12
T 0 τ∈Th
h1τ s j Uh 1
Zh s
H 1 τ
.
3.8
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9
Proof. It follows from 3.3 and 3.4 that
T
T
0
F u, u − v ≤ 0,
∀v ∈ K,
0
3.9
Fh Uh , Uh − vh ≤ 0,
∀vh ∈
Khi
⊂ K.
Then it follows from assumptions 3.5, 3.6 and Schwartz inequality that
cu − Uh 2L2
≤
T
J;L2 Ω
F u − F Uh , u − Uh
0
≤
T
Fh Uh , vh − u
0
≤C
T 0
τ∈Th
Fh Uh − F Uh , u − Uh
1 s
h1τ s Fh Uh H s τ
F Uh − F Uh 2 2
h
L Ω
3.10
c
u − Uh 2L2 J;L2 Ω .
2
It is not difficult to show
Fh Uh j Uh F Uh Zh ,
j Uh zUh ,
3.11
where zUh is defined in 2.39–2.44. Thanks to 3.11, it is easy to derive
F Uh − F Uh 2
≤
C
Z
−
zU
h
h
h
L Ω
L2 Ω
.
3.12
Then by estimates 3.10 and 3.12 we can prove the requested result 3.7.
In order to estimate the a posteriori error of the mixed finite element approximation
solution, we will use the following dual equations:
−ϕt − div A∇ϕ
G,
x ∈ Ω, t ∈ 0, T ,
ϕ|∂Ω
0,
t ∈ J,
3.13
ϕx, T 0, x ∈ Ω,
ψt − div A∗ ∇ψ
G, x ∈ Ω, t ∈ 0, T ,
ψ|∂Ω
0,
t ∈ J,
ψx, 0
0,
x ∈ Ω.
The following well-known stability results are presented in 28.
3.14
10
Mathematical Problems in Engineering
Lemma 3.2. Let ϕ and ψ be the solutions of 3.13, and 3.14, respectively. Then, for v
ϕ or v
ψ,
vL∞ J;L2 Ω ≤ CGL2 J;L2 Ω ,
∇vL2 J;L2 Ω ≤ CGL2 J;L2 Ω ,
2 ≤ CGL2 J;L2 Ω ,
D v 2 2
3.15
L J;L Ω
vt L2 J;L2 Ω ≤ CGL2 J;L2 Ω ,
where D2 v
∂2 v/∂xi ∂xj , 1 ≤ i, j ≤ 2.
Now, we are able to derive the main result.
Theorem 3.3. Let Yh , Ph , Zh , Qh , Uh and yUh , pUh , zUh , qUh , Uh be the solutions of
2.32–2.38 and 2.39–2.44. Then,
Yh − yUh 2 2 2
L J;L Ω
Ph − pUh 2L2 J;L2 Ω ≤ Cη22 ,
3.16
where
η22
2
div Ph − f − Uh 2
Yht
L J;L2 Ω
Yh − yUh x, 02 2
L Ω
2
Yh − Y h 2
t L
3.17
L J;L2 Ω
2
Ph − Ph 2
J;L2 Ω
Proof. Letting ϕ be the solution of 3.13 with G
2
f − f 2
L J;Hdiv;Ω
.
Yh − yUh , we infer
Yh − yUh 2 2
L J;L2 Ω
T
Yh − yUh , F dt
0
T
Yh − yUh , −ϕt − div A∇ϕ dt
0
T
Yh − yUh t , ϕ − Ph − pUh , ∇ϕ dt
0
Yh − yUh x, 02 2
L Ω
T
Yht − yt Uh , ϕ
divPh − pUh , ϕ dt
0
Yh − yUh x, 02 2
L Ω
.
3.18
Mathematical Problems in Engineering
11
Then it follows from 2.39-2.40 that
Yh − yUh 2 2 2
L J;L Ω
T
Yht , ϕ
div Ph , ϕ − yt Uh , ϕ − div pUh , ϕ
div Ph − Ph , ϕ
dt
0
Yh − yUh x, 02 2
L Ω
T
Yht
div Ph − f − Uh , ϕ
f − f, ϕ
div Ph − Ph , ϕ
dt
f − f, ϕ
div Ph − Ph , ϕ
dt
0
Yh − yUh x, 02 2
L Ω
T
Yht
div Ph − f − Uh , ϕ
0
Yh − yUh x, 02 2
L Ω
≤ CδYht
2
div Ph − f − Uh 2
L J;L2 Ω
2
CδPh − Ph 2
L J;Hdiv;Ω
Yh − yUh x, 02 2
L Ω
2
Cδf − f 2
L J;L2 Ω
2
CδYh − Yh 2
L J;L2 Ω
2
CδϕL2 J;L2 Ω ,
3.19
where and after, δ is an arbitrary positive number, Cδ is the constant dependent on δ−1 .
Now, we are in the position of estimating the error Ph − pUh 2L2 J;L2 Ω . First, we
derive from 2.32-2.33 and 2.39-2.40 the following useful error equations:
A−1 Ph − pUh , vh − Yh − yuh , div vh
Yh − yUh , wh
t
div Ph − pUh , wh
f − f, wh −
where vh ∈ Vh , wh ∈ Wh . Choose vh Ph − pUh and wh
and add the two relations of 3.20-3.21 such that
A−1 Ph − pUh , Ph − pUh f − f, Yh − yUh −
3.20
0,
Yh − Yh , wh ,
t
3.21
Yh − yUh as the test functions
Yh − yUh , Yh − yUh t
Yh − Yh , Yh − yUh .
t
3.22
12
Mathematical Problems in Engineering
Then, using -Cauchy inequality, we can find an estimate as follows:
2
cPh − pUh 2
Yh − yUh , Yh − yUh L Ω
t
2
≤ CYh − yUh L2 Ω
3.23
2
C Yh − Yh 2
2
Cf − f L2 Ω
t L Ω
.
Note that
2
1 ∂
Yh − yUh 2 ,
L Ω
2 ∂t
Yh − yUh , Yh − yUh t
3.24
then, using the assumption on A, we verify that
2
1 ∂
Yh − yUh 2
L Ω
L Ω
2 ∂t
2
2
2
≤ CYh − yUh 2
Cf − f 2
C Yh − Yh 2
2
cPh − pUh 2
L Ω
L Ω
Integrating 3.25 in time and since Yh 0 − yUh 0
easily obtain the following error estimate:
Ph − pUh L2 J;L2 Ω
≤ Cf − f 3.25
t L Ω
.
0, applying Gronwall’s lemma, we can
Yh − yUh L∞ J;L2 Ω
L2 J;L2 Ω
3.26
C Yh − Yh t L2 J;L2 Ω
.
Using the triangle inequality and 3.26, we deduce that
Ph − pUh L2 J;L2 Ω
≤ C f − f 2
L J;L2 Ω
Y h − Yh t L2 J;L2 Ω
Ph − Ph L2 J;L2 Ω
3.27
.
Then letting δ be small enough, it follows from 3.18–3.27 that
Yh − yUh 2 2 2
L J;L Ω
Ph − pUh 2L2 J;L2 Ω ≤ Cη22 .
3.28
This proves 3.16.
Similarly, letting ψ be the solution of 3.14 with G
2.27, 2.42-2.43, we can conclude the following.
Zh − zUh , with the aid of 2.26-
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13
Theorem 3.4. Let Yh , Ph , Zh , Qh , Uh and yUh , pUh , zUh , qUh , Uh be the solutions of
2.32–2.38 and 2.39–2.44. Then,
Zh − zUh 2L2 J;L2 Ω
Qh − qUh 2L2 J;L2 Ω ≤ C η22
η32 ,
3.29
where
2
Zht − div Qh 2
g2 Yh
η32
L J;L2 Ω
2
Zh − Zh 2
2
Zh − Zh 2
2
Yh − Yh 2
2
Qh − Qh 2
L J;L2 Ω
L J;L2 Ω
t L J;L2 Ω
L J;Hdiv;Ω
3.30
.
Let p, y, q, z, u and Ph , Yh , Qh , Zh , Uh be the solutions of 2.8–2.14 and 2.32–
2.38, respectively. We decompose the errors as follows:
p − Ph
p − pUh pUh − Ph :
1
ε1 ,
y − Yh
y − yUh yUh − Yh : r1
e1 ,
q − Qh
q − qUh qUh − Qh :
ε2 ,
z − Zh
z − zUh zUh − Zh : r2
2
3.31
e2 .
From 2.8–2.13 and 2.39–2.44, we derive the error equations:
A−1 1 , v − r1 , div v
r1t , w
div
1 , w
A−1 2 , v − r2 , div v
r2t , w
div
2 , w
0,
∀v ∈ V,
u − Uh , w,
3.32
∀w ∈ W,
− g1 p − g1 pUh v ,
g2 y − g2 yUh , w ,
3.33
∀v ∈ V,
3.34
∀w ∈ W.
3.35
Theorem 3.5. There is a constant C > 0, independent of h, such that
1 L2 J;L2 Ω
r1 L2 J;L2 Ω ≤ Cu − Uh L2 J;L2 Ω ,
3.36
2 L2 J;L2 Ω
r2 L2 J;L2 Ω ≤ Cu − Uh L2 J;L2 Ω .
3.37
14
Mathematical Problems in Engineering
Proof. Part I
Choose v
we have
1
and w
r1 as the test functions and add the two relations of 3.32-3.33, then
A−1 1 ,
1
r1t , r1 u − Uh , r1 .
3.38
Then, using the Cauchy inequality, we can find an estimate as follows:
A−1 1 ,
r1t , r1 ≤ C r1 2L2 Ω
1
u − Uh 2L2 Ω .
3.39
Note that
r1t , r1 1 ∂
r1 2L2 Ω ,
2 ∂t
3.40
then, using the assumption on A, we can obtain that
1 2L2 Ω
1 ∂
r1 2L2 Ω ≤ C r1 2L2 Ω
2 ∂t
Integrating 3.41 in time and since r1 0
obtain the following error estimate
1 2L2 J;L2 Ω
3.41
u − Uh 2L2 Ω .
0, applying the Gronwall’s lemma, we can easily
r1 2L2 J;L2 Ω ≤ Cu − Uh 2L2 J;L2 Ω .
3.42
This implies 3.36.
Part II
Similarly, choose v
2 and w
3.35, then we can obtain that
A−1 2 ,
r2t , r2 2
r2 as the test functions and add the two relations of 3.34-
g2 y − g2 yUh , r2 − g1 p − g1 pUh ,
2
.
3.43
Then, using the Cauchy inequality, we can find an estimate as follows:
A−1 2 ,
2
r2t , r2 ≤ C r1 2L2 Ω
r2 2L2 Ω
1 2L2 Ω
c
2 2L2 Ω .
2
3.44
Note that
r2t , r2 1 ∂
r2 2L2 Ω ,
2 ∂t
3.45
Mathematical Problems in Engineering
15
then, using the assumption on A, we verify that
2 2L2 Ω
1 ∂
r2 2L2 Ω ≤ C r1 2L2 Ω
2 ∂t
Integrating 3.46 in time and since r2 T obtain the following error estimate
2 2L2 J;L2 Ω
r2 2L2 Ω
1 2L2 Ω .
3.46
0, applying the Gronwall’s lemma, we can easily
r2 2L2 J;L2 Ω ≤ Cu − Uh 2L2 J;L2 Ω .
3.47
Then 3.37 follows from 3.47 and the previous statements immediately.
Collecting Theorems 3.1–3.5, we can derive the following results.
Theorem 3.6. Let p, y, q, z, u and Ph , Yh , Qh , Zh , Uh be the solutions of 2.8–2.14 and
2.32–2.38, respectively. In addition, assume that j Uh Zh |τ ∈ H s τ, for all τ ∈ Th , s
0, 1, and that there is a vh ∈ Kh such that
j Uh hτ j Uh Zh , vh − u ≤ C
τ∈Th
Zh H s τ
u − Uh sL2 τ .
3.48
Then one has that, for all t ∈ 0, T ,
u − Uh 2L2 J;L2 Ω
y − Yh 2 2
z − Zh 2L2 J;L2 Ω
L J;L2 Ω
p − Ph 2L2 J;L2 Ω
q − Qh 2L2 J;L2 Ω ≤ C
3
ηi2 ,
3.49
i 1
where η1 , η2 , and η3 are defined in Theorems 3.1–3.4.
4. An Adaptive Algorithm
In the section, we introduce an adaptive algorithm to guide the mesh refine process. A
posteriori error estimates which have been derived in Section 3 are used as an error indicator
to guide the mesh refinement in adaptive finite element method.
Now, we discuss the adaptive mesh refinement strategy. The general idea is to refine
the mesh such that the error indicator like η is equally distributed over the computational
2
mesh. Assume that an a posteriori error estimator η has the form of η2
τi ητi , where τi is
the finite elements. At each iteration, an average quantity of all ητ2i is calculated, and each ητ2i
is then compared with this quantity. The element τi is to be refined if ητ2i is larger than this
quantity. As ητ2i represents the total approximation error over τi , this strategy makes sure that
higher density of nodes is distributed over the area where the error is higher.
Based on this principle, we define an adaptive algorithm of the optimal control problems 2.1 as follows.
16
Mathematical Problems in Engineering
Starting from initial triangulations Th0 of Ω, we construct a sequence of refined triangulation Thj as follows. Given Thj , we compute the solutions Ph , Yh , Qh , Zh , Uh of the system
2.32–2.38 and their error estimator
T ητ2
0 τ∈Th
h1τ s j Uh 1
Zh H 1 τ
Yh − yUh x, 02 2
L τ
2
Yh − Y h 2
L J;L2 τ
2
f − f L2 J;L2 τ
2
Ph − Ph t L J;L2 τ
L2 J;Hdiv;τ
2
Zht − div Qh 2
g2 Yh
2
Zh − Zh 2
J;L2 τ
2
Zh − Zh 2
2
Yh − Yh 2
2
Qh − Qh 2
L
4.1
L
L J;L2 τ
Ej
2
div Ph − f − Uh 2
Yht
s
J;L2 τ
t L J;L2 τ
L J;Hdiv;τ
,
ητ2 .
τ∈Th
Then we adopt the following mesh refinement strategy. All the triangles τ ∈ Thj
satisfying ητ2 ≥ αEj /n are divided into four new triangles in Thj 1 by joining the midpoints
of the edges, where n is the numbers of the elements of Thj , α is a given constant. In order to
maintain the new triangulation Thj 1 to be regular and conformal, some additional triangles
need to be divided into two or four new triangles depending on whether they have one or
more neighbor which have refined. Then we obtain the new mesh Thj 1 . The above procedure
will continue until Ej ≤ tol, where tol is a given tolerance error.
5. Numerical Example
The purpose of this section is to illustrate our theoretical results by numerical example. Our
numerical example is the following optimal control problem:
T
1
min
p − pd 2
u∈K⊂U 2 0
yt
−zt
div q
div p
y − yd ,
f
u,
p
q
−∇z
y − yd 2
−∇y,
5.1
u − u0 dt ,
x ∈ Ω, y|∂Ω
p − pd ,
2
0, yx, 0
x ∈ Ω, z|∂Ω
5.2
0,
0, zx, T 0.
5.3
Mathematical Problems in Engineering
17
In our example, we choose the domain Ω 0, 1 × 0, 1. Let Ω be partitioned into Th
as described Section 2. For the constrained optimization problem,
5.4
min Fu,
u∈K
{u ∈ L2 Ω : u ≥ 0 a.e. in Ω × J}, the
where Fu is a convex functional on U and K
iterative scheme reads n 0, 1, 2, . . .
b un
1/2 , v
bun , v − ρn F un , v ,
un 1 PKb un 1/2 ,
∀v ∈ U,
5.5
where b·, · is a symmetric and positive definite bilinear form such that there exist constants
c0 and c1 satisfying
∀u, v ∈ U,
|bu, v| ≤ c1 uU vU ,
5.6
bu, u ≥ c0 u2U ,
and the projection operator PKb U → K is defined. For given w ∈ U find PKb w ∈ K such that
b PKb w − w, PKb w − w
min bu − w, u − w.
5.7
u∈K
The bilinear form b·, · provides suitable preconditioning for the projection algorithm. An
application of 5.5 to the discretized parabolic optimal control problem yields the following
algorithm:
b un
1/2 , vh
T
T
T
bun , vh − ρn
T
un
T
pn , vh − yn , div vh
div pn , wh yn 0, w0
qn , vh − zn , div vh −
0
−znt , wh zn , vh ,
∀vh ∈ Uh ,
0
T
0
un
1
1/2
f
un , w h ,
pn − pd , vh ,
zn T , wh T PKb un
T
∀wh ∈ Wh ,
0
0
div qn , wh ∀vh ∈ Vh ,
0,
0
ynt , wh
0
T
5.8
∀vh ∈ Vh ,
yn − yd , wh ,
∀wh ∈ Wh ,
0
,
un
1/2 , un
∈ Uh .
The main computational effort is to solve the four state and costate equations and to compute
the projection PKb un 1/2 . In this paper we use a fast algebraic multigrid solver to solve the
18
Mathematical Problems in Engineering
state and costate equations. Then it is clear that the key to saving computing time is how to
compute PKb un 1/2 efficiently. If one uses the C0 finite elements to approximate to the control,
then one has to solve a global variational inequality, via, for example, semismooth Newton
method. The computational load is not trivial. For the piecewise constant elements, Kh
{uh : uh ≥ 0} and bu, v u, vU , then
PKb un
1/2 |τ
max 0, avg un
1/2
|τ ,
5.9
where avgun 1/2 |τ is the average of un 1/2 over τ.
In solving our discretized optimal control problem, we use the preconditioned projec0.8. We now briefly
tion gradient method with bu, v
u, vU and a fixed step size ρ
describe the solution algorithm to be used for solving the numerical examples in this section.
5.1. Algorithm
1 Solve the discretized optimization problem with the projection gradient method on
the current meshes, and calculate the error estimators ηi .
2 Adjust the meshes using the estimators, and update the solution on new meshes, as
described.
Now, we present a numerical example to illustrate our theoretical results.
Example 5.1. We choose the state function by
yx1 , x2 and the function fx1 , x2 px1 , x2 yt
sin πx1 sin πx2 sin πt
5.10
div p − u with
−π cos πx1 sin πx2 sin πt, π sin πx1 cos πx2 sin πt,
qx1 , x2 pd x1 , x2 px1 , x2 .
5.11
The costate function can be chosen as
zx1 , x2 sin πx1 sin πx2 sin πt.
5.12
It follows from 5.2-5.3 that
yd x1 , x2 y
zt − div q.
5.13
Mathematical Problems in Engineering
19
Table 1: Numerical results on uniform and adaptive meshes.
Nodes
Sides
Elements
Dofs
Total L2 error
u
8097
23968
15872
15872
6.915e − 03
On uniform mesh
y
z
8097
8097
23968
23968
15872
15872
15872
15872
4.065e − 3
4.018e − 3
u
1089
2825
6348
6348
6.527e − 03
On adaptive mesh
y
z
1393
1393
3819
3819
2423
2423
2423
2423
4.346e − 3
4.323e − 3
1
0.8
0.6
0.4
0.2
0
0
0.5
1
0
0.2
0.4
0.8
0.6
Figure 1: The profile of the control solution u at t
1
0.25.
We assume that
λ
u0 x1 , x2 ⎧
⎨0.5,
x1
x2 > 1.0,
⎩0.0,
x1
x2 ≤ 1.0,
1 − sin
πx2
πx1
− sin
2
2
5.14
λ.
Thus, the control function is given by
ux1 , x2 maxu0 − z, 0.
5.15
In this example, the optimal control has a strong discontinuity, introduced by u0 .
The exact solution for the control u is plotted in Figure 1. The control function u is discretized by piecewise constant functions, whereas the state y, p and the costate z, q were
approximation by the lowest-order Raviart-Thomas mixed finite elements. In Table 1, numerical results of u, y, and z on uniform and adaptive meshes are presented. It can be
founded that the adaptive meshes generated using our error indicators can save substantial
computational work, in comparison with the uniform meshes. At the same time, for the
discontinuous control variable u, the accuracy has been improved obviously from the
uniform meshes to the adaptive meshes in Table 1.
20
Mathematical Problems in Engineering
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
Figure 2: The adaptive meshes of the control solution u at t
1
0.25.
In Figure 2, the adaptive mesh for u at t 0.25 is shown. In the computing, we use
η1 − η3 as the error indicators in the adaptive finite element method. It can be founded that
the mesh adapts well to be neighborhood of the discontinuity, and a higher density of node
points is indeed distributed along them.
Acknowledgments
The authors express their thanks to the referees for their helpful suggestions, which led to
improvement of the presentation. This work was supported by National Nature Science
Foundation under Grant 10971074 and Hunan Provincial Innovation Foundation For Postgraduate CX2009B119.
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