Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods

Finite Elements in Analysis and Design 46 (2010) 957–965

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Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel

Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods$ Yanping Chen a,, Zuliang Lu b a b

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, PR China College of Mathematics and Computer Sciences, Chongqing Three Gorges University, Chongqing 404000, PR China

a r t i c l e in fo

abstract

Article history: Received 16 April 2009 Received in revised form 19 June 2010 Accepted 27 June 2010

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. 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. By applying some error estimates techniques of standard mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present some numerical examples which confirm our theoretical results. & 2010 Elsevier B.V. All rights reserved.

Keywords: A priori error estimates Parabolic equations Optimal control problem Fully discrete mixed finite element methods Gronwall’s Lemma

1. Introduction In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by parabolic equations. The finite element method is undoubtedly the most widely numerical method in computing optimal control problem. The literature in this aspect is huge. There have been extensive studies in convergence of the finite element approximation of optimal control problems, see, for example, [1,2,6,20,33]. For optimal control problems governed by linear elliptic equations, a priori error estimates of the standard finite element discretization were established long ago, see, for example, Falk [16]. Then, Malanowski in [29] established a priori error estimates for the finite element approximations to convex constrained optimal control systems. While the a priori error analysis for finite element discretization of optimal control problems governed by elliptic equations is discussed in many publications, see, e.g., [32], there are only few published results on this topic for parabolic problems, see [34]. A priori error estimates have been obtained for the finite element approximations to state

$ This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University, Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (2008), National Science Foundation of China 10971074.  Corresponding author. E-mail addresses: [email protected] (Y. Chen), [email protected] (Z. Lu).

0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.06.011

constrained parabolic time optimal control problems in [21]. Some recent progress in a priori error estimates can be found in [31], but there are only few published results on this topic for parabolic optimal control problems. In many control problems, the objective functional contains gradient of the state variables. Thus the accuracy of gradient is important in numerical approximation of the state equations. In finite element method, mixed finite elements are widely used to approximate flux variables, although there is only very limited research work on analyzing such elements for optimal control problem. The literature in the aspect is huge, see, for instance, [3–5,7,22,30]. Recently, we have derived a priori error estimates, a posteriori error estimates and superconvergence for quadratic optimal control problems using mixed finite element methods in [8–11,26–28]. In this paper we develop a priori error analysis of the fully discrete mixed finite element discretizations for quadratic optimal control problem governed by parabolic partial differential equations. We focus our attention on the following quadratic parabolic optimal control problem:  Z T  1 min ðJppd J2 þ Jyyd J2 þ aJuJ2 Þ dt ð1:1Þ uðtÞ A K 2 0 subject to the state equation yt ðx,tÞ þ div pðx,tÞ ¼ f ðx,tÞ þuðx,tÞ, pðx,tÞ ¼ AðxÞryðx,tÞ,

x A O,

x A O, t A J,

ð1:2Þ ð1:3Þ

958

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

yðx,tÞ ¼ 0, x A @O, t A J,

yðx,0Þ ¼ y0 ðxÞ, x A O,

ð1:4Þ

where O be a bounded open set in R2 with a smooth boundary @O, J¼[0,T], f A L2 ðJ; L2 ðOÞÞ, and a is a positive constant. Here, AðxÞ A L1 ðJ,H1 ðOÞÞ and K denotes the admissible set of the control variable, defined by 2

2

K ¼ fuðx,tÞ A L ðJ; L ðOÞÞ : uðx,tÞ Z0 a:e: x A O, t A Jg:

For ease of exposition we will assume that O is convex polygon. Let T h be regular triangulation of O. They are assumed to satisfy the angle condition which means that there is a positive constant C such that 2

C 1 h2t r jtj rCht ,

8t A T h ,

For 1 r p o1 and m any nonnegative integer let W m,p ðOÞ ¼ fvA Lp ðOÞ; Db v A Lp ðOÞ if jbj r mg denote the Sobolev P spaces endowed with the norm JvJpm,p ¼ jbj r m JDb vJpLp ðOÞ , and P the semi-norm jvjpm,p ¼ jbj ¼ m JDb vJpLp ðOÞ . We set W0m,p ðOÞ ¼

where jtj is the area of t and ht is the diameter of t. Let h ¼ maxht . Let V h  Wh  V  W denote the Raviart–Thomas space [19,12] of the lowest order associated with the triangulation T h of O. 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 A T h is a triangle, VðtÞ ¼ fv A P02 ðtÞ þ x  P0 ðtÞg. We define

fvA W m,p ðOÞ : vj@O ¼ 0g. For p ¼2, we denote Hm ðOÞ ¼ W m,2 ðOÞ,

V h :¼ fvh A V : 8t A T h , vh jt A VðtÞg,

ð1:5Þ

H0m ðOÞ ¼ W0m,2 ðOÞ, and J  Jm ¼ J  Jm,2 , J  J ¼ J  J0,2 .In addition C or c denotes a general positive constant independent of h. The outline of this paper is as follows. In Section 2, we construct the fully discrete mixed finite element discretization for quadratic optimal control problem governed by parabolic equations. Then, we derive a priori error estimates for the lowest order Raviart–Thomas mixed finite element approximation for the parabolic optimal control problem in Section 3. Finally, numerical examples are presented in Section 4.

2. Mixed methods of optimal control problem

8v A V,

ðyt ,wÞ þðdiv p,wÞ ¼ ðf þu,wÞ, yðx,0Þ ¼ y0 ðxÞ,

8wA W,

8x A O:

8v A V,

ðyt ,wÞ þðdiv p,wÞ ¼ ðf þu,wÞ, yðx,0Þ ¼ y0 ðxÞ,

8x A O,

8vh A Vh ,

ðyht ,wh Þ þ ðdiv ph ,wh Þ ¼ ðf þuh ,wh Þ, yh ðx,0Þ ¼ Yðx,0Þ,

ð2:13Þ

8wh A Wh ,

8x A O,

ð2:14Þ ð2:15Þ

ð2:2Þ ð2:3Þ

ðA1 ph ,vh Þðyh ,div vh Þ ¼ 0,

ð2:4Þ

ðyht ,wh Þ þ ðdiv ph ,wh Þ ¼ ðf þuh ,wh Þ,

ð2:5Þ

8wA W,

By the definition of finite element subspace, the mixed finite element discretization of (2.1)–(2.4) is as follows: compute ðph ,yh ,uh Þ A V h  Wh  Kh such that  Z T  1 min ðJph pd J2 þ Jyh yd J2 þ aJuh J2 Þ dt , ð2:12Þ uh A Kh 2 0

where Y(x,0) is the elliptic mixed methods projection (to be defined below) into the finite dimensional space Wh of the initial data function y0(x). It is well known that the optimal control problem (2.12)–(2.15) again has a unique solution ðph ,yh ,uh Þ, and that a triplet ðph ,yh ,uh Þ is the solution of (2.12)–(2.15) if and only if there is a co-state ðqh ,zh Þ A V h  Wh such that ðph ,yh ,qh ,zh ,uh Þ satisfies the following optimality conditions:

It is well known (see, e.g., [17,24]) that the optimal control problem (2.1)–(2.4) has a unique solution ðp,y,uÞ, and that a triplet ðp,y,uÞ is the solution of (2.1)–(2.4) if and only if there is a co-state ðq,zÞ A V  W such that ðp,y,q,z,uÞ satisfies the following optimality conditions: ðA1 p,vÞðy,div vÞ ¼ 0,

Kh :¼ fu~ h A K : 8t A T h , u~ h jt A P0 ðtÞg:

ðA1 ph ,vh Þðyh ,div vh Þ ¼ 0,

In this section we shall describe the fully discrete mixed finite element approximation of parabolic optimal control problem (1.1)–(1.4). Let V ¼ HðdivÞ ¼ fv A ðL2 ðOÞÞ2 ,div v A L2 ðOÞg endowed with the norm given by JvJHðdivÞ ¼ ðJvJ20, O þ Jdiv vJ20, O Þ1=2 . We denote W ¼ U ¼ L2 ðOÞ. We recast (1.1)–(1.4) as the following weak form: find ðp,y,uÞ A V  W  K such that  Z T  1 ðJppd J2 þ Jyyd J2 þ aJuJ2 Þ dt ð2:1Þ min uðtÞ A K 2 0 ðA1 p,vÞðy,div vÞ ¼ 0,

Wh :¼ fwh A W : 8t A T h , wh jt A P0 ðtÞg,

ð2:6Þ ð2:7Þ

yh ðx,0Þ ¼ Yðx,0Þ,

8vh A V h , 8wh A Wh ,

8x A O,

ðzht ,wh Þ þðdiv qh ,wh Þ ¼ ðyh yd ,wh Þ,

8vh A V h ,

ð2:19Þ

8wh A Wh ,

ð2:20Þ

8x A O,

ðzh þ auh , u~ h uh Þ Z0,

ð2:17Þ ð2:18Þ

ðA1 qh ,vh Þðzh ,div vh Þ ¼ ðph pd ,vh Þ,

zh ðx,TÞ ¼ 0,

ð2:16Þ

ð2:21Þ 8u~ h A Kh :

ð2:22Þ

Let Dt 4 0, N ¼ T=Dt A Z, and t n ¼ nDt, n A R. Also, let

cn cn1 n n n1 , dc ¼ c c : Dt

ðA1 q,vÞðz,div vÞ ¼ ðppd ,vÞ,

8v A V,

ð2:8Þ

cn ¼ cn ðxÞ ¼ cðx,tn Þ, dt cn ¼

ðzt ,wÞ þ ðdiv q,wÞ ¼ ðyyd ,wÞ,

8w A W,

ð2:9Þ

We define for 1 r p o1 the discrete time dependent norms !1=p N X n ;c;Lp ð0,T;Hs ðOÞÞ :¼ Dt Jc Jps ,

zðx,TÞ ¼ 0, Z

8x A O,

ð2:10Þ

n¼1

T 0

~ ðz þ au, uuÞ U dt Z 0,

8u~ A K,

ð2:11Þ

where ð,ÞU is the inner product of U. In the rest of the paper, we shall simply write the product as ð,Þ whenever no confusion should be caused.

and the standard modification for p ¼ 1. By the above definitions, we define the fully discrete finite ,zn1 ,unh Þ satisfies element solution ðpnh ,ynh ,qn1 h h ðA1 pnh ,vÞðynh ,div vÞ ¼ 0,

8v A V h ,

ð2:23Þ

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

ðdt ynh ,wÞ þðdiv pnh ,wÞ ¼ ðf þ unh ,wÞ, y0h ðxÞ ¼ Yðx,0Þ,

8w A Wh ,

8x A O,

,vÞðzn1 ,div vÞ ¼ ðpnh pd ,vÞ, ðA1 qn1 h h ,wÞ ¼ ðynh yd ,wÞ, ðdt znh ,wÞ þ ðdiv qn1 h

8v A V h , 8w A Wh ,

959

ð2:24Þ

following relations:

ð2:25Þ

ðA1 Pn ðunh Þ,vÞðY n ðunh Þ,div vÞ ¼ 0,

ð2:26Þ

ðdiv Pn ðunh Þ,wÞ ¼ ðf þ unh ynt ðunh Þ,wÞ,

ð2:27Þ

ðA1 Q n1 ðunh Þ,vÞðZ n1 ðunh Þ,div vÞ ¼ ðpn ðunh Þpd ,vÞ,

8v A V h ,

ð3:17Þ

8w A Wh ,

ð3:18Þ 8v A V h , ð3:19Þ

zN h ðxÞ ¼ 0,

8x A O,

n ~ ðznh þ aunh , uu h Þ Z 0,

ð2:28Þ 8u~ A Kh :

ð2:29Þ

Now, we define the standard L -orthogonal projection

ph : K-Kh , which satisfies [13,14]: for any u~ A K ~ ph u, ~ u~ h ÞU ¼ 0, 8u~ h A Kh , ðu

ð2:30Þ

~ t,r,U r Cjuj ~ 1,r,U h1 þ t , ~ ph uJ Ju

ð2:31Þ

t ¼ 0,1 for u~ A W 1,r ðOÞ:

In the rest of the paper, we shall use some intermediate variables. For any control function u~ A K, we first define the ~ n ðuÞ,q ~ n1 ðuÞ,z ~ n1 ðuÞÞ ~ associated with u~ that state solution ðpn ðuÞ,y satisfies n

~ ~ p ðuÞ,vÞðy ðuÞ,div vÞ ¼ 0,

8v A V,

~ ~ ~ ðynt ðuÞ,wÞ þðdiv pn ðuÞ,wÞ ¼ ðf þ u,wÞ, y0 ðxÞ ¼ Yðx,0Þ,

xn1 ¼ yn ðunh ÞY n ðunh Þ, zn1 ¼ pn ðunh ÞPn ðunh Þ,

ð3:21Þ

xn2 ¼ zn ðunh ÞZ n ðunh Þ, zn2 ¼ qn ðunh ÞQ n ðunh Þ:

ð3:22Þ

n

8wA W,

ð3:2Þ

8x A O, 8v A V,

8w A W,

8x A O:

n

n

n

n

n

n

Jz1 J0 þ Jx1 J0 þ Jx1 J0,1 rCh,

n ~ ~ ðA1 pnh ðuÞ,vÞðy h ðuÞ,div vÞ ¼ 0,

8v A V h ,

~ ~ ~ ðdt ynh ðuÞ,wÞ þðdiv pnh ðuÞ,wÞ ¼ ðf þ u,wÞ,

n

8w A Wh ,

n

Jdiv z1 J0 þ Jdiv z2 J0 rCh:

ð3:25Þ

n

n

n

n

Estimates for x1t , x1t , z2t , and z2t are given in [18]. We state them here without proof.

ð3:5Þ

Jz1t J0 þ Jx1t J0 þ Jx1t J0,1 rCh,

ð3:6Þ

Jz2t J0 þ Jx2t J0 þ Jx2t J0,1 rCh,

ð3:8Þ ð3:9Þ

n1 ~ ~ ~ ðuÞ,vÞðz ðuÞ,div vÞ ¼ ðpnh ðuÞp ðA1 qn1 d ,vÞ, h h

~ ~ ~ ðdt znh ðuÞ,wÞ þ ðdiv qn1 ðuÞ,wÞ ¼ ðynh ðuÞy d ,wÞ, h

8v A V h , 8wA Wh ,

8x A O:

n

n

n

n

n

n

ð3:26Þ ð3:27Þ

n

Jdiv z1t J0 þJdiv z2t J0 r Ch:

ð3:28Þ

Combining Lemmas 3.1 and 3.2, we can also derive the following error estimates: Theorem 3.1. Assume that the mesh is such that Dt  h1 ¼ Oð1Þ. There is a positive constant C 40, independent of h, such that

ð3:10Þ

;pðunh Þph ;L1 ðJ;HðdivÞÞ þ;yðunh Þyh ;L1 ðJ;L2 ðOÞÞ rCðDt þ hÞ,

ð3:29Þ

ð3:11Þ

;qðunh Þqh ;L1 ðJ;HðdivÞÞ þ ;zðunh Þzh ;L1 ðJ;L2 ðOÞÞ r CðDt þhÞ:

ð3:30Þ

ð3:12Þ

Now, we introduced a linear mixed elliptic projection from V  W onto Vh  Wh . For introducing the elliptic projection, we shall assume that the following boundary value problem: x A O,

ð3:13Þ

F ¼ 0, x A @O, divðAðrC þ pn ðunh Þpd ÞÞ ¼ yn ðunh Þyd þ znt ðunh Þ,

C ¼ 0, x A @O

ð3:24Þ

Lemma 3.2. There is a positive constant C independent of h such that

ð3:7Þ

8x A O,

divðArFÞ ¼ f þ unh ynt ðunh Þ,

ð3:23Þ

Jz2 J0 þ Jx2 J0 þ Jx2 J0,1 rCh,

n

zN h ðxÞ ¼ 0,

n

ð3:4Þ

Then, we define the fully discrete state solution ~ nh ðuÞ,q ~ n1 ~ n1 ~ associated with u~ A K that satisfies ðpnh ðuÞ,y ðuÞ,z ðuÞÞ h h

Yðx,0Þ,

n

ð3:3Þ

~ ~ ~ þ ðdiv qn1 ðuÞ,wÞ ¼ ðyn ðuÞy ðznt ðuÞ,wÞ d ,wÞ,

y0h ðxÞ ¼

n

Estimates for x1 , x2 , z1 , and z2 are given in [17] and are presented in Lemma 3.1 without proof.

ð3:1Þ

n1 ~ ~ ~ ðuÞ,div vÞ ¼ ðpn ðuÞp ðA1 qn1 ðuÞ,vÞðz d ,vÞ,

zN ðxÞ ¼ 0,

ð3:20Þ

Lemma 3.1. For t A J and for h sufficiently small, there is a positive constant C independent of h such that

3. A priori error estimates

ðA

8w A Wh :

Let 2

1 n

ðdiv Q n1 ðunh Þ,wÞ ¼ ðyn ðunh Þyd þznt ðunh Þ,wÞ,

ð3:14Þ x A O,

ð3:15Þ ð3:16Þ

is solvable. Note that (y,z) is the solution of (3.13)–(3.16). Then, define the elliptic mixed method projection of ðpn ðunh Þ,yn ðunh Þ, qn1 ðunh Þ,zn1 ðunh ÞÞ to be (Pn(unh),Yn(unh),Qn  1(unh),Zn  1(unh)) by the

Proof. Let Zn1 ¼ Y n ðunh Þynh , Zn2 ¼ Pn ðunh Þpnh . From (3.2) we see that ðynt ðunh Þ,wÞ þ ðdiv pn ðunh Þ,wÞ ¼ ðf þunh ,wÞ,

8w A W:

ð3:31Þ

From the elliptic mixed method projection, we have ðYtn ðunh Þ,wh Þ þ ðdiv Pn ðunh Þ,wh Þ n

¼ ðynt ðunh Þ,wh Þ þðdiv pn ðunh Þ,wh Þðx1t ,wh Þ,

8wh A Wh :

ð3:32Þ

Using (3.31) and Wh  W we have n

ðYtn ðunh Þ,wh Þ þ ðdiv Pn ðunh Þ,wh Þ ¼ ðf þ unh ,wh Þðx1t ,wh Þ:

ð3:33Þ

Subtract (2.24) from (3.33) to get n

ðYtn ðunh Þdt ynh ,wh Þ þ ðdiv Zn2 ,wh Þ ¼ ðx1t ,wh Þ:

ð3:34Þ

960

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

Substituting (2.23) in (3.17) and using the discrete version of the elliptic projection we get ðA1 ðPn ðunh Þpnh Þ,vh ÞðY n ðunh Þynh ,div vh Þ ¼ 0:

Substituting (3.50) in (3.48) we have n

ðdt Zn1 ,dt Zn1 Þ þðA1 dt Zn2 , Zn2 Þ ¼ ðdt Y n ðunh ÞYtn ðunh Þ,dt Zn1 Þðx1t ,dt Zn1 Þ:

ð3:35Þ

Take wh ¼ Zn1 and vh ¼ Zn2 as test functions in (3.34) and (3.35), respectively, and add those equations to get n

ðdt Zn1 , Zn1 Þ þ ðA1 Zn2 , Zn2 Þ ¼ ðdt Y n ðunh ÞYtn ðunh Þ, Zn1 Þðx1t , Zn1 Þ:

ð3:36Þ

ð3:51Þ By the inequality (3.39), we have C

2 ðZn2 Þ2 ðZn1 2 Þ r ðA1 dt Zn2 , Zn2 Þ, 2Dt

ð3:52Þ

We shall estimate the left-hand side of (3.36). First, using the coercivity property,

CJdt Zn1 J20 rðdt Zn1 ,dt Zn1 Þ:

ðA1 Zn2 , Zn2 Þ Z CJZn2 J20 :

Next, let us bound each term in the right hand side of (3.51):

ð3:37Þ

Observe that ðdt Zn1 , Zn1 Þ ¼

1 1 2 2 ð1,ðZn1 Þ2 ðZn1 ð1,ðZn1 Zn1 1 Þ Þþ 1 Þ Þ: 2Dt 2Dt

ð3:38Þ

ð3:53Þ

ðynh yn ðunh Þ,dt Zn1 Þ rJynh yn ðunh ÞJ0  Jdt Zn1 J0 rCðDt þ hÞ  Jdt Zn1 J0 rCðDt þ hÞ2 þCJdt Zn1 J20 :

ð3:54Þ

ðdt Y n ðunh ÞYtn ðunh Þ,dt Zn1 Þ r CI2n þ CJdt Zn1 J20 ,

ð3:55Þ

Note that (3.38) can be rewritten as

Zn1 Zn1 1 2Dt

r

Zn1 Zn1 1 2Dt

þ

Next,

1 2 n n ð1,ðZn1 Zn1 1 Þ Þ ¼ ðdt Z1 , Z1 Þ: 2Dt

ð3:39Þ

Now, we shall estimate the right hand side of (3.36): jðdt Y n ðunh ÞYtn ðunh Þ, Zn1 Þj rCI2n þ CJZn1 J20 , n

n

ð3:40Þ

2

ð3:41Þ

where Z ¼

ð3:56Þ

I2n

jðx1t , Zn1 Þj r Ch þCJZn1 J20 ,

In2

2

ðx1t ,dt Zn1 Þ rCh þ CJdt Zn1 J20 ,

 2  !2 @ Y  2    2 ð,sÞ ds r CðDtÞ : tn1 @t

where be defined in (3.42). Now, multiply the resultant equations by Dt, sum in n, for n¼1,2,y,l, and apply the bounds for each term of the sum in the right hand side and left hand side to obtain

tn

ð3:42Þ

l X

Jdt Zn1 J20 Dt þ

n¼1

Next, by the triangle inequality, we have

l X JZl2 J20 rCððDtÞ2 þh2 ÞDt þC Jdt Zn1 J20 Dt þ CJZ12 J20 : 2 n¼1

It follows from the Gronwall Lemma that

ðynh yn ðunh Þ, Zn1 Þ r Jynh yn ðunh ÞJ0  JZn1 J0 n rðJx1 J0 þ JZn1 J0 Þ  JZn1 J0

l X

2

rCh þ CJZn1 Jn0 :

ð3:43Þ

Jdt Zn1 J20 þJZl1 J20 þJZl2 J20 rCððDtÞ2 þh2 Þ þ CðJZ11 J20 þ JZ12 J20 Þ:

n¼1

ð3:57Þ We next multiply each one of the estimates of the terms of (3.36)–(3.43), by Dt and sum on n, n¼1, 2,y,l for 2r l r N to obtain

Zl1

þC

2

l X

2 n 2 2 1 J0 Dt r CððDtÞ þ h ÞDt þ C

JZ

n¼1

l X

n 2 1 J0 Dt:

JZ

ð3:44Þ

l 2 1 J0 þ

JZ

1 2.

Now, we wish to get some bounds on Z and Z (3.45), we obtain JZ11 J0 þ JZ12 J0 rCðDt þ hÞ:

By applying

ð3:58Þ

Combining (3.57) and (3.58), we have

n¼1 l X

Using the discrete Gronwall Lemma that l X

1 1

2 n 2 n 2 2 1 J0 þJ 2 J0 Þ rCððDtÞ þh Þ:

ðJZ

Jdt Zn1 J20 þJZl1 J20 þJZl2 J20 rCððDtÞ2 þh2 Þ:

Z

ð3:45Þ

n¼1

By Lemma 3.1 and (3.59) to obtain ;pðunh Þph ;L1 ðJ;L2 ðOÞÞ r CðDt þhÞ:

So, we have ;Yðunh Þyh ;L1 ðJ;L2 ðOÞÞ rCðDt þ hÞ:

ð3:46Þ

Using the triangle inequality with Lemma 3.1 will give ;yðunh Þyh ;L1 ðJ;L2 ðOÞÞ r CðDt þhÞ:

ð3:47Þ

n

ðdt Zn1 ,dt Zn1 Þ þ ðdiv Zn2 ,dt Zn1 Þ ¼ ðdt Y n ðunh ÞYtn ðunh Þ,dt Zn1 Þðx1t ,dt Zn1 Þ: ð3:48Þ

ðdivðpn ðunh Þpnh Þ,div Zn2 Þ ¼ ðynt ðunh Þdt ynh ,div Zn2 Þ

8vh A V h :

ð3:49Þ

ð3:61Þ

So, we have Jdiv Zn2 J0 r CðDt þhÞ:

ð3:62Þ

By the triangle inequality, (3.62), and Lemma 3.1 we obtain ;pðunh Þph ;L1 ðJ;HðdivÞÞ r CðDt þ hÞ:

Combining (2.23) and (3.1), we obtain that

ð3:60Þ

Subtract (2.24) from (3.2) and take wh ¼ div Zn2 as a test function we have that rCððDtÞ2 þh2 Þ þ eJdiv Zn2 J20 :

Now, we estimate ;pðunh Þph ;L1 ðJ;HðdivÞÞ r CðDt þhÞ. Using dt Zn1 as a test function in (3.34), we get the following equation:

ðA1 Zn2 ,vh ÞðZn1 ,div vh Þ ¼ 0,

ð3:59Þ

n¼1

ð3:63Þ

So, we have proved (3.29). The proof of (3.30) is quite similar with above and we omitted here. & Set some intermediate errors:

Now, take the difference in time of (3.49) and take vh ¼ Zn2 as test function to get

en1 ¼ pn pn ðunh Þ,

r1n ¼ yn yn ðunh Þ,

ð3:64Þ

ðA1 dt Zn2 , Zn2 Þðdt Zn1 ,div Zn2 Þ ¼ 0,

en2 ¼ qn qn ðunh Þ,

r2n ¼ zn zn ðunh Þ:

ð3:65Þ

8vh A V h :

ð3:50Þ

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

From (2.5)–(2.10) and (3.1)–(3.6), we derive the following error equations: ðA1 en1 ,vÞðr1n ,div vÞ ¼ 0,

8v A V,

ð3:66Þ

ðynt dt yn ðunh Þ,wÞ þðdiv en1 ,wÞ ¼ ðun unh ,wÞ, n1 ðA1 en1 ,div vÞ ¼ ðen1 ,vÞ, 2 ,vÞðr2

8w A W,

ð3:67Þ

8v A V,

n ðznt dt zn ðunh Þ,wÞ þ ðdiv en1 2 ,wÞ ¼ ðr1 ,wÞ,

ð3:68Þ

8wA W:

ð3:69Þ

Theorem 3.2. There is a constant C 40, independent of h and Dt, such that ;ppðunh Þ;L1 ðJ;HðdivÞÞ þ;yyðunh Þ;L1 ðJ;L2 ðOÞÞ rCðDt þ h þ;uuh ;L2 ðJ;L2 ðOÞÞ Þ,

961

Then, we have proved (3.71). Thus, we have completed the proof of the theorem. & Now we study a priori error estimates for the fully discrete mixed finite element approximation to the quadratic parabolic optimal control problem. Let SðÞ : K-R be a G-differential uniform convex functional near the solution u which satisfies the following form: Sðun Þ ¼

1 n 1 a Jp pd J2 þ Jyn yd J2 þ Jun J2 , 2 2 2

Sðunh Þ ¼

1 n n 1 a Jp ðuh Þpd J2 þ Jyn ðunh Þyd J2 þ Junh J2 : 2 2 2

It can be shown that ðSuðun Þ,vÞ ¼ ðaun þ zn ,vÞ,

ð3:70Þ ;qqðunh Þ;L1 ðJ;HðdivÞÞ þ ;zzðunh Þ;L1 ðJ;L2 ðOÞÞ r CðDt þ h þ ;uuh ;L2 ðJ;L2 ðOÞÞ Þ:

ð3:71Þ

Proof. Part I: Choose v ¼ en1 and w¼rn1 as the test functions and add the two relations of (3.66)–(3.67), then we obtain that ðA1 en1 ,en1 Þ ¼ ðun unh ,r1n Þðynt dt yn ðunh Þ,r1n Þ: Then, using e-Cauchy inequality, we can find an estimate as follows: Jen1 J20 þ Jr1n J20 r CððDtÞ2 þ h2 þ Jun unh J20 Þ þ eJr1n J20 :

ð3:72Þ

So, we have

ðSuðunh Þ,vÞ ¼ ðaunh þ zn ðunh Þ,vÞ, where ðpn ðunh Þ,yn ðunh Þ,qn1 ðunh Þ,zn1 ðunh ÞÞ is the solution of (3.1)–(3.6) with u~ ¼ unh . In many applications, SðÞ is uniform convex near the solution u [25]. The convexity of SðÞ is closely related to the second order sufficient conditions of the optimal control problem, which are assumed in many studies on numerical methods of the problem. In the following we estimate ;uuh ;L2 ðJ;L2 ðOÞÞ and then obtain the results: Theorem 3.3. Let ðp,y,q,z,uÞ A ðV  WÞ2  K and ðpnh ,ynh ,qn1 ,zn1 , h h unh Þ A ðV h  Wh Þ2  Kh be the solution of (2.5)–(2.11) and (2.23)– (2.29), respectively. Assume that the mesh is such that Dt  h1 ¼ Oð1Þ and zn þ aun A H1 ðOÞ. Then, we have ;uuh ;L2 ðJ;L2 ðOÞÞ r CðDt þ hÞ,

ð3:80Þ

Now, take w ¼ div en1 as a test function in (3.67) and using e-Cauchy inequality, then we get

;pph ;L1 ðJ;HðdivÞÞ þ ;yyh ;L1 ðJ;L2 ðOÞÞ rCðDt þ hÞ,

ð3:81Þ

Jdiv en1 J20 ¼ ðun unh ,div en1 Þðynt dt yn ðunh Þ,div en1 Þ

;qqh ;L1 ðJ;HðdivÞÞ þ ;zzh ;L1 ðJ;L2 ðOÞÞ rCðDt þ hÞ:

ð3:82Þ

Jen1 J0 þ Jr1n J0 r CðDt þh þ Jun unh J0 Þ:

ð3:73Þ

r CJynt dt yn ðunh ÞJ20 þCJun unh J20 þ CJr1n J20 þ

e

Jdiv en1 J20 , ð3:74Þ

Proof. We choose u~ ¼ unh in (2.11) and u~ h ¼ ph un in (2.29) to get that Z T ðzn þ aun ,unh un Þ Z 0 ð3:83Þ

ð3:75Þ

and Z T ðznh þ aunh , ph un unh Þ Z0:

then, using the estimate (3.73), we have Jdiv en1 J0 r CJynt dt ynh ðun ÞJ0 þCJun unh J0 þ CJr1n J0 r CðDt þh þ Jun unh J0 Þ:

0

Then (3.70) follows from (3.72) and (3.75). and w¼rn2  1 as the test functions and Part II: Choose v ¼ en1 2 add the two relations of (3.68)–(3.69), then we obtain that n1 n n1 Þ þðznt dt znh ðun Þr2n1 Þ: ðA1 en1 2 ,e2 Þ ¼ ðr1 ,r2

Then, using e-Cauchy inequality, we can find an estimate as follows: 2 n1 2 J0 r CððDtÞ2 þ h2 þ Jun unh J20 Þ þ eJr2n1 J20 : Jen1 2 J0 þJr2

ð3:76Þ

So, we have n1 J0 r CðDt þ h þ Jun unh J0 Þ: Jen1 2 J0 þJr2

ð3:77Þ

¼ div en1 2

as a test function in (3.69) and using Now, take w e-Cauchy inequality, then we get 2 Jdiv en1 2 J0

¼ ðr1n ,div en1 Þþ ðznt dt znh ðun Þ,div en1 2 2 Þ n n n 2 r CJzt dt zh ðu ÞJ0 þCJr1n J20 þCJr2n J20 þ

Note that ph un unh ¼ ph un un þ un unh in (3.84) and add the two inequalities (3.83)–(3.84), we have Z T Z T ðznh þ aunh zn aun ,un unh Þ þ ðznh þ aunh , ph un un Þ Z 0: ð3:85Þ 0

0

By applying the uniform convexity of SðÞ and (3.85), we obtain Z T Z T 2 ðSuðun Þ,un unh Þ ðSuðunh Þ,un unh Þ c;uuh ;L2 ðJ;L2 ðOÞÞ r 0 0 Z T Z T ðzn þ aun ,un unh Þ ðzn ðunh Þ þ aunh ,un unh Þ ¼ 0 0 Z T Z T ðznh zn ðunh Þ,un unh Þ þ aðunh un , ph un un Þ r 0

0

Z 2 eJdiv en1 2 J0 ,

T

þ ð3:78Þ

0

Z

T

þ 0

Z

then, using the estimate (3.77) and (3.73), we have n n Jdiv en1 2 J0 rCðDt þ h þJu uh J0 Þ:

ð3:79Þ

ð3:84Þ

0

þ 0

T

ðznh zn ðunh Þ, ph un un Þ ðzn ðunh Þzn , ph un un Þ ðzn þ aun , ph un un Þ:

ð3:86Þ

962

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

Now, we estimate all terms at the right side of (3.86). By the

Substituting (3.87)–(3.92) to (3.86), we have

dCauchy inequality and Theorem 3.1, we obtain Z

T 0

;uuh ;L2 ðJ;L2 ðOÞÞ rCðDt þ hÞ:

Then we derive the result (3.80). Finally, we will give the estimates for the state and the co-state variables. By using the triangle inequality, (3.93), and Theorems 3.1 and 3.2 we obtain that

ðznh zn ðunh Þ,un unh Þ r CJznh zn ðunh ÞJL2 ðJ;L2 ðOÞÞ  Jun unh JL2 ðJ;L2 ðOÞÞ 2

rCðDt þ hÞ2 þ d;uuh ;L2 ðJ;L2 ðOÞÞ ,

ð3:87Þ

;pph ;L1 ðJ;HðdivÞÞ þ ;yyh ;L1 ðJ;L2 ðOÞÞ

for any small d 4 0. From the d-Cauchy inequality and the approximation property (2.31), it is clear that Z

T

r ;ppðunh Þ;L1 ðJ;HðdivÞÞ þ ;pðunh Þph ;L1 ðJ;HðdivÞÞ þ ;yyðunh Þ;L1 ðJ;L2 ðOÞÞ þ;yðunh Þyh ;L1 ðJ;L2 ðOÞÞ r CðDt þ h þ;uuh ;L2 ðJ;L2 ðOÞÞ Þ

ðunh un ,

a 0

ð3:93Þ

n

n

ph u u Þ r Ch

2

2 þ d;uuh ;L2 ðJ;L2 ðOÞÞ :

r CðDt þ hÞ

ð3:88Þ

ð3:94Þ

and Using the approximation property (2.31) of the projection ph , we have Z

T 0

;qqh ;L1 ðJ;HðdivÞÞ þ ;zzh ;L1 ðJ;L2 ðOÞÞ r ;qqðunh Þ;L1 ðJ;HðdivÞÞ þ ;qðunh Þqh ;L1 ðJ;HðdivÞÞ þ ;zzðunh Þ;L1 ðJ;L2 ðOÞÞ þ ;zðunh Þzh ;L1 ðJ;L2 ðOÞÞ

ðznh zn ðunh Þ, ph un un Þ rCðDt þ hÞ2 :

ð3:89Þ

r CðDt þ h þ;uuh ;L2 ðJ;L2 ðOÞÞ Þ r CðDt þ hÞ:

From the d-Cauchy inequality, Theorem 3.2, and the approximation property (2.31), it can be seen that Z

T 0

2

2

ðzn ðunh Þzn , ph un un Þ rCh þ d;uuh ;L2 ðJ;L2 ðOÞÞ :

Collecting the above (3.81)–(3.82). &

(3.94)

and

(3.95)

yields

4. Numerical tests

ðo, RÞ , JoJk, O ¼ sup R A 0, R a 0 JRJk, O

In this section, we are going to validate the a priori error estimates for the error in the control, state, and co-state numerically. The optimization problem were dealt numerically with codes developed based on AFEPACK. The package is freely available and the details can be found at [23]. Our numerical example is the following optimal control problem:  Z T  1 ðJppd J2 þ Jyyd J2 þ JuJ2 Þ dt ð4:1Þ min uðtÞ A K 2 0

ð3:91Þ

and using the approximation property (2.31), then we obtain T 0

inequality

ð3:90Þ

As we can see,

Z

ð3:95Þ

2

ðzn þ aun , ph un un Þ rCJzn þ aun JL1 ðJ;H1 ðOÞÞ  Jph un un J1, O r Ch : ð3:92Þ

Table 1 The numerical error for state and control functions. Resolution

16  16 32  32 64  64 128  128

Error ;uuh ;

;pph ;

;yyh ;

;qqh ;

;zzh ;

1.74910E-02 8.80422E-03 4.41559E-03 2.22418E-03

4.11371E-01 2.06597E-01 1.04768E-01 5.29067E-02

2.09124E-02 1.04931E-02 5.31972E-03 2.70019E-03

1.74071E-02 8.70516E-03 4.40326E-03 2.23764E-03

2.08135E-02 1.04167E-02 5.21364E-03 2.64844E-03

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 1

0.8

0.6 0.4 y 0.2

0

0

0.6 0.4 x 0.2

0.8

1

0 0

0.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

Fig. 1. The profile of the exact (left) and approximation (right) control solution at t ¼0.25.

0.8

1

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

yt þdiv p ¼ u þ f ,

p ¼ ry,

yðx,tÞ ¼ 0, x A @O, t A J, zt þ div q ¼ yyd ,

x A O, t A J,

q ¼ rzp þ pd ,

zðx,tÞ ¼ 0, x A @O, t A J,

ð4:4Þ

where i responds to the spatial partition, and Ei denote the L1 norm for the state and co-state approximation and L2 norm for the control approximation. For simplifying the statement, theoretical analysis and numerical computation have used the same backward scheme for solving the state equations.

ð4:5Þ

Example 1. We set the known functions as follows:

ð4:2Þ

yðx,0Þ ¼ 0, x A O,

ð4:3Þ

x A O, t A J,

zðx,0Þ ¼ 0, x A O:

963

u ¼ maxð0:4z,0Þ, In our examples, we choose the domain O ¼ ½0,1  ½0,1 and T¼1. We adopt the same mesh partition for the state and the control. For simplicity, we also take Dt ¼ h in our test. We present below three examples to illustrate the theoretical results of the quadratic optimal control problem. The convergence order is computed by the following formula: order C

z ¼ sinpx1 sinpx2 sinpt, f ¼ yt þ 2p2 yu,

yd ¼ y þzt ,

q ¼ pd ¼ ð0,0Þ,

logðEi =Ei þ 1 Þ , logðhi =hi þ 1 Þ

p ¼ ðpcospx1 sinpx2 sinpt, psinpx1 cospx2 sinptÞ:

0

In this example, the errors ;uuh ;L2 ðJ;L2 ðOÞÞ , ;pph ;L1 ðJ;HðdivÞÞ , ;yyh ;L1 ðJ;L2 ðOÞÞ , ;qqh ;L1 ðJ;HðdivÞÞ , ;zzh ;L1 ðJ;L2 ðOÞÞ obtained on the lowest order Raviart–Thomas mixed finite element approximation for state functions and piecewise constant approximation for control function are presented in Table 1. We also show the profiles of the solutions for control in Fig. 1, where the first graph is the profile of the exact solution and the second one is that of the approximations. We also show the convergence orders by slopes in Fig. 2. This confirms our analysis again. We plot the errors log10(error) vs. log10(sqrt(dofs)), where dofs denotes degree of freedoms.

u−uhL2norm p−phL∞norm y−yhL∞norm q−qhL∞norm z−zhL∞norm

−0.5 −1 log10(error)

y ¼ sinpx1 sinpx2 sinpt,

−1.5 −2 −2.5

Example 2. For the second example, the data are as follows: u ¼ maxð1:0z,0Þ,

−3 1.3

1.4

1.5

1.6 1.7 1.8 1.9 log10(sqrt(dofs))

2

2.1

2.2

y ¼ x1 sin2px1 sin2px2 sinpt, z ¼ x1 sin2px1 sin2px2 sinpt,

Fig. 2. Convergence orders of u  uh, pph , y yh, qqh , and z  zh in different norms.

f ¼ yt þ div pu,

Table 2 The numerical error for state and control functions. Resolution

16  16 32  32 64  64 128  128

Error ;uuh ;

;pph ;

;yyh ;

;qqh ;

;zzh ;

2.42231E-02 1.20487E-02 6.01613E-03 3.00431E-03

2.02624E + 00 1.01775E + 00 5.09541E-01 2.55026E-01

2.40604E-02 1.20334E-02 6.01823E-03 3.00761E-03

2.20071E + 00 1.10366E + 00 5.57406E-01 2.82231E-01

2.42231E-02 1.20487E-02 6.01612E-03 3.00431E-03

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4 0

0.2

0.4 x

0.6

0.8

1 0

0.5 1 0.40 y

0.2

0.4

0.6

0.8

1 0

Fig. 3. The profile of the exact (left) and approximation (right) control solution at t ¼0.25.

0.5

1

964

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

yd ¼ yþ zt div q,

yd ¼ y þ zt div q,

pd ¼ ððsin2px1 þ 2px1 cos2px1 Þsin2px2 sinpt,2px1 sin2px1 cos2px2 sinptÞ,

u ¼ maxð0:01z,0Þ,

p ¼ q ¼ ððsin2px1 þ 2px1 cos2px1 Þsin2px2 sinpt,2px1 sin2px1 cos2px2 sinptÞ:

y ¼ ð1x1 Þ2 x21 x2 ð1x2 Þsinpt, z ¼ 2ð1x1 Þ2 x21 x2 ð1x2 Þsinpt,

In this example, we investigate the convergence order for the solutions. The solutions are computed on a series of triangular meshes. Table 2 displays the errors for state and control functions. Fig. 3 is surfaces of the exact solution u and the approximation solution uh at t ¼0.25. In Fig. 4, we show the convergence orders by slopes.

q ¼ ð4x1 ð1x1 Þð12x2 Þx2 ð1x2 Þsinpt,2ð1x2 Þ2 x21 ð12x2 ÞsinptÞ, p ¼ pd ¼ ð2x1 ð1x1 Þð12x2 Þx2 ð1x2 Þsinpt,ð1x2 Þ2 x21 ð12x2 ÞsinptÞ: In Table 3 numerical results are presented for the third example, and we also show the profiles of the solutions and the convergence orders by slopes for control in Fig. 6. From the error data on the uniformly refined triangle meshes, as listed in our

Example 3. For the third example, the data are as follows (Fig. 5): f ¼ yt þdiv pu,

0.5

−1

0

−1.5 −2 2

u−uhL norm

−1

log10(error)

log10(error)

−0.5

u−uh L2 norm p−ph L∞ norm y−yh L∞ norm ∞ q−qh L norm z−zhL∞ norm

p−phL∞norm y−yhL∞norm

−1.5

q−qhL∞norm z−zhL∞norm

−2

−2.5 −3 −3.5 −4

−2.5

−4.5

−3 1.3

1.4

1.5

1.6 1.7 1.8 1.9 log10(sqrt(dofs))

2

2.1

1.3

2.2

Fig. 4. Convergence orders of u  uh, pph , y yh, qqh , and z  zh in different norms.

1.4

1.5

1.6 1.7 1.8 1.9 log10(sqrt(dofs))

2

2.1

2.2

Fig. 6. Convergence orders of u  uh, pph , y  yh, qqh , and z  zh in different norms.

Table 3 The numerical error for state and control functions. Resolution

16  16 32  32 64  64 128  128

Error ;uuh ;

;pph ;

;yyh ;

;qqh ;

;zzh ;

5.15922E-04 2.58417E-04 1.30013E-04 6.60402E-05

1.47400E-02 7.38208E-03 3.70237E-03 1.86372E-03

3.22948E-04 1.61540E-04 8.18333E-05 4.22101E-05

3.28588E-02 1.65314E-02 8.40923E-03 4.31532E-03

6.84465E-04 3.44522E-04 1.83291E-04 9.35753E-05

0.01

0.01

0.005

0.005

0 1

1 0.8 0.8

0.6 y

0.4

0.6 0.4 x 0.2

0.2 00

1

0 0

0.8 0.2

0.6 0.4

0.6

0.4 0.8

0.2 01

Fig. 5. The profile of the exact (left) and approximation (right) control solution at t ¼0.25.

Y. Chen, Z. Lu / Finite Elements in Analysis and Design 46 (2010) 957–965

examples, it can be seen that the a priori estimates remains in our numerical tests.

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