Error estimates of a new lowest order mixed finite element approximation for semilinear optimal cont

Mathematical Computation September 2013, Volume 2, Issue 3, PP.62-67

Error Estimates of a New Lowest Order Mixed Finite Element Approximation for Semilinear Optimal Control Problems Zuliang Lu 1, 2#, Dayong Liu 3 1. School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, P.R.China 2. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, P.R.China 3. Chongqing Wanzhou Senior Midddle School, Chongqing 404000, P.R.China #Email: zulianglux@126.com

Abstract This document gives a priori error estimates for the semilinear elliptic optimal control problems by using a new mixed finite element method with the lowest order. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. A priori error estimates for the new mixed finite element approximation of semilinear optimal control problems is derived. Two numerical examples are presented to confirm the theoretical results.

Keywords: A Priori Error Estimates; Semilinear Optimal Control Problems; A New Mixed Finite Element Method

1 INTRODUCTION In this paper, a priori error estimates of a new mixed finite element method with the lowest order for semilinear optimal control problems has been devised. The following semilinear optimal control problems have been taken into consideration: 1  2 2 2 1 min  p  pd  y  yd  u  (1) uK 2 2 2  divp   ( y )  f  Bu, x  ,

p   Ay,

x  ,

y  0,

x  ,

(2)

where  R2 is a convex polygon with the boundary  , pd and yd are two known functions, p , y are the state variables, u is the control variable, and  is fixed constant. It can be assumed that f  H 1 () and B is a continuous linear operator from L2 () to H 1 () . For any r  0 the function  ( y) W 2 (r, r ) ,  ' ( y)  L2 () for any y  H 1 () , and  ' ( y)  0 .

Furthermore, it is also supposed that the coefficient matrix A( x)   aij ( x) 22  L (, R22 ) is a symmetric 2  2 matrix and there is a constant c  0 satisfying any vector x  R2 , X T AX  c X of the control variable, defined by

K  u  L2 () : a  u  b .

2 R2

. Here, K denotes the admissible set (3)

Optimal control problems governed by partial differential equations that are significant in mathematics arise in many science and engineering applications. Efficient numerical methods are critical for those optimal control problems.

Recently, the finite element method of optimal control problems plays an important role in numerical method for these problems, see, for example, [1].

Mixed finite element discretization is an efficient method for many problems, particularly for those problems of the - 62 www.ivypub.org/mc

objective functional contains gradient of the state variables. But the mixed finite element method is not as widely used as in engineering simulations for optimal control. More recently, many researchers have done some preliminary work on such a posteriori error estimates, error estimates and superconvergence of mixed finite element method for optimal control problems, see, for example, [2-3]. However, it doesn't seem to be straightforward to extend these existing techniques to the semilinear elliptic optimal control problems.

The paper is organized as follows. In Section 2, the scheme of the new Raviart-Thomas mixed finite element with the lowest order has been depicted for the optimal control problems governed by semilinear elliptic equations. In Section 3, a priori error estimates for the mixed finite element solutions has been proved. Finally, two numerical examples have been presented to illustrate the theoretical results in Section 4.

2 A NEW MIXED FINITE ELEMENT SCHEMES In this section, the mixed finite element discretization of semilinear elliptic optimal control problem is illustrated.

Let

V  H (div; )  {v  L2 ()2 , divv  L2 ()}. be endowed with the norm given by

v

H ( div ;  )



 v

2 0, 

 divv



1/ 2

2 0, 

.

We denote W  L2 () and U  L2 () . A subspace V0 of V is defined by

V0  {v V , divv  0}.

(1)-(2) are recast as the following weak form, and it is found out that ( p, y, u) V0  W  K

1 min  p  pd uK 2

2



1 y  yd 2

2





( A1 p, v )  ( y, divv )  0,

2 u  2 

v V0 ,

(divp, w)  ( ( y), w)  ( f  Bu, w), w W .

(4) (5)

It is well known (see e.g., [4-5]) that the optimal control problem (4)-(5) has a solution ( p, y, u) , and that a triplet

( p, y, u) is the solution of (4)-(5) then there is a co-state (q, z) V0  W such that ( p, y, q, z, u) satisfies the following optimality conditions (Problem 1):

( A1 p, v )  ( y, divv )  0, (divp, w)  ( ( y ), w)  ( f  Bu, w), ( A1q , v)  ( z, divv )  ( p  pd , v ), (divq , w)  ( ' ( y ) z, w)  ( y  yd , w), ( B* z  u , u  u )U  0,

v V0 , w  W , v V0 , w  W , u  K ,

where (, )U is the inner product of U , B* is the adjoint operator of B .

Introducing a projection [ a ,b] ( f ( x))  max(a, min(b, f ( x))), the above inequality can be denoted by

u( x)  [ a ,b] ( B* z / ) .

Then the optimality conditions (Problem 1) are equivalent to

( A1 p, v )  ( y, divv )  0, (divp, w)  ( ( y ), w)  ( f  Bu, w), ( A1q , v )  ( z, divv )  ( p  pd , v), (divq , w)  ( ' ( y ) z, w)  ( y  yd , w), u ( x)  [ a ,b ] ( B* z /  ).

v V0 , w  W , v V0 , w  W ,

Now the two linear bounded operators L1 , L2 can be denoted by

L1 : L2 ()  W , L1 : f  Bu  y, - 63 www.ivypub.org/mc

L2 : L2 ()  W , L2 : y  yd  z.

Then the optimality conditions can be written as

y  L1 ( f  Bu), p  y ,

z  L2 ( y  yd ), q  z  p  pd .

u( x)  [ a ,b] ( B* z / ) .

For ease of exposition  is assumed to be polygon. Let  h be regular triangulation or rectangulation of  . They are assumed to satisfy the angle condition, which means that there is a positive constant C such that for all T h , C 1hT2  T  ChT2 , where T is the area of T and hT is the diameter of T . Let h  max(hT ) .

Let Vh  Wh  V0  W denote the Raviart-Thomas space [6] of the lowest order associated with the triangulation or rectangulation  h of  . Pk denotes the space of polynomials of total degree at most k , Qm , n indicates the space of polynomials of degree no more than m and n in x and

y , respectively. If T

is a triangle,

V (T )  v  P (T )  x  P0 (T ) and if T is a rectangle, V (T )  v  Q1,0 (T )  Q0,1 (T ) . We define 2 0





Vh : vh  V0 : T   h , vh |T  V (T ) , Wh : wh  W : T   h , wh |T  P0 (T ) , K h : uh  K : T   h , uh |T  P0 (T ).

By the definition of finite element subspace, the mixed finite element approximation of (4)-(5) is as follows: compute ( ph , yh , uh ) Vh  Wh  Kh such that

1 2 1 min  ph  pd  yh  yd u K 2 2 1 ( A ph , v )  ( yh , divv )  0, h

h

2





2 uh  2  v Vh ,

(divph , w)  ( ( yh ), w)  ( f  Buh , w), w Wh .

(6) (7)

It is well known that the optimal control problem (6)-(7) has a solution ( ph , yh , uh ) , and that a triplet ( ph , yh , uh ) is the solution of (6)-(7) then there is a co-state (qh , zh ) Vh  Wh such that ( ph , yh , qh , zh , uh ) satisfies the following optimality conditions (Problem 2): ( A1 ph , v )  ( yh , divv )  0, v Vh ,

(divph , w)  ( ( yh ), w)  ( f  Buh , w), w  Wh , ( A1qh , v )  ( zh , divv )  ( ph  pd , v), v Vh , ' (divqh , w)  ( ( yh ) zh , w)  ( yh  yd , w), w Wh , ( B* zh  uh , u  uh )U  0, u  K h .

The above inequality can be denoted by

uh  [ a ,b] ( B* zh / ) .

Then the optimality conditions (Problem 2) are equivalent to

( A1 ph , v )  ( yh , divv )  0, v Vh , (divph , w)  ( ( yh ), w)  ( f  Buh , w), w  Wh , ( A1qh , v )  ( zh , divv )  ( ph  pd , v ), v Vh , ' (divqh , w)  ( ( yh ) zh , w)  ( yh  yd , w), w Wh , uh  [ a ,b ] ( B* zh /  ).

Next the linear bounded operators L1h , L2h is defined by

L1h : L2 ()  W , L1h : f  Buh  yh , L2 h : L2 ()  W , L2 h : yh  yd  zh .

- 64 www.ivypub.org/mc

Then the optimality conditions can be written as

yh  L1h ( f  Buh ), ph  yh ,

zh  L2 h ( yh  yd ), qh  zh  ph  pd , uh  [ a ,b ] ( B* zh / ).

For  Wh , we shall write [7]

 ( )   ( p)   ( )( p   )   ' ( p)( p   )   '' ( )( p   )2 , where

 ' ( )  01 ' (  t ( p   ))dt,  '' ( )  01 (1  t ) '' ( p  t (  p))dt, are bounded functions on  .

3 A PRIORI ERROR ESTIMATES In the rest of the paper, some intermediate variables are in use. For any control function u  K , we first define the state solution ( p(u), y(u), q(u), z(u)) associated with u that satisfies

( A1 p(u ), v )  ( y (u ), divv )  0, (divp(u ), w)  ( ( y (u )), w)  ( f  Bu, w), ( A1q (u ), v )  ( z (u ), divv )  ( p(u )  pd , v ), (divq (u ), w)  ( ' ( y (u )) z (u ), w)  ( y (u )  yd , w).

Correspondingly, the discrete state solution ( ph (u), yh (u), qh (u), zh (u)) is defined associated with u that satisfies

( A1 ph (u ), v )  ( yh (u ), divv )  0, (divph (u ), w)  ( ( yh (u )), w)  ( f  Bu, w), ( A1qh (u ), v )  ( zh (u ), divv )  ( ph (u )  pd , v ), (divqh (u ), w)  ( ' ( yh (u )) zh (u ), w)  ( yh (u )  yd , w).

By Lemma 2.1 in [5], the following error estimates can be established: Theorem 1 There is a positive constant C independent of h such that

y  yh (u)

L2 (  )

 z  zh (u)

L2 (  )

 p  ph (u)

L2 (  )

 q  qh (u)

L2 (  )

 Ch.

Lemma 1 If f  H 1 () , then there hasa constant C such that

|| ( L1  L1h )( f ) ||L ( )  Ch || f ||H 2

More, if L1 f W

2, 

1

( )

.

() , then there hasa constant C such that

|| ( L1  L1h )( f ) ||L



()

 Ch(|| L1 f ||W

2,

()

 || f ||H

1

()

).

Let ( p(u), y(u)) and ( ph (uh ), yh (uh )) be the solutions of (Problem 1) and (Problem 2), respectively. Let J () : U  R be a G-differential convex functional which satisfies the following form: 1 1  2 2 2 J (u )  p  pd  y  yd  u , 2 2 2 1 1  2 2 2 J h (uh )  ph  pd  yh  yd  uh .

2 2 2 It can be shown that ( J ' (u), v)  (u  B* z, v),

( J h ' (u), v)  (u  B* zh (u), v), ( J h ' (uh ), v)  (uh  B* zh , v).

In the following, we estimate u  uh

L2 (  )

and then obtain the results:

Theorem 2 Let ( p, y, q, z, u)  (V  W )2  K and ( ph , yh , qh , zh , uh )  (Vh  Wh )2  Kh be solutions of (Problem 1) and (Problem 2), respectively. If u, z, f , yd  H 1 (). Then there is a positive constant C independent of h such that - 65 www.ivypub.org/mc

u  uh

L2 (  )

 y  yh

L2 (  )

 z  zh

L2 (  )

 Ch.

If u, z, f , yd  L (). Then there is a positive constant C independent of h such that 

u  uh

L (  )

 y  yh

L (  )

 z  zh

L (  )

 Ch.

4 NUMERICAL EXAMPLES In this section, the priori error estimates for the errors in the control, state, and co-state variables have been validated.

The optimization problem was dealt numerically with codes developed based on AFEPACK. The discretization was already simplified: the control function u was discretized by piecewise constant functions, whereas the state ( y, p) and the co-state ( z, q ) were approximated by the lowest order Raviart-Thomas mixed finite element functions.

Our numerical examples are the following optimal control problems: 1 1 2 2 2 1 min  p  pd  y  yd  u  uK 2 2 2  5 divp  y  u  f , p  y, x , y |  0,

divq  5 y 4 z  y  yd , q  z  p  pd , x , z |  0.

Now, the control constraint K is assumed to be defined by K  u  L2 () : 1  u  2 .

Example 1. In this example, let

y  sin  x1 sin  x2 , z  2sin  x1 sin  x2 ,   cos  x1 sin  x2  u  max(min(2, f ( x)),1), p    ,   sin  x1 cos  x2    cos  x1 sin  x2   2 cos  x1 sin  x2  pd    , q   ,   sin  x1 cos  x2   2 sin  x1 cos  x2 

f  2 2 y  y 5  u, yd  (1  4 2 ) y  5 y 4 z.

We can obtain that TABLE 1 ERROR ESTIMATES ON A SEQUENCE OF UNIFORMLY TRIANGLE MESH

h 1 / 16 1 / 32 1 / 64 1 / 128

u  uh

y  yh

L2

2.237E-01 1.135E-01 5.298E-02 2.667E-02

L2

4.485E-01 2.286E-01 1.063E-01 5.342E-02

z  zh

L2

2.054E-01 1.136E-01 5.299E-02 2.668E-02

Example 2. In this example, let y  sin 2 x1 sin 2 x2 e x  x , z  sin 2 x1 sin 2 x2 e x  x , 1

2

1

2

 2 cos 2 x1 sin 2 x2 e x  x  y  u  max(min(2, f ( x)),1), p    , x x  y  2 sin 2 x1 cos 2 x2 e  2 cos 2 x1 sin 2 x2 e x  x  y   2 cos 2 x1 sin 2 x2 e x  x  y  pd    , q     , x x x x  y  y  2 sin 2 x1 cos 2 x2 e  2 sin 2 x1 cos 2 x2 e yd  (3  8 2 ) y  4 sin 2 ( x1  x2 )  ex1  x2  5 y 4 z, f  (8 2  2) y  4 sin 2 ( x1  x2 )  ex1  x2  y5  u.

1

2

1

2

1

2

1

2

1

2

1

2

We can obtain that TABLE 2 ERROR ESTIMATES ON A SEQUENCE OF UNIFORMLY TRIANGLE MESH

h 1 / 16 1 / 32 1 / 64 1 / 128

u  uh

L

1.311E-01 6.491E-02 3.247E-02 1.629E-02

y  yh

L

2.123E-01 1.061E-01 5.135E-02 2.572E-02

- 66 www.ivypub.org/mc

z  zh

L

2.132E-01 1.067E-01 5.138E-02 2.574E-02

In the two examples, the priori error estimates for mixed finite element solutions of semilinear elliptic optimal control problems has been shown in Tables 1-2. This is consistent with the results proved in the previous.

5 CONCLUSIONS In this paper, a priori error estimates for mixed finite element solutions of semilinear elliptic optimal control problems has been studied. The state and the co-state are approximated by Raviart-Thomas mixed finite element spaces with the lowest order and the control is approximated by piecewise constant functions. A priori error estimates for the new mixed finite element approximation of semilinear elliptic optimal control problems is obtained.

ACKNOWLEDGMENTS This work is supported by National Science Foundation of China (11201510), Mathematics TianYuan Special Funds of the National Natural Science Foundation of China (11126329), China Postdoctoral Science Foundation funded project (2011M500968), Natural Science Foundation Project of CQ CSTC (cstc2012jjA00003), and Natural Science Foundation of Chongqing Municipal Education Commission (KJ121113). ·

REFERENCES [1]

W. B. Liu, N. N. Yan. “Adaptive finite element method for optimal control governed by PDEs.” Science press, Beijing, 2008

[2]

Y. Chen and Z. Lu. “Error estimates of fully discrete mixed finite element method for semilinear quadratic parabolic optimal control problems.” Comput. Method Appl. Mech. Eng., 199(2010): 1415-1423

[3]

Y. Chen and Z. Lu. “Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods.” Finite Elem. Anal. Des., 46 (2010): 957-965

[4]

Z. Lu and Y. Chen. “A posteriori error estimates of triangular mixed finite element method for semilinear optimal control problems.” Adv. Appl. Math. Mech., 1 (2009): 242-256

[5]



Z. Lu and Y. Chen. “ L -error estimates of triangular mixed finite element method for optimal control problem govern by semilinear elliptic equation.” Numer. Anal. Appl., 12 (2009): 74-86

[6]

Y. Chen , W. B. Liu. “Error estimates and superconvergence of mixed finite elements for quadratic optimal control.” Internat. J. Numer. Anal. Modeling, 3 (2006): 311-321

[7]

F. A. Miliner. “Mixed finite element method for quasilinear second-order elliptic problems.” Math. Comp., 44(1985): 303-320

AUTHORS 1Zuliang

Lu, Ph.D, Associate Professor..

Science, Changsha University of Science and Technology. He

Born in 1980, in Changde, Hunan, China.

has published more than thirty papers in many famous journal,

His current job is in School of Mathematics

for example, “Computer Methods in Applied Mechanics and

and Statistics, Chongqing Three Gorges

Engineering”, “Finite Elements in Analysis and Design”,

University, Chongqing 404000 and College

“Mathematical Problems in Engineering.” and so on.

of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105,

Dr. Zuliang is Chongqing Mathematical Institute director.

P.R.China. Current and previous research interests are mixed

2Dayong

finite element methods and optimal control problems.

Chongqing 404000, P.R. China

Education: (1) Ph.D: Department of Mathematics, Xiangtan University; (2) M.Sc: College of Mathematics and Computing

- 67 www.ivypub.org/mc

Liu, Chongqing Wanzhou Senior Midddle School,